0, a new round is played as follows.\nFirst, Spoiler chooses one of the two nodes *x*0, *x*1, i.e. he chooses an index $i \\in \\set{0,1}$. The idea is that Spoiler thinks that the node *x**i* has some property that the other node *x*1 − *i* does not have. He then chooses to make either a descendant move (in this case, Spoiler thinks that *x**i* has a descendant unlike all descendants of *x*1 − *i*) or an ancestor move (Spoiler thinks that *x**i* has an ancestor unlike all ancestors of *x*1 − *i*). If Spoiler chooses a descendant (respectively, ancestor) move, then he must choose a proper descendant (respectively, proper ancestor) *y**i* of *x**i* in the forest *s**i*. To this, Duplicator must respond by choosing a proper descendant (respectively, proper ancestor) *y*1 − *i* of *x*1 − *i* in the other forest *s*1 − *i*. The idea is that Duplicator thinks that *y*1 − *i* is similar to *y**i*, at least as far as the remaining *n* − 1 rounds are concerned. Formally, the new configuration becomes (*y*0, *y*1, *s*0, *s*1) and the game continues with *n* − 1 rounds left.\nWe also define how the *n*-round game is played on two forests *s*0, *s*1 in case when the nodes *x*0, *x*1 are not specified. In this case, there is a special introductory round, where Spoiler chooses $i\n\\in \\set{0,1}$ and a root node *x**i* in *s**i*; Duplicator responds with a root node *x*1 − *i* in the other forest. Then the standard *n*-round game continues from this configuration.\n[prop:ef-char] A forest language is forest-definable in EF + F− 1 if and only if for some *n*, Spoiler wins the *n*-round game for any pair of forests *s*0 ∈ *L* and *s*1 ∉ *L*.\nThe proof is standard, and omitted here. The idea is that *n* is the nesting depth of the formulas used to forest-define *L*. The nesting depth counts the maximal nesting of EF and F− 1 in a formula, while boolean operations are for free.\nMorphic images\n--------------\nIn this section, we show that languages forest-definable in EF + F− 1 are closed under morphic preimages. Actually, we show a slightly more general result. The more general setting will be used in Section [sec:empty-forests], where we show that our characterization also works for a different model of forest algebra, where empty forests are allowed.\nWe first describe the more general setting. The generalization is twofold. First, we allow empty forests. Second\\*[1](#fn1), we consider forests over a single alphabet (unlike the two-sorted alphabet *A*, *B* considered before, with *A* allowed only in leaves and *B* allowed only in inner nodes). The new type of forests will be called *one-sorted forests*, to distinguish them from the *two-sorted forests* considered before. The one-sorted forests are more general in the following sense: the two-sorted forests over an alphabet (*A*, *B*) are a subset of the one-sorted forests over the alphabet *A* ∪ *B*. Of course, the difference is not that big: the one-sorted forests over *A* are the two-sorted forests over (*A*, *A*), plus the empty forest.We also have an analogous concept of *one-sorted contexts*. A *one-sorted morphism*, with *source alphabet *A** and *target alphabet *B** is given by a function that assigns to each letter of *A* a one-sorted context, possibly empty, over *B*. A one-sorted morphism uniquely extends to one-sorted forests and one-sorted contexts. To avoid confusion, in this section we use the name *two-sorted morphism* for the morphisms introduced previously in the paper.\n[thm:ef] Let *α* be a one-sorted morphism. If a forest language *L* over the target alphabet *B* is forest-definable in EF + F− 1, then so is its inverse image *α*− 1(*L*).\nThe version of this theorem for two-sorted morphisms is a special case of the one-sorted version, since every for two-sorted morphism there is a one-sorted morphism that gives the same results over all legal two-sorted forests.\nTo show this theorem, we will use the Ehrenfeucht-Fraïssé game. We fix the forest-language *L* and the (one-sorted) morphism *α* from the theorem for the rest of this section. Let *n* be the number of rounds obtained by applying Proposition [prop:ef-char] to the forest *L* in the statement of the theorem. By invoking Proposition [prop:ef-char] a second time, to establish that the inverse image *α*− 1(*L*) is forest-definable in EF + F− 1, it suffices to show that Spoiler can win the *n*-round game over any two preimages, one taken from the preimage *α*− 1(*L*), and the other taken from its complement. The proof will be by showing how a strategy of Duplicator over the preimage can be lifted to a strategy over the image, as stated in the following proposition.\n[prop:ef-transfer] If Duplicator wins the *n*-round game over *s*0, *s*1, then Duplicator also wins the *n*-round game over *α*(*s*0), *α*(*s*1).\nTo prove this transfer of strategies, we will be switching back and forth between the Ehrenfeucht-Fraïssé games on *s*0, *s*1 and on *α*(*s*0), *α*(*s*1). To avoid confusion, we use the name *preimage game* for the former and we use the name *image game* for the latter. We will be comparing configurations of the two games in the following way. Every node *x* in a morphic image *α*(*s*) can be uniquely identified by two pieces of information: its *preimage* $\\bar x$, which is a node in the preimage forest *s*, and its *offset*, which is a node of the context assigned by *α* to the label in $\\bar x$. These concepts are illustrated below, in an example where both the source and target alphabets are $\\set{a,b}$, and the one-sorted morphism is defined by *α*(*a*) = *a*(□ + *b*) and *α*(*b*) = □.\n\nNote that some nodes in the preimage forest *s* are not the preimage of any node in *α*(*s*), these are the nodes whose labels are mapped to an empty context by *α*.\nArmed with the definitions of offset and preimage, we now prove the strategy transfer from Proposition [prop:ef-transfer]. We only give the main invariant, which is described below. The missing part of the proof, for the introductory round of the game where the root nodes are chosen, is done in a similar way.\nLet *m* ≤ *n*. Let *x*0, *x*1 be nodes with the same offset such that $\\bar x\\_0, \\bar x\\_1$ have the same label. If Duplicator can win the *n*-round preimage game in configuration $(\\bar\n x\\_0,\\bar x\\_1, s\\_0,s\\_1)$, then he can also win the *m*-round image game in configuration (*x*0, *x*1, *α*(*s*0), *α*(*s*1)).\nThe proof is by induction on *n*. Consider first the case of *n* = 0. By assumption on the preimage game, the nodes $\\bar x\\_0$ and $\\bar x\\_1$ have the same labels in *s*0, *s*1. Since the two nodes *x*0, *x*1 have the same offsets, they must also have the same labels in the images *α*(*s*0), *α*(*s*1), and therefore Duplicator wins.\nConsider now the induction step. We only do the case when Spoiler chooses a descendant move, the ancestor move is done the same way. Assume then that Spoiler chooses *x**i* and indicates a proper descendant *y**i* of *x**i* in *α*(*s**i*). How should Duplicator respond? There are two possible cases:\n1. The preimage $\\bar y\\_i$ is a proper descendant of $\\bar x\\_i$. We now go to the preimage game, and make Spoiler play a descendant move where he chooses $\\bar y\\_i$. By assumption on Duplicator winning the preimage game, there is a proper descendant of $\\bar x\\_{1-i}$, call it $\\bar y\\_{1-i}$, such that Duplicator wins the (*n* − 1)-round preimage game from configuration $(\\bar y\\_0,\\bar y\\_1, s\\_0,s\\_1)$. In particular, the nodes $\\bar y\\_0,\\bar y\\_1$ have the same labels in the preimage, and therefore the same possible offsets in the image. Therefore, there exists a node *y*1 − *i* in *α*(*s*1 − *i*) such that its preimage is $\\bar y\\_{1-i}$, and this node can be chosen to have the same offset as $\\bar y\\_i$. We now use the induction assumption to show that Duplicator wins the rest of the image game from configuration (*y*0, *y*1, *α*(*s*0), *α*(*s*1)).\n2. If the preimage $\\bar y\\_i$ is not a proper descendant of $\\bar\n x\\_i$, then $\\bar y\\_i = \\bar x\\_i$ and the only difference between *y**i* and *x**i* is in the offset. Duplicator’s response is to choose in the forest *s*1 − *i* a node *y*1 − *i* that has the same offset as *y**i*, and such that $\\bar y\\_{1-i} = \\bar x\\_{1-i}$. We then use the induction assumption to show that Duplicator wins the rest of the image game.\nCorrectness of the identities\n-----------------------------\nWe are now ready to show the easier implication in Theorem [thm:main], namely that the syntactic forest algebra of a language forest-definable in EF + F− 1 satisfies the three identities. Validity of ([eq:apcom]) can easily be shown. We omit the proof of ([eq:da]) for two reasons: first, it is the same as in the word case, see e.g. ; and second, it follows along similar lines as the proof of ([eq:special]).\nThe rest of this section is devoted to showing the validity of identity ([eq:special]). Let *L* be a forest language forest-definable in EF + F− 1. We need to show that the syntactic algebra of *L* satisfies identity ([eq:special]). Recall that elements of the syntactic algebra are equivalence classes of the Myhill-Nerode equivalence relation. Therefore, in order to show the validity of ([eq:special]), we have to show that for any formula *φ* of EF + F− 1, for all contexts *p*1 ⊣ *p*2 and *q*1 ⊣ *q*2, every context *p* and every nonempty forest *t*, for almost all $n \\in \\Nat$ the formula *φ* is true in some tree of either both or neither of the forests\n \n*s*0 = *p*(*p*1*q*1)*n*(*p*2*q*2)*n**t* *s*1 = *p*(*p*1*q*1)*n**p*1*q*2(*p*2*q*2)*n**t*. \nWe will use the Ehrenfeucht-Fraïssé game, and show that Duplicator can win the *n*-round game over the above two forests. To keep notation simple, we assume the following *simplifying assumptions* are met.\n1. The context *p* is a single node *b* (in particular, *s*0 and *s*1 are trees).\n2. The forest *t* is a single node *a*.\n3. The contexts *p*1, *p*2, *q*1, *q*2 are \n$$\\eqalign{\n p\\_1 &= b\\_1 \\cdots b\\_k\\cr\n q\\_1 &= b\\_{k+1} \\cdots b\\_m \\cr\n }\n \\qquad\n \\eqalign{ \n p\\_2 &= b\\_1(a\\_1 + {\\Box}) \\cdots b\\_k(a\\_k+ {\\Box})\\cr\n q\\_2 &= b\\_{k+1}(a\\_{k+1} + {\\Box}) \\cdots b\\_m(a\\_m+ {\\Box})\\cr\n }$$ \n for some *k* < *m* and *b*1, …, *b**m* ∈ *B*, *a*1, …, *a**m* ∈ *A*.\n4. The labels *a*, *a*1, …, *a**m*, *b*, *b*1, …, *b**m* and *a* are all distinct.\nThe trees *s*0 and *s*1 are shown in Figure [fig:pumping]. Why can we make these simplifying assumptions? The reason is that the general case follows from this special case by way of homomorphic images. More specifically, consider the two forests *s*0, *s*1 in the general case, as given in ([eq:non-diff]). We want to show that Duplicator wins the *n*-round game over these two forests. The key observation is that any two forests *s*0, *s*1 as in ([eq:non-diff]) can obtained as homomorphic images *s*0 = *α*(*t*0) and *s*1 = *α*(*t*1) from trees *t*0, *t*1 that satisfy the simplifying assumption, for some (two-sorted) morphism *α*. As long as we know how Duplicator can win the game over the simpler trees *t*0, *t*1, we can use Proposition [prop:ef-transfer] to transfer this result to the forests *s*0, *s*1.\n![The trees s_0 and s_1](Graphics/ef \"fig:\") [fig:pumping]\nWe now proceed to describe a winning strategy for Duplicator over trees *s*0, *s*1 that satisfy the simplifying assumptions. We use the term *main path* for the ancestors of the node *a*. The *projection* of a node onto the main path is its closest ancestor (not necessarily proper) that is on the main path. For a node in either *s*0 or *s*1, the *ancestor block count* (respectively, *descendant block count*) is the number of ancestors with label *b**m* (respectively, descendants with *b*1) of the node’s projection onto the main path. For *m* ≤ *n*, we say that two nodes *x*0, *x*1 in the trees *s*0, *s*1 are **m*-similar* if their labels are the same and moreover one of the conditions in the following invariant holds:\n1. The trees *s*0, *s*1 agree on nodes in the subtrees of *y*0, *y*1; or\n2. The trees *s*0, *s*1 agree on nodes not in the subtrees of *y*0, *y*1; or\n3. The ancestor and descendant block counts of *x*0, *x*1 are both at least *m*.\nLet *m* ≤ *n*. If the nodes *x*0, *x*1 are *m*-similar, then Duplicator wins the *m*-round game from configuration (*x*0, *x*1, *s*0, *s*1).\nThe proof is by induction on *m*. For the base case *m* = 0 we use the assumption that the labels are the same. Consider now the induction step. We only do one case, when Spoiler chooses a descendant move to go from *x*1 to a node *x*ʹ1 in the “new block” of *s*1 (the new block is the context *p*1*q*2). This Spoiler move means that *x*0, *x*1 are *m*-similar for reason (2) or (3), since item (1) forbids a descendant of *x*1 in the new block. What is Duplicator’s response? Note that for all nodes in the new block, both the ancestor and descendant block counts are at least *n* ≥ *m* − 1. Duplicator goes to any node *x*ʹ0 in the tree *s*0 where the ancestor and descendant block counts are both at least *m* − 1. This must be possible, since either one of items (2) or (3) of the invariant was true for *x*0, *x*1. The rest of the game is played according the induction assumption, since *x*ʹ0 and *x*ʹ1 are (*m* − 1)-similar.\nBy taking *m* = *n* in the above lemma, we get the desired result. This is because the two roots of *s*0, *s*1 have the same (empty) prefixes, thus they are *m*-similar, and must therefore satisfy the same formulas of size *m* = *n*.\nCompleteness\n============\nThis section is devoted to showing:\n[prop:completeness] Any forest language recognized by a forest algebra satisfying ([eq:apcom]), ([eq:da]) and ([eq:special]) can be forest-defined.\nThe above statement immediately implies the more difficult “if” part of Theorem [thm:main]. Indeed, if *L* is recognized by an algebra satisfying ([eq:apcom]), ([eq:da]) and ([eq:special]), then its syntactic algebra satisfies these identities. This is because the syntactic algebra is a morphic image of any algebra recognizing the language, and identities are preserved by morphic images.\nLet *X* ⊆ *H* be a set of forest types. We say a forest *t* is **X*-trimmed* if the only subtrees of *t* that have a type in *X* are leaves. We say a tree language *L* is tree-definable *modulo *X** if there is a formula *φ* such that \n*t*satisfies *φ* iff *t* ∈ *L* \n holds for all *X*-trimmed trees (for other trees, *φ* may disagree with *L*). In a similar fashion, we define a forest language that is forest-definable modulo *X*.\nInstead of Proposition [prop:completeness], we show the slightly more general result below, which contains the induction parameters that appear in the proof.\n[induction:prop] Let *α* : (*A*, *B*)Δ → (*H*, *V*) be a morphism, with (*H*, *V*) satisfying identities ([eq:apcom]), ([eq:da]) and ([eq:special]). Let *X* ⊆ *H* be a set of forest types, and let *v* ∈ *V* be a context type. For each forest type *h* ∈ *H* the following forest language is forest-definable modulo *X*:\n \n$$\\label{eq:main-ind}\n \\set{ t : v(\\alpha( t))= h}\\.$$ \nFor the rest of Section [sec:completeness], we fix *α* : (*A*, *B*)Δ → (*H*, *V*), *h* ∈ *H*, *v* ∈ *V* and *X* ⊆ *H* from Proposition [induction:prop]. Clearly Proposition [prop:completeness] follows from the above result, taking *X* = ∅, *v* to be the empty context type □, and doing a disjunction over all forest types *h* ∈ *α*(*L*). The rest of Section [sec:completeness] is devoted to a proof of Proposition [induction:prop]. The proof is by induction on four parameters:\n1. The size of *H*, i.e. the number of all forest types.\n2. The size of *H* \\ *X*, i.e. the number of forest types that can be found outside leaves.\n3. The size of *v**V*, i.e. the number of context types reachable from *v*.\n4. The size of *B*, i.e. the number of inner node labels.\nThe order of these parameters is important: first we try to minimize *H*, then the other three parameters (the order for the other three is not important). Note that the last parameter depends on the alphabet *B*, and the notion “modulo *X*” depends on the morphism.\nWe say a morphism *α* into (*H*, *V*) is *leaf saturated* if for every *h* ∈ *H*, there is a representative leaf label *a* whose type *α*(*a*) is *h*. In the rest of this section, we will only consider such morphisms. By adding leaf labels, any morphism can be extended to one that is leaf saturated, without affecting the target forest algebra.\nWe begin by outlining our proof strategy for Proposition [induction:prop]. We will consider three possible cases. First, in Section [sec:not-all-labels-preserve], we see what happens when some inner node label *b* ∈ *B* has the property that *v* cannot be reached from *v**α*(*b*). Then, in Section [sec:one-forest-comp], we see what happens if *H* \\ *X* intersects more than one forest component, i.e. contains at least two forest types that are not mutually reachable. Finally, in Section [sec:otherwise], we show that if neither of the above holds, then the formula *φ* in Proposition [induction:prop] can basically be replaced by either “true” or “false”.\nFor some inner node label *b* ∈ *B*, *v* is not reachable from *v**α*(*b*)\n--------------------------------------------------------------------------\nWe begin with this case, which is the easiest of the three. The basic idea is that we cut the forest into two parts, by looking at the first occurrence of *b* on each path, beginning at the root. Since after reading the label *b*, the context type *v* is no longer reachable, we can use the induction assumption to calculate the subtree below each such first *b*. These subtrees can then be squashed into single leafs using the antichain composition principle, and therefore the induction assumption can be used on a smaller alphabet of inner node labels, which now no longer contains *b*.\nWe say that two forest types *h*, *g* ∈ *H* are **v*-equivalent* if *v**u**h* = *v**u**g* holds whenever *v* is not reachable from *v**u*.\n[lemma:conteqplus] For each *h*, the set of forests whose type is *v*-equivalent to *h* is forest-definable modulo *X*.\nFix some context type *u* such that *v* is not reachable from *v**u*. By induction assumption—the third parameter is decreased—the set of forests *s* satisfying *v**u**α*(*s*) = *v**u**h* is forest-definable modulo *X*. The set in the statement of the lemma is the intersection, over *u*, of all these sets.\n[lemma:rgreen-preservers] If *v*, *w*, *v**u* ∈ *V* are in the same context component, then so is *w**u*.\nBy assumption there must be context types *v*ʹ, *w*ʹ with *v**u**w*ʹ = *w* and *w**v*ʹ = *v*. But then we have *w**v*ʹ*u**w*ʹ = *w*. In particular, *w*(*v*ʹ*u**w*ʹ)*ω**v*ʹ = *v*. Using identity ([eq:da]), we get \n*v* = *w*(*v*ʹ*u**w*ʹ)*ω**v*ʹ = *w*(*v*ʹ*u**w*ʹ)*ω**u**w*ʹ(*v*ʹ*u**w*ʹ)*ω**v*ʹ = *w**u**w*ʹ(*v*ʹ*u**w*ʹ)*ω**v*ʹ, \n which shows *v* can be reached from *w**u*.\nLet *γ*1, …, *γ**n* be all the equivalence classes of *v*-equivalence. For each such class *γ**i*, let *L**i* be the set of trees $\\set{bt : \\alpha(t) \\in \\gamma\\_i}$. Thanks to Lemma [lemma:conteqplus], each set *L**i* is tree-definable. For any *i* = 1, …, *n*, let *h**i* be an arbitrarily chosen forest type in the class *γ**i*, and let *a**i* be a leaf label whose type is *α*(*b*)*h**i*. The label *a**i* exists by assumption on leaf saturation. Note that *a**i* may have a different type than some of the trees in *L**i*, since *h**i* need not be the only forest type in *γ**i*. However, we will show that no information is lost by squashing subtree in *L**i* into a single leaf with label *a**i*, at least as long as the resulting forest is going to be an argument of *v*. More formally, we show:\n[lemma:v-squash] Let *φ* = *b* ∧ ¬F− 1*b*, i.e. “a *b* without *b* ancestors”. For any forest *t* we have \n*v**α*(*t*) = *v**α*(*t*[(*L*1, *φ*) → *a*1, …, (*L**n*, *φ*) → *a**n*]). \nBefore we show this lemma, we show how it concludes the case considered in this section. Recall that we want to show that the following language is forest-definable modulo *X*: \n$$L= \\set{t : v\\alpha(t)=h}\\.$$ \n By Lemma [lemma:v-squash], this is the same language as \n$$\\set{t : t[ (L\\_1,\\varphi) \\to a\\_1,\\ldots,\n (L\\_n,\\varphi) \\to a\\_n] \\in L}\\.$$ \n Since the substitution operation removes all letters *b* from the forest, we get \n$$L= \\set{t : t[ (L\\_1,\\varphi) \\to a\\_1,\\ldots,\n (L\\_n,\\varphi) \\to a\\_n] \\in K}\\,$$ \n where *K* is the set of trees in *L* that do not use the letter *b*. To *K* we can apply the induction assumption on a smaller alphabet, and then use the antichain composition principle to transfer definability from *K* to *L*.\nWe now resume with the proof of Lemma [lemma:v-squash].\nNote first that the tree on the right hand side of the equation is well defined, since the languages *L**i* are disjoint, and *φ* is an antichain formula. The proof is by induction on the number of *b* nodes in the forest *t*. The induction base, where there are no *b*’s, is immediate since the substitution on the right hand side does not change the forest. Otherwise, let *t* be of the form *p**b**s*, with the context *p* not containing any *b*’s on the main path, and let *L**i* be such that *b**s* ∈ *L**i*. By induction assumption, we have \n*α*(*p**a**i*[(*L*1, *φ*) → *a*1, …, (*L**n*, *φ*) → *a**n*]) = *α*(*p**a**i*). \n By definition of the substitution we have \n*t*[(*L*1, *φ*) → *a*1, …, (*L**n*, *φ*) → *a**n*] = *p**a**i*[(*L*1, *φ*) → *a*1, …, (*L**n*, *φ*) → *a**n*], \n it therefore remains to show that *v**α*(*p**b**a**i*) = *v**α*(*p**b**s*).\nFirst, we claim that *v* is not reachable from *v**α*(*p**b*). Indeed, if *v* is not reachable from *v**α*(*p*) then we are done. Otherwise, *v* and *v**α*(*p*) are in the same context component. If this context component would also contain *v**α*(*p**b*), then by Lemma [lemma:rgreen-preservers] it would also contain *v**α*(*b*), a contradiction with the assumption on *b*.\nRecall now the forest type *h**i* that represented the equivalence class *γ**i* ∋ *α*(*s*). By assumption on *α*(*s*) and *h**i* being *v*-equivalent, we get \n$$v\\alpha(pbs)=v \\alpha(pb)\\alpha(s)=v \\alpha(pb) h\\_i = v\n \\alpha(p)\\alpha(b)h\\_i = v \\alpha(p) \\alpha(a\\_i) = v \\alpha(pa\\_i)\\ \n.\\eqno{\\qEd}$$ \nThere is more than one forest component in *H* \\ *X*\n----------------------------------------------------\nWe now turn to the second case in the proof of Proposition [induction:prop]. Let *G* ⊆ *H* be a forest component not included in *X*. We pick *G* so that no forest type in *G* can be reached from a forest type outside *X* ∪ *G*. Intuitively speaking, forest types from *G* are close to the leaves. The essential idea in this section is that we will add *G* to *X*, by squashing each subtree of type *g* to a single leaf with the *g* written in its label. This is done by applying the antichain composition.\nLet *W* ⊆ *V* be the set of context types that preserve *G*, i.e. context types *w* such *g* is reachable from *w**g* for some *g* ∈ *G*. The following lemma, proved the same way as Lemma [lemma:rgreen-preservers], shows that “some” in the above definition can be replaced by “all”.\n[lemma:reach-preservers] If *g*, *h*, *v**g* are in the same forest component, then so is *v**h*.\nLet *F* ⊆ *H* be the set of those forest types *f* from which a forest type in *G* can be reached. In particular, we have \n*G* ⊆ *F* ⊆ *H*. \n Note that all forest types in *F* \\ *X* are from *G* by choice of *G*. Furthermore, the inclusion *F* ⊆ *H* is proper, since *H* \\ *X* contains more than one forest component by assumption. The inclusion *G* ⊆ *F* may also be proper, however all forest types in the difference *F* \\ *G* are from *X*.\nWe say *f* ∈ *H* is a *bad brother* if for all *g* ∈ *G*, we have *f* + *g* ∉ *G*, i.e. *g* is not reachable from *f* + *g*. Likewise, we say *f* ∈ *H* is a *good brother* if for all *g* ∈ *G*, we have *f* + *g* ∈ *G*, i.e. *g* is reachable from *f* + *g*. Note that by definition of *F*, all good brothers are in *F*. Clearly *f* is a bad brother if and only if the context type *f* + □ is outside *W*. Therefore by Lemma [lemma:reach-preservers], every forest type in *H* is either a good brother or a bad brother. In particular, all forest types in *G* are good brothers, since they cannot be bad brothers by *g* + *g* = *g*. Furthermore, since *W* is closed under context composition, good brothers are closed under forest concatenation, i.e. form a subsemigroup of *H*.\nWe fix the sets *G*, *F* and *W* for the rest of Section [sec:one-forest-comp].\nA *twig* is a tree of depth exactly two, i.e. a root and some leaves. A *twig node* is a node whose subtree is a twig.\n[lemma:find-h] There is a formula *ψ* such that in any *X*-trimmed tree, *ψ* holds in nodes with a subtree of type in *G*.\nLet *t* be an *X*-trimmed tree, and *x* a node in this tree. If the node is a leaf, then the type of its subtree can be read from the label. Otherwise, the type of the subtree must be either in *G* or outside *F*, by assumption on the tree being *X*-trimmed. We claim that the following condition is necessary and sufficient for the subtree of *x* to have a type outside *F*, and can furthermore be tested by a formula of EF + F− 1. The condition is that some descendant *y* of *x*, not necessarily proper, is either\n1. A leaf or twig node with a type outside *F*; or\n2. A non-twig inner node with a label *b* ∈ *B* whose type *α*(*b*) is outside *W*; or\n3. An inner node whose brother has a leaf label *a* whose type *α*(*a*) is a bad brother.\nWe begin by showing that these conditions can be tested by an EF + F− 1 formula. Testing for 1) is simple. Using EF, we search for a candidate *y* for the node. If *y* is a leaf, we just test its label. Otherwise, we test if *y* is a twig node (no path of length at least two). Then we read the label of *y* and the set of labels in descendants of *y*, which uniquely determine the type of the subtree of *y*, thanks to idempotency and commutativity, i.e. identities ([eq:apcom]). Condition 2 is tested in a similar way. For condition 3 we use EF to go into a leaf *y* with a label *a* whose type is a bad brother. We then test if *y* has a sibling that is an inner node (all ancestors of *y* have an inner node descendant).\nWe now show that these conditions are sufficient. The first one is clearly sufficient. For the other two, note that every inner node has a subtree with type outside *X* by assumption on the tree being *X*-trimmed. This type must then be either in *G* ⊇ *F* \\ *X* or outside *F*. For the second condition, let *b**s* be the subtree of a non-twig inner node, with *s* a forest. Since *s* has depth at least two, its type must be outside *X*, and therefore either outside *F*, or in *G* ⊇ *F* \\ *X*. In either case, the type of *b**s* is outside *F*. The last condition is shown in a similar way.\nIt remains to show that the conditions are necessary. Indeed, assume that the subtree of *x* has a type outside *G*. Let *s* be a minimal subtree below *x* that has a type outside *F*. If *s* is a leaf or a twig, then item 1 must hold. Otherwise *s* is of the form *b*(*s*1 + ⋯*s**n*), for some label *b* ∈ *B* and trees *s*1, …, *s**n*, with at least one tree *s**i* not being a leaf. By assumption on the tree being *X*-trimmed, the type of *s**i* is outside *X*. Since *F* \\ *X* ⊆ *G*, the type of this *s**i* is in *G*. If all the types of *s**j*, for *j* ≠ *i*, are good brothers, then the type of *s*1 + ⋯ + *s**n* must belong to *G* by closure of good brothers under composition, and therefore case 2 must hold. Finally, we consider the case when the type of some tree *s**j* is a bad brother. Since all forest types from *G* are good brothers, the type of *s**j* is in *F* \\ *G* ⊆ *X*. Since the tree is *X*-trimmed, *s**j* is a single leaf, and thus 3 holds.\n[lemma:describe-h] For each *g* ∈ *G*, the set of trees with type *g* is tree-definable modulo *X*.\nThe general idea is that (*G*, *W*) is a (smaller) forest algebra, and therefore the induction assumption can be applied to languages recognized by (*G*, *W*). However, thanks to bad brothers and such, (*G*, *W*) does not recognize the language in the lemma. Before we solve this problem, we show how Lemmas [lemma:find-h] and [lemma:describe-h] along with the antichain composition principle conclude the case considered in this section. The idea is that we add all forest types from *G* to *X*.\nLet *h*, *v* be as in the statement of Proposition [induction:prop]. We need to show that the language \n$$L = \\set{ t : v(\\alpha( t)) = h}$$ \n is forest-definable modulo *X*. By induction assumption, we know that this language is forest-definable modulo *X* ∪ *G*. In other words, there is some forest-definable set of forests *K* that agrees with *L* over (*X* ∪ *G*)-trimmed forests. To describe *L* modulo *X*, we will use the antichain composition principle.\nLet *ψ* be the formula from Lemma [lemma:find-h]. Let \n*φ* = *ψ* ∧ ¬F− 1*ψ*. \n This formula holds in a node whose subtree has a type in *G*, and the node is closest to the root for this property. Thanks to the last clause, *φ* is an antichain formula. Let $G=\\set{g\\_1,\\ldots,g\\_n}$. By assumption that *α* is leaf saturated, for each *g**i* there is a leaf label *a**i* ∈ *A* with *α*(*a**i*) = *g**i*. For each *g**i*, let *L**i* be the set of trees with type *g**i*. Thanks to Lemma [lemma:describe-h], each tree language *L**i* is tree-definable modulo *X*.\nIt is easy to see that squashing a subtree with type *g**i* into a single leaf with label *a**i* does not change the type of the whole tree. More precisely, a forest *t* has the same value as \n*t*[(*L*1, *φ*) → *a*1, …, (*L**n*, *φ*) → *a**n*]. \n Furthermore, the above forest is (*X* ∪ *G*)-trimmed, at least as long as *t* was *X*-trimmed. It follows that over *X*-trimmed forests, *L* agrees with \n$$\\set{ t : t[ (L\\_1,\\varphi) \\to a\\_1,\\ldots, (L\\_n,\\varphi)\n \\to a\\_n] \\in K}\\,$$ \n which is forest-definable thanks to the antichain composition principle. It now remains to show Lemma [lemma:describe-h], which we do in the next section.\n### Trees with type in *G*.\nFix some forest type *g* ∈ *G*. Our goal is to show that the set of trees with type *g* is tree-definable modulo *X*.\n[lemma:loss-of-generality] Without loss of generality, we may assume that all forest types in *F* are good brothers and all inner node labels *b* satisfy *α*(*b*) ∈ *W*.\nRecall that all forest types from *G* are good brothers. In particular, all bad brothers in *F* are from *X*, and can therefore only appear in leaves, as long as we are working over *X*-trimmed forests. Let *A*ʹ ⊆ *A* be the set of leaf labels that are mapped by *α* to a good brother in *F*. Let *B*ʹ ⊆ *B* be the set of inner node labels *b* with *α*(*b*) ∈ *W*.\nLet *β* be the restriction of *α* to this smaller alphabet: \n*β* : (*A*ʹ, *B*ʹ)Δ → (*H*, *V*). \n Note that over *X*-trimmed forests, the only forest types from *F* in the image of *β* are good brothers, and all inner node labels *b* satisfy *α*(*b*) ∈ *W*. Assume now, that we have shown Lemma [lemma:describe-h] for the morphism *β*, i.e. the set *K* of trees that have type *g* under *β* is tree-definable modulo *X*. We will use the antichain composition principle to extend this result to *α*. The idea is that we squash twig nodes into leaves, thus eliminating labels outside *A*ʹ, *B*ʹ.\nLet *φ* be a formula that is true in twig nodes (the node is not a leaf, but all of its proper descendants are leaves); this is clearly an antichain formula. Let $G=\\set{g\\_1,\\ldots,g\\_n}$. By assumption that *α* is leaf saturated, for each *g**i* there is a leaf label *a**i* ∈ *A* with *α*(*a**i*) = *g**i*. For each *g**i*, let *L**i* be the set of twig trees with value *g**i* (under *α*). Each *L**i* is tree-definable, since the type of a twig tree is determined by its root label and the set of its leaf labels by ([eq:apcom]). It is easy to see that a tree *t* over (*A*, *B*) has the same type under *α* as the tree \n*t*[(*L*1, *φ*) → *a*1, …, (*L**n*, *φ*) → *a**n*]. \n Furthermore, if the type of *t* under *α* is *g*, then the latter forest belongs to the domain of *β*, since all nodes with a label outside *A*ʹ or *B*ʹ are covered by *φ*. Therefore, we can use the antichain composition principle to conclude that the forests with value *g* under *α* can be defined in EF + F− 1.\nFrom now on, we use the assumptions stated in the previous lemma. Recall that good brothers are closed under concatenation, and therefore *F* is a subsemigroup of *H*. This allows us to define a semigroup automaton $\\Aa$, whose semigroup is *F*. The input alphabet of this automaton is:\n1. The inner node labels are *B*\n2. The leaf labels are $A'=\\set{a \\in A : \\alpha(a) \\in F}$.\nFor *a* ∈ *A*ʹ, we define *β**A*(*a*) to be *α*(*a*). For *b* ∈ *B*, we would like the associated function *β**B*(*b*) to be *α*(*b*). Even though Lemma [lemma:loss-of-generality] guarantees that *α*(*b*) belongs to *W*, this context type cannot be used since it need not generate a function *F* → *F*. The reason is that *α*(*b*)*h* may be outside *F* for types *h* outside *G*. To solve this problem, we artificially redefine the function:\n \n$$\\label{eq:redefine}\n \\beta\\_B(b)(h) = \\left\\{\n \\begin{array}{rcl}\n \\alpha(b)h & & \\text{ if }\\alpha(b)h \\in F\\\\\n g\\_0 & & \\text{ otherwise}.\n \\end{array}\\right.$$ \nIn the above, *g*0 is an arbitrarily chosen forest type from *G*.\nBy the proof of Theorem [thm:equiv-with-regul], this automaton induces a forest algebra morphism \n*β* : (*A*ʹ, *B*ʹ)Δ → (*F*, *F**F*). \n This morphism is not the same as *α*, due to the second clause in ([eq:redefine]). However, it agrees with *α* over the forests that are relevant to Lemma [lemma:describe-h]:\n[lemma:new-agrees-with-old] For any *g* ∈ *G*, and forest *t*, if *α*(*t*) = *g* then *β*(*t*) = *g*.\nIf *t* has a type in *G* under *α*, then all of its leaf labels belong to *A*ʹ by definition of *F*. Therefore, *t* belongs to the domain of *β*. The lemma is proved by induction on the size of *t*. If *α*(*t*) = *g*, then the “bad” second case in ([eq:redefine]) is never used while calculating *β*(*t*).\n[lemma:new-is-ok] The image of *β* satisfies identities ([eq:apcom]), ([eq:da]) and ([eq:special]).\nWe only focus on identity ([eq:special]), the others are easy to show. The key idea is that *α* and *β* only disagree in twig nodes, and these are not important for the identity ([eq:special]).\nLet then *p*1 ⊣ *p*2, *q*1 ⊣ *q*2 be contexts. We need to show that \n*β*((*p*1*q*1)*ω*(*p*2*q*2)*ω*) = *β*((*p*1*q*1)*ω**p*1*q*2(*p*2*q*2)*ω*). \n Thanks to the faithfulness of contexts in forest algebra, it suffices to show that both sides induce the same transformations on forests, i.e. \n*β*((*p*1*q*1)*ω*(*p*2*q*2)*ω**t*) = *β*((*p*1*q*1)*ω**p*1*q*2(*p*2*q*2)*ω**t*) \n holds for every forest *t*.\nConsider first the case when both *p*2, *q*2 have the hole in the root, and therefore so do *p*1, *q*2. In this case the equality above becomes: \n*β*(*ω*(*s*1 + *t*1) + *ω*(*s*2 + *t*2) + *t*) = *β*(*ω*(*s*1 + *t*1) + *s*1 + *t*2 + *ω*(*s*2 + *t*2) + *t*). \n The above equality follows by commutativity of the horizontal monoid *F*, and aperiodicity of *H*, i.e. *ω**h* + *h* = *ω**h*. The latter is a consequence of aperiodicity of *V*, itself a consequence of ([eq:da]), by iterating \n*ω**h* + *h* = (*h* + □)*ω**h* = (*h* + □)*ω*(*h* + □)*ω**h* = *ω**h* + *h* + *h*. \n We can therefore now assume that in the context *p*2*q*2, at least one inner node is an ancestor of the hole. Thanks to the assumption on leaf saturation, in the contexts *p*1, *q*1, *p*2, *q*2 every subtree that does not contain the hole can be squashed to a single node, without affecting the image under *β*. We therefore assume that in the contexts above, all nodes outside the main path are leaves. As remarked above, a consequence of equation ([eq:da]) is that *V* is aperiodic, i.e. *v**ω* = *v**ω**v* holds for every context type *v*. Therefore, it is sufficient to show\n \n$$\\label{eq:redefined-ok}\n \\begin{array}{rcl}\n \\beta((p\\_1 q\\_1 )^\\omega (p\\_2 q\\_2 )^\\omega(p\\_2 q\\_2 ) t)=\n \\beta( (p\\_1 q\\_1 )^\\omega p\\_1 q\\_2 (p\\_2 q\\_2 )^\\omega (p\\_2 q\\_2 )\n t)\\.\n \\end{array}$$ \nThe only part where *α* and *β* disagree are twig nodes. Thanks to our assumption on the form of *p*1, *p*2, *q*1, *q*2, the only place where the forests in ([eq:redefined-ok]) contain twig nodes is *p*2*q*2*t*. Therefore, we have \n*β*((*p*1*q*1)*ω*(*p*2*q*2)*ω*(*p*2*q*2)*t*) = *α*((*p*1*q*1)*ω*(*p*2*q*2)*ω*)*β*((*p*2*q*2)*t*). \n In the same way we can decompose the right side of ([eq:redefined-ok]). Applying the assumption that the image of *α* satisfies ([eq:special]), we get the desired result.\n[Proof of Lemma [lemma:describe-h]] By Lemma [lemma:new-agrees-with-old], a tree has type *g* under *α* if and only if a) its type under *α* belongs to *G*; and b) it has type *g* under *β*. Condition a) can be tested by thanks to Lemma [lemma:find-h]. Since *F* is a proper subset of *H*, we can use the induction assumption to test condition b).\nThe induction base\n------------------\nIn this section, we assume that the techniques from the previous two sections cannot be applied. That is:\n1. All forest types from *H* \\ *X* are in a single forest component.\n2. For all inner node labels *b* ∈ *B*, *v* is reachable from *v**α*(*b*).\nNote that the second assumption does not necessarily mean that any context type reachable from *v* is in the same context component. Indeed, it is possible that for some forest type *g*, the context type *v* is no longer reachable from *v*(□ + *g*).\nWe will show\n \n*v**f* = *v**g* for all *f*, *g* ∈ *H* \\ *X*. \nBefore we do this, we show how Proposition [induction:prop] follows. For every every forest type *h* ∈ *H*, we need to show that the forest language \n$$L = \\set{ t : v(\\alpha( t)) = h}$$ \n is forest definable modulo *X*. By assumption ([eq:one-component]), there is some forest type *h*0 ∈ *H* such that *v**f* = *h*0 holds for all *f* ∈ *H* \\ *X*.\n1. If an *X*-trimmed forest *t* contains an inner node label—which can easily be tested by the logic—then *α*(*t*) must be in the single forest component *H* \\ *X*. In particular, *v**α*(*t*) = *h*0. So in this case, *φ* is either “true” or “false” depending on whether *h*0 = *h* or not.\n2. Otherwise, the forest *t* is the concatenation of some leaves *a*1 + ⋯ + *a**n*. In this case, the type of *v**α*(*t*) can be calculated based on the set of leaf labels in *t*.\nThe rest of this section is devoted to showing ([eq:one-component]). The following lemma is the key step in our proof ([eq:one-component]). It says that not only any two forest types *h*, *g* ∈ *H* \\ *X* can be reached from each other—which is the assumption on there being one forest component—but they can also be reached from each other by only using contexts without any branching. Furthermore, the context type that goes from *g* to *h* can be chosen independently of *g*. However, all these statements are relative to context types from the context component of *v*.\n[lemma:letter-reach] Let *h* ∈ *H* \\ *X*. There are inner node labels *b*1, …, *b**n* ∈ *B* such that *w**h* = *w**α*(*b*1⋯*b**n*)*g* holds for each forest type *g* ∈ *H* \\ *X* and context type *w* in the context component of *v*.\nLet *h* be a forest type outside *X*. We first show that there is a context type *u**g* such that *h* = *u**h**f* holds for every forest type *f* ∈ *H*. By assumption on there being only one forest component outside *X*, the forest type *h* can be reached from every forest type. In particular, there is some context type *u* such that *h* = *u*(*h*1 + ⋯ + *h**n*), where *h*1, …, *h**n* are all the forest types in *H*. Let \n*u**h* = *u*(*h*1 + ⋯ + *h**n* + □). \n Thanks to idempotency and commutativity of *H*, i.e. identity ([eq:apcom]), \n*h*1 + ⋯ + *h**n* = *h*1 + ⋯ + *h**n* + *f* \n holds for any forest type *f*, and therefore also *h* = *u**h**f*.\nWe can decompose the context *u**h* as \n*u**h* = (*f*1 + *α*(*b*1))⋯(*f**n* + *α*(*b**n*)) \n for some *n* and *f*1, …, *f**n* ∈ *H* and *b*1, …, *b**n* ∈ *B*. (In general, some of the *f**i* may be empty; but the proof follows the same lines.) Let us denote *α*(*b**i*) by *v**i*. We will show that \n*w**h* = *w**v*1⋯*v**n**g* \n holds for any forest type *g* and any context type *w* in the context component of *v*, thus proving the lemma.\nLet then *g*, *w* be as above. As for *h*, we can define a context type *u**g* such that *g* = *u**g**f* holds for any forest type *f*. This context can also be decomposed as \n*u**g* = (*f**n* + 1 + *α*(*b**n* + 1))⋯(*f**m* + *α*(*b**m*)) \n for some *m* ≥ *n* + 1 and *f**n* + 1, …, *f**m* ∈ *H* and *b**n* + 1, …, *b**m* ∈ *B*. As previously, we denote *α*(*b**i*) by *v**i*. By definition, we have\n \n$$\\begin{aligned}\n\\label{eq:mainle}\n v\\_1 \\cdots v\\_n {\\dashv}u\\_h \\qquad v\\_{n+1} \\cdots v\\_m {\\dashv}u\\_g\n \\end{aligned}$$ \nLet now *w* ∈ *V* be in the same context component as *v*. By assumption on *w* and Lemma [lemma:rgreen-preservers], also the context type *w**v*1⋯*v**m* is in the same context component as *v*. In particular, there is some $ \\bar w \\in V$ such that \n$$w v\\_1 \\cdots v\\_m \\bar w = w\\.$$ \n By iterating the above *ω* times, and appending *h*, we get \n$$wh = w(v\\_1 \\cdots v\\_m \\bar w)^\\omega h\\.$$ \n Since *u**h**f* = *h* holds for all forest types *f*, the above can be rewritten as \n$$w(v\\_1 \\cdots v\\_m \\bar w)^\\omega (u\\_hu\\_g \\bar w)^\\omega h\\.$$ \n Using the property from identity ([eq:special]), we get \n$$\\eqalign{\n w(v\\_1\\cdots v\\_m\\bar w)^\\omega (u\\_h u\\_g\\bar w)^\\omega h\n&= w(v\\_1\\cdots v\\_m\\bar w)^\\omega v\\_1 \\cdots v\\_n u\\_g\\bar w (u\\_hu\\_g \n v)^\\omega h\\cr\n&= w(v\\_1 \\cdots v\\_m \\bar w)^\\omega v\\_1 \\cdots v\\_n g=\n\\bar wv\\_1\\cdots v\\_n g\\,\n }$$ \n which concludes the proof of the lemma.\nWe now use the above Lemma to conclude the proof of ([eq:one-component]). Indeed, let *f*, *g* be forest types outside *X*. By the above lemma, there are inner node labels *b*1, …, *b**m* ∈ *B* such that \n*f* = *w**α*(*b*1⋯*b**n*)*h* *g* = *w**α*(*b**n* + 1⋯*b**m*)*h* \n holds for all *w* in the context component of *v* and all forest types *h* outside *X*. Let *v**i* = *α*(*b**i*). By assumption on the equivalence class of *v* and by Lemma [lemma:rgreen-preservers], there must be some *v* ∈ *V* such that \n$$v v\\_1 \\cdots v\\_m \\bar v = v\\.$$ \n But then we have \n$$vf = v(v\\_1 \\cdots v\\_m \\bar v)^\\omega f = \n v(v\\_1 \\cdots v\\_m \\bar v)^\\omega v\\_{n+1} \\cdots v\\_m \\bar v (v\\_1\n \\cdots v\\_m \\bar v)^\\omega f = \n v(v\\_1 \\cdots v\\_m \\bar v)^\\omega g = vg\\.$$ \n The second equality follows from ([eq:da]).\nEmpty forests\n=============\nThe forest algebra setting used in this paper does not allow empty forests. There is also a two-sorted alphabet (*A*, *B*), where letters from *A* are only allowed in leaves, and letters from *B* are only allowed in inner nodes. A different, and arguably more elegant, setting is considered in , where empty forests are allowed, and only one alphabet is used.\nWhy do we not use the forest algebra with empty forests here? The reason is that the completeness proof in Proposition [induction:prop] uses an induction on the size of the leaf alphabet, so it helps that the leaf alphabet is part of the definition of the forest algebra. The assumption on nonempty forests follows, since if we want a separate alphabet for leaves, there are algebraic reasons to consider forest algebras without the empty forest. A natural question emerges: does our characterization also work for forest algebra with empty forests? In this section, we give an informal argument that the answer to this question is yes.\nWe will not give a detailed discussion of forest algebra with empty forests here. We define only define the syntactic object. The interested reader is referred to . Let *A* be an alphabet. We define *A**H*Δ (respectively, *A**V*Δ) to be the set of (possibly) empty forests (respectively, contexts) labeled by *A*, without any restriction on labels in leaves or inner nodes. We write *A*Δ for the pair (*A**H*Δ, *A**V*Δ). The only difference between *A*Δ and (*A*, *A*)Δ is that the second does not allow the empty forest on its first coordinate. It is not hard to see that *A*Δ is a forest algebra, as defined in Section [sec:forest-algebra]. Given a set *L* of forests, possibly including the empty forest, the *syntactic forest algebra with empty forests of *L** is defined to be the quotient of *A*Δ under the two-sorted equivalence relation defined below. \n$$\\eqalign{\n t &\\simeq t'\\cr\n p &\\simeq p'\\cr\n }\n \\qquad\n \\eqalign{\n {\\text{ iff }}&\\qquad \\forall p \\in A^\\Delta\\_V \\ \\ pt \\in\n L\\ \\iff\\ pt' \\in L\\cr\n {\\text{ iff }}&\\qquad \\forall q \\in A^\\Delta\\_V \\forall s\n \\in A^\\Delta\\_H \\ \\ qps \\in L\\ \\iff\\ qp's \\in L\\cr\n }$$ \n This equivalence relation is a refinement of the Myhill-Nerode equivalence introduced in Section [sec:forest-algebra] (for the case when *A* = *B*). It may possibly distinguish more contexts because the variable *s* can also quantify over the empty forest.\nLet *L* be a forest language. Let (*H*, *V*) be its syntactic forest algebra, and let (*H*ʹ, *V*ʹ) be its syntactic forest algebra with empty forests. If (*H*, *V*) satisfies the identities from Theorem [thm:main], then so does (*H*ʹ, *V*ʹ), and vice versa.\nWe begin with the right to left implication. Since (*A*, *A*)Δ is a subalgebra of *A*Δ, and since the equivalence relation defining (*H*ʹ, *V*ʹ) is a refinement of the equivalence relation defining (*H*, *V*), it follows that (*H*, *V*) is a subalgebra of (*H*ʹ, *V*ʹ). In particular, any identities that hold in the latter must also hold in the former.\nFor the left to right implication, assume that (*H*, *V*) satisfies the identities from Theorem [thm:main]. By the theorem, the recognized language *L* is forest-definable in EF + F− 1. To conclude, we will show that if a language *L* is forest-definable in EF + F− 1, then its syntactic forest algebra with empty forests (*H*ʹ, *V*ʹ) satisfies the identities from Theorem [thm:main]. This follows by the correctness argument presented in Section [sec:correctness]. The reason why we can use that argument is that it relied on Proposition [prop:ef-transfer] to transfer Duplicator strategies, and this proposition also works for the more general one-sorted morphisms that are appropriate for forest algebras with empty forests.\nOne quantifier alternation\n==========================\nIn , it was shown that over words, the temporal logic F + F− 1 has the same expressive power as Σ2 ∩ Π2, where\n1. Σ2 are word properties definable by a first-order formula with quantifier prefix ∃\\*∀\\*; the signature contains label tests and the left-to-right order on word positions.\n2. Π2 are complements of Σ2.\nFor instance, consider the word language *b*\\**a**A*\\* over the alphabet $A=\\set{a,b,c}$. This language can be defined in F + F− 1 by the formula \nF(*a* ∧ ¬F− 1¬*b*). \n This language can also be defined both in Σ2 and Π2, as witnessed by the formulas: \n$$\\begin{array}{lll}\n \\exists x \\forall y \\ &\\ a(x) \\ \\land \\ (y < x\\ \\Rightarrow \\ b(y)) & \\qquad \\in \\Sigma\\_2\\\\\n \\forall x \\exists y \\ &\\ c(x)\\ \\Rightarrow \\ (y < x \\ \\land \\ a(y))& \\qquad \\in \\Pi\\_2\\.\n \\end{array}$$ \n Both classes Σ2 and Π2 can be extended to trees using the descendant order on tree nodes. We show here that the result from fails for trees:\nOver trees, the classes EF + F− 1 and Σ2 ∩ Π2 have incomparable expressive power. Likewise for forests.\nA mentioned in the introduction, the class Σ2 ∩ Π2 was given an effective characterization in . We prove the above proposition for forests, the case for trees is done the same way. The inequality \n$${{\\mathsf{EF}}+ {\\mathsf{F}}^{-1}}\\supsetneq \\Sigma\\_2 \\cap \\Pi\\_2$$ \n is witnessed by the language “three nodes with label *a*”, which cannot be defined in EF + F− 1 by virtue of ([eq:apcom]). To show the remaining inequality \n$${{\\mathsf{EF}}+ {\\mathsf{F}}^{-1}}\\subsetneq \\Sigma\\_2 \\cap \\Pi\\_2\\,$$ \n we will demonstrate in the following lemma that the forest property “no root node is a leaf” cannot be defined in Σ2, although it is forest-definable in EF + F− 1.\n[lemma:delta2-pumping] Let *a* be a leaf label, *b* an inner node label, and *φ* be a formula of the form \n∃*x*1…*x**i*∀*y*1…*y**j**ψ*(*x*1…*x**i*, *y*1…*y**j*) ∈ Σ2, \n with *ψ* quantifier-free. Let *n* > *i* + *j*. If *n*(*b**a*) satisfies *φ*, then so does *n*(*b**a*) + *a*.\nAssume then that *n*(*b**a*) satisfies *φ*. We need to show that *n*(*b**a*) + *a* does too. For *x*1, …, *x**i*, we pick the same nodes in *n*(*b**a*) + *a* as the nodes in *n*(*b**a*) that witnessed *φ*. We need to show that for any assignment of the nodes *y*1, …, *y**j* in *n*(*b**a*) + *a* that makes *ψ* false, we also can find an assignment in *n*(*b**a*) that makes *ψ* false. The key point is that any assignment of *x*1, …, *x**i*, *y*1, …, *y**j* in *n*(*b**a*) + *a* must leave at least one copy of *b**a* without any variables; this copy can be used in *n*(*b**a*) to simulate *a*.\nClosing remarks\n===============\nThe contribution of this paper is a characterization of languages definable in EF + F− 1. This characterization is expressed in terms of identities that must be satisfied in the syntactic algebra. A corollary of this characterization is an algorithm for deciding if a given regular language can be expressed in EF + F− 1. The algorithm runs in polynomial time if the input is given as a forest algebra.\nAs mentioned in the introduction, there are many open problems waiting to be solved in this field. Of those closely related to EF + F− 1, the following look interesting:\n1. What are the identities for two-variable first-order logic with the descendant relation? The question boils down to: what identity should replace idempotency *h* + *h* = *h*? Here is one candidate: *v*(*h* + *h*) + *v**h* = *v**h* + *v**h*.\n2. What are the identities for an extension of EF + F− 1, where we allow operators of the form EF*k**φ*, with the meaning: “the current node has *k* incomparable descendants where *φ* holds”. This seems to be a reasonable extension of EF + F− 1 that is capable of counting in a proper way (recall that two-variable logic could express the property “there are two *a*’s”, but not the property “there are three *a*’s”).\nIt is conceivable that a modification of the techniques developed in this paper can be sufficient to solve the above two logics. For other logics mentioned in this paper, such as full first-order logic, or even variants of EF + F− 1 with horizontal order, new techniques need to be developed.\n10\nM. Benedikt and L. Segoufin. Regular languages definable in FO. In *Symposium on Theoretical Aspects of Computer Science*, volume 3404 of *Lecture Notes in Computer Science*, pages 327 – 339, 2005.\nM. Bojańczyk.. PhD thesis, Warsaw University, 2004.\nM. Bojańczyk and L. Segoufin. Tree languages definable with one quantifier alternation. Submitted.\nM. Bojańczyk, L. Segoufin, and H. Straubing. Piecewise testable tree languages. Logic in Computer Science, 2008.\nM. Bojańczyk and I. Walukiewicz. Characterizing EF and EX tree logics., 358(2-3):255–273, 2006.\nM. Bojańczyk and I. Walukiewicz. Forest algebras. In *Automata and Logic: History and Perspectives*, pages 107 – 132. Amsterdam University Press, 2007.\nU. Heuter. First-order properties of trees, star-free expressions, and aperiodicity. In *Symposium on Theoretical Aspects of Computer Science*, volume 294 of *Lecture Notes in Computer Science*, pages 136–148, 1988.\nR. McNaughton and S. Papert.. MIT Press, 1971.\nJ.-É. Pin. Logic, semigroups and automata on words., 16:343–384, 1996.\nA. Potthoff. First-order logic on finite trees. In *Theory and Practice of Software Development*, volume 915 of *Lecture Notes in Computer Science*, pages 125–139, 1995.\nM. P. Schützenberger. On finite monoids having only trivial subgroups., 8:190–194, 1965.\nP. Tesson and D. Thérien. Diamonds are forever: the variety da. In *Semigroups, Algorithms, Automata and Languages*, pages 475–500, 2002.\nD. Thérien and T. Wilke. Over words, two variables are as powerful as one quantifier alternation. In *ACM Symposium on the Theory of Computing*, pages 256–263, 1998.\nW. Thomas. On chain logic, path logic, and first-order logic over infinite trees. In *Logic in Computer Science*, pages 245–256, 1987.\nT. Wilke. Classifying discrete temporal properties. In *Symposium on Theoretical Aspects of Computer Science*, volume 1563 of *Lecture Notes in Computer Science*, pages 32–46, 1999.\n---\n1. \\* It turns out that in forest algebra, the first generalization entails the second.[↩](#fnref1)\n"},"data_id":{"kind":"string","value":"arxiv_0002030"}}},{"rowIdx":492,"cells":{"content":{"kind":"string","value":"Sparse random graphs with clustering\n====================================\n### October 21, 2009\nIn 2007 we introduced a general model of sparse random graphs with (conditional) independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with (conditionally) independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (non-Poisson) multi-type branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with power-law degree sequences with a wide range of degree exponents and clustering coefficients.\nIntroduction and results\n========================\nIn , a very general model for sparse random graphs was introduced, corresponding to an inhomogeneous version of *G*(*n*, *c*/*n*), and many properties of this model were determined, in particular, the critical point of the phase transition where the giant component emerges. Part of the motivation was to unify many of the new random graph models introduced as approximations to real-world networks. Indeed, the model of includes many of these models as exact special cases, as well as the ‘mean-field’ simplified versions of many of the more complicated models. (The original forms are frequently too complex for rigorous mathematical analysis, so such mean-field versions are often studied instead.) Unfortunately, there are many models with key features that are not captured by their mean-field versions, and hence not by the model of . The main problem is that many real-world networks exhibit *clustering*: for example, while there are *n* vertices and only 5*n* edges, there may be 10*n* triangles, say. In contrast, the model of , like *G*(*n*, *c*/*n*), produces graphs that contain essentially no triangles or short cycles.\nMost models introduced to approximate particular real-world networks turn out to be mathematically intractable, due to the dependence between edges. Nevertheless, many such models have been studied; as this is not our main focus, let us just list a few examples of early work in this field. One of the starting points in this area was the (homogeneous) ‘small-world’ model of Watts and Strogatz . Another was the observation of power-law degree sequences in various networks by Faloutsos, Faloutsos and Faloutsos , among others. Of the new inhomogeneous models, perhaps the most studied is the ‘growth with preferential attachment’ model introduced in an imprecise form by Barabási and Albert , later made precise as the ‘LCD model’ by Bollobás and Riordan . Another is the ‘copying’ model of Kumar, Raghavan, Rajagopalan, Sivakumar, Tomkins and Upfal , generalized by Cooper and Frieze , among others. For (early) surveys of work in this field see, for example, Barabási and Albert , Dorogovtsev and Mendes , or Bollobás and Riordan .\nRoughly speaking, any sparse model with clustering must include significant dependence between edges, so one might expect it to be impossible to construct a general model of this type that is still mathematically tractable. However, it turns out that one can do this. The model that we shall define is essentially a generalization of that in , although we shall handle certain technicalities in a different way here.\nThroughout this paper we use standard graph theoretic notation as in . For example, if *G* is a graph then *V*(*G*) denotes its vertex set, *E*(*G*) its edge set, ∣*G*∣ the number of vertices, and *e*(*G*) the number of edges. We also use standard notation for probabilistic asymptotics as in : a sequence ${\\mathcal E}\\_n$ of events holds *with high probability*, or *whp*, if ${{\\mathop{\\mathbb P{}}\\nolimits}}({\\mathcal E}\\_n)\\to 1$ as *n* → ∞. If (*X**n*) is a sequence of random variables and *f* is a deterministic function, then *X**n* = *o*p(*f*(*n*)) means $X\\_n/f(n){\\overset{\\mathrm{p}}{\\to}}0$, where ${\\overset{\\mathrm{p}}{\\to}}$ denotes convergence in probability.\nThe model\n---------\nLet us set the scene for our model. By a *type space* we simply mean a probability space $({{\\mathcal S}},\\mu)$. Often, we shall take ${{\\mathcal S}}=[0,1]$ or (0, 1] with *μ* Lebesgue measure. Sometimes we consider ${{\\mathcal S}}$ finite. As will become clear, any model with ${{\\mathcal S}}$ finite can be realized as a model with type space [0, 1], but sometimes the notation will be simpler with ${{\\mathcal S}}$ finite. More generally, as shown in , every instance of the random graph model we are going to describe can be realized as an equivalent model with type space [0, 1]. Hence, when it comes to proofs, we lose no generality by taking ${{\\mathcal S}}=[0,1]$, but we usually prefer allowing an arbitrary type space, which is more flexible for applications. For example, as with the model in , type spaces such as ${{\\mathcal S}}=[0,1]^2$ are likely to be useful for geometric applications, as in .\nLet ${{\\mathcal F}}$ consist of one representative of each isomorphism class of finite connected graphs, chosen so that if $F\\in {{\\mathcal F}}$ has *r* vertices then *V*(*F*) = [*r*] = {1, 2, …, *r*}. Given $F\\in {{\\mathcal F}}$ with *r* vertices, let *κ**F* be a measurable function from ${{\\mathcal S}}^{r}$ to [0, ∞); we call *κ**F* the *kernel* corresponding to *F*. A sequence ${{\\undertilde{{\\kappa}}}}=({\\kappa}\\_F)\\_{F\\in{{\\mathcal F}}}$ is a *kernel family*. In our results we shall impose an additional integrability condition on ${{\\undertilde{{\\kappa}}}}$, but this is not needed to define the model.\nLet ${{\\undertilde{{\\kappa}}}}$ be a kernel family and *n* an integer; we shall define a random graph $G(n,{{\\undertilde{{\\kappa}}}})$ with vertex set [*n*] = {1, 2, …, *n*}. First let $x\\_1,x\\_2,\\ldots,x\\_n\\in {{\\mathcal S}}$ be i.i.d. (independent and identically distributed) with the distribution *μ*. Given ${{\\bf x}}=(x\\_1,\\ldots,x\\_n)$, construct $G(n,{{\\undertilde{{\\kappa}}}})$ as follows, starting with the empty graph. For each *r* and each $F\\in {{\\mathcal F}}$ with ∣*F*∣ = *r*, and for every *r*-tuple of distinct vertices (*v*1, …, *v**r*) ∈ [*n*]*r*, add a copy of *F* on the vertices *v*1, …, *v**r* (with vertex *i* of *F* mapped to *v**i*) with probability\n \n$$\\label{pdef}\n p=\np(v\\_1,\\dots,v\\_{r};F)=\n\\frac{{\\kappa}\\_F(x\\_{v\\_1},\\ldots,x\\_{v\\_{r}})}{ n^{{r}-1}},$$ \nall these choices being independent. If *p* > 1, then we simply add a copy with probability 1. We shall often call the added copies of the various *F* that together form $G(n,{{\\undertilde{{\\kappa}}}})$ *atoms* since, in our construction of $G(n,{{\\undertilde{{\\kappa}}}})$, they may be viewed as indivisible building blocks. Sometimes we refer to them as *small graphs*, although there is in general no bound on their sizes. Usually we think of $G(n,{{\\undertilde{{\\kappa}}}})$ as a simple graph, in which case we simply replace any multiple edges by single edges. Typically there will be very few multiple edges, so this makes little difference.\nNote that we assume that the atoms of $G(n,{{\\undertilde{{\\kappa}}}})$ are connected. The extension to the case where some atoms may be disconnected is discussed in Section [Sdisconnected].\nThe reason for dividing by *n**r* − 1 in is that we wish to consider sparse graphs; indeed, our main interest is the case when $G(n,{{\\undertilde{{\\kappa}}}})$ has *O*(*n*) edges. As it turns out, we can be slightly more general; however, when *κ**F* is integrable (which we shall always assume), the expected number of added copies of each graph *F* is *O*(*n*). Note that all incompletely specified integrals are with respect to the appropriate *r*-fold product measure *μ**r* on ${{\\mathcal S}}^r$.\nThere are several plausible choices for the normalization in. The one we have chosen means that if *κ**F* = *c* is constant, then (asymptotically) there are on average *c**n* copies of *F* in total, and each vertex is on average in *r**c* copies of *F*. An alternative is to divide the expression in by *r*; then (asymptotically) each vertex would on average be in *c* copies of *F*. Another alternative, natural when adding cliques only but less so in the general case, would be to divide by *r*!; this is equivalent to considering unordered sets of *r* vertices instead of ordered *r*-tuples. When there is only one kernel, corresponding to adding edges, this would correspond to the normalization used in, and in particular to that of the classical model *G*(*n*, *c*/*n*); the normalization we use here differs from this by a factor of 2. Yet another normalization would be to divide by aut(*F*), the number of automorphisms of *F*; this is equivalent to considering the distinct copies of *F* in *K**n*, which is natural but leads to extra factors aut(*F*) in many formulae, and we do not find that the advantages outweigh the disadvantages.\nAs in , there are several minor variants of $G(n,{{\\undertilde{{\\kappa}}}})$; perhaps the most important is the *Poisson multi-graph* version of $G(n,{{\\undertilde{{\\kappa}}}})$. In this variant, for each *F* and each *r*-tuple, we add a Poisson Po(*p*) number of copies of *F* with this vertex set, where *p* is given by, and we keep multiple edges.\nAlternatively, we could add a Poisson number of copies and delete multiple edges, which is the same as adding one copy with probability 1 − *e*− *p* and no copy otherwise. More generally, we could add one copy of *F* with probability *p* + *o*(*p*), and two or more copies with probability *o*(*p*). As long as the error terms are uniform over graphs *F* and *r*-tuples (*v*1, …, *v**r*), all our results will apply in this greater generality. Since this will follow by simple sandwiching arguments (after reducing to the ‘bounded’ case; see Definition [Dbounded]), we shall consider whichever form of the model is most convenient; usually this turns out to be the Poisson multi-graph form.\nUnder certain mild conditions, the results of imply a strong form of asymptotic equivalence between the various versions of the model. For example, if we add copies of *F* with probability *p* + *O*(*p*2), where the implied constant is uniform over *F* and (*v*1, …, *v**r*), and\n \n$$\\label{ea}\n {{\\mathop{\\mathbb E{}}\\nolimits}}\\sum\\_F \\sum\\_{v\\_1,\\dots,v\\_{|F|}} p(v\\_1,\\dots,v\\_{|F|};F)^3 =o(1),$$ \nthen the resulting model is equivalent to that with probability *p*, in that the two random graphs can be coupled to agree whp; this is a straightforward modification of. Extending the argument in, it can be shown that holds if\n \n$$\\sum\\_{F\\in{{\\mathcal F}}} \\int\\_{{{\\mathcal S}}^{|F|}} {\\kappa}\\_F^{{|F|}/({|F|}-1)} <\\infty.$$ \nThis certainly holds for the bounded kernel families (see Definition [Dbounded]) that we consider in most of our proofs, although is easy to verify directly for such kernel families.\nIn the special case where all *κ**F* are zero apart from *κ**K*2, the kernel corresponding to an edge, we recover (essentially) a special case of the model of ; we call this the *edge-only* case, since we add only edges, not larger graphs. We write *κ*2 for *κ**K*2. Note that in the edge-only case, given ${{\\bf x}}$, two vertices *i* and *j* are joined with probability\n \n$$\\label{2sym}\n \\frac{{\\kappa}\\_2(x\\_i,x\\_j)+{\\kappa}\\_2(x\\_j,x\\_i)}{n}\n +O\\left(\\frac{({\\kappa}\\_2(x\\_i,x\\_j)+{\\kappa}\\_2(x\\_j,x\\_i))^2}{n^2}\\right).$$ \nThe correction term will never matter, so we may as well replace *κ*2 by its symmetrized version. In fact, we shall always assume that *κ**F* is invariant under the action of the automorphism group Aut(*F*) of the graph *F*. In other words, if *ϕ* : [*r*] → [*r*] is a permutation such that *ϕ*(*i*)*ϕ*(*j*) ∈ *E*(*F*) if and only if *i**j* ∈ *E*(*F*), then we assume that *κ**F*(*ϕ*(*x*1), …, *ϕ*(*x**r*)) = *κ**F*(*x*1, …, *x**r*) for all $x\\_1,\\ldots,x\\_r\\in{{\\mathcal S}}$. In the Poisson version, or if we add copies of graphs *F* with probability 1 − *e*− *p*, the correction terms in and its generalizations disappear: in the edge-only case, given ${{\\bf x}}$, vertices *i* and *j* are joined with probability $1-\\exp{\\bigl(-({\\kappa}\\_2(x\\_i,x\\_j)+{\\kappa}\\_2(x\\_j,x\\_i))/n\\bigr)}$, and in general we obtain exactly the same random graph if we symmetrize each *κ**F* with respect to Aut(*F*).\nFor any kernel family ${{\\undertilde{{\\kappa}}}}$, let ${{{\\kappa}\\_{\\mathrm e}}}$ be the corresponding *edge kernel*, defined by\n \n$$\\label{kaedef}\n {{{\\kappa}\\_{\\mathrm e}}}(x,y) = \\sum\\_F \\sum\\_{ij\\in E(F)}\n\\int\\_{{{\\mathcal S}}^{V(F)\\setminus \\{i,j\\}}}\n {\\kappa}\\_F(x\\_1,\\ldots,x\\_{i-1},x,x\\_{i+1},\\ldots,x\\_{j-1},y,x\\_{j+1},\\ldots,x\\_{|F|}),$$ \nwhere the second sum runs over all 2*e*(*F*) ordered pairs (*i*, *j*) with *i**j* ∈ *E*(*F*), and we integrate over all variables apart from *x* and *y*. Note that the sum need not always converge; since every term is positive this causes no problems: we simply allow ${{{\\kappa}\\_{\\mathrm e}}}(x,y)=\\infty$ for some *x*, *y*. Given *x**i* and *x**j*, the probability that *i* and *j* are joined in $G(n,{{\\undertilde{{\\kappa}}}})$ is at most ${{{\\kappa}\\_{\\mathrm e}}}(x\\_i,x\\_j)/n$, and this upper bound is typically quite sharp. For example, if ${{\\undertilde{{\\kappa}}}}$ is bounded in the sense of Definition [Dbounded] below, then the probability is ${{{\\kappa}\\_{\\mathrm e}}}(x\\_i,x\\_j)/n+O(1/n^2)$. In other words, ${{{\\kappa}\\_{\\mathrm e}}}$ captures the edge probabilities in $G(n,{{\\undertilde{{\\kappa}}}})$, but not the correlations.\nBefore proceeding to deeper properties, let us note that the expected number of added copies of *F* is $(1+O(n{^{-1}}))n\\int\\_{{{\\mathcal S}}^{|F|}} {\\kappa}\\_F$. Unsurprisingly, the actual number turns out to be concentrated about this mean. Let\n \n$${{\\xi}({{\\undertilde{{\\kappa}}}})}= \\sum\\_{F\\in{{\\mathcal F}}} e(F)\\int\\_{{{\\mathcal S}}^{|F|}} {\\kappa}\\_F \n=\\frac12\\int\\_{{{\\mathcal S}}^2} {{{\\kappa}\\_{\\mathrm e}}}\\le\\infty$$ \nbe the *asymptotic edge density* of ${{\\undertilde{{\\kappa}}}}$. Since every copy of *F* contributes *e*(*F*) edges, the following theorem is almost obvious, provided we can ignore overlapping edges. A formal proof will be given in Section [secsubgraphs]. (A similar result for the total number of atoms is given in Lemma [badv].)\n[Tedges] As *n* → ∞, $e({\\ensuremath{G(n,{{\\undertilde{{\\kappa}}}})}})/n$ converges in probability to the asymptotic edge density ${{\\xi}({{\\undertilde{{\\kappa}}}})}$. In other words, if ${{\\xi}({{\\undertilde{{\\kappa}}}})}<\\infty$ then $e({\\ensuremath{G(n,{{\\undertilde{{\\kappa}}}})}})={{\\xi}({{\\undertilde{{\\kappa}}}})}n+{{o\\_{\\mathrm{p}}}(n)}$, and if ${{\\xi}({{\\undertilde{{\\kappa}}}})}=\\infty$ then, for every constant *C*, we have $e({\\ensuremath{G(n,{{\\undertilde{{\\kappa}}}})}})>Cn$ whp. Moreover, ${{\\mathop{\\mathbb E{}}\\nolimits}}e({\\ensuremath{G(n,{{\\undertilde{{\\kappa}}}})}})/n\\to{{\\xi}({{\\undertilde{{\\kappa}}}})}\\le\\infty$\nAs in , our main focus will be the emergence of the giant component. By the *component structure* of a graph *G*, we mean the set of vertex sets of its components, i.e., the structure encoding only which vertices are in the same component, not the internal structure of the components themselves. When studying the component structure of $G(n,{{\\undertilde{{\\kappa}}}})$, the model can be simplified somewhat. Recalling that the atoms $F\\in {{\\mathcal F}}$ are connected by definition, when we add an atom *F* to a graph *G*, the effect on the component structure is simply to unite all components of *G* that meet the vertex set of *F*, so only the vertex set of *F* matters, not its graph structure. We say that ${{\\undertilde{{\\kappa}}}}$ is a *clique kernel family* if the only non-zero kernels are those corresponding to complete graphs; the corresponding random graph model $G(n,{{\\undertilde{{\\kappa}}}})$ is a *clique model*. For questions concerning component structure, it suffices to study clique models. For clique kernels we write *κ**r* for *κ**K**r*; as above, we always assume that *κ**r* is symmetric, here meaning invariant under all permutations of the coordinates of ${{\\mathcal S}}^{r}$. Given a general kernel family ${{\\undertilde{{\\kappa}}}}$, the corresponding (symmetrized) clique kernel family is given by ${\\bar{{{\\undertilde{{\\kappa}}}}}}=({\\kappa}\\_{r})\\_{{r}\\ge2}$ with\n \n$$\\label{hk}\n {\\kappa}\\_{r}(x\\_1,\\ldots,x\\_{r}) =\n \\sum\\_{F\\in {{\\mathcal F}}: |F|={r}} \n \\frac{1}{{r}!}\n \\sum\\_{\\pi\\in {\\mathfrak S\\_r}}\n {\\kappa}\\_F(x\\_{\\pi(1)},\\ldots,x\\_{\\pi({r})}),$$ \nwhere ${\\mathfrak S\\_r}$ denotes the symmetric group of permutations of [*r*]. (This is consistent with our notation *κ*2 = *κ**K*2.) In the Poisson version, with or without merging of parallel edges, the probability of adding some connected graph *F* on a given set of *r* vertices is exactly the same in $G(n,{{\\undertilde{{\\kappa}}}})$ and $G(n,{\\bar{{{\\undertilde{{\\kappa}}}}}})$, so there is a natural coupling of these random graphs in which they have exactly the same component structure. In the non-Poisson version, the probabilities are not quite the same, but close enough for our results to transfer from one to the other. Thus, when considering the size (meaning number of vertices) of the giant component in $G(n,{{\\undertilde{{\\kappa}}}})$, we may always replace ${{\\undertilde{{\\kappa}}}}$ by the corresponding clique kernel family.\nIt is often convenient to think of a clique model as a random hypergraph, with the cliques as the hyperedges; for this reason we call a clique kernel family a *hyperkernel*. Note that each unordered set of *r* vertices corresponds to *r*! *r*-tuples, so the probability that we add a *K**r* on a given set of *r* vertices is *r*!*κ**r*(*x**v*1, …, *x**v**r*)/*n**r* − 1. (More precisely, this is the expected number of *K**r*s added with this vertex set.)\nA branching process\n-------------------\nAssociated to each hyperkernel ${{\\undertilde{{\\kappa}}}}=({\\kappa}\\_{r})\\_{{r}\\ge 2}$, there is a branching process ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$ with type space ${{\\mathcal S}}$, defined as follows. We start with generation 0 consisting of a single particle whose type is chosen randomly from ${{\\mathcal S}}$ according to the distribution *μ*. A particle *P* of type *x* gives rise to children in the next generation according to a two-step process: first, for each *r* ≥ 2, construct a Poisson process *Z**r* on ${{\\mathcal S}}^{{r}-1}$ with intensity\n \n*r**κ**r*(*x*, *x*2, …, *x**r*) *d**μ*(*x*2)⋯ *d**μ*(*x**r*). \nWe call the points of *Z* = ⋃*r* ≥ 2*Z**r* the *child cliques* of *P*. There are *r* − 1 children of *P* for each child clique $(x\\_2,\\ldots,x\\_{r})\\in {{\\mathcal S}}^{{r}-1}$, one each of types *x*2, …, *x**r*. Thus the types of the children of *P* form a multiset on ${{\\mathcal S}}$, with a certain compound Poisson distribution we have just described. As usual, the children of different particles are independent of each other, and of the history.\nConsidering the relationship to the graph $G(n,{{\\undertilde{{\\kappa}}}})$, the initial factor *r* in arises because a particular vertex *v* may be any one of the *r* vertices in an *r*-tuple (*v*1, …, *v**r*) on which we add a *K**r*.\nWe also consider the branching processes ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}(x)$, $x\\in{{\\mathcal S}}$, defined exactly as ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$, except that we start with a single particle of the given type *x*.\nTwo integral operators\n----------------------\nWe shall consider two integral operators naturally associated to ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$. Given any (measurable) $f:{{\\mathcal S}}\\to [0,1]$, define ${S\\_{{{\\undertilde{{\\kappa}}}}}}(f)$ by\n \n$$\\begin{gathered}\n\\label{Sk}\n {S\\_{{{\\undertilde{{\\kappa}}}}}}(f)(x) \n\\\\\n= \\sum\\_{{r}=2}^\\infty \\int\\_{{{\\mathcal S}}^{{r}-1}} {r}{\\kappa}\\_{r}(x,x\\_2,x\\_3,\\ldots,x\\_{r}) \n \\left(1-\\prod\\_{i=2}^{{r}} (1-f(x\\_i))\\right) {\\,d}\\mu(x\\_2)\\cdots{\\,d}\\mu(x\\_{r}), \\end{gathered}$$ \nand let \n$${\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}(f)(x) = 1-e^{-{S\\_{{{\\undertilde{{\\kappa}}}}}}(f)(x)}.$$ \n (The factors *r* in and in the definition of ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$ are unfortunate consequences of our choice of normalization.)\nLet *P* be a particle of ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$ in generation *t* with type *x*, and suppose that each particle in generation *t* + 1 of type *y* has some property ${\\mathcal Q}$ with probability *f*(*y*), independently of the other particles. Given a child clique (*x*2, …, *x**r*) of *P*, the bracket in the definition of ${S\\_{{{\\undertilde{{\\kappa}}}}}}$ expresses the probability that one or more of the *r* − 1 corresponding child particles has property ${\\mathcal Q}$. Hence ${S\\_{{{\\undertilde{{\\kappa}}}}}}(f)(x)$ is the expected number of child cliques containing a particle with property ${\\mathcal Q}$, and, from the Poisson distribution of the child cliques, ${\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}(f)(x)$ is the probability that there is at least one such clique, i.e., the probability that at least one child of *P* has property ${\\mathcal Q}$.\nLet $\\rho({{\\undertilde{{\\kappa}}}})$ denote the survival probability of the branching process ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$, and $\\rho\\_{{\\undertilde{{\\kappa}}}}(x)$ the survival probability of ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}(x)$. Assuming for the moment that the function $\\rho\\_{{\\undertilde{{\\kappa}}}}:{{\\mathcal S}}\\to[0,1]$ is measurable, from the comments above and the independence built into the definition of ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$, we see that the function $\\rho\\_{{\\undertilde{{\\kappa}}}}$ satisfies \n$$\\rho\\_{{\\undertilde{{\\kappa}}}}= {\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}(\\rho\\_{{\\undertilde{{\\kappa}}}}).$$ \n Using simple standard arguments as in , for example, it is easy to check that $\\rho\\_{{\\undertilde{{\\kappa}}}}$ is given by the maximum solution to this equation, i.e., the pointwise supremum of all solutions $f:{{\\mathcal S}}\\to [0,1]$ to\n \n$$\\label{fS}\n f = 1-e^{-{S\\_{{{\\undertilde{{\\kappa}}}}}}(f)};$$ \nsee Lemma [lmax] below. From the definitions of ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$ and ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}(x)$, it is immediate that \n$$\\rho({{\\undertilde{{\\kappa}}}}) = \\int\\_{{\\mathcal S}}\\rho\\_{{\\undertilde{{\\kappa}}}}(x) {\\,d}\\mu(x).$$ \nIn our analysis we shall also consider the linear operator ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}$ defined by\n \n$$\\label{Tkdef}\n {T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}(f)(x) = \\int\\_{{\\mathcal S}}{{{\\kappa}\\_{\\mathrm e}}}(x,y)f(y){\\,d}\\mu(y),$$ \nwhere ${{{\\kappa}\\_{\\mathrm e}}}$ is defined by. For a hyperkernel ${{\\undertilde{{\\kappa}}}}$ (which is the only type of kernel family for which we define the branching process), we have\n \n$$\\label{ce}\n {{{\\kappa}\\_{\\mathrm e}}}(x,y) = \\sum\\_{{r}\\ge 2}{r}({r}-1)\n \\int\\_{{{\\mathcal S}}^{{r}-2}}{\\kappa}\\_{r}(x,y,x\\_3,x\\_4,\\ldots,x\\_{r})\n {\\,d}\\mu(x\\_3)\\cdots{\\,d}\\mu(x\\_{r}),$$ \nfrom which it is easy to check that ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}$ is the linearized form of ${S\\_{{{\\undertilde{{\\kappa}}}}}}$: more precisely, ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}$ is obtained by replacing 1 − ∏*i* = 2*r*(1 − *f*(*x**i*)) by ∑*i* = 2*r**f*(*x**i*) in the definition of ${S\\_{{{\\undertilde{{\\kappa}}}}}}$.\nLet us note two simple consequences of this fact. For any sequence (*y**i*)*i* in [0, 1] we have 1 − ∏*i*(1 − *y**i*) ≤ ∑*i**y**i*, so\n \n$$\\label{st}\n0\\le{S\\_{{{\\undertilde{{\\kappa}}}}}}(f)\\le{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}(f)$$ \nfor any $f:{{\\mathcal S}}\\to [0,1]$. Also, 1 − ∏*i*(1 − *y**i*) > 0 if and only if ∑*i**y**i* > 0. Since the integral of a non-negative function is positive if and only if the function is positive on a set of positive measure, it follows that for any $f:{{\\mathcal S}}\\to [0,1]$ we have\n \n$$\\label{SzTz}\n {S\\_{{{\\undertilde{{\\kappa}}}}}}(f)(x)>0 \\iff {T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}(f)(x)>0.$$ \nIn the edge-only case, when only *κ*2 is non-zero, ${{{\\kappa}\\_{\\mathrm e}}}=2{\\kappa}\\_2$ and ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}={S\\_{{{\\undertilde{{\\kappa}}}}}}$. When translating results from , it is sometimes ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}$ and sometimes ${S\\_{{{\\undertilde{{\\kappa}}}}}}$ that plays the role of the linear operator *T**κ* appearing there.\nMain results\n------------\nIn most of our results we shall need to impose some sort of integrability condition on our kernel family; the exact condition depends on the context.\n[Dint]\n[Dwint] A kernel family ${{\\undertilde{{\\kappa}}}}=({\\kappa}\\_F)\\_{F\\in{{\\mathcal F}}}$ is *integrable* if\n \n$$\\label{wint}\n {\\int {{\\undertilde{{\\kappa}}}}} = \\sum\\_{F\\in{{\\mathcal F}}} |F|\\int\\_{{{\\mathcal S}}^{|F|}} {\\kappa}\\_F <\\infty.$$ \nThis means that the expected number of atoms containing a given vertex is bounded.\n[Deint] A kernel family ${{\\undertilde{{\\kappa}}}}=({\\kappa}\\_F)\\_{F\\in{{\\mathcal F}}}$ is *edge integrable* if\n \n$$\\sum\\_{F\\in{{\\mathcal F}}} e(F)\\int\\_{{{\\mathcal S}}^{|F|}} {\\kappa}\\_F <\\infty;$$ \nequivalently, ${{\\xi}({{\\undertilde{{\\kappa}}}})}<\\infty$ or $\\int\\_{{{\\mathcal S}}^2}{{{\\kappa}\\_{\\mathrm e}}}<\\infty$. This means that the expected number of edges in ${\\ensuremath{G(n,{{\\undertilde{{\\kappa}}}})}}$ is *O*(*n*), see Theorem [Tedges], and thus the expected degree of a given vertex is bounded.\nNote that a hyperkernel (*κ**r*) is integrable if and only if $ \\sum\\_{r\\ge2} r\\int\\_{{{\\mathcal S}}^{r}} {\\kappa}\\_r <\\infty$, and edge integrable if and only if $ \\sum\\_{r\\ge2} r^2\\int\\_{{{\\mathcal S}}^{r}} {\\kappa}\\_r <\\infty$.\nSince we only consider connected atoms *F*, it is clear that\n \nedge integrable $\\implies$ integrable. \nOur main result is that if ${{\\undertilde{{\\kappa}}}}$ is an integrable kernel family satisfying a certain extra assumption, then the normalized size of the giant component in $G(n,{{\\undertilde{{\\kappa}}}})$ is simply $\\rho({{\\undertilde{{\\kappa}}}})+{o\\_{\\mathrm{p}}}(1)$. The extra assumption is essentially that the graph does not split into two pieces. As in, we say that a symmetric kernel ${{{\\kappa}\\_{\\mathrm e}}}:{{\\mathcal S}}^2\\to [0,\\infty)$ is *reducible* if\n \n$$\\text{$\\exists A\\subset {{\\mathcal S}}$ with\n$0<\\mu(A)<1$ such that ${{{\\kappa}\\_{\\mathrm e}}}=0$ a.e.\\ on $A\\times({{\\mathcal S}}\\setminus\nA)$};$$ \notherwise ${{{\\kappa}\\_{\\mathrm e}}}$ is *irreducible*. Thus ${{{\\kappa}\\_{\\mathrm e}}}$ is irreducible if\n \n$$\\text{$A\\subseteq {{\\mathcal S}}$ and ${{{\\kappa}\\_{\\mathrm e}}}=0$ a.e.\\ on $A\\times({{\\mathcal S}}\\setminus A)$\nimplies $\\mu(A)=0$ or $\\mu({{\\mathcal S}}\\setminus A)=0$}.$$ \nA kernel family $({\\kappa}\\_F)\\_{F\\in{{\\mathcal F}}}$ or hyperkernel (*κ**r*)*r* ≥ 2 is *irreducible* if the corresponding edge kernel ${{{\\kappa}\\_{\\mathrm e}}}$ is irreducible. It is easy to check that a kernel family $({\\kappa}\\_F)\\_{F\\in{{\\mathcal F}}}$ is irreducible if and only if for every $A\\subset {{\\mathcal S}}$ with 0 < *μ*(*A*) < 1 there exists an $F\\in{{\\mathcal F}}$ such that, with *r* = ∣*F*∣, if *x*1, …, *x**r* are chosen independently at random in ${{\\mathcal S}}$ with distribution *μ*, then there is a positive probability that {*x**i*} ∩ *A* ≠ ∅, ${\\ensuremath{\\{x\\_i\\}}}\\cap({{\\mathcal S}}\\setminus A)\\neq\\emptyset$ and *κ**F*(*x*1, …, *x**r*) > 0. Informally, $({\\kappa}\\_F)\\_{F\\in{{\\mathcal F}}}$ is irreducible if, whenever we partition the type space into two non-trivial parts, edges between vertices with types in the two parts are possible.\nNote that a kernel family ${{{\\undertilde{{\\kappa}}}}'}$ and the corresponding hyperkernel ${{\\undertilde{{\\kappa}}}}$ do *not* have the same edge kernel: replacing each atom by a clique in general adds edges, so ${{{\\kappa}\\_{\\mathrm e}'}}\\le {{{\\kappa}\\_{\\mathrm e}}}$ with strict inequality possible. If ${{{\\kappa}\\_{\\mathrm e}'}}$ is irreducible, then so is ${{{\\kappa}\\_{\\mathrm e}}}$; using the characterization of irreducibility above, it is easy to check that the reverse implication also holds.\nWe are now ready to state our main result; we write *C**i* for the number of vertices in the *i*th largest component of a graph *G*.\n[th1] Let ${{{\\undertilde{{\\kappa}}}}'}=({\\kappa}'\\_F)\\_{F\\in {{\\mathcal F}}}$ be an irreducible, integrable kernel family, and let ${{\\undertilde{{\\kappa}}}}=({\\kappa}\\_{r})\\_{{r}\\ge2}$ be the corresponding hyperkernel, given by. Then \n$$C\\_1{\\bigl(G(n,{{{\\undertilde{{\\kappa}}}}'})\\bigr)} = \\rho({{\\undertilde{{\\kappa}}}})n +{o\\_{\\mathrm{p}}}(n),$$ \n and $C\\_2{\\bigl(G(n,{{{\\undertilde{{\\kappa}}}}'})\\bigr)} ={o\\_{\\mathrm{p}}}(n)$.\nThe reducible case reduces to the irreducible one; see Remark [Rred].\nUnsurprisingly, part of the proof of Theorem [th1] involves showing that (in the hyperkernel case) the branching process captures the ‘local structure’ of $G(n,{{\\undertilde{{\\kappa}}}})$; see Section [secloc] and in particular Lemma [nkint]. So Theorem [th1] can be seen as saying that *within this broad class of models* the local structure determines the size of the giant component. Of course, the restriction is important, as shown by the fact that the global assumption of irreducibility is necessary.\nOf course, for Theorem [th1] to be useful we would like to know something about the survival probability $\\rho({{\\undertilde{{\\kappa}}}})$; as noted earlier, $\\rho({{\\undertilde{{\\kappa}}}})$ can be calculated from $\\rho\\_{{\\undertilde{{\\kappa}}}}$, which is in turn the largest solution to a certain functional equation . Of course, the main thing we would like to know is when $\\rho({{\\undertilde{{\\kappa}}}})$ is positive; as in , it turns out that the answer depends on the *L*2-norm ${\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}\\|}}\\le\\infty$ of the operator ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}$ defined by. (Since this operator is symmetric, its *L*2-norm is the same as its spectral radius. In other contexts, it may be better to work with the latter.)\n[th2] Let ${{\\undertilde{{\\kappa}}}}$ be an integrable hyperkernel. Then $\\rho({{\\undertilde{{\\kappa}}}})>0$ if and only if ${\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}\\|}}>1$. Furthermore, if ${{\\undertilde{{\\kappa}}}}$ is irreducible and ${\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}\\|}}>1$, then $\\rho\\_{{\\undertilde{{\\kappa}}}}(x)$ is the unique non-zero solution to the functional equation, and $\\rho\\_{{\\undertilde{{\\kappa}}}}(x)>0$ holds for a.e. *x*.\nIn general, ${\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}\\|}}$ may be rather hard to calculate; a non-trivial example where we can calculate the norm easily is given in Subsection [ssegg]. Let us give a trivial example here: suppose that each *κ**r* is constant, say *κ**r* = *c**r*. Then ${{{\\kappa}\\_{\\mathrm e}}}(x,y) = \\sum\\_r r(r-1)c\\_r = 2{\\xi}({\\kappa})$ for all *x* and *y*, so\n \n$$\\label{unif}\n {\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}\\|}}=2{\\xi}({\\kappa}).$$ \nThis is perhaps surprising: it tells us that for such uniform hyperkernels, the critical point where a giant component emerges is determined only by the total number of edges added; it does not matter what size cliques they lie in, even though, for example, the third edge in every triangle is ‘wasted’. This is not true for arbitrary kernel families: we must first replace each atom by a clique.\nNote that for any hyperkernel, \n$${\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}\\|}}\\ge \\langle 1,{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}1\\rangle = \\int {{{\\kappa}\\_{\\mathrm e}}}= 2{\\xi}({\\kappa}),$$ \n with equality if and only if 1 is an eigenfunction, i.e., if the asymptotic expected degrees ${\\lambda}(x)=\\int\\_{{\\mathcal S}}{{{\\kappa}\\_{\\mathrm e}}}(x,y){\\,d}\\mu(y)$ are the same (ignoring sets of measure 0); c.f. .\nRelationship to the results in\n------------------------------\nIn the edge-only case, the present results are almost (see below) special cases of those . The set-up here is much simpler, as we choose to insist that the vertex types *x*1, …, *x**n* are i.i.d. This avoids many of the complications arising in . In one way, the present set-up is, even in the edge-only case, more general than that considered in : with the types i.i.d., there is no need to restrict the kernels other than to assume integrability (in we needed them continuous a.e.), and one does not need to impose the ‘graphicality’ assumption of . Thus the edge-only case here actually complements the results in . We could form a common generalization, but we shall not do this in detail; we believe that it is just a question of combining the various technicalities here and in , and that no interesting new difficulties arise. Of course, these technicalities are rather beside the point of the present paper; our interest is the extension from kernels to hyperkernels. This turns out not to be as straightforward as one might perhaps expect. The problem is that the correlation between edges forces us to deal with a non-linear operator, namely ${S\\_{{{\\undertilde{{\\kappa}}}}}}$.\nThe rest of the paper is organized as follows. In the next section we prove the results about the non-Poisson branching process ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$ that we shall need later, the most important of which is Theorem [th2]. In Section [secloc] we consider the local coupling between the graph and the branching process, showing in particular that the ‘right’ number of vertices are in components of any fixed size. In Section [secgiant] we complete the proof of Theorem [th1], showing that whp there is at most one ‘large’ component, which is then a ‘giant’ component of the right size. We briefly discuss percolation on the graphs $G(n,{{\\undertilde{{\\kappa}}}})$ in Section [Sdisconnected]. In Sections [secdegrees] and [secsubgraphs] we consider simpler properties of $G(n,{{\\undertilde{{\\kappa}}}})$, namely the asymptotic degree distribution and the number of subgraphs isomorphic to a given graph. Our results in Section [secsubgraphs] include Theorem [Tedges] as a simple special case. In Section [secpl] we illustrate the flexibility of the model by carrying out explicit calculations for a special case, giving graphs with power-law degree sequences with a range of exponents and a range of clustering and mixing coefficients; see Section [secpl] for the definitions of these coefficients. Finally, in Section [seclim] we discuss connections between our model and various notions of graph limit, and state two open questions.\nAnalysis of the branching process\n=================================\nIn this section, which is the heart of the paper, we forget about graphs, and study the (compound Poisson) branching process ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$. One might expect the arguments of to carry over *mutatis mutandis* to the present context, but in the branching process analysis this is very far from the truth; this applies especially to the proof of Theorem [unique] below.\nThroughout the section we work with an integrable hyperkernel ${{\\undertilde{{\\kappa}}}}=({\\kappa}\\_{r})\\_{{r}\\ge 2}$, i.e., we assume that ${\\int {{\\undertilde{{\\kappa}}}}} = \\sum\\_r r\\int{\\kappa}\\_r<\\infty$. Our main aim in this section is to prove Theorem [th2].\nFor $x\\in {{\\mathcal S}}$ let \n$${\\lambda}(x) = ({S\\_{{{\\undertilde{{\\kappa}}}}}}(1))(x) \n= \\sum\\_{{r}=2}^\\infty \\int\\_{{{\\mathcal S}}^{{r}-1}} {r}{\\kappa}\\_{r}(x,x\\_2,x\\_3,\\ldots,x\\_{r}) \n {\\,d}\\mu(x\\_2)\\cdots{\\,d}\\mu(x\\_{r}),$$ \n so *λ*(*x*) is the expected number of child cliques of a particle of type *x*. We have \n$$\\int\\_{{\\mathcal S}}{\\lambda}(x) {\\,d}\\mu(x) \n= \\sum\\_{{r}\\ge 2} \\int\\_{{{\\mathcal S}}^{r}} {r}{\\kappa}\\_{r}= {\\int {{\\undertilde{{\\kappa}}}}},$$ \n which is finite by our integrability assumption. It follows that *λ*(*x*) < ∞ holds almost everywhere. Changing each kernel *κ**r* on a set of measure zero, we may assume that *λ*(*x*) is finite for all *x*. (Such a change is irrelevant for the branching process and for the graph.) From now on, we thus assume that *λ*(*x*) < ∞ holds for all *x*, for any hyperkernel ${{\\undertilde{{\\kappa}}}}$ we consider.\nSince a Poisson random variable with finite mean is always finite, any particle in ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$ has a finite number of child cliques, and hence a finite number of children, even though the expected number of children may perhaps be infinite. Hence, the event that the branching process dies out (i.e., that some generation is empty) coincides with the event that it is finite.\nUsing this fact, we have the following, standard result. Recall that $\\rho\\_{{\\undertilde{{\\kappa}}}}(x)$ denotes the survival probability of the branching process ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}(x)$ that starts with a single particle of type *x*, and $\\rho\\_{{\\undertilde{{\\kappa}}}}$ denotes the function $x\\mapsto \\rho\\_{{\\undertilde{{\\kappa}}}}(x)$.\n[lmax] The function $\\rho\\_{{\\undertilde{{\\kappa}}}}$ satisfies the functional equation. Furthermore, if $f:{{\\mathcal S}}\\to[0,1]$ is any other solution to, then $0\\le f(x)\\le \\rho\\_{{\\undertilde{{\\kappa}}}}(x)<1$ holds for every *x*.\nLet *ρ**t*(*x*) be the probability that ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}(x)$ survives for at least *t* generations, so *ρ*0 is identically 1. Conditioned on the set of child cliques, and hence children, of the root, each child of type *y* survives for *t* further generations with probability *ρ**t*(*y*). These events are independent for different children by the definition of the branching process, so $\\rho\\_{t+1}={\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}(\\rho\\_t)$. The result follows from the monotonicity of ${\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}$ and the fact that $\\rho\\_t(x){\\searrow}\\rho\\_{{\\undertilde{{\\kappa}}}}(x)$, noting that ${\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}(1)(x)=1-e^{-\\lambda(x)}<1$ for the strict inequality.\nLet us remark for the last time on the measurability of the functions we consider: in the proof above, *ρ*0 is measurable by definition. From the definition of ${\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}$ and the measurability of each *κ**k*, it follows by induction that each *ρ**t* is measurable, and hence that $\\rho\\_{{\\undertilde{{\\kappa}}}}$ is. Similar arguments apply in many places later, but we shall omit them.\nWe next turn to the uniqueness of the non-zero solution (if any) to. The key ingredient in establishing this is the following simple inequality concerning the non-linear operator ${S\\_{{{\\undertilde{{\\kappa}}}}}}$.\n[lfSg] Let ${{\\undertilde{{\\kappa}}}}$ be an integrable hyperkernel, and let *f* and *g* be measurable functions on ${{\\mathcal S}}$ with 0 ≤ *f* ≤ *g* ≤ 1. Then \n$$\\int\\_{{\\mathcal S}}f {S\\_{{{\\undertilde{{\\kappa}}}}}}g \\le \\int\\_{{\\mathcal S}}g {S\\_{{{\\undertilde{{\\kappa}}}}}}f.$$ \nWe may write ${S\\_{{{\\undertilde{{\\kappa}}}}}}$ as ∑*r* ≥ 2*S**r*, where *S**r* is the non-linear operator corresponding to the single kernel *κ**r*, so *S**r*(*f*) is defined by the summand in. It suffices to prove that\n \n$$\\label{fSkg}\n \\int\\_{{\\mathcal S}}f S\\_{r}g \\le \\int\\_{{\\mathcal S}}g S\\_{r}f.$$ \nWe shall in fact show that for any (distinct) $x\\_1,\\ldots,x\\_{r}\\in {{\\mathcal S}}$ we have\n \n$$\\label{pointwise}\n \\sum\\_{\\pi\\in {\\mathfrak S\\_r}} \n f(x\\_{\\pi(1)})\\left(1- \\prod\\_{i=2}^{r}{\\bigl(1-g(x\\_{\\pi(i)})\\bigr)}\\right)\n\\le\n \\sum\\_{\\pi\\in {\\mathfrak S\\_r}} \n g(x\\_{\\pi(1)})\\left(1- \\prod\\_{i=2}^{r}{\\bigl(1-f(x\\_{\\pi(i)})\\bigr)}\\right)$$ \nSince *κ**r* is symmetric, follows. (In fact, can be true in general only if always holds, considering the symmetrization of a delta function.) Now can be viewed as an inequality in 2*r* variables *f*(*x*1), …, *f*(*x**r*), *g*(*x*1), …, *g*(*x**r*). This inequality is linear in each variable. Furthermore, it is linear in each pair (*f*(*x**i*), *g*(*x**i*)). In proving for any 0 ≤ *f* ≤ *g* ≤ 1, we may thus assume that for each *i* one of three possibilities holds: 0 = *f*(*x**i*) = *g*(*x**i*), *f*(*x**i*) = *g*(*x**i*) = 1, or *f*(*x**i*) = 0 and *g*(*x**i*) = 1. In other words, we may assume that *f* and *g* are {0, 1}-valued.\nSuppose then for a contradiction that fails for some {0, 1}-valued *f* and *g* with *f* ≤ *g*. Then there must be some permutation *π* such that\n \n$$\\label{pw2}\n f(x\\_{\\pi(1)})\\left(1- \\prod\\_{i=2}^{r}{\\bigl(1-g(x\\_{\\pi(i)})\\bigr)}\\right)\n>\n g(x\\_{\\pi(1)})\\left(1- \\prod\\_{i=2}^{r}{\\bigl(1-f(x\\_{\\pi(i)})\\bigr)}\\right),$$ \nwhich we may take without loss of generality to be the identity permutation. Since both sides of are {0, 1}-valued, the left must be 1 and the right 0. Since the left is 1, we have *f*(*x*1) = 1, so, using *f* ≤ *g*, *g*(*x*1) = 1. But now for the right hand side of to be 0 the final product in must be 1, so *f*(*x**i*) = 0 for *i* = 2, …, *r*, i.e., *f* takes the value 1 only once. Of course, *g* must take the value 1 at least twice, otherwise we have equality. But now the left hand side of is exactly (*r* − 1)!, coming from terms with *π*(1) = 1 and hence *f*(*x**π*(1)) = 1. The right hand side is at least (*r* − 1)!, from any *π* mapping 1 to some *j* ≠ 1 with *g*(*x**j*) = 1. Hence holds after all, giving a contradiction and completing the proof.\nIf ${{\\undertilde{{\\kappa}}}}$ is reducible, then may in general have several non-zero solutions. To prove uniqueness in the irreducible case we need to know what irreducibility tells us about ${S\\_{{{\\undertilde{{\\kappa}}}}}}$.\n[lred2] If there exists a measurable $f:{{\\mathcal S}}\\to [0,1]$ with 0 < *μ*{*f* > 0} < 1 and ${\\ensuremath{\\{{S\\_{{{\\undertilde{{\\kappa}}}}}}f>0\\}}}\\subseteq {\\ensuremath{\\{f>0\\}}}$, then ${{\\undertilde{{\\kappa}}}}$ is reducible.\nLet *A* = {*f* > 0}, so by assumption ${S\\_{{{\\undertilde{{\\kappa}}}}}}f=0$ on $A^{{\\mathrm{c}}}={{\\mathcal S}}\\setminus A$. From we have ${\\ensuremath{\\{{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}f=0\\}}}={\\ensuremath{\\{{S\\_{{{\\undertilde{{\\kappa}}}}}}f=0\\}}}$, so ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}f=0$ on *A*c. From the definition of ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}$ it follows that ${{{\\kappa}\\_{\\mathrm e}}}=0$ a.e. on *A*c × *A*, so ${{{\\kappa}\\_{\\mathrm e}}}$ is reducible. But this is what it means for ${{\\undertilde{{\\kappa}}}}$ to be reducible.\nIn fact, taking *f* to be a suitable indicator function, one can check that the converse of Lemma [lred2] also holds.\nUsing Lemmas [lfSg] and [lred2] it is easy to deduce uniqueness of any non-zero solution to .\n[unique] Let ${{\\undertilde{{\\kappa}}}}$ be an irreducible, integrable hyperkernel, and let *f* and *g* be solutions to with 0 ≤ *f*(*x*) ≤ *g*(*x*) ≤ 1 for every *x*. Then either *f* = 0 or *f* = *g*. In particular, the only solutions to are $\\rho\\_{{\\undertilde{{\\kappa}}}}$ and the zero function, which may or may not coincide.\nWe may suppose that *f* is not 0 a.e.; otherwise, $f={\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}(f)$ would be identically zero. Since *f* solves, we have ${\\ensuremath{\\{f=0\\}}}={\\ensuremath{\\{{S\\_{{{\\undertilde{{\\kappa}}}}}}f=0\\}}}$, so by Lemma [lred2] we cannot have 0 < *μ*{*f* > 0} < 1. The only possibility left is that *μ*{*f* > 0} = 1, i.e., *f* > 0 a.e. Turning to *g*, since ${{\\undertilde{{\\kappa}}}}$ is integrable, we have ${S\\_{{{\\undertilde{{\\kappa}}}}}}(g)(x)\\le{S\\_{{{\\undertilde{{\\kappa}}}}}}(1)(x)={\\lambda}(x)<\\infty$ for a.e. *x*, and thus $g={\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}(g)<1$ a.e.\nSince *f* and *g* solve, we have ${S\\_{{{\\undertilde{{\\kappa}}}}}}(f)(x)=-\\log(1-f(x))$ and ${S\\_{{{\\undertilde{{\\kappa}}}}}}(g)(x)=-\\log(1-g(x))$. Hence,\n \n$$\\begin{aligned}\n f{S\\_{{{\\undertilde{{\\kappa}}}}}}(g) = -f\\log(1-g) &= f(g+g^2/2+g^3/3+\\cdots) \\\\\n &\\ge g(f+f^2/2+f^2/3+\\cdots) = g{S\\_{{{\\undertilde{{\\kappa}}}}}}(f)\\end{aligned}$$ \nwhenever 0 ≤ *f* ≤ *g* ≤ 1, with strict inequality whenever 0 < *f* < *g*. Since ${{\\undertilde{{\\kappa}}}}$ is integrable, it is immediate from the definition that ${S\\_{{{\\undertilde{{\\kappa}}}}}}f$ and ${S\\_{{{\\undertilde{{\\kappa}}}}}}g$ are integrable, and it follows that \n$$\\int\\_{{\\mathcal S}}f {S\\_{{{\\undertilde{{\\kappa}}}}}}g \\ge \\int\\_{{\\mathcal S}}g {S\\_{{{\\undertilde{{\\kappa}}}}}}f,$$ \n with strict inequality unless *f* = *g* a.e. Since Lemma [lfSg] gives the reverse inequality, we have *f* = *g* a.e., and thus $f={\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}f={\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}g=g$. The second statement then follows from Lemma [lmax].\nTheorem [unique] generalizes the corresponding result in , namely Lemma 5.9. Indeed, in the edge-only case (when only *κ*2 is non-zero), the operators ${S\\_{{{\\undertilde{{\\kappa}}}}}}$ and ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}$ coincide, and Lemma [lfSg] holds trivially, using the symmetry of ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}$. This shows that, with hindsight, the proof of Lemma 5.9 in may be simplified considerably, by considering $\\int\\_{{\\mathcal S}}fTg$ instead of $\\int\\_{{\\mathcal S}}fTh$, *h* = (*g* − *f*)/2. This is significant, since the proof in does not adapt readily to the present context.\nAlthough simple, the proof of Theorem [unique] above is a little mysterious from a branching process point of view. It is tempting to think that the result is ‘obvious’, and indeed that a corresponding result should hold for any Galton–Watson process. However, some conditions are certainly necessary, and it is not clear what the right conditions are for a general process. (Irreducibility is always needed, of course.) In , a corresponding result is proved for a general branching process satisfying a certain continuity assumption; the proof uses the convexity property Φ(*λ**f*) ≥ *λ*Φ(*f*) for any function 0 ≤ *f* ≤ 1 and any 0 ≤ *λ* ≤ 1, which holds for all Galton–Watson branching processes. In Theorem [unique], continuity is not needed, but some kind of symmetry is; there does not seem to be an obvious common generalization of these results.\nIndeed, the next example shows that the situation is not that simple: in the compound Poisson case (as opposed to the simple Poisson case), symmetry of the relevant linear operator is not enough.\nLet ${{\\mathcal S}}=\\{1,2,3,\\ldots\\}$ with *μ*{*i*} = 2− *i* for each *i*, and consider the branching process ${{\\mathfrak X}}={{\\mathfrak X}}(x)$ with type space $({{\\mathcal S}},\\mu)$ defined as follows. Start with a single particle of some given type *x*. Each particle of type *i* has a Poisson number of children of type *i* + 1 with mean 2 = 2*i* + 2*μ*{*i* + 1}; we call these ‘forward children’. Also, for *i* ≥ 2, a particle of type *i* has ‘backward children’ of type *i* − 1: the number of these is 4*i* + 1 times a Poisson with mean 4− *i*. Note that the expected number of backward children is 4 = 2*i* + 1*μ*{*i* − 1}. Defining the ‘edge-kernel’ ${{{\\kappa}\\_{\\mathrm e}}}$ so that the expected number of children of type *j* that each particle of type *i* has is given by ${{{\\kappa}\\_{\\mathrm e}}}(i,j)\\mu\\{j\\}$, we have ${{{\\kappa}\\_{\\mathrm e}}}(i,j)=2^{1+\\max\\{i,j\\}}$ if ∣*i* − *j*∣ = 1 and ${{{\\kappa}\\_{\\mathrm e}}}(i,j)=0$ otherwise, so ${{{\\kappa}\\_{\\mathrm e}}}$ is symmetric and irreducible.\nDefine the non-linear operator Φ associated to ${{\\mathfrak X}}$ in the natural way, so Φ(*f*)(*x*) is the probability that at least one child of the root of type *x* has a certain property, if each child of type *y* has this property independently with probability *f*(*y*). As before, the survival probability *ρ*(*x*) satisfies *ρ* = Φ(*ρ*).\nLet *τ*(*x*) denote the probability that the process *survives transiently*, i.e., survives forever, but, for each *i*, contains in total only finitely many particles of type *i*. Consider the ‘forward process’ given by ignoring backward children. This is simply a Poisson Galton–Watson process with on average 2 offspring, and so survives with some positive probability. Also, given that it survives, there is a positive probability that for every *t*, generation *t* contains at most 3*t* particles, say. But since the particles in generation *t* have type *x* + *t*, the expected number of *sets* of backwards children of all particles in the forward process is at most ∑*t* ≥ 03*t*4− *t* − 1 < ∞, and with positive probability the particles in the forward process have no backwards children. But in this case, the forward process is the whole process, and the process survives transiently. Hence *τ*(*x*) > 0 for every *x*.\nLet *σ*(*x*) = *ρ*(*x*) − *τ*(*x*) be the probability that the process survives recurrently. Considering the children of the initial particle, we see that *σ* = Φ(*σ*). The process restricted to any two consecutive types is already supercritical, and so has positive probability of surviving by alternating between these types. Thus *σ*(*x*) > 0 for all *x*. We showed above that *τ*(*x*) = *ρ*(*x*) − *σ*(*x*) > 0 for all *x*, so 0 < *σ*(*x*) < *ρ*(*x*), and *f* = Φ(*f*) has (at least) two non-zero solutions, namely *σ* and *ρ*.\nLet us turn to the analysis of the solution $\\rho\\_{{\\undertilde{{\\kappa}}}}$ to, and in particular the question of when *ρ* > 0, i.e., when the branching process ${{\\mathfrak X}}\\_{{\\undertilde{{\\kappa}}}}$ is supercritical. Throughout we consider an integrable hyperkernel ${{\\undertilde{{\\kappa}}}}$, with corresponding edge kernel ${{{\\kappa}\\_{\\mathrm e}}}$.\nRecall that we may assume that ${\\lambda}(x)={S\\_{{{\\undertilde{{\\kappa}}}}}}(1)(x)$ is finite everywhere. Hence, for any *f* satisfying, we have *f*(*x*) < 1 for all *x*. On the other hand, we cannot assume that ${{{\\kappa}\\_{\\mathrm e}}}$ is integrable, or indeed finite. For one natural example, consider the integrable hyperkernel with each *κ**r* constant, and *κ**r* = 1/*r*3. In this case ${{{\\kappa}\\_{\\mathrm e}}}(x,y)=\\infty$ for all *x* and *y*. If ${{{\\kappa}\\_{\\mathrm e}}}$ is infinite on a set of positive measure, then we take ${\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}\\|}}$ to be infinite.\n[lsc] If ${\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}\\|}}\\le 1$, then $\\rho({{\\undertilde{{\\kappa}}}})=0$.\nSuppose that *f* is a solution to that is not 0 a.e. Since − log(1 − *t*) > *t* for 0 < *t* < 1, we have ${S\\_{{{\\undertilde{{\\kappa}}}}}}(f)(x)\\ge f(x)$, with strict inequality on a set of positive measure. But ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}(f)(x)\\ge {S\\_{{{\\undertilde{{\\kappa}}}}}}(f)(x)$ by, so ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}(f)(x)\\ge f(x)$, with strict inequality on a set of positive measure. Hence ${\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}f\\|\\_2}}>{\\ensuremath{\\|f\\|\\_2}}$, so ${\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}\\|}}> 1$.\nLemmas 5.12 and 5.13 of carry over to the present context, with only minor modifications. Given functions *f*1, *f*2, … and *f*, we write $f\\_n{\\nearrow}f$ if the sequence (*f**n*) is monotone increasing and converges to *f* pointwise.\nIf 0 ≤ *f* ≤ 1 and ${\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}(f)\\ge f$, then ${\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}^m(f){\\nearrow}g$ as *m* → ∞, for some 1 ≥ *g* ≥ *f* with ${\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}(g)=g$.\nSince $f\\le{\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}(f)$, monotonicity of ${\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}$ gives ${\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}(f)\\le {\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}^2(f)$ and, by induction, ${\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}^m(f)\\le {\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}^{m+1}(f)$ for all *m* ≥ 0. Since $0\\le {\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}^m(f)\\le 1$, it follows that $g(x)=\\lim\\_{m\\to\\infty} {\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}^m(f)(x)$ exists for every *x*, and 0 ≤ *g* ≤ 1. From monotone convergence we have ${S\\_{{{\\undertilde{{\\kappa}}}}}}(g)=\\lim\\_{m\\to\\infty} {S\\_{{{\\undertilde{{\\kappa}}}}}}(\\Phi\\_k^m(f))$, from which it follows that ${\\Phi\\_{{{\\undertilde{{\\kappa}}}}}}(g)=g$.\n[lfup] If there is a function $f:{{\\mathcal S}}\\to [0,1]$, not a.e. 0, such that ${S\\_{{{\\undertilde{{\\kappa}}}}}}(f)\\ge (1+\\delta)f$ for some *δ* > 0, then $\\rho({{\\undertilde{{\\kappa}}}})>0$.\nThe proof is the same as that of Lemma 5.13 in , using ${S\\_{{{\\undertilde{{\\kappa}}}}}}$ in place of *T**κ*.\nThe next step is to show that if ${\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}\\|}}>1$, then there is a function *f* with the property described in Lemma [lfup]. In we did this by considering a bounded kernel. Here we have to be a little more careful, as we are working with the non-linear operator ${S\\_{{{\\undertilde{{\\kappa}}}}}}$ rather than with ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}$; this is no problem if we truncate our kernels suitably.\n[Dbounded] We call a hyperkernel ${{\\undertilde{{\\kappa}}}}=({\\kappa}\\_r)\\_{r\\ge2}$ *bounded* if two conditions hold: only finitely many of the *κ**r* are non-zero, and each *κ**r* is bounded.\nSimilarly (for later use), a general kernel family $({\\kappa}\\_F)\\_{F\\in{{\\mathcal F}}}$ is *bounded* if only finitely many of the *κ**F* are non-zero, and each *κ**F* is bounded.\nIn other words, ${{\\undertilde{{\\kappa}}}}$ is bounded if there are constants *R* and *M* such that *κ**r* = 0 for *r* > *R*, and *κ**r* is pointwise bounded by *M* for *r* ≤ *R*. Note that if ${{\\undertilde{{\\kappa}}}}$ is bounded, then the corresponding edge kernel ${{{\\kappa}\\_{\\mathrm e}}}$ is bounded in the usual sense.\nGiven a hyperkernel ${{\\undertilde{{\\kappa}}}}=({\\kappa}\\_r)$, for each *M* > 0 we let ${{{{\\undertilde{{\\kappa}}}}^M}}$ be the bounded hyperkernel obtained from *κ* by truncating each *κ**r*, *r* ≤ *M*, at *M*, and replacing *κ**r* by a zero kernel for *r* > *M*. Thus\n \n$$\\label{trunc}\n {{\\kappa}^M}\\_r=\n \\begin{cases}\n {\\kappa}\\_r\\wedge M, & r\\le M,\n\\\\\n0, & r>M.\n \\end{cases}$$ \nThe truncation ${{{{\\undertilde{{\\kappa}}}}^M}}=({{\\kappa}^M}\\_F)\\_{F\\in{{\\mathcal F}}}$ of a general kernel family $({\\kappa}\\_F)\\_{F\\in{{\\mathcal F}}}$ is defined similarly, replacing the condition *r* ≤ *M* by ∣*F*∣ ≤ *M*.\n[lef] If ${\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}\\|}}>1$ then there is a *δ* > 0 and an $f:{{\\mathcal S}}\\to [0,1]$, not a.e. 0, such that ${S\\_{{{\\undertilde{{\\kappa}}}}}}(f)\\ge (1+\\delta)f$.\nWe slightly modify the proof of Lemma 5.16 of .\nConsider the truncated hyperkernels ${{{{\\undertilde{{\\kappa}}}}^M}}$ defined in. From and monotone convergence, the corresponding edge kernels ${{{\\kappa}\\_{\\mathrm e}^M}}$ tend up to ${{{\\kappa}\\_{\\mathrm e}}}$ (which may be infinite in some places) pointwise. Arguing as in the proof of Lemma 5.16 of , since ${\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}\\|}}>1$ there is some positive *f* with ∥*f*∥2 = 1 and $1<{\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}f\\|\\_2}}\\le\\infty$. By monotone convergence, $T\\_{{{{\\kappa}\\_{\\mathrm e}^M}}}f{\\nearrow}{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}f$, so ${\\ensuremath{\\|T\\_{{{{\\kappa}\\_{\\mathrm e}^M}}}f\\|\\_2}}{\\nearrow}{\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}f\\|\\_2}}$, and there is some *M* with ${\\ensuremath{\\|T\\_{{{{\\kappa}\\_{\\mathrm e}^M}}}\\|}}\\ge {\\ensuremath{\\|T\\_{{{{\\kappa}\\_{\\mathrm e}^M}}}f\\|\\_2}}>1$.\nSince ${{{\\kappa}\\_{\\mathrm e}^M}}$ is bounded, setting $\\delta=({\\ensuremath{\\|T\\_{{{{\\kappa}\\_{\\mathrm e}^M}}}\\|}}-1)/2>0$, by Lemma 5.15 of it follows that there is a bounded *f* ≥ 0 with *f* not 0 a.e. such that \n$$T\\_{{{{\\kappa}\\_{\\mathrm e}^M}}} f = {\\ensuremath{\\|T\\_{{{{\\kappa}\\_{\\mathrm e}^M}}}\\|}}f = (1+2\\delta) f.$$ \n We may assume that 0 ≤ *f* ≤ 1. If 0 ≤ *y**i* ≤ *γ* < 1, *i* = 1, …, *r*, then (by induction) 1 − ∏*i* = 1*r*(1 − *y**i*) ≥ (1 − *γ*)*r* − 1∑*i* = 1*r**y**i*, and it follows that if *γ* > 0 is chosen small enough, then \n$$S\\_{{{{{\\undertilde{{\\kappa}}}}^M}}} (\\gamma f) \n\\ge(1-{\\gamma})^{M-1} T\\_{{{{\\kappa}\\_{\\mathrm e}^M}}}({\\gamma}f)\n\\ge (1+\\delta) (\\gamma f).$$ \n Since ${S\\_{{{\\undertilde{{\\kappa}}}}}}(\\gamma f)\\ge S\\_{{{{{\\undertilde{{\\kappa}}}}^M}}}(\\gamma f)$, the result follows.\nTheorem [th2] follows by combining the results above.\n[Proof of Theorem [th2]] Together Lemmas [lsc], [lfup] and [lef] show that $\\rho({{\\undertilde{{\\kappa}}}})>0$ if and only if ${\\ensuremath{\\|{T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}\\|}}>1$. Uniqueness is given by Theorem [unique]. The final statement is immediate from Lemma [lred2].\nHaving proved Theorem [th2], our next aim is to prove Theorem [th1]. The basic strategy will involve comparing the neighbourhoods of a vertex in the random graph $G(n,{{\\undertilde{{\\kappa}}}})$ with the branching process ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$. As in , it will be convenient to carry out the comparison only for certain restricted hyperkernels. In order to deduce results about $G(n,{{\\undertilde{{\\kappa}}}})$ in general, one needs approximation results both for the graph and for the branching process. We now turn to such results for branching processes.\nLemma 6.3 and Theorems 6.4 and 6.5 of carry over to the present context, *mutatis mutandis*, using the results above about $\\rho({{\\undertilde{{\\kappa}}}})$ instead of the equivalents in , and replacing *T**κ* by ${S\\_{{{\\undertilde{{\\kappa}}}}}}$ or ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}$ as appropriate: ${S\\_{{{\\undertilde{{\\kappa}}}}}}$ when considering Φ*κ*, and ${T\\_{{{{\\kappa}\\_{\\mathrm e}}}}}$ when arguing using *L*2-norms. In these results $\\rho\\_{{\\undertilde{{\\kappa}}}}$ denotes the function $x\\mapsto \\rho\\_{{\\undertilde{{\\kappa}}}}(x)$, and $\\rho\\_{\\ge k}({{\\undertilde{{\\kappa}}}},x)$ and $\\rho\\_{\\ge k}({{\\undertilde{{\\kappa}}}})$ denote respectively the probabilities that ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}(x)$ and ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$ have total size at least *k*, where the *size* of a branching process is the total number of particles in all generations.\n[L12] If ${{\\undertilde{{\\kappa}}}}\\le{{{\\undertilde{{\\kappa}}}}'}$, then $\\rho({{\\undertilde{{\\kappa}}}})\\le \\rho({{{\\undertilde{{\\kappa}}}}'})$.\n[TappB]\nLet ${{\\undertilde{{\\kappa}}\\_n}}$, *n* = 1, 2, …, be a sequence of hyperkernels on $({{\\mathcal S}},\\mu)$ increasing a.e.=1000 to an integrable hyperkernel ${{\\undertilde{{\\kappa}}}}$. Then $\\rho\\_{{{\\undertilde{{\\kappa}}\\_n}}}{\\nearrow}\\rho\\_{{\\undertilde{{\\kappa}}}}$ a.e.=1000 and $\\rho({{\\undertilde{{\\kappa}}\\_n}}){\\nearrow}\\rho({{\\undertilde{{\\kappa}}}})$.\nLet ${{\\undertilde{{\\kappa}}\\_n}}$, *n* = 1, 2, …, be a sequence of integrable hyperkernels on $({{\\mathcal S}},\\mu)$ decreasing a.e.=1000 to ${{\\undertilde{{\\kappa}}}}$. Then $\\rho\\_{{{\\undertilde{{\\kappa}}\\_n}}}{\\searrow}\\rho\\_{{\\undertilde{{\\kappa}}}}$ a.e.=1000 and $\\rho({{\\undertilde{{\\kappa}}\\_n}}){\\searrow}\\rho({{\\undertilde{{\\kappa}}}})$.\n[TappC]\nLet ${{\\undertilde{{\\kappa}}\\_n}}$, *n* = 1, 2, …, be a sequence of hyperkernels on $({{\\mathcal S}},\\mu)$ increasing a.e.=1000 to a hyperkernel ${{\\undertilde{{\\kappa}}}}$. Then, for every *k* ≥ 1, ${{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}\\_n}};x){\\nearrow}{{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}}};x)$ for a.e.=1000 *x* and ${{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}\\_n}}){\\nearrow}{{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}}})$.\nLet ${{\\undertilde{{\\kappa}}\\_n}}$, *n* = 1, 2, …, be a sequence of integrable hyperkernels on $({{\\mathcal S}},\\mu)$ decreasing a.e.=1000 to ${{\\undertilde{{\\kappa}}}}$. Then, for every *k* ≥ 1, ${{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}\\_n}};x){\\searrow}{{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}}};x)$ for a.e.=1000 *x* and ${{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}\\_n}}){\\searrow}{{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}}})$.\nThe assumption that ${{\\undertilde{{\\kappa}}\\_n}}$ be integrable in Theorems [TappB](ii) and [TappC](ii) can be weakened to ${\\lambda}\\_{{{\\undertilde{{\\kappa}}\\_n}}}(x)<\\infty$ for a.e. *x*, where ${\\lambda}\\_{{{\\undertilde{{\\kappa}}\\_n}}}(x)$ is the expected number of child cliques in ${{\\mathfrak X}}\\_{{{\\undertilde{{\\kappa}}\\_n}}}$ of a particle of type *x*; see.\nLocal coupling\n==============\nWe now turn to the local coupling between our random graph and the corresponding branching process, relating the distribution of small components in $G(n,{{\\undertilde{{\\kappa}}}})$ to the branching process ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$. In , we were essentially forced to condition on the vertex types, since these were allowed to be deterministic to start with. Here, with i.i.d. vertex types, there is no need to do so. This allows us to couple directly for all bounded hyperkernels, rather than simply for finite type ones.\nWe shall consider a variant of the usual component exploration process, designed to get around the following problem. When we test edges from a given vertex *v* to all other vertices, the probability of finding a given edge *v**w* depends on the type of *w* as well as that of *v*. Hence, *not* finding such an edge changes the conditional distribution of the type of *w*. If the kernel is well behaved, it is easy to see that this is a small effect. Rather than quantify this, it is easier to embed $G(n,{{\\undertilde{{\\kappa}}}})$ inside a larger random graph with uniform kernels. Testing edges in the larger graph does not affect the conditional distribution of the vertex types; we make this precise below. In doing so, it will be useful to take the hypergraph viewpoint: given a hyperkernel ${{\\undertilde{{\\kappa}}}}$, let $H(n,{{\\undertilde{{\\kappa}}}})$ be the hypergraph on [*n*] constructed according to the same rules as $G(n,{{\\undertilde{{\\kappa}}}})$, except that instead of adding a *K**r* we add a hyperedge with *r* vertices. In fact, we consider the Poisson version of the model, allowing multiple copies of the same hyperedge.\nLet ${{\\undertilde{{\\kappa}}}}$ be a bounded hyperkernel, and let ${{{\\undertilde{{\\kappa}}}}^+}$ be a corresponding upper bound, so *κ*+*r* is the constant kernel *M* for *r* ≤ *R*, and zero for *r* > *R*, while *κ**r* ≤ *κ*+*r* holds pointwise for all *r*.\nTaking, as usual, our vertex types $x\\_1,\\ldots,x\\_n\\in {{\\mathcal S}}$ to be independent, each having the distribution *μ*, we construct coupled random (multi-)hypergraphs *H**n* and *H**n*+ on [*n*] as follows: first construct $H^+\\_n=H(n,{{\\undertilde{{\\kappa}}}}^+)$ by taking, for every 2 ≤ *r* ≤ *R*, a Poisson Po(*r*!*M*/*n**r* − 1) number of copies of each possible *r*-element hyperedge, with all these numbers independent. Although in our formal definition of *H**n*+ we first decide the vertex types, *H**n*+ is clearly independent of these types. Hence, given *H**n*+, the types are (still) i.i.d. with distribution *μ*.\nGiven *H**n*+ and the i.i.d. types *x*1, …, *x**n* of the vertices, we may form *H**n* by selecting each hyperedge {*v*1, …, *v**r*} of *H**n*+ to be a hyperedge of *H**n* with probability *κ**r*(*x**v*1, …, *x**v**r*)/*M*, independently of all other hyperedges. It is easy to see that this gives the right distribution for $H\\_n=H(n,{{\\undertilde{{\\kappa}}}})$. (If we disallowed multiple copies of an edge, there would be an irrelevant small correction here.)\nTurning to the branching processes, there is an analogous coupling of ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$ and ${{\\mathfrak X}}\\_{{{\\undertilde{{\\kappa}}}}^+}$: first construct ${{\\mathfrak X}}\\_{{{\\undertilde{{\\kappa}}}}^+}$, which may be viewed as a single-type process, according to our two-step construction via child cliques. Then assign each particle a type according to the distribution *μ*, independently of the other particles and of the branching process. Then form the child cliques in ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$ by keeping each child clique in ${{\\mathfrak X}}\\_{{{\\undertilde{{\\kappa}}}}^+}$ with an appropriate probability depending on the types, deleting not only the children corresponding to deleted child cliques, but also all their descendants.\nLet *v* ∈ [*n*] be chosen uniformly at random, independently of *H**n* and *H**n*+. Let Γ*d* denote the *d*-neighbourhood of *v* in *H**n*, and Γ*d*+ that in *H**n*+. Counting the expected number of cycles shows that for any fixed *d*, the hypergraph Γ*d*+ is whp treelike. Furthermore, standard arguments as for *G*(*n*, *c*/*n*) show that one may couple Γ*d*+ and the first *d* generations of ${{\\mathfrak X}}\\_{{{\\undertilde{{\\kappa}}}}^+}$ so as to agree in the natural sense whp. When Γ*d*+ is treelike, then Γ*d* ⊂ Γ*d*+ may be constructed using exactly the same random deletion process that gives (the first *d* generations of) ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$ as a subset of ${{\\mathfrak X}}\\_{{{\\undertilde{{\\kappa}}}}^+}$. It follows that Γ*d* and the first *d* generations of ${{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}$ may be coupled to agree whp.\nRecalling that $G(n,{{\\undertilde{{\\kappa}}}})$ and *H**n* have the same components, for any fixed *k* ≥ 1 one can determine whether the component containing *v* has exactly *k* vertices by examining Γ*k* + 1. Writing *N**k*(*G*) for the number of vertices of a graph *G* that are in components of size *k*, it follows that \n$${{\\mathop{\\mathbb E{}}\\nolimits}}N\\_k(G(n,{{\\undertilde{{\\kappa}}}})) = n{{\\mathop{\\mathbb P{}}\\nolimits}}(|{{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}|=k) +o(n).$$ \n As in , starting from two random vertices easily gives a corresponding second moment bound, giving convergence in probability.\n[nkbdd] Let ${{\\undertilde{{\\kappa}}}}$ be a bounded hyperkernel. Then \n$$\\frac{1}{n} N\\_k(G(n,{{\\undertilde{{\\kappa}}}})) {\\overset{\\mathrm{p}}{\\to}}{{\\mathop{\\mathbb P{}}\\nolimits}}(|{{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}|=k)$$ \n for any fixed *k*.\nOf course it makes no difference whether we work with *N**k* or *N*≥ *k* = *n* − ∑*j* = 1*k* − 1*N**j*: Lemma [nkbdd] also tells us that\n \n$$\\label{ngek}\n \\frac{1}{n} N\\_{\\ge k}(G(n,{{\\undertilde{{\\kappa}}}})) {\\overset{\\mathrm{p}}{\\to}}{{\\mathop{\\mathbb P{}}\\nolimits}}(|{{\\mathfrak X}\\_{{{\\undertilde{{\\kappa}}}}}}|\\ge k).$$ \nThe extension to arbitrary hyperkernels is easy from Theorem [TappC].\n[nkint] Let ${{\\undertilde{{\\kappa}}}}$ be an integrable hyperkernel. Then for each fixed *k* we have \n$$\\frac{1}{n} N\\_{\\ge k}(G(n,{{\\undertilde{{\\kappa}}}})) {\\overset{\\mathrm{p}}{\\to}}{{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}}}).$$ \nAs in , we simply approximate ${{\\undertilde{{\\kappa}}}}$ by bounded hyperkernels. For *M* > 0 let ${{{{\\undertilde{{\\kappa}}}}^M}}$ be the truncated hyperkernel defined by.\nLet *k* ≥ 1 be fixed, and let *ɛ* > 0 be arbitrary. From monotone convergence and integrability, \n$$\\lim\\_{M\\to\\infty} \\sum\\_{{r}\\ge 2} \\int\\_{{{\\mathcal S}}^{r}}r{{\\kappa}^M}\\_{r}= \n \\sum\\_{{r}\\ge 2} \\int\\_{{{\\mathcal S}}^{r}}r{\\kappa}\\_{r}< \\infty,$$ \n so for *M* large enough we have \n$$\\Delta = \\sum\\_{{r}\\ge 2} \\int\\_{{{\\mathcal S}}^{r}}r({\\kappa}\\_{r}-{{\\kappa}^M}\\_{r}) \\le {\\varepsilon}^2/(6k),$$ \n say. By Theorem [TappC](i), increasing *M* if necessary, we may also assume that\n \n$$\\label{rcl}\n {{\\ensuremath{\\rho\\_{\\ge k}}}}({{{{\\undertilde{{\\kappa}}}}^M}})\\ge {{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}}})-{\\varepsilon}/3.$$ \nSince ${{{{\\undertilde{{\\kappa}}}}^M}}\\le {{\\undertilde{{\\kappa}}}}$ holds pointwise, we may couple the hypergraphs *H**n*ʹ and *H**n* associated to $G(n,{{{{\\undertilde{{\\kappa}}}}^M}})$ and $G(n,{{\\undertilde{{\\kappa}}}})$ so that *H**n*ʹ ⊆ *H**n*. Recall that $G(n,{{\\undertilde{{\\kappa}}}})$ is produced from *H**n* by replacing each hyperedge *E* with *r* vertices by an *r*-clique. However, as noted earlier, if we form *G**n* from *H**n* by replacing each *E* by any connected simple graph on the same set of vertices, then *G**n* and $G(n,{{\\undertilde{{\\kappa}}}})$ will have exactly the same component structure, and in particular $N\\_{\\ge k}(G\\_n)=N\\_{\\ge k}(G(n,{{\\undertilde{{\\kappa}}}}))$. Let us form *G**n* and *G**n*ʹ in this way from *H**n* and *H**n*ʹ, replacing any hyperedge with *r* vertices by some tree on the same set of vertices. Recalling that *H**n*ʹ ⊆ *H**n*, we may of course assume that *G**n*ʹ ⊆ *G**n*.\nWriting *e**r*(*H*) for the number of *r*-vertex hyperedges in a hypergraph *H*,\n \n$$\\begin{split}\n {{\\mathop{\\mathbb E{}}\\nolimits}}{\\bigl( |E(G\\_n)\\setminus E(G\\_n')| \\bigr)} &\\le \\sum\\_{r\\ge 2} (r-1){{\\mathop{\\mathbb E{}}\\nolimits}}{\\bigl(e\\_r(H\\_n)-e\\_r(H\\_n')\\bigr)} \n\\\\\n &\\le \\sum\\_{r\\ge 2} {(r-1)n}\\int\\_{{{\\mathcal S}}^r} ({\\kappa}\\_r-{{\\kappa}^M}\\_r)\n\\le n\\Delta.\n \\end{split}$$ \nHence, \n$${{\\mathop{\\mathbb P{}}\\nolimits}}{\\bigl(|E(G\\_n)\\setminus E(G\\_n')| \\ge {\\varepsilon}n/6k \\bigr)} \\le n\\Delta/({\\varepsilon}n/6k) \\le {\\varepsilon}.$$ \n Recalling that *G**n*ʹ ⊆ *G**n* and noting that adding one edge to a graph cannot change *N*≥ *k* by more than 2*k*, we see that with probability at least 1 − *ɛ* we have \n$${\\bigl| N\\_{\\ge k}(G(n,{{{{\\undertilde{{\\kappa}}}}^M}})) - N\\_{\\ge k}(G(n,{{\\undertilde{{\\kappa}}}})) \\bigr|}\n = {\\bigl| N\\_{\\ge k}(G\\_n')-N\\_{\\ge k}(G\\_n) \\bigr|}\n \\le 2k({\\varepsilon}n/6k) = {\\varepsilon}n/3.$$ \n Applying Lemma [nkbdd] (or rather ) to the bounded hyperkernel ${{{{\\undertilde{{\\kappa}}}}^M}}$, we have $\\frac{1}{n} N\\_{\\ge k}(G(n,{{{{\\undertilde{{\\kappa}}}}^M}})){\\overset{\\mathrm{p}}{\\to}}{{\\ensuremath{\\rho\\_{\\ge k}}}}({{{{\\undertilde{{\\kappa}}}}^M}})$. Using it follows that when *n* is large enough, with probability at least 1 − 2*ɛ*, say, we have $|\\frac{1}{n} N\\_{\\ge k}(G(n,{{\\undertilde{{\\kappa}}}}))- {{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}}})|\\le {\\varepsilon}$. Since *ɛ* > 0 was arbitrary, we thus have $\\frac{1}{n} N\\_{\\ge k}(G(n,{{\\undertilde{{\\kappa}}}})) {\\overset{\\mathrm{p}}{\\to}}{{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}}})$ as required.\nThe giant component\n===================\nThe local coupling results of the previous section easily give us the ‘right’ number of vertices in large components. As usual, we will pass from this to a giant component by using the ‘sprinkling’ method of Erdos and Rényi , first uncovering the bulk of the edges, and then using the remaining ‘sprinkled’ edges to join up the large components. The following lemma gathers together the relevant consequences of the results in the previous section.\n[leasypart] Let ${{\\undertilde{{\\kappa}}}}=({\\kappa}\\_r)$ be an integrable hyperkernel, and let $G\\_n=G(n,{{\\undertilde{{\\kappa}}}})$. Then $C\\_1(G\\_n)\\le \\rho({{\\undertilde{{\\kappa}}}})n +{o\\_{\\mathrm{p}}}(n)$. Furthermore, given any *ɛ* > 0, there is a *δ* > 0 and a function *ω* = *ω*(*n*) → ∞ such that\n \n$$\\label{nbp}\n N\\_{\\ge \\omega}(G\\_n') \\ge (\\rho({{\\undertilde{{\\kappa}}}})-{\\varepsilon})n$$ \nholds whp, where $G\\_n'=G(n,(1-\\delta){{\\undertilde{{\\kappa}}}})$.\nFrom Lemma [nkint] we have $\\frac{1}{n} N\\_{\\ge k}(G\\_n){\\overset{\\mathrm{p}}{\\to}}{{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}}})$ for each fixed *k*. Since ${{\\ensuremath{\\rho\\_{\\ge k}}}}({{\\undertilde{{\\kappa}}}})\\to \\rho({{\\undertilde{{\\kappa}}}})$ as *k* → ∞, it follows that for some *ω* = *ω*(*n*) → ∞ we have\n \n$$\\label{nbig}\n \\frac{1}{n} N\\_{\\ge \\omega}(G\\_n){\\overset{\\mathrm{p}}{\\to}}\\rho({{\\undertilde{{\\kappa}}}});$$ \nwe may and shall assume that *ω* = *o*(*n*). Since *C*1(*G**n*) ≤ max{*ω*, *N*≥ *ω*(*G**n*)}, the first statement of the lemma follows.\nFor the second, we may of course assume that $\\rho({{\\undertilde{{\\kappa}}}})>{\\varepsilon}$; otherwise, there is nothing to prove. As *δ* → 0, from Theorem [TappB](i) we have $\\rho((1-\\delta){{\\undertilde{{\\kappa}}}})\\to \\rho({{\\undertilde{{\\kappa}}}})$. Fix *δ* > 0 with $\\rho((1-\\delta){{\\undertilde{{\\kappa}}}})>\\rho({{\\undertilde{{\\kappa}}}})-{\\varepsilon}/2$, and let $G\\_n'=G(n,(1-\\delta){{\\undertilde{{\\kappa}}}})$. Applying to *G**n*ʹ, there is some *ω* = *ω*(*n*) → ∞ such that \n$$N=N\\_{\\ge\\omega}(G\\_n')= \\rho((1-\\delta){{\\undertilde{{\\kappa}}}})n+{{o\\_{\\mathrm{p}}}(n)}\\ge (\\rho({{\\undertilde{{\\kappa}}}})-{\\varepsilon}/2)n+{{o\\_{\\mathrm{p}}}(n)},$$ \n which implies.\nIn the light of Lemma [leasypart], and writing *G**n* for $G(n,{{\\undertilde{{\\kappa}}}})$, to prove Theorem [th1] it suffices to show that if ${{\\undertilde{{\\kappa}}}}$ is irreducible, then for any *ɛ* > 0 we have\n \n$$\\label{low}\n C\\_1(G\\_n) \\ge (\\rho({{\\undertilde{{\\kappa}}}})-2{\\varepsilon})n$$ \nwhp; then $C\\_1(G\\_n)/n{\\overset{\\mathrm{p}}{\\to}}\\rho({{\\undertilde{{\\kappa}}}})$ as required. Also, from and the fact that *C*1(*G**n*) + *C*2(*G**n*) ≤ max{2*ω*, *N*≥ *ω*(*G**n*) + *ω*}, we obtain *C*2(*G**n*) = *o*p(*n*) as claimed.\nSince $(1-\\delta){{\\undertilde{{\\kappa}}}}\\le {{\\undertilde{{\\kappa}}}}$, there is a natural coupling of the graphs *G**n*ʹ and *G**n* appearing in Lemma [leasypart] in which *G**n*ʹ ⊆ *G**n* always holds. Our aim is to show that, whp, in passing from *G**n*ʹ to *G**n*, the extra ‘sprinkled’ edges join up almost all of the *N* vertices of *G**n*ʹ in ‘large’ components (those of size at least *ω*) into a single component.\nUnfortunately, we have to uncover the vertex types before sprinkling, so we do not have the usual independence between the bulk and sprinkled edges. A similar problem arose in Bollobás, Borgs, Chayes and Riordan in the graph context, as opposed to the present hypergraph context. It turns out that we can easily reduce to the graph case, and thus apply a lemma from . This needs a little setting up, however. Here it will be convenient to take ${{\\mathcal S}}=[0,1]$ with *μ* Lebesgue measure; as noted in Section [secir], this loses no generality.\nLet *f* be a bounded symmetric measurable function $f:[0,1]^2\\to {{\\mathbb R}}$. Following Frieze and Kannan , the *cut norm* ∥*f*∥▫ of *f* is defined by \n∥*f*∥▫ = sup*S*, *T* ⊆ [0, 1]∣∫*S* × *T**f*(*x*, *y*) *d**x* *d**y*∣, \n where the supremum is taken over all pairs of measurable sets. Note that ∥*f*∥▫ ≤ ∥*f*∥1, since the integral above is bounded by ∫*S* × *T*∣*f*∣ ≤ ∫[0, 1]2∣*f*∣.\nGiven a kernel *κ* on [0, 1] and a measurable function *φ* : [0, 1] → [0, 1], let *κ*(*φ*) be the kernel defined by \n*κ*(*φ*)(*x*, *y*) = *κ*(*φ*(*x*), *φ*(*y*)). \n If *φ* is a measure-preserving bijection, then *κ*(*φ*) is a *rearrangement* of *κ*. (One can also consider measure-preserving bijections between subsets of [0, 1] with full measure; it makes no difference.) We write *κ* ∼ *κ*ʹ if *κ*ʹ is a rearrangement of *κ*.\nGiven two kernels *κ*, *κ*ʹ on [0, 1], the *cut metric* of Borgs, Chayes, Lovász, Sós and Vesztergombi is defined by \n*δ*▫(*κ*, *κ*ʹ) = inf*κ*ʺ ∼ *κ*ʹ∥*κ* − *κ*ʺ∥▫. \n Note that this is a pseudo-metric rather than a metric, as we can have *δ*▫(*κ*, *κ*ʹ) = 0 for different kernels. (Probabilistically, it is probably more natural to consider couplings between kernels as in , rather than rearrangements, but this is harder to describe briefly and turns out to make no difference.)\nLet *A**n* be a symmetric *n*-by-*n* matrix with non-negative entries *a**i**j*, which we may think of as a (dense) weighted graph. There is a piecewise-constant kernel *κ**A**n* associated to *A**n*; this simply takes the value *a**i**j* on the square ((*i* − 1)/*n*, *i*/*n*] × ((*j* − 1)/*n*, *j*/*n*], 1 ≤ *i*, *j* ≤ *n*. There is also a sparse random graph *G*(*A**n*) associated to *A**n*; this is the graph on [*n*] in which edges are present independently, and the probability that *i**j* is an edge is *a**i**j*/*n*. (If *A**n* has non-zero diagonal entries then *G*(*A**n*) may contain loops. These are irrelevant here.)\nThe main result of Bollobás, Borgs, Chayes and Riordan is that if *κ* is an irreducible bounded kernel and (*A**n*) is a sequence of matrices with uniformly bounded entries such that *δ*▫(*κ**A**n*, *κ*) → 0, then the normalized size of the giant component in *G*(*A**n*) converges in probability to *ρ*(*κ*). The sprinkling argument there relies on the following lemma concerning the graph *G*(*A**n*), in which edges are present independently.\n[lBBCR] Let *κ* be an irreducible bounded kernel on [0, 1], and *δ* and *β*max positive constants. There is a constant *c* = *c*(*κ*, *β*max, *δ*) > 0 such that whenever *A**n* is a sequence of symmetric matrices with entries in [0, *β*max] with *δ*▫(*κ**A**n*, *κ*) → 0, then for sufficiently large *n* we have \n$${{\\mathop{\\mathbb P{}}\\nolimits}}(V\\_n\\sim\\_{G({A\\_n})} V\\_n') \\ge 1-\\exp(-cn)$$ \n for all disjoint *V**n*, *V**n*ʹ ⊂ [*n*] with ∣*V**n*∣, ∣*V**n*ʹ∣ ≥ *δ**n*, where *V**n* ∼ *G*(*A**n*)*V**n*ʹ denotes the event that *G*(*A**n*) contains a path starting in *V**n* and ending in *V**n*ʹ.\nIn fact, this lemma is not stated explicitly in , but this is exactly the content of the end of Section 3 there; for an explicit statement and proof of (a stronger version of) this lemma see .\nWe shall apply Lemma [lBBCR] to graphs *G*(*A**n*) corresponding to (subgraphs of) $G(n,\\delta{{\\undertilde{{\\kappa}}}})$, where *δ* is as in Lemma [leasypart]. To achieve independence between edges, we shall simply take only one edge from each hyperedge. Unfortunately, the problem of conditioning on the *x**i* still remains; we shall return to this shortly.\n[Gnkdef] Let ${{\\undertilde{{\\kappa}}}}$ be an integrable hyperkernel and let *H**n* be the Poisson (multi-)hypergraph corresponding to $G(n,{{\\undertilde{{\\kappa}}}})$. Given the sequence ${{\\bf x}}=(x\\_1,\\ldots,x\\_n)$, let ${{\\widetilde G}}(n,{{\\undertilde{{\\kappa}}}},{{\\bf x}})$ be the random (multi-)graph formed from *H**n* by replacing each *r*-vertex hyperedge *E* by a single edge, chosen uniformly at random from the $\\binom{r}{2}$ edges corresponding to *E*.\nWith ${{\\bf x}}$ fixed, the numbers of copies of each edge *E* in *H**n* are independent Poisson random variables. From basic properties of Poisson processes, it follows that, with ${{\\bf x}}$ fixed, the number of copies of each edge *i**j* in ${{\\widetilde G}}(n,{{\\undertilde{{\\kappa}}}},{{\\bf x}})$ are also independent Poisson random variables. Our next aim is to calculate the edge probabilities in ${{\\widetilde G}}(n,{{\\undertilde{{\\kappa}}}},{{\\bf x}})$.\nAs usual, we write *a*(*b*) for the *falling factorial* *a*(*a* − 1)⋯(*a* − *b* + 1). Given *x*1, …, *x**n* and distinct *i*, *j* ∈ [*n*], for *r* ≥ 2 let\n \n*a**r*, *i*, *j* = *n*− (*r* − 2)∑*κ**r*(*x**i*, *x**j*, *x**k*3, …, *x**k**r*), \nwhere the sum runs over all (*n* − 2)(*r* − 2) sequences *k*3, …, *k**r* of distinct indices in [*n*] \\ {*i*, *j*}, and let *A* be the *n*-by-*n* matrix with entries\n \n*a**i**j* = 2∑*r* ≥ 2*a**r*, *i*, *j* \nfor *i* ≠ *j* and *a**i**j* = 0 if *i* = *j*.\nWith ${{\\bf x}}$ given, the expected number of *r*-vertex hyperedges in *H**n* containing *i**j* is *r*(*r* − 1)*a**r*, *i*, *j*/*n*. Hence the expected number of *i**j* edges in ${{\\widetilde G}}(n,{{\\undertilde{{\\kappa}}}},{{\\bf x}})$ is exactly *a**i**j*/*n*. Now *a**i**j* clearly depends on *x**i* and *x**j*. Unfortunately, it also depends on all the other *x**k*. The next lemma will show that the latter dependence can be neglected.\nSet \n$${\\kappa}\\_{r}(x,y,\\*) = \\int\\_{{{\\mathcal S}}^{{r}-2}}{\\kappa}\\_{r}(x,y,x\\_3,x\\_4,\\ldots,x\\_{r}) {\\,d}\\mu(x\\_3)\\cdots{\\,d}\\mu(x\\_{r}),$$ \n and let *τ* be the ‘re-scaled’ edge kernel defined by\n \n*τ*(*x*, *y*) = 2∑*r* ≥ 2*κ**r*(*x*, *y*, \\* ). \nComparing with the formula for ${{{\\kappa}\\_{\\mathrm e}}}(x,y)$, note that we have divided each term in the sum in by $\\binom{{r}}{2}$, the number of edges in *K**r*. Note that\n \n$$\\label{tkae}\n \\tau(x,y)=0 \\iff {{{\\kappa}\\_{\\mathrm e}}}(x,y)=0.$$ \nRecall that *a**r*, *i*, *j* and *a**i**j* depend on the random sequence ${{\\bf x}}$. In the next lemma, the expectation is over the random choice of ${{\\bf x}}$; no graphs appear at this stage.\n[LA] Let ${{\\undertilde{{\\kappa}}}}=({\\kappa}\\_r)\\_{r\\ge2}$ be an integrable hyperkernel. Then\n \n$$\\label{lak}\n{{\\mathop{\\mathbb E{}}\\nolimits}}\\frac1{n^2}\\sum\\_{i\\neq j} |{a\\_{r,i,j}}-{\\kappa}\\_r(x\\_i,x\\_j,\\*)|\n=o(1)$$ \nfor every *r*, and\n \n$$\\label{lat}\n{{\\mathop{\\mathbb E{}}\\nolimits}}\\frac1{n^2}\\sum\\_{i\\neq j} |{a\\_{ij}}-\\tau(x\\_i,x\\_j)|\n=o(1).$$ \nWe have\n \n$${{\\mathop{\\mathbb E{}}\\nolimits}}{\\bigl({a\\_{r,i,j}}\\mid x\\_i,x\\_j\\bigr)}\n=(n-2){\\_{(r-2)}}\\,n^{-(r-2)}{\\kappa}\\_r(x\\_i,x\\_j,\\*).$$ \nSuppose first that *κ**r* is bounded. Let\n \n$$\\begin{split}\n Y\\_{ij}\n&=\n{a\\_{r,i,j}}-(n-2){\\_{(r-2)}}\\, n^{-(r-2)}{\\kappa}\\_r(x\\_i,x\\_j,\\*)\n\\\\&\n=n^{-(r-2)}\n\\sum({\\kappa}\\_r(x\\_i,x\\_j,x\\_{k\\_3},\\dots,x\\_{k\\_r})-{\\kappa}\\_r(x\\_i,x\\_j,\\*)), \n \\end{split}$$ \nwhere the sum again runs over all (*n* − 2)(*r* − 2) sequences *k*3, …, *k**r* of distinct indices in [*n*] \\ {*i*, *j*}. Given *x**i* and *x**j*, each term in the sum has mean 0, and any two terms with disjoint index sets {*k*3, …, *k**r*} are independent. Since there are *O*(*n*2*r* − 5) pairs of terms with overlapping index sets, and *κ**r* is bounded, we have\n \n$${{\\mathop{\\mathbb E{}}\\nolimits}}(Y\\_{ij}^2\\mid x\\_i,x\\_j) =O(n^{2r-5-2(r-2)})=O(n^{-1}).$$ \nThus ${{\\mathop{\\mathbb E{}}\\nolimits}}Y\\_{ij}^2 =O(n^{-1})$. Hence, by the Cauchy–Schwarz inequality, ${{\\mathop{\\mathbb E{}}\\nolimits}}|Y\\_{ij}|\\le({{\\mathop{\\mathbb E{}}\\nolimits}}Y\\_{ij}^2)^{1/2}=O(n^{-1/2})$, and so\n \n$${{\\mathop{\\mathbb E{}}\\nolimits}}\\frac1{n^2}\\sum\\_{i\\neq j} |{a\\_{r,i,j}}-{\\kappa}\\_r(x\\_i,x\\_j,\\*)|\n=\n\\frac1{n^2}\\sum\\_{i\\neq j}{{\\mathop{\\mathbb E{}}\\nolimits}}|Y\\_{ij}|+O(1/n)\n=O(n^ {-1/2}).$$ \nThis proves and thus for bounded hyperkernels.\nFor general hyperkernels, we use truncation and define *κ**M**r* by. For the corresponding *a**r*, *i*, *j*(*M*), *A*(*M*) and *τ**M*,\n \n$$\\begin{gathered}\n{{\\mathop{\\mathbb E{}}\\nolimits}}\\frac1{n^2}\\sum\\_{i\\neq j} |{a\\_{r,i,j}}-{a^{(M)}\\_{r,i,j}}|\n\\le\\int({\\kappa}\\_r-{{\\kappa}^M}\\_r),\n\\\\\n{{\\mathop{\\mathbb E{}}\\nolimits}}\\frac1{n^2}\\sum\\_{i\\neq j} |{\\kappa}\\_r(x\\_i,x\\_j,\\*)-{{\\kappa}^M}\\_r(x\\_i,x\\_j,\\*)| \n\\le\\int({\\kappa}\\_r-{{\\kappa}^M}\\_r),\n\\intertext{and thus}\n{{\\mathop{\\mathbb E{}}\\nolimits}}\\frac1{n^2}\\sum\\_{i\\neq j} |{a\\_{ij}}-{a^{(M)}\\_{ij}}|\n\\le2\\sum\\_r\\int({\\kappa}\\_r-{{\\kappa}^M}\\_r),\n\\\\\n{{\\mathop{\\mathbb E{}}\\nolimits}}\\frac1{n^2}\\sum\\_{i\\neq j} |\\tau(x\\_i,x\\_j)-{\\tau^M}(x\\_i,x\\_j)| \n\\le2\\sum\\_r\\int({\\kappa}\\_r-{{\\kappa}^M}\\_r).\\end{gathered}$$ \nSince (*κ**r*) is integrable, given any *ɛ* > 0 we can make these expected differences less than *ɛ* by choosing *M* large enough, and the result follows from the bounded case.\nWith the preparation above we are now ready to prove Theorem [th1].\n[Proof of Theorem [th1]] We assume without loss of generality that ${{\\mathcal S}}=[0,1]$, with *μ* Lebesgue measure.\nLet ${{{\\undertilde{{\\kappa}}}}'}=({\\kappa}'\\_F)\\_{F\\in {{\\mathcal F}}}$ be an irreducible, integrable kernel family, let ${{\\undertilde{{\\kappa}}}}=({\\kappa}\\_r)\\_{r\\ge2}$ be the corresponding hyperkernel, given by, and let *ɛ* > 0. As noted after Lemma [leasypart], in the light of this lemma, it suffices to prove the lower bound on $C\\_1(G(n,{{\\undertilde{{\\kappa}}}}))$. We may and shall assume that $\\rho({{\\undertilde{{\\kappa}}}})>0$ and ${\\varepsilon}<\\rho({{\\undertilde{{\\kappa}}}})/10$, say.\nLet *δ* > 0 and *ω* = *ω*(*n*) be as in Lemma [leasypart], and let *H**n*, *H**n*ʹ and ${{\\widetilde H}}\\_n$ be the Poisson multi-hypergraphs associated to the hyperkernels ${{\\undertilde{{\\kappa}}}}$, $(1-\\delta){{\\undertilde{{\\kappa}}}}$ and $\\delta{{\\undertilde{{\\kappa}}}}$, respectively. Using the same vertex types ${{\\bf x}}=(x\\_1,\\ldots,x\\_n)$ for all three hypergraphs, there is a natural coupling in which $H\\_n = H\\_n'\\cup {{\\widetilde H}}\\_n$, with *H**n*ʹ and ${{\\widetilde H}}\\_n$ *conditionally* independent given ${{\\bf x}}$.\nDefine *A* and *τ* by and, respectively, starting from the integrable hyperkernel $\\delta{{\\undertilde{{\\kappa}}}}$. Note that *τ* is a kernel on [0, 1], while *A* is an *n*-by-*n* matrix *that depends on ${{\\bf x}}$*. Recall from that *τ*(*x*, *y*) = 0 if and only if ${{{\\kappa}\\_{\\mathrm e}}}(x,y)=0$, so *τ* is irreducible. In order to be able to apply Lemma [lBBCR], we would like to work with a bounded kernel and matrices that are bounded uniformly in *n*. We achieve this simply by considering ${{\\overline {A}}}=({{\\overline a}\\_{ij}})$ and ${{\\overline\\tau}}$ defined by \n$${{\\overline a}\\_{ij}}= \\min\\{{a\\_{ij}},1\\} \\quad\\hbox{ and }\\quad{{\\overline\\tau}}(x,y)=\\min\\{\\tau(x,y),1\\}.$$ \nLet *B* be the (random) ‘sampled’ matrix corresponding to *τ*, defined by \n*b**i**j* = *τ*(*x**i*, *x**j*), \n and let ${{\\overline{B}}}$ be the corresponding matrix associated to ${{\\overline\\tau}}$. The second statement of Lemma [LA] tells us exactly that \n$${{\\mathop{\\mathbb E{}}\\nolimits}}\\frac{1}{n^2} \\sum\\_{i\\ne j} |{a\\_{ij}}-{b\\_{ij}}| =o(1),$$ \n where the expectation is over the random choice of ${{\\bf x}}$. Since $|{{\\overline a}\\_{ij}}-{{\\overline b}\\_{ij}}|\\le |{a\\_{ij}}-{b\\_{ij}}|$ for *i* ≠ *j*, while $|{{\\overline a}\\_{ii}}-{{\\overline b}\\_{ii}}|\\le 1$, it follows that \n$${{\\mathop{\\mathbb E{}}\\nolimits}}\\frac{1}{n^2} \\sum\\_{i,j} |{{\\overline a}\\_{ij}}-{{\\overline b}\\_{ij}}| =o(1),$$ \n or, equivalently, that ${{\\mathop{\\mathbb E{}}\\nolimits}}{\\ensuremath{\\|{\\kappa}\\_{{{\\overline {A}}}}-{\\kappa}\\_{{\\overline{B}}}\\|}}\\_1=o(1)$, where we write *κ**M* for the piecewise constant kernel ${\\kappa}\\_M:[0,1]^2\\to{{\\mathbb R}}$ associated to a matrix *M*.\nSince *δ*▫(*κ*1, *κ*2) ≤ ∥*κ*1 − *κ*2∥▫ ≤ ∥*κ*1 − *κ*2∥1, it follows that ${{\\mathop{\\mathbb E{}}\\nolimits}}{{\\delta\\_{\\square}}}({\\kappa}\\_{{{\\overline {A}}}},{\\kappa}\\_{{\\overline{B}}})\\to 0$, and hence that ${{\\delta\\_{\\square}}}({\\kappa}\\_{{{\\overline {A}}}},{\\kappa}\\_{{\\overline{B}}}){\\overset{\\mathrm{p}}{\\to}}0$. Coupling the random sequences ${{\\bf x}}$ for different *n* appropriately, we may and shall assume that\n \n$$\\label{c1}\n {{\\delta\\_{\\square}}}({\\kappa}\\_{{{\\overline {A}}}},{\\kappa}\\_{{\\overline{B}}})\\to 0$$ \nalmost surely.\nSince ${{\\overline\\tau}}$ is a bounded kernel on [0, 1], i.e., a ‘graphon’ in the terminology of , Theorem 4.7 of Borgs, Chayes, Lovász, Sós and Vesztergombi tells us that with probability at least 1 − *e*− *n*2/(2log2*n*) = 1 − *o*(1), we have ${{\\delta\\_{\\square}}}({\\kappa}\\_{{\\overline{B}}},{{\\overline\\tau}})\\le 10\\sup{{\\overline\\tau}}/\\sqrt{\\log\\_2 n}=o(1)$. It follows that ${{\\delta\\_{\\square}}}({\\kappa}\\_{{\\overline{B}}},{{\\overline\\tau}})\\to 0$ both in probability and almost surely. Using, we see that\n \n$$\\label{c2}\n {{\\delta\\_{\\square}}}({\\kappa}\\_{{{\\overline {A}}}}, {{\\overline\\tau}}) \\to 0$$ \nalmost surely. Note that ${\\kappa}\\_{{{\\overline {A}}}}$ depends on the sequences ${{\\bf x}}$.\nLet *G**n*ʹ and *G**n* be the simple graphs underlying *H**n*ʹ and *H**n*. From Lemma [leasypart], holds whp. For the rest of the proof, *we condition on ${{\\bf x}}$ and on *H**n*ʹ*. We assume, as we may, that holds for all large enough *n*, and that holds. It suffices to show that with *conditional* probability 1 − *o*(1) we have $C\\_1(G\\_n)\\ge (\\rho({{\\undertilde{{\\kappa}}}})-2{\\varepsilon}) n$. Recall that, given ${{\\bf x}}$, the (multi-)hypergraphs *H**n*ʹ and ${{\\widetilde H}}\\_n$ are independent, so after our conditioning (on ${{\\bf x}}$ and *H**n*ʹ), the hypergraph ${{\\widetilde H}}\\_n$ is formed by selecting each *r*-tuple *v*1, …, *v**r* to be an edge independently, with probability *δ**κ**r*(*x**v*1, …, *x**v**r*)/*n**r* − 1.\nLet ${{\\widetilde G}}\\_n={{\\widetilde G}}(n,\\delta{{\\undertilde{{\\kappa}}}},{{\\bf x}})$ be the random (multi-)graph defined from ${{\\widetilde H}}\\_n$ by taking one edge from each hyperedge as in Definition [Gnkdef], noting that $G\\_n'\\cup {{\\widetilde G}}\\_n\\subseteq G\\_n$. Since we have conditioned on ${{\\bf x}}$ (and *G**n*ʹ), as noted after Definition [Gnkdef], each possible edge *i**j* is present in ${{\\widetilde G}}\\_n$ independently. In the multi-graph version, the number of copies of *i**j* is Poisson with mean *a**i**j*/*n*. Passing to a subgraph, we shall take instead the number of copies to be Poisson with mean ${{\\overline a}\\_{ij}}/n$. Since this mean is *O*(1/*n*), the probability that one or more copies of *i**j* is present is *a*ʹ*i**j*/*n*, where ${a'\\_{ij}}={{\\overline a}\\_{ij}}+O(1/n)$. Since ${{\\delta\\_{\\square}}}({\\kappa}\\_{{A'}},{\\kappa}\\_{{{\\overline {A}}}})=O(1/n)=o(1)$, we have ${{\\delta\\_{\\square}}}({\\kappa}\\_{{A'}},{{\\overline\\tau}})\\to 0$. Since ${{\\overline\\tau}}$ is an irreducible bounded kernel, the (simple graphs underlying) ${{\\widetilde G}}\\_n$ satisfy the assumptions of Lemma [lBBCR], so there is a constant *c* > 0 such that for any two set *V**n*, *V**n*ʹ of at least *ɛ**n*/2 vertices of ${{\\widetilde G}}\\_n$, the probability that *V**n* and *V**n*ʹ are *not* joined by a path in ${{\\widetilde G}}\\_n$ is at most *e*− *c**n*.\nRecall that we have conditioned on *G**n*ʹ, assuming. Suppose also that $C\\_1(G\\_n)\\le (\\rho({{\\undertilde{{\\kappa}}}})-2{\\varepsilon})n$. Then there is a partition (*V*1, *V*2) of the set of vertices of *G**n*ʹ in large components in *G**n*ʹ with ∣*V*1∣, ∣*V*2∣ ≥ *ɛ**n* such that there is no path in *G**n* from *V*1 to *V*2. Let us call such partition (*V*1, *V*2) a *bad partition*. Having conditioned on *G**n*ʹ, noting that in any potential bad partition *V*1 must be a union of large components of *G**n*ʹ, the number of possible choices for (*V*1, *V*2) is at most 2*n*/*ω* = *e**o*(*n*). On the other hand, since ${{\\widetilde G}}\\_n\\subseteq G\\_n$, the probability that any given partition is bad is at most *e*− *c**n*, so the expected number of bad partitions is *o*(1), and whp there is no bad partition. Thus $C\\_1(G\\_n) > (\\rho({{\\undertilde{{\\kappa}}}})-2{\\varepsilon})n$ holds whp, as required.\n[Rred] The restriction to irreducible kernel families in Theorem [th1] is of course necessary; roughly speaking, if ${{\\undertilde{{\\kappa}}}}$ is reducible, then our graph $G(n,{{\\undertilde{{\\kappa}}}})$ falls into two or more parts. Lemma [leasypart] still applies to show that we have $\\rho({{\\undertilde{{\\kappa}}}})n+{o\\_{\\mathrm{p}}}(n)$ vertices in large components, but it may be that two or more parts have giant components, each of smaller order than $\\rho({{\\undertilde{{\\kappa}}}})n$.\nMore precisely, let ${{\\undertilde{{\\kappa}}}}$ be a reducible, integrable kernel family. Thus the edge kernel ${{{\\kappa}\\_{\\mathrm e}}}$ is reducible. By Lemma 5.17 of , there is a partition ${{\\mathcal S}}=\\bigcup\\_{i=0}^N {{\\mathcal S}}\\_i$, *N* ≤ ∞, of our ground space ${{\\mathcal S}}$ (usually [0, 1]) such that each ${{\\mathcal S}}\\_i$ is measurable, the restriction of ${{{\\kappa}\\_{\\mathrm e}}}$ to ${{\\mathcal S}}\\_i$ is irreducible (in the natural sense), and, apart from a measure zero set, ${{{\\kappa}\\_{\\mathrm e}}}$ is zero off $\\bigcup\\_{i=1}^N {{\\mathcal S}}\\_i\\times {{\\mathcal S}}\\_i$.\nSuppressing the dependence on *n*, let *G**i* be the subgraph of $G(n,{{\\undertilde{{\\kappa}}}})$ induced by the vertices with types in ${{\\mathcal S}}\\_i$. Since the vertex types are i.i.d., the probability that $G(n,{{\\undertilde{{\\kappa}}}})$ contains any edges other than those of ⋃*i* ≥ 1*G**i* is 0. Now *G**i* has a random number *n**i* of vertices, with a binomial ${\\operatorname{Bi}}(n,\\mu({{\\mathcal S}}\\_i))$ distribution, which is concentrated around its mean. Given *n**i*, the graph *G**i* is another instance of our model.\nLet $a\\_i=\\int\\_{{{\\mathcal S}}\\_i} \\rho\\_{{\\undertilde{{\\kappa}}}}(x){\\,d}\\mu(x)$, so that $\\sum\\_i a\\_i=\\rho({{\\undertilde{{\\kappa}}}})<1$. From the remarks above it is easy to check that Theorem [th1] gives $C\\_1(G\\_i)/n{\\overset{\\mathrm{p}}{\\to}}a\\_i$ and *C*2(*G**i*) = *o*p(*n*); we omit the details. Sorting the *a**i* into decreasing order ${{\\hat a}}\\_1,{{\\hat a}}\\_2,\\ldots$, it follows that $C\\_i(G(n,{{\\undertilde{{\\kappa}}}}))={{\\hat a}}\\_i n+{o\\_{\\mathrm{p}}}(n)$ for each fixed (finite) 1 ≤ *i* ≤ *N*, in particular, for *i* = 1 and *i* = 2.\nDisconnected atoms and percolation\n==================================\nOne of the most studied features of the various inhomogeneous network models is their ‘robustness’ under random failures, and in particular, the critical point for site or bond percolation on these random graphs. For example, this property of the Barabási–Albert model was studied experimentally by Barabási, Albert and Jeong , heuristically by Callaway, Newman, Strogatz and Watts (see also"},"data_id":{"kind":"string","value":"arxiv_0002031"}}},{"rowIdx":493,"cells":{"content":{"kind":"string","value":"All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces\n===================================================================================\nWe demonstrate that the linear quotient singularity for the exceptional subgroup *G* in $\\Sp(4,{\\mathbb{C}})$ of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of C4/*G* as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the *n*-pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension 2*n* − 6; for example, we show that there are 1684 projective crepant resolutions when *n* = 6. We also prove that the resulting affine cones are *not* quotient singularities for *n* ≥ 6.\nIntroduction\n============\nThis paper has three main goals: (a) to establish a remarkable coincidence of four-dimensional symplectic cones, namely a finite quotient singularity and a hyperpolyon space; (b) to describe completely the birational geometry of hyperpolygon spaces in *all* even dimensions, including all their projective crepant resolutions; and (c) to show that the phenomenon from (a) does not occur for hyperpolygon spaces in dimension greater than four.\nGoal (a) concerns, on one hand, the symplectic quotient singularity C4/*G* associated to an exceptional subgroup *G* in $\\Sp(4,{\\mathbb{C}})$ of order 32 that is known to admit 81 projective crepant resolutions. On the other hand, we consider the affine hyperpolygon space *X*5(0) that admits projective crepant resolutions by variation of GIT quotient for quiver varieties.\nGoal (b) is achieved by exploiting the identification between hyperpolygon spaces *X**n*(0) and quiver varieties for a star-shaped quiver. This enables us to give a complete, uniform description of the birational geometry of hyperpolygon spaces *X**n*(*θ*) for any *n* ≥ 3, and in the case *n* = 5, we use the semi-invariant ring of the quiver to identify C4/*G* with *X*5(0). As a consequence, we describe C4/*G* and its crepant resolutions in terms of quiver varieties, leading to a transparent, uniform construction of all 81 projective crepant resolutions of C4/*G*. This provides a conceptual, quiver-theoretic explanation for the main results of that bypasses computer calculations.\nNote that *X*4(0) is well known to be isomorphic to the du Val singularity C2/*Q*8 for *Q*8 the quaternionic group of order eight. As a result, *X**n*(0) is a finite quotient singularity for *n* ≤ 5. It is therefore an obvious question to ask whether any of these are finite quotient singularities for larger *n*. Goal (c) answers this question in the negative: *X**n*(0) is not a finite quotient singularity for any *n* > 5, or equivalently, for hyperpolygon spaces in dimension greater than four.\nThe phenomenon that we reveal in goal (a) can be viewed as special in two ways: it establishes one more finite quotient singularity which is a quiver variety; and it reveals another quiver variety which is a finite quotient singularity.\nThis is very unusual since, unlike the case of quiver varieties, *very few* finite quotient singularities admit crepant resolutions. In fact, the classification of finite, symplectic quotient singularities admitting a symplectic resolution is nearly complete (there are up to 45 remaining finite symplectic quotients in dimension four that are not expected to admit crepant resolutions, see ): they fall into one infinite family, namely symmetric powers of du Val singularities, and two exceptional cases in dimension four, one of which is the quotient singularity C4/*G* that we study in goal (a). The infinite family is well known to be isomorphic to quiver varieties for extended Dynkin quivers, with crepant resolutions given by variation of GIT quotient (after framing the quiver). In this context, then, our results for C4/*G* are consistent with the behaviour in the infinite family. Note that, in the remaining exceptional case, although it is unknown whether the resolution of singularities can be constructed by variation of GIT quotient, the singularity itself was recently identified to be a symplectic quotient of a vector space by a reductive group, also known as a Higgs branch variety.\nThus a consequence of our result is that *all* finite symplectic quotient singularities admitting crepant resolutions are such quotients (up to the 45 remaining cases mentioned above). The four-dimensional example of this paper, like the symmetric powers of du Val singularities, is a quiver variety with resolution given by variation of GIT. In this sense the phenomenon established in goal (a) is not just a special case, but completes a general pattern.\nAn exceptional quotient singularity\n-----------------------------------\nThe group $G = Q\\_8 \\times\\_{{\\mathbb{Z}}\\_2} D\\_8 < \\Sp(4,{\\mathbb{C}})$ of order 32 was identified in as one for which the linear quotient C4/*G* admits a crepant (equivalently, a symplectic) resolution of singularities; here $Q\\_8 < \\Sp(2,{\\mathbb{C}})$ is the quaternionic group of order 8, *D*8 < O(2, R) is the dihedral group of order 8, and Z2 is identified with their respective centres. Subsequently, it was established in that C4/*G* admits precisely 81 projective crepant resolutions; simultaneously, these resolutions were all constructed explicitly in by variation of GIT quotient.\nMore recently, Mekareeya discovered that this quotient singularity should be isomorphic to a Higgs branch variety. More precisely, showed that the Hilbert series of the ring of functions on the singularity C4/*G* coincides with that of the conical quiver variety ${\\mathfrak{M}}\\_{0}({\\boldsymbol{v}},0)$ associated to a star-shaped quiver *Q* with a central vertex and five external vertices, where the components of the dimension vector ${\\boldsymbol{v}}$ equal 2 on the central vertex and 1 elsewhere (see Section [ss:quiver] for precise definitions). The present work was motivated by our desire to give a conceptual geometric proof of Mekareeya’s numerical observation:\n[t:isom] There is a Poisson isomorphism ${\\mathbb{C}}^4 / G {{\\;\\stackrel{\\_\\sim}{\\longrightarrow}\\;}}X\\_5(0):= {\\mathfrak{M}}\\_{0}({\\boldsymbol{v}},0)$.\nWhile comparing these two varieties directly seems to be difficult, it is natural to compare instead the Cox ring of the singularity C4/*G*, as studied in, with a semi-invariant ring ${\\mathbb{C}}[\\mu^{-1}(0)]^{\\SL\\_2}$, where *μ* is the moment map arising in the description of ${\\mathfrak{M}}\\_{0}({\\boldsymbol{v}},0)$ as a GIT quotient. We prove that a (C×)5-invariant subring of ${\\mathbb{C}}[\\mu^{-1}(0)]^{\\SL\\_2}$ is isomorphic to the coordinate ring of ${\\mathfrak{M}}\\_{0}({\\boldsymbol{v}},0)$, and Theorem [t:isom] follows by passing to the torus-invariant subrings. This semi-invariant ring is closely related to the Cox ring of a crepant resolution of ${\\mathfrak{M}}\\_{0}({\\boldsymbol{v}},0)$; the relationship between these two rings will be discussed in the more general context of quiver varieties in.\nA hyperplane arrangement\n------------------------\nIt turns out that studying quiver varieties for the star-shaped quiver with *n* external vertices (for *n* ≥ 3, see Figure [fig:quivers]) and dimension vector ${\\boldsymbol{v}}=(2,1,1,\\dots,1)$ is not substantially harder than the case when *n* = 5. We consider this more general setting, where for any stability parameter *θ* ∈ Θ : = Q*n*, the quiver variety \n$$X\\_n(\\theta):= {\\mathfrak{M}}\\_{0}({\\boldsymbol{v}},\\theta) = \\mu^{-1}(0)^{\\theta\\text{-ss}}{\\ensuremath{/\\!\\!/\\!}}\\operatorname{\\mathrm{GL}}\\_{{\\boldsymbol{v}}}$$ \n is obtained as a GIT quotient (see Section [ss:quiver]). For any sufficiently general stability parameter *θ* ∈ Θ satisfying *θ**i* > 0, this *hyperpolygon space* provides a hyperkähler analogue of a moduli space of polygons with prescribed edge lengths. Hyperpolygon spaces have proved to be an effective testing ground for conjectures on quiver varieties; see, for example,.\nTo study these spaces, consider the following hyperplane arrangement in Θ: \n$$\\mc{A} = \n \\big\\{ L\\_i, H\\_I \\mid 1\\leq i\\leq n, \\{1\\}\\subseteq I\\subseteq \\{1,\\dots,n\\}\\big\\},$$ \n where *L**i* = {*θ* ∈ Θ ∣ *θ**i* = 0} is a coordinate hyperplane and $H\\_I= \\{ \\theta \\in \\Theta \\mid \\sum\\_{i \\in I} \\theta\\_i = \\sum\\_{j \\notin I} \\theta\\_j\\}$. This arrangement is known in the theory of (hyper-)polygon spaces: avoiding the hyperplanes *H**I* is precisely the condition that your polygon should not contain a line. Our interest lies largely with stability parameters lying in the complement to these hyperplanes: a *chamber* of $\\Theta^{\\reg} = \\Theta \\smallsetminus \\bigcup\\_{H\\in \\mathcal{A}} H$ is the intersection with Θ of a connected component of the real hyperplane arrangement complement in Θ ⊗ ZR. We prove (see Proposition [prop:Hnthetasmooth]) that $\\Hyp\\_n(\\theta)$ is smooth if and only if $\\theta \\in \\Theta^{\\reg}$ and, moreover, the chambers in the complement of A are precisely the GIT chambers arising in the GIT construction of the spaces *X**n*(*θ*).\nThe birational geometry of hyperpolygon spaces\n----------------------------------------------\nObtaining an explicit understanding of the GIT chamber decomposition proved to be a key step in the recent quiver variety description of all projective crepant resolutions for symmetric powers of Du Val singularities. Here, we apply similar arguments to show that variation of GIT quotient produces all projective crepant resolutions of the affine hyperpolygon space $X\\_n(0)={\\mathfrak{M}}\\_0({\\boldsymbol{v}},0)$. The resolutions *X**n*(*θ*) for $\\theta\\in \\Theta^{\\reg}$ all share the same movable cone of line bundle classes, with different resolutions corresponding to different ample cones. We use the linearisation map for the GIT construction to identify each GIT chamber in Θ with the ample cone of the appropriate hyperpolygon space (up to the action of Namikawa’s Weyl group, in this case Z2*n*). We also show that every projective crepant resolution arises in this way, giving us a precise count of the resolutions.\nTo state the results in more detail, observe that the positive orthant \n*F* = {*θ* ∈ Θ ∣ *θ**i* ≥ 0, ∀*i* ≠ 0} \n is the union of the closures of a collection of GIT chambers in Θ. In we fix once and for all a chamber *C*+ in *F*, define *X* : = *X**n*(*θ*) for *θ* ∈ *C*+ and write *Y* : = *X**n*(0). Let *N*1(*X*/*Y*) denote the relative Néron–Severi space, defined as the rational vector space spanned by equivalence classes of line bundles on *X* up to numerical equivalence relative to curves contracted to a point in *Y*.\n[thm:resolutionsofstar] Let *n* ≥ 5. There is an isomorphism *L**F*: Θ → *N*1(*X*/*Y*) of rational vector spaces such that:\n1. *L**F* identifies the closed polyhedral cone *F* with the movable cone $\\Mov(X/Y)$; and\n2. *L**F* induces a bijection between the chambers in *F* and the ample cones $\\mr{Amp}(X\\_i/Y)$ of all projective crepant resolutions of *Y*.\nThus, for each *θ* ∈ *F*, the space *X**n*(*θ*) is isomorphic to the birational model of *X* determined by the line bundle *L**F*(*θ*), and conversely, every partial crepant resolution of *Y* = *X**n*(0) is of the form *X**n*(*θ*) for some *θ* ∈ *F*.\nIn the course of proving Theorem [thm:resolutionsofstar], we show that crossing any wall contained in the interior of *F* induces a Mukai flop between the corresponding hyperpolygon spaces. This result was originally established by Godinho–Mandini ; here, we bypass the study of wall crossing for moduli spaces of parabolic Higgs bundles from , deducing the result instead from an étale local description of wall crossing for quiver varieties.\nBeyond the movable cone\n-----------------------\nThere is an action of the group $\\Z\\_2^n$ on Θ, where the generator of the *i*th factor acts by reflection in the supporting hyperplane {*θ* ∈ Θ ∣ *θ**i* = 0} of the cone *F*. Notice that $\\Z\\_2^n$ permutes the hyperplanes in $\\mc{A}$, so it acts on the set of all chambers.\n[prop:beyondF] Let *n* ≥ 5. The isomorphism *L**F* from Theorem [thm:resolutionsofstar] identifies the action of $\\Z\\_2^n$ on Θ with the action of the Namikawa Weyl group on *N*1(*X*/*Y*). In particular, *θ* ∈ *C* and *θ*ʹ ∈ *C*ʹ satisfy $\\Hyp\\_n(\\theta) \\cong \\Hyp\\_n({\\theta'})$ as spaces over $\\Hyp\\_n(0)$ if and only if there exists $w \\in \\Z\\_2^n$ such that *w*(*C*) = *C*ʹ.\nThe spaces *X**n*(*θ*) have traditionally been studied only for those $\\theta\\in \\Theta^{\\reg}$ satisfying *θ**i* > 0 for 1 ≤ *i* ≤ *n*, i.e. for parameters in the interior of *F*, in which case the values of the components *θ**i* record the edge-lengths of polygons in the class parametrised by an irreducible component in the core of *X**n*(*θ*). Proposition [prop:beyondF] extends this observation to all $\\theta\\in \\Theta^{\\reg}$, because we have $\\Hyp\\_n(\\theta) \\cong \\Hyp\\_n(w(\\theta))$ for the unique *w* ∈ Z2*n* satisfying *w*(*θ*) ∈ *F*. In this case, the absolute values |*θ**i*| = *w*(*θ*)*i* for 1 ≤ *i* ≤ *n* record the appropriate edge-lengths. In short, the space *X**n*(*θ*) can legitimately be described as a *hyperpolygon space* for any $\\theta\\in \\Theta^{\\reg}$.\nTheorem [thm:resolutionsofstar](ii) establishes the following result.\n[cor:count] For *n* ≥ 5, the number of projective crepant resolutions of $\\Hyp\\_n(0)$ is equal to the number of chambers in *F*.\nCombining this result with Theorems [t:isom] and [thm:resolutionsofstar] when *n* = 5 recovers all 81 projective crepant resolutions of C4/*G* that were constructed explicitly by Donten-Bury–Wiśniewski , but here we demonstrate in addition that they are all hyperpolygon spaces. In this case, the hyperplanes in A are those appearing in. When *n* = 6, we compute that there are precisely 1684 resolutions. After the first draft of this paper appeared, Alastair King observed that the sequence recording the number of chambers in *F* for *n* ≥ 4 appears in the OEIS and, in particular, the number of projective crepant resolutions of *X**n*(0) for small values of *n* is as follows (see Remark [rem:adk]):\nc||c|c|c|c|c *n* & 5 & 6 & 7 & 8 & 9 \nresolution count & 81 & 1,684 & 122,921 & 33,207,256 & 34,448,225,389.\nIn general, counting the chambers in *F* for large values of *n* appears to be computationally difficult.\nQuotient singularities and Hamiltonian reduction.\n-------------------------------------------------\nIt is a classical fact that, for a finite subgroup $\\Gamma \\subset \\SL(2,{\\mathbb{C}})$, the symplectic quotient singularity ${\\mathbb{C}}^{2n} / (\\s\\_n \\wr \\Gamma)$ can be realised as a quiver variety, and it was shown in that all projective crepant resolutions of this quotient are obtained as quiver varieties by variation of GIT quotient. Therefore, it is natural to ask if every symplectic quotient singularity that admits a projective crepant resolution can be realised as a quiver variety (or, more generally, as a Hamiltonian reduction by a connected group). The only known example not already mentioned is the quotient C4/*G*4 that was studied in and. Though the singularity can be described as a Hamiltonian reduction, we do not know of a construction of this quotient as a quiver variety.\nMotivated by the behaviour of the hyperkähler metrics on *X**n*(*θ*) (see below) it is natural to ask, conversely, if *X**n*(0) is isomorphic to a symplectic quotient singularity for some *n* > 5. Our final main result answers this question:\n[thm:notisoquotientsing] For *n* > 5, the space *X**n*(0) is not isomorphic to a symplectic quotient singularity.\nThe proof depends on the (incomplete) classification of symplectic quotient singularities admitting a symplectic resolution.\nThe hyperkähler geometry of hyperpolygon spaces\n-----------------------------------------------\nBy virtue of the construction of the hyperpolygon space *X**n*(*θ*) as a hyperkähler quotient, in the sense of, it carries a complete hyperkähler metric for any generic choice of *θ*. Recently, there has been much interest in the asymptotic geometry of such spaces. The resolution of singularities *X**n*(*θ*) → *X**n*(0) identifies the singularity *X**n*(0) with the tangent cone at infinity of the hyperkähler manifold *X**n*(*θ*). The full description of the asymptotic geometry of *X**n*(*θ*) requires precise estimates on the decay of the true hyperkähler metric on *X**n*(*θ*) to the (singular) cone metric on *X**n*(0).\nIt is known that *X*4(*θ*) is asymptotically locally Euclidean (ALE). As a further generalisation of ALE, a smooth noncompact hyperkähler manifold is called *quasi-asymptotically locally Euclidean* (QALE) if its asymptotic geometry is modelled on C2*k*/*G* for a finite subgroup *G* in $\\Sp(k)$. Both ALE and QALE metrics have maximal, that is, Euclidean volume growth. Theorem [t:isom] shows that the hyperpolygon space *X*5(0) is the orbifold cone C4/*G* for the group *G* = *Q*8 × Z2*D*8. For generic *θ*, we expect that *X*5(*θ*) is quasi-asymptotically locally Euclidean; see Conjecture [conj:QALE].\nWe prove that such a result does *not* hold for polygons of more than five sides:\n[cor:QALEnot] For *n* > 5, *X**n*(*θ*) is not quasi-asymptotically locally Euclidean for any *θ*.\nWe now provide an outline of the paper, making reference to goals (a), (b) and (c) introduced above. Section [sec:2] provides the combinatorial framework that allows us to study hyperpolygon spaces and the associated GIT wall-and-chamber structure. Section [sec:theresofhyperpoly] achieves goal (b) by describing the birational geometry of the hyperpolygon spaces *X**n*(*θ*) for *n* ≥ 5 (in Theorem [thm:movable]). In Section [sec:4], we specialise to the case *n* = 5 to achieve goal (a) (see Theorem [thm:4foldsing]) by marrying our description of the $\\SL\\_2$-invariant ring (see Proposition [p:gens-repsl2]) with the Cox ring C[*V*][*G*, *G*] of the quotient singularity C4/*G* (see Proposition [p:rels-qt]). Section [sec:5] achieves goal (c) by showing that *X**n*(0) is not isomorphic to a symplectic quotient singularity for any *n* > 5; the key ingredient here describes the movable cone of a product (see Proposition [prop:movableproduct]) and hence reduces the problem to the study of irreducible symplectic reflections groups. Finally, Section [sec:6] outlines the implications of our results for metrics on hyperpolygon spaces, leading to a proof of Corollary [cor:QALEnot].\n**Acknowledgements.** We thank Amihay Hanany for pointing out and for multiple discussions, and Alastair King for the observation that forms Remark [rem:adk]. We also thank the referees for their comments and corrections. SR thanks Hiraku Nakajima and Laura Schaposnik for helpful conversations. SR and HW thank Laura Fredrickson, Rafe Mazzeo, and Jan Swoboda for useful discussions. GB and AC were partially supported by a Research Project Grant from the Leverhulme Trust. SR acknowledges the support of an Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant. HW acknowledges the support of the Deutsche Forschungsgemeinschaft (DFG) within SPP 2026 “Geometry at infinity”.\nHyperpolygon spaces\n===================\nIn this section we define hyperpolygon spaces as quiver varieties, and we recall some results that hold for any quiver variety. We also describe the GIT chamber decomposition for these spaces, and describe several invariant rings associated to the quiver GIT construction.\nHyperpolygon spaces as quiver varieties\n---------------------------------------\nThe hyperpolygon spaces are the hyperkähler analogues of polygon spaces, first introduced by Konno.\nTo construct these spaces, let *n* ≥ 3 and let *Q* = *Q*(*n*) be the star-shaped quiver with vertex set *Q*0 : = {0, 1, …, *n*} and arrow set *Q*1 : = {*a*1, …, *a**n*}, where for 1 ≤ *i* ≤ *n*, the arrow *a**i* has tail at 0 and head at *i*. Let ${\\overline{Q}}$ be the doubled quiver of *Q*, defined by ${\\overline{Q}}\\_0=Q\\_0$ and ${\\overline{Q}}\\_1 = Q\\_1 \\sqcup \\{a\\_1^\\*, \\ldots, a\\_n^\\*\\}$, where for 1 ≤ *i* ≤ *n*, the arrow *a**i*\\* has tail at *i* and head at 0. Consider the dimension vector ${\\boldsymbol{v}}= (2,1,1,\\dots,1) \\in {\\mathbb{N}}^{Q\\_0}$ with ${\\boldsymbol{v}}\\_0=2$ and ${\\boldsymbol{v}}\\_i=1$ for *i* ≠ 0. Let *V*0 = C2 and *V**i* = C for 1 ≤ *i* ≤ *n*.\n(0L) at (0,0) 2; (0R) at (5,0) 2; in 45, 90,..., 359 (Lv) at ($(0L)+(\\a:1.75)$) 1; (0L) edge (Lv); (Rv) at ($(0R)+(\\a:1.75)$) 1; (0R) edge[bend right=10] (Rv); (Rv) edge[bend right=10] (0R); ; (Ld) at ((0*L*) + (1.5, 0.1)) ⋮; (Rd) at ((0*R*) + (1.5, 0.1)) ⋮;\n[fig:quivers]\nThe representation space $\\operatorname{\\mathrm{Rep}}(\\overline{Q},{\\boldsymbol{v}}) := \\bigoplus\\_{i = 1}^n\\operatorname{\\mathrm{Hom}}(V\\_0, V\\_i)\\oplus \\bigoplus\\_{i = 1}^n\\operatorname{\\mathrm{Hom}}(V\\_i, V\\_0)$ admits an action by $\\operatorname{\\mathrm{GL}}\\_{{\\boldsymbol{v}}}:= \\operatorname{\\mathrm{GL}}\\_2 \\times ({\\mathbb{C}}^\\times)^n$, where $g\\cdot \\big((B\\_{a\\_i}),(B\\_{a\\_i^\\*})\\big) = \\big((g\\_iB\\_{a\\_i}g\\_0^{-1}), (g\\_0B\\_{a\\_i}g\\_i^{-1})\\big)$ for $g\\in \\operatorname{\\mathrm{GL}}\\_{{\\boldsymbol{v}}}$. The moment map $\\mu\\colon \\operatorname{\\mathrm{Rep}}(\\overline{Q},{\\boldsymbol{v}}) \\to\n\\mathfrak{gl}\\_{{\\boldsymbol{v}}} = \\mathfrak{gl}\\_2 \\oplus \\mathfrak{gl}\\_1^n$ for this action takes the form \n$$\\mu\\big((B\\_{a\\_i}),(B\\_{a\\_i^\\*})\\big) = \\left(\\sum\\_{j=1}^n B\\_{a\\_j^\\*}B\\_{a\\_j}, -B\\_{a\\_1}B\\_{a\\_1^\\*}, \\dots, -B\\_{a\\_n}B\\_{a\\_n^\\*}\\right).$$ \n Define $\\Theta:=\\{\\theta\\in \\mathbb{Q}^{n+1} \\mid \\theta\\cdot {\\boldsymbol{v}}= 0\\}$. For any *θ* ∈ Θ and any central $\\lambda\\in \\mathfrak{gl}\\_{{\\boldsymbol{v}}}$, let *μ*− 1(*λ*)*θ*-ss denote the locus of *θ*-semistable representations parametrised by points in *μ*− 1(*λ*). The associated (Nakajima) *quiver variety* is the GIT quotient \n$$\\mathfrak{M}\\_\\lambda({\\boldsymbol{v}},\\theta) = \\mu^{-1}(\\lambda){\\ensuremath{/\\!\\!/\\!}}\\_\\theta \\operatorname{\\mathrm{GL}}\\_{{\\boldsymbol{v}}} := \\mu^{-1}(\\lambda)^{\\theta\\text{-ss}}/\\operatorname{\\mathrm{GL}}\\_{{\\boldsymbol{v}}}$$ \n (we use the framing vector **w** = 0). We are particularly interested in the special case *λ* = 0, and to emphasise the dependence on the number *n* of external vertices in that case, we write \n$$X\\_n(\\theta) := \\mathfrak{M}\\_0({\\boldsymbol{v}},\\theta) = \\mu^{-1}(0)^{\\theta\\text{-ss}}/\\operatorname{\\mathrm{GL}}\\_{{\\boldsymbol{v}}}$$ \n for the resulting GIT quotient. It is easy to see that the variety *X**n*(*θ*) contains the GIT quotient $M\\_n(\\theta):= \\operatorname{\\mathrm{Rep}}(Q,{\\boldsymbol{v}}){\\ensuremath{/\\!\\!/\\!}}\\_\\theta \\operatorname{\\mathrm{GL}}\\_{{\\boldsymbol{v}}}$ that is a *polygon space* for parameters *θ* satisfying *θ**i* > 0 for *i* > 0. See for more on polygon and hyperpolygon spaces.\nCombinatorics of star shaped root systems\n-----------------------------------------\nLet *e*0, …, *e**n* denote the standard basis of R*Q*0, so *e**i* ∈ N*Q*0. The Cartan form ( − , − ) : Z*Q*0 × Z*Q*0 → Z for the quiver $\\overline{Q}$ is given by \n(*α*, *β*) = 2∑*i* = 0*n**α**i**β**i* − ∑*j* = 1*n*(*α*0*β**j* + *α**j**β*0) \n for *α* = *α*0*e*0 + ⋯ + *α**n**e**n* and *β* = *β*0*e*0 + ⋯ + *β**n**e**n*. Define \n$$p(\\alpha) = 1 - \\frac{1}{2} (\\alpha, \\alpha) = 1 - \\sum\\_{i = 0}^n \\alpha\\_i^2 + \\sum\\_{j = 1}^n \\alpha\\_0 \\alpha\\_j.$$ \nFor *i* ∈ *Q*0, the simple reflection *s**i*: Z*Q*0 → Z*Q*0 is the linear map given by *s**i*(*α*) : = *α* − (*α*, *e**i*)*e**i*. The *real roots* are the elements of N*Q*0 obtainable from the *e**i* by applying arbitrary sequences of simple reflections. A vector *α* ∈ N*Q*0 is said to be in the *fundamental region* if (*α*, *e**i*) ≤ 0 for all *i* and the support (= the collection of vertices *i* ∈ *Q*0 such that *α**i* ≠ 0) is connected. The (positive) *imaginary roots* are the elements of N*Q*0 obtainable from elements of the fundamental region by applying arbitrary sequences of simple reflections. An imaginary root *α* is *isotropic* if (*α*, *α*) = 0 and *anisotropic* if (*α*, *α*) < 0.\nLet *R**λ*, *θ*+ denote the set of all positive roots *α* such that *α* ⋅ *λ* = 0 = *α* ⋅ *θ*. The set Σ*λ*, *θ* consists of all roots *α* ∈ *R**λ*, *θ*+ such that, for every proper decomposition *α* = *α*(1) + ⋯ + *α*(*m*) with *α*(*i*) ∈ *R**λ*, *θ*+, we have *p*(*α*) > *p*(*α*(1)) + ⋯ + *p*(*α*(*m*)).\nIn terms of representation theory, the set Σ*λ*, *θ* consists of the dimension vectors of *θ*-stable representations of the deformed preprojective algebra Π*λ*.\nRecall that ${\\boldsymbol{v}}= (2,1,1,\\ds,1)$. The simple roots *e**i* are all, by definition, real. Let ${\\boldsymbol{v}}^{(i)} = {\\boldsymbol{v}}- e\\_i$ for $i = 1, \\ds, n$. We note that $({\\boldsymbol{v}},e\\_0) = 4 - n$ and $({\\boldsymbol{v}},e\\_i) = 0$ for *i* > 0, which implies that ${\\boldsymbol{v}}$ is in the fundamental region when *n* ≥ 4. Moreover, it is an anisotropic root when *n* ≥ 5 since $p({\\boldsymbol{v}}) = n - 3$. We also note that $p({\\boldsymbol{v}}- e\\_0) = p({\\boldsymbol{v}}) + ({\\boldsymbol{v}},e\\_0) - 1 = 0$.\n[lem:ve0realroot] The dimension vector ${\\boldsymbol{v}}- e\\_0$ is a real root.\nThis is by induction on *n*, and it is convenient to allow any *n* ≥ 0. When *n* = 0, ${\\boldsymbol{v}}- e\\_0= e\\_0$ is a (real) root. Assume *n* > 0. Notice that for any *i* > 0, in particular for *i* = *n*, we have $s\\_i({\\boldsymbol{v}}- e\\_0) = {\\boldsymbol{v}}- e\\_{0} - e\\_i$ since $({\\boldsymbol{v}}-e\\_0,e\\_i) = 1$. If we define ${\\boldsymbol{v}}\\_n = {\\boldsymbol{v}}- e\\_0$ then this shows that $s\\_n({\\boldsymbol{v}}\\_n) = {\\boldsymbol{v}}\\_{n-1}$. By induction, ${\\boldsymbol{v}}\\_{n-1}$ is a real root, so ${\\boldsymbol{v}}\\_n$ is a real root.\nFor each subset *I* ⊂ [1, *n*] define ${\\boldsymbol{v}}^I = {\\boldsymbol{v}}- \\sum\\_{i \\in I} e\\_i$.\n[lem:rootsstar]\n1. A dimension vector $\\alpha \\le {\\boldsymbol{v}}$ is a root of *Q* if and only if it has connected support and is not equal to ${\\boldsymbol{v}}^I$ for ∣*I*∣ ≥ *n* − 2.\n2. A positive root $\\alpha \\le {\\boldsymbol{v}}$ is (a) anisotropic if $\\alpha={\\boldsymbol{v}}^I$ with *n* − ∣*I*∣ > 4 (b) isotropic if $\\alpha={\\boldsymbol{v}}^I$ with *n* − ∣*I*∣ = 4 (c) real otherwise.\nPart (i). Every dimension vector $\\alpha \\le {\\boldsymbol{v}}$ with connected support is of the form ${\\boldsymbol{v}}^I$ or ${\\boldsymbol{v}}^I - e\\_0$, for *I* ⊂ [1, *n*], or *e**j* for some *j* > 0. We have $p({\\boldsymbol{v}}^I) = n - |I| - 3$. Therefore, the dimension vectors ${\\boldsymbol{v}}^I$, where ∣*I*∣ ≥ *n* − 2, cannot be roots because $p({\\boldsymbol{v}}^I) < 0$ in this case.\nWe have explained previously that ${\\boldsymbol{v}}^I$ for ∣*I*∣ < *n* − 2 is a root, and Lemma [lem:ve0realroot] implies that ${\\boldsymbol{v}}^I - e\\_0$ is always a real root. Similarly, the *e**j* are clearly roots.\nPart (ii) follows from the above computations.\nIf $\\Sigma\\_{\\theta,\\lambda}({\\boldsymbol{v}}) = \\{ \\alpha \\in \\Sigma\\_{\\theta,\\lambda} \\mid \\alpha \\le {\\boldsymbol{v}}\\}$, then we deduce from Lemma [lem:rootsstar] that\n \n$$\\label{eq:singma0bv}\n \\Sigma\\_0({\\boldsymbol{v}}) = \\{ e\\_0, \\ds, e\\_n, {\\boldsymbol{v}}\\} \\cup \\{ {\\boldsymbol{v}}^I \\, | \\, n - |I| \\ge 4 \\}.$$ \n[lem:stablegenericrephyperpoly] For any *θ* ∈ Θ and *λ* with $\\lambda \\cdot {\\boldsymbol{v}}= 0$, we have that:\n1. ${\\boldsymbol{v}}\\in \\Sigma\\_{\\theta,\\lambda}$, and hence there exists a *θ*-stable Π*λ*-representation of dimension vector ${\\boldsymbol{v}}$; and\n2. the morphism $\\mf{M}\\_\\lambda({\\boldsymbol{v}},\\theta) \\to \\mf{M}\\_{\\lambda}({\\boldsymbol{v}},0)$ is a projective, birational Poisson morphism.\nIn particular, $\\Hyp\\_n(\\theta)$ is an irreducible variety of dimension $2p({\\boldsymbol{v}}) = 2n - 6$.\nSince ${\\boldsymbol{v}}\\in \\Sigma\\_0$, it follows that ${\\boldsymbol{v}}\\in \\Sigma\\_{\\theta,\\lambda}$ for all *θ* ∈ Θ and *λ* with $\\lambda \\cdot {\\boldsymbol{v}}= 0$. This means that the canonical decomposition of ${\\boldsymbol{v}}$ is $({\\boldsymbol{v}},1)$ for all *θ*, *λ*; equivalently, for any *θ* ∈ Θ there exists a *θ*-stable Π*λ*-representation of dimension vector ${\\boldsymbol{v}}$. Part (ii) follow from part (i) by and the proof of. Finally, since ${\\boldsymbol{v}}\\in \\Sigma\\_{\\theta,0}$, the results imply that $\\Hyp\\_n(\\theta)$ is an irreducible variety of dimension $2p({\\boldsymbol{v}}) = 2n - 6$.\nSymplectic leaves\n-----------------\nSince the hyperpolygon space is constructed by Hamiltonian reduction, it is a Poisson variety. In fact, by, it is an example of a *symplectic singularity* in the sense of Beauville. This means that it is a normal variety equipped with a symplectic structure on the smooth locus, whose pullback to any resolution of singularities extends to a regular (globally-defined) two-form. This condition implies that it has a finite stratification by symplectic leaves, which are smooth locally closed subvarieties. Since *Y* is a cone and its generic symplectic form has positive weight for the contracting C× action, it is moreover a *conical symplectic singularity*.\nBy, the symplectic leaves of $\\mathfrak{M}\\_\\lambda({\\boldsymbol{v}},\\theta)$ are labelled by the representation types *τ* of ${\\boldsymbol{v}}$. These are tuples $\\tau = (\\beta^{(1)},n\\_1; \\ds ; \\beta^{(k)},n\\_k)$, where *β*(*i*) ∈ Σ*θ*, *λ*, *n**i* ∈ N and ${\\boldsymbol{v}}= \\sum\\_{i = 1}^k n\\_i \\beta^{(i)}$. Below, we consider the case *λ* = 0 and *θ* = 0. For *I* ⊂ [1, *n*] with ∣*I*∣ < *n* − 2, let $\\mc{L}\\_I$ be the leaf of type $({\\boldsymbol{v}}^I,1; e\\_j, 1 : j \\in I)$.\n[lem:sympleavesH0]\n1. The symplectic leaves of $\\Hyp\\_n(0)$ comprise $\\mc{L}\\_I$ for all subsets $I \\subset [1\\ds n]$ satisfying |*I*| ≤ *n* − 4, together with the leaf {0}. The codimension of $\\mc{L}\\_I$ is 2∣*I*∣.\n2. Étale locally, a slice to the leaf $\\mc{L}\\_I$ is given by $({\\mathbb{C}}^2 / \\Z\\_2)^{\\times |I|}$.\nPart (i). It follows from the description of $\\Sigma\\_0({\\boldsymbol{v}})$ in that the representations types are $({\\boldsymbol{v}},1)$ (the open leaf $\\mc{L}\\_{\\emptyset}$), $({\\boldsymbol{v}}^I,1;e\\_j,1 \\colon j \\notin I)$ (the codimension 2∣*I*∣ leaves $\\mc{L}\\_{I}$) and $(e\\_0,2;e\\_1,1; \\ds )$ (the zero-dimensional leaf).\nPart (ii). Let ℓ : = *n* − 3 − ∣*I*∣. Let *Q*ʹ be the quiver with vertices *f*0 and *f**j* for $j \\notin I$, two arrows from *f**j* to *f*0 and ℓ loops at *f*0. We let *α* be the dimension vector $(1,1,\\ds,1)$ for *Q*ʹ. Since $({\\boldsymbol{v}}^I,e\\_j) = -2$ and $p({\\boldsymbol{v}}^I) = \\ell$, says that $\\Hyp\\_n(0)$ is étale locally isomorphic at $x \\in \\mc{L}\\_I$ to 0 in the quiver variety $\\mf{M}\\_0(\\alpha,0)$ associated to *Q*ʹ. Since *α*0 = 1, the action of $\\operatorname{\\mathrm{GL}}\\_\\alpha$ factors through (C×)∣*I*∣ acting at the external vertices. As a result the quotient factors as a product of quiver varieties: one for the quiver having only one vertex with ℓ loops and dimension (1), and ∣*I*∣ quivers having two vertices and two arrows (in the same direction), with dimension (1, 1). This implies that $\\mf{M}\\_0(\\alpha,0)$ is isomorphic to ${\\mathbb{C}}^{2\\ell} \\times ({\\mathbb{C}}^2 / \\Z\\_2)^{\\times |I|}$.\nLemma [lem:sympleavesH0] says that the codimension two leaves in $\\Hyp\\_n(0)$ are labelled by the representation type $({\\boldsymbol{v}}^{(i)},1;e\\_i,1)$; the leaf is $\\mc{L}\\_{ \\{ i \\} }$. The singularity transverse to the leaf $\\mc{L}\\_{\\{ i\\} }$ is ${\\mathbb{C}}^2 / \\Z\\_2$.\nThe closure $\\overline{\\mc{L}}\\_I$ is isomorphic to the affine hyperpolygon space $\\Hyp\\_m(0)$, where *m* = *n* − ∣*I*∣. This is clear set-theoretically, but can also be shown to hold on the level of Poisson schemes using Proposition [p:gens-repsl2]. The details are left to the interested reader.\nThe hyperplane arrangement\n--------------------------\nIdentify $\\Theta = \\{ \\theta \\in {\\mathbb{Q}}^{n+1} \\mid \\theta \\cdot {\\boldsymbol{v}}= 0 \\}$ with Q*n* by projection away from the *θ*0 component. As in the introduction, consider the hyperplane arrangement\n \n$$\\label{eq:hyperpolygonarrangementintro}\n \\mc{A} = \n \\big\\{ L\\_i, H\\_I \\mid 1\\leq i\\leq n, \\{1\\}\\subseteq I\\subseteq \\{1,\\dots,n\\}\\big\\},$$ \nwhere *L**i* = {*θ* ∈ Θ ∣ *θ**i* = 0} is a coordinate hyperplane and $H\\_I= \\{ \\theta \\in \\Theta \\mid \\sum\\_{i \\in I} \\theta\\_i = \\sum\\_{j \\notin I} \\theta\\_j\\}$. It is sometimes convenient to identify A with the subset of Q*n* comprising the set of points lying in the union of all hyperplanes in A, and we do so without comment from now on.\n[prop:Hnthetasmooth] Let *θ* ∈ Θ. The following statements are equivalent:\n1. the space *X**n*(*θ*) is non-singular;\n2. every *θ*-semistable Π-module of dimension vector ${\\boldsymbol{v}}$ is *θ*-stable; and\n3. *θ* does not lie in $\\mc{A}$.\nWe say *θ* is *generic* if it satisfies one, and hence any, of these conditions.\nSince ${\\boldsymbol{v}}$ belongs to Σ0, the equivalence of (i) and (ii) follows from, which says that in this case a closed point is singular if and only if it corresponds to a non-stable point.\nNote that, if there is a semistable representation which is not stable, then there is also a polystable one (by taking a filtration of semistable representations and taking the associated graded representation). This exists if and only if there is a proper decomposition of ${\\boldsymbol{v}}$ into a sum of vectors from Σ*θ*. We must show that this is true if and only if $\\theta \\in \\mc{A}$. Assume that ${\\boldsymbol{v}}= \\beta^{(1)} + \\cdots + \\beta^{(r)}$ with *r* > 1 and *β*(*i*) ∈ Σ*θ*. If there exists *β*(*i*) with *β*0(*i*) = 2 then the other *β*(*j*) are all of the form *e**k* for some *k* ≥ 1 since their support is connected. In particular, there exists a *k* such that *θ* ⋅ *e**k* = 0, giving *θ* ∈ *L**k*.\nOtherwise, we assume that *β*0(1) = *β*0(2) = 1. Then all other *β*(*j*) are of the go form *e**k* for some *k* ≥ 1, again since their support is connected. In this case, we can write ${\\boldsymbol{v}}= \\beta^{(1)} + (\\beta^{(2)} + \\cdots + \\beta^{(r)})$. Then both *α* : = *β*(1) and *β* : = (*β*(2) + ⋯ + *β*(*r*)) are real roots by Lemma [lem:rootsstar] since *β* will be connected. The hyperplane in Θ corresponding to *θ* ⋅ *α* = *θ* ⋅ *β* = 0 is of the form *H**I*, for some *I*. Thus, $\\theta \\in \\mc{A}$.\nConversely, if $\\theta \\in \\mc{A}$ then one can check, by considering the cases *θ* ∈ *L**k* and *θ* ∈ *H**I* separately, that ${\\boldsymbol{v}}= \\alpha^{(1)} + \\cdots+ \\alpha^{(j)}$ admits a decomposition into a (proper) sum of positive roots. Each *α*(*i*) can, by definition, be written as a sum of roots in Σ*θ*. It follows that ${\\boldsymbol{v}}$ can be decomposed into a proper sum of roots in Σ*θ*. By, this implies that there exists a *θ*-semistable Π-module of dimension vector ${\\boldsymbol{v}}$ that is not *θ*-stable.\nProposition [prop:Hnthetasmooth] shows that the hyperplane arrangement A determines the GIT chamber decomposition of Θ. Nakajima also introduces a chamber decomposition which, for *λ* = 0, is determined by the arrangement comprising hyperplanes $\\{\\theta\\in \\mathbb{Q}^{n+1} \\mid \\theta\\cdot {\\boldsymbol{v}}=\\theta\\cdot u =0\\}$ associated to all vectors *u* ∈ Z*n* + 1 satisfying $0 < u < {\\boldsymbol{v}}$ and (*u*, *u*) ≤ 2. Each hyperplane from A appears in this arrangement, so the chamber decomposition of Nakajima refines the GIT chamber decomposition. However, these chamber decompositions do not coincide since, for example, the hyperplane of vectors perpendicular to *u* = (2, 1, …, 1, 0, 0) does not lie in A.\nLemma [lem:stablegenericrephyperpoly] and Proposition [prop:Hnthetasmooth] together imply the following result; see for the definition of symplectic resolution, noting here that it is equivalent to the morphism being crepant.\n[cor:symplecticreshyperpoly] The morphism *f**θ*: *X**n*(*θ*) → *X**n*(0) obtained by variation of GIT quotient is a projective, symplectic resolution if and only if $\\theta\\in \\Theta^{\\reg}$.\nFor all *λ*, we have that:\n1. *μ*− 1(*λ*) is a reduced, irreducible, complete intersection; and\n2. $\\mu^{-1}(\\lambda) {\\ensuremath{/\\!\\!/\\!}}\\, \\SL\\_2$ is reduced and irreducible.\nSince ${\\boldsymbol{v}}$ belongs to Σ*λ*, 0, part (i) is due to Crawley-Boevey. Part (ii) follows from (i).\nThe semi-invariant ring\n-----------------------\nWe now introduce a presentation of the ring ${\\mathbb{C}}[\\mu^{-1}(\\lambda)]^{\\SL\\_2}$. We refer to Figure [fig:quivers] for notation of the quiver. Given a vertex *i*, let $V\\_i = {\\mathbb{C}}^{{\\boldsymbol{v}}\\_i}$ be the vector space at vertex *i*. Given an arrow *a*, let $\\ev\\_a: \\operatorname{\\mathrm{Rep}}({\\overline{Q}},{\\boldsymbol{v}}) \\to \\operatorname{\\mathrm{Hom}}(V\\_{a\\_s}, V\\_{a\\_t})$ be the corresponding function. We fix a symplectic form on *V*0 = C2 and view $\\SL\\_2 = \\Sp(V\\_0)$. This gives an isomorphism *V*0 ≅ *V*0\\* which is $\\SL\\_2$-equivariant. We also fix a trivialisation *V**i* ≅ C for each *i*. Put together we can view $\\ev\\_a$ as valued in *V*0 for every arrow *a*. We can therefore define the $\\SL\\_2$-invariant functions $\\varphi\\_{a,b} := \\ev\\_a \\wedge \\ev\\_b: \\operatorname{\\mathrm{Rep}}({\\overline{Q}},{\\boldsymbol{v}}) \\to {\\mathbb{C}}$. Note that *φ**a*, *b* = − *φ**b*, *a*.\n[p:gens-repsl2]\n1. The ring ${\\mathbb{C}}[\\operatorname{\\mathrm{Rep}}({\\overline{Q}},{\\boldsymbol{v}})]^{\\SL\\_2}$ is generated by the *φ**a*, *b*, subject to the relations *φ**a*, *b**φ**c*, *d* − *φ**a*, *c**φ**b*, *d* + *φ**a*, *d**φ**b*, *c* = 0 (and *φ**a*, *b* = − *φ**b*, *a*).\n2. The ring ${\\mathbb{C}}[\\mu^{-1}(\\lambda)]^{\\SL\\_2}$ is the quotient of ${\\mathbb{C}}[\\operatorname{\\mathrm{Rep}}({\\overline{Q}},{\\boldsymbol{v}})]^{\\SL\\_2}$ by the ideal generated by the functions *φ**a**i*, *a**i*\\* − *λ**i* for 1 ≤ *i* ≤ *n* and ∑*i**φ**a**i*, *b**φ**a**i*\\*, *c* − *λ*0*φ**b*, *c* for all $b,c \\in {\\overline{Q}}\\_1$.\n3. The subring ${\\mathbb{C}}[\\mu^{-1}(\\lambda)]^{\\operatorname{\\mathrm{GL}}\\_{{\\boldsymbol{v}}}}$ is spanned by those monomials in *φ**a*, *b* in which the total number of occurrences of the index *a**i* is the same as of *a**i*\\* for all *i*.\nFor (i), generation is formally a direct consequence of first fundamental theorem of invariant theory for $\\Sp(V\\_0)=\\SL\\_2$ by Weyl . Indeed, applying this theorem as in with *p* = 2*n*, *q* = 0 and *n* = 2 shows that ${\\mathbb{C}}[\\operatorname{\\mathrm{Rep}}({\\overline{Q}},{\\boldsymbol{v}})]^{\\SL\\_2}$ is generated by the determinants *φ**a*, *b* (denoted $[\\ev\\_a,\\ev\\_b]$ in *loc. cit.*). The relations are a direct consequence of the second theorem of invariant theory. Equivalently, the relations can be interpreted as the Plücker relations between the determinants *φ**a*, *b* listed above, or as diagrammatic Pfaffians as in ; see also for an ideal generating set.\nFor (ii), the first set of equations is precisely the set defined by the moment map at the external vertices. For the second, the moment map condition at the internal vertex says that $\\sum\\_i \\ev\\_{a\\_i^\\*} \\otimes \\ev\\_{a\\_i}$ is the constant function valued at *λ*0Id*V*0 ∈ *V*0 ⊗ *V*0\\*, where we use the symplectic form *ω* to identify $\\ev\\_{a\\_i}$ with $\\omega(\\ev\\_{a\\_i},-)\\in V\\_0^\\*$ (otherwise replace Id*V*0 by the inverse of the symplectic form in *V*0 ⊗ *V*0). We pair this element with $\\ev\\_c \\otimes \\ev\\_b$ using the symplectic form *ω* on the left-hand factor and the natural pairing of *V*0 and its dual on the right-hand factor to obtain\n \n$$\\label{eqn:pairings}\n\\sum\\_i \\omega(\\ev\\_{a\\_i^\\*}, \\ev\\_c) \\cdot \\omega(\\ev\\_{a\\_i},\\ev\\_b) = \\sum\\_i \\varphi\\_{a\\_i^\\*,c} \\cdot \\varphi\\_{a\\_i,b}\\quad \\text{ and }\\quad \\lambda\\_0 \\omega(\\ev\\_b,\\ev\\_c) = \\lambda\\_0 \\varphi\\_{b,c},$$ \nwhere in the second equality, the symplectic form identifies Id*V*0 with *ω*( − , − ) ∈ *V*0\\* ⊗ *V*0\\*. The functions in agree, so we obtain the given equations. Conversely, note that an invariant which is in the ideal generated by the coefficients of $\\sum\\_a \\ev\\_a \\otimes \\ev\\_{a^\\*} - \\lambda\\_0 {\\mathrm{Id}}\\_{V\\_0}$ is precisely one such that it becomes zero after we successively eliminate pairs of indices *a*1, *a*1\\* via the relation. This will be generated by the given equations together with ∑*φ**a**i*, *a**i*\\* − 2*λ*0, but the latter element is in the ideal generated by the first set of equations since $\\lambda \\cdot {\\boldsymbol{v}}= 0$.\nFor (iii), note that $\\operatorname{\\mathrm{GL}}\\_{{\\boldsymbol{v}}}/ \\SL\\_{2} = ({\\mathbb{C}}^{\\times})^{n+1}$, with the diagonal C× acting trivially. The elements *φ**a**i*, *a**j*, *φ**a**i*, *a**j*\\*, and *φ**a**i*\\*, *a**j*\\* are weight functions of weights *e**i* + *e**j*, *e**i* − *e**j*, and − *e**i* − *e**j*, respectively. The result follows immediately.\nNote that the equations in (ii) allow us to eliminate generators *φ**a**i*, *a**i*\\*. Together with skew-symmetry, we can restrict generators to *φ**a**i*, *a**j*, *φ**a**i*\\*, *a**j*\\* for *i* < *j*, and *φ**a**i*, *a**j*\\* for *i* ≠ *j*.\nSuppose that *λ* ≠ 0. Then, the first set of relations in (ii) imply, together with (i), that $\\frac{\\varphi\\_{a\\_i^\\*, b}}{\\varphi\\_{a\\_i, b}}$ does not depend on $b \\notin\\{a\\_i, a\\_i^\\*\\}$. Call this element *u**i*, so that *φ**a**i*\\*, *a**j* = *u**i**φ**a**i*, *a**j*. Then we can rewrite our generators as:\n \n*φ**a**i*, *a**j**u**i**ε*1*u**j**ε*2, for *ε*1, *ε*2 ∈ {0, 1}and 1 ≤ *i* < *j* ≤ *n*. \nWorking over C[*u*1, …, *u**n*], this allows us to eliminate all relations involving *a**i*\\*, leaving:\n \n$$\\begin{gathered}\n\\label{e:red-rels1}\n \\varphi\\_{a, b} \\varphi\\_{c,d} - \\varphi\\_{a,c} \\varphi\\_{b,d} + \\varphi\\_{a,d} \\varphi\\_{b,c}, \\\\\n \\label{e:red-rels2} \\sum\\_i \\varphi\\_{a\\_i, b} \\varphi\\_{a\\_i, c} u\\_i,\\end{gathered}$$ \nfor all *a*, *b*, *c*, *d* ∈ {*a*1, …, *a**n*}, with the conventions *φ**a**i*, *a**i* : = 0 and *φ**a**j*, *a**i* : = − *φ**a**i*, *a**j* for *j* > *i*. The $\\operatorname{\\mathrm{GL}}\\_{{\\boldsymbol{v}}}$ invariant functions in these terms are those linear combinations of monomials in which each index *i* occurs as a subscript of *φ* twice as many times as it does as a subscript of *u*.\nAll 81 symplectic resolutions are quiver varieties\n==================================================\nThe results of generalise to arbitrary quiver varieties, as explained in. Here we introduce the set-up and highlight only the computations that are specific to our example $\\Hyp\\_n(0)$.\nThe birational geometry of hyperpolygon spaces\n----------------------------------------------\nConsider the cone\n \n$$\\label{eqn:C\\_+}\n C\\_+ = \\big\\{ \\theta \\in \\Theta \\mid \\theta\\_1 > \\theta\\_2+\\cdots + \\theta\\_n \\textrm{ and } \\theta\\_i > 0 \\text{ for all }i>0 \\big\\}$$ \nand let *θ* ∈ *C*+. Since *θ*1 > *θ*2 + ⋯ + *θ**n*, the inequality $\\sum\\_{i \\in I} \\theta\\_i > \\sum\\_{j \\notin I} \\theta\\_j$ holds for every set *I* satisfying {1} ⊆ *I* ⊆ {1, …*n*}. It follows that every such *θ* satisfies a strict inequality determined by each hyperplane in A, so *C*+ is a GIT chamber. For *θ*+ ∈ *C*+, write \n*f*: *X* : = *X**n*(*θ*+) → *Y* : = *X**n*(0) \n for the corresponding projective symplectic resolution.\nRecall the relative Néron–Severi space *N*1(*X*/*Y*) defined in the introduction. For a resolution of singularities *X* → *Y*, the exceptional locus is the union of all fibres which are not a single point.\n[prop:dimN1] The rational vector space *N*1(*X*/*Y*) has dimension *n*. Moreover, *Y* is Q-factorial and the exceptional locus of *f* is the union of *n* irreducible divisors.\nWe claim that the number of irreducible exceptional divisors is exactly *n*. Consider the symplectic leaves of codimension-two in *Y*. By Lemma [lem:sympleavesH0], and the discussion thereafter, these are the leaves L{*i*} for 1 ≤ *i* ≤ *n*, which locally are transverse to type *A*1 du Val singularities. The restriction of *f* to a neighbourhood of each codimension-two singularity is the minimal resolution (isomorphic to the blow up of the singular locus) with exceptional fibres P1. Now, let *Y*≤ 2 ⊆ *Y* denote the union of the open symplectic leaf and the symplectic leaves of codimension two. Since P1 is irreducible, the exceptional locus of *f* over *Y*≤ 2 has exactly *n* irreducible components: the preimages of the leaves L{*i*}. Finally, since *f* is semismall, and *f* is an isomorphism over the smooth locus of *Y*, it follows that *X* \\ *f*− 1(*Y*≤ 2) has codimension at least two. We have proved the claim.\nIt is a well-known fact that, for any projective resolution *X* → *Y*, the classes of the exceptional divisors are linearly independent in *N*1(*X*/*Y*): this follows from the Hodge index theorem in the surface case, and by induction on dimension using a generic hypersurface section in general; see for a more detailed argument. As a result, dim*N*1(*X*/*Y*) ≥ *n*.\nFor the opposite inequality, Konno computes the Poincaré polynomial to be \n$$P\\_t(X) \n = \\frac{(1+t^2)^{n-1}}{1-t^2} -\\frac{2^{n-1}t^{2(n-2)}}{1-t^2} + t^{2(n-2)}\n = 1+(n-1)t^2 + t^2 + \\cdots,$$ \n giving *n* = *b*2(*X*) = dim*H*2(*X*, Q). Since the cycle map is an isomorphism, we may identify *H*2(*X*, Z) with *A*1(*X*), and hence with $\\Pic(X)$ since *X* is smooth. It follows that $\\dim N^1(X/Y) \\leq \\dim \\Pic(X) \\otimes\\_{{\\mathbb{Z}}} \\mathbb{Q}=n$, so the map $\\Pic(X) \\otimes\\_{{\\mathbb{Z}}} \\mathbb{Q} \\twoheadrightarrow N^1(X/Y)$ is an isomorphism.\nLet *E*1, …, *E**n* be the *n* irreducible exceptional divisors identified above, and set *U* : = *X* \\ (⋃*E**i*). The standard exact sequence on divisor class groups implies that $\\Pic(X)$ surjects onto $\\Pic(U)$, with kernel spanned by the $\\mc{O}\\_X(E\\_i)$. Since we have shown that $\\Pic(X) \\otimes\\_{{\\mathbb{Z}}} \\mathbb{Q}$ is spanned by the $\\mc{O}\\_X(E\\_i)$, the group $\\Pic(U)$ is finite. The locus over which *f* is an isomorphism is an open subset of *U*, whose complement in *U* has codimension at least two. Therefore its Picard group also identifies with the preceding finite abelian group. The same is therefore true for its image *f*(*U*), which is the smooth locus of *Y*. Note that the complement of *f*(*U*) in *Y*≤ 2 has codimension at least two, so *Y* \\ *f*(*U*) has codimension two in *Y*. Since *Y* is normal, the Weil divisor class group of *Y* is isomorphic to the Picard group of *f*(*U*), hence of *U*. As this is finite, *Y* is Q-factorial. It follows that the exceptional locus of any resolution of *Y* is pure of codimension one, hence in this case it is precisely the union of the *n* exceptional divisors.\nMore generally, any conical symplectic resolution (i.e. C×-equivariant symplectic resolution of a cone) satisfies $\\Pic(X) \\otimes\\_{{\\mathbb{Z}}} \\mathbb{Q} \\cong N^1(X/Y) \\cong H^2(X, \\mathbb{Q})$. Namely, Namikawa has explained that the algebraic and analytic rational Picard groups of *X* are isomorphic. This implies the statement via the exponential exact sequence and Matsusaka’s Theorem that any numerically trivial bundle has torsion first Chern class.\nLet *C* ⊂ Θ be any chamber. Each *θ* ∈ *C* is generic by Proposition [prop:Hnthetasmooth] and the dimension vector ${\\boldsymbol{v}}$ is indivisible, so King implies $X\\_n(\\theta)=\\mathfrak{M}\\_{0}({\\boldsymbol{v}},\\theta)$ is the fine moduli space of *θ*-stable Π-modules of dimension vector ${\\boldsymbol{v}}$. It carries a tautological vector bundle *T**n*(*θ*) : = *T*0 ⊕ ⨁1 ≤ *i* ≤ *n**T**i* whose fibre over each closed point is the corresponding Π-module. Note that $\\rank(T\\_0)=2$, $\\rank(T\\_i)=1$ for 1 ≤ *i* ≤ *n*, and we choose our normalisation of *T**n*(*θ*) such that det(*T*0) ≅ O*X**n*(*θ*). When *n* = 4, the bundle *T*4(*θ*) is tilting, but this is far from true when *n* = 5.\nSince *θ* ∈ Θreg, the birational map $f\\_{\\theta}^{-1}\\circ f\\colon X\\dashrightarrow X\\_n(\\theta)$ is an isomorphism in codimension-one, so we may identify *N*1(*X**n*(*θ*)/*Y*) with *N*1(*X*/*Y*) by taking the strict transform. As a result, we may define the *linearisation map* *L**C*: Θ → *N*1(*X*/*Y*) to be the Q-linear map given by \n*L**C*(*θ*) = *T*1*θ*1 ⊗ ⋯ ⊗ *T**n**θ**n*. \n For *θ* ∈ *C*, we have $L\\_C(\\theta)\\in \\Amp(X\\_n(\\theta)/Y)$ and hence $L\\_C(C)\\subseteq \\Amp(X\\_n(\\theta)/Y)$. The following result establishes that this inclusion is in fact equality.\n[prop:ample] Let *C* ⊂ Θ be a chamber. For *θ* ∈ *C*, the map *L**C* is an isomorphism of rational vector spaces that identifies *C* and $\\overline{C}$ with $\\Amp(X\\_n(\\theta)/Y)$ and $\\Nef(X\\_n(\\theta)/Y)$ respectively.\nGiven Proposition [prop:Hnthetasmooth] and Proposition [prop:dimN1], the result follows as in.\nConsider next the closed cone\n \n*F* = {*θ* ∈ Θ ∣ *θ**i* ≥ 0, for all *i* ≠ 0}. \nThis is the union of closures of a collection of GIT chambers in Θ. The maps *L**C* and *L**C*′ associated to a pair of adjacent chambers do not agree in general, but the next result shows that for *n* ≥ 5, they agree when *C*, *C*′ both lie in *F*. To state the result, recall that *L* ∈ *N*1(*X*/*Y*) is said to be *movable* if the stable base locus of *L* over *Y* has codimension at least two in *X*. The relative *movable cone* $\\Mov(X/Y)$ is the closure in *N*1(*X*/*Y*) of the cone generated by all movable divisor classes over *Y*.\n[thm:movable] Let *n* ≥ 5. There is an isomorphism *L**F*: Θ → *N*1(*X*/*Y*) of rational vector spaces such that\n1. *L**F* = *L**C* for each chamber *C* ⊆ *F*; and\n2. *L**F* identifies *F* with the movable cone $\\Mov(X/Y)$ in such a way that each chamber *C* ⊆ *F* is identified with the ample cone $\\Amp(X\\_n(\\theta)/Y)$ of the hyperpolygon space *X**n*(*θ*) for *θ* ∈ *C*.\nIn particular, every projective crepant resolution of *Y* = *X**n*(0) is a hyperpolygon space *X**n*(*θ*) for some generic *θ* ∈ *F*.\nDefine *L**F* : = *L**C*+. Proposition [prop:ample] shows that *L**F* identifies *C*+ with $\\Amp(X/Y)$. To study the other chambers in *F* we analyse how *X**n*(*θ*) and its tautological bundle *T**n*(*θ*) change as *θ* crosses (or simply touches) walls of the GIT chamber decomposition using the étale-local description from. There are two cases.\nFirst, consider a wall in the interior of *F*. Any such wall is of the form $H\\_I\\cap \\overline{C}\\cap \\overline{C'}$ for some subset {1} ⊆ *I* ⊆ {1, …, *n*}, where *C*, *C*′ ⊂ *F* are chambers separated by the wall. If *θ*0 ∈ *H**I* is generic then the two real roots *α* : = *e*0 + ∑*i* ∈ *I**e**i* and $\\beta := e\\_0 + \\sum\\_{j \\notin I} e\\_j$ satisfy *θ*0(*α*) = *θ*0(*β*) = 0 and *p*(*α*) = *p*(*β*) = 0. By Lemma [lem:rootsstar], these are the only positive roots $< {\\boldsymbol{v}}$ that pair to zero with *θ*0. Therefore, the set $\\Sigma\\_{\\theta\\_0}({\\boldsymbol{v}})$ equals $\\{ \\alpha,\\beta, {\\boldsymbol{v}}\\}$. As explained in Section [sec:sympleaves], this implies that $\\Hyp\\_{n}(\\theta\\_0)$ contains exactly two symplectic leaves, labelled by the representation types $({\\boldsymbol{v}},1)$ and (*α*, 1; *β*, 1) respectively. Since *α*, *β* are real, the closed leaf labelled by (*α*, 1; *β*, 1) is zero-dimensional, so $\\Hyp\\_n(\\theta\\_0)$ has a single isolated singularity. Since *α*0 = *β*0 = 1, we have \n(*α*, *β*) = 2 + ∑*i* = 1*n**α**i**β**i* − ∑*j* = 1*n*(*β**j* + *α**j*) = 2 − *n* = − (*n* − 2). \n Thanks to, we see that in a étale neighbourhood of the singular point, *X**n*(*θ*0) looks locally like the zero point in an affine quiver variety for the quiver with 2 vertices and *n* − 2 arrows from vertex 0 to vertex 1. This is precisely the quiver encoding $\\overline{\\mc{O}}\\_{\\mr{min}}$ in $\\mf{sl}(n-2)$, and *ibid*. shows that crossing this wall induces a birational map $f\\_{C, C'}\\colon X\\_n(\\theta)\\dashrightarrow X\\_n(\\theta')$ for *θ* ∈ *C* and *θ*ʹ ∈ *C*ʹ. Étale-locally around the singular point, this is a Mukai flop of the form \n$$\\begin{tikzcd}\n T^\\* \\mathbb{P}(V) \\ar[dr] \\ar[rr,dashed] & & T^\\* \\mathbb{P}(V^\\*) \\ar[dl] \\\\\n & \\overline{\\mc{O}}\\_{\\mr{min}} & \n\\end{tikzcd}$$ \n where dim*V* = *n* − 2 ≥ 3. The morphisms from both *X**n*(*θ*) and *X**n*(*θ*ʹ) to *X**n*(*θ*0) are semi-small, so the unstable locus for each morphism has codimension greater than one. It follows as in the proof of that *L**C* = *L**C*ʹ and, moreover, the birational map *f**C*, *C*ʹ is a flop. In particular, the linearisation maps *L**C* agree for all chambers *C* in *F*, proving part (i).\nOtherwise, the wall lies in the boundary of *F*. Any such wall is $L\\_i\\cap \\overline{C}$ for some 1 ≤ *i* ≤ *n* and chamber *C* ⊆ *F*. If *θ* ∈ *C* and *θ*0 is generic in the wall, then there is a morphism *X**n*(*θ*) → *X**n*(*θ*0). Then $\\Hyp\\_n(\\theta\\_0)$ has two symplectic leaves, labelled by $({\\boldsymbol{v}},1)$ and $({\\boldsymbol{v}}- e\\_i,1;e\\_i,1)$. Theorem 3.3 of says that, étale locally on the closed leaf labelled by $({\\boldsymbol{v}}- e\\_i,1;e\\_i,1)$, the variety *X**n*(*θ*0) is isomorphic to a neighbourhood of zero in the affine quiver variety $\\mf{M}\\_0((1,1),0)$, where the underlying quiver has *n* − 4 loops at one vertex and two arrows from one vertex to the other. That is, locally we get ${\\mathbb{C}}^{2(n-4)} \\times {\\mathbb{C}}^2 / \\Z\\_2$. Thus, it follows from that the morphism *X**n*(*θ*) → *X**n*(*θ*0) is a divisorial contraction, so *L**F*(*θ*0) = *L**C*(*θ*0) lies in the boundary of $\\Nef(X\\_n(\\theta)/Y)$. This is true for all walls in the boundary of *F*, so *L**F* identifies the boundary of *F* with the boundary of $\\Mov(X/Y)$.\nIt follows that the relative ample cone of every projective crepant resolution *X*ʹ → *Y* is identified via strict transform with a cone in $\\Mov(X/Y)$, so *X*ʹ = *X**n*(*θ*ʹ) for $\\theta'\\in L\\_F^{-1}(\\Amp(X'/Y))$.\n[rem:surfaceCase] For *n* = 4, the proof of Theorem [thm:movable] goes through verbatim, except that the birational map $f\\_{C,C'} \\colon X\\_n(\\theta)\\dashrightarrow X\\_n(\\theta')$ obtained by crossing any interior wall in *F* is an isomorphism because dim*V* = 2. Thus, every wall induces a divisorial contraction *X**n*(*θ*) → *X**n*(*θ*0). In this case, the hyperplane arrangement decomposes *F* into the union of 12 GIT chamber closures, each of which is isomorphic to the relative Nef cone by Proposition [prop:ample], which in turn is the relative movable cone because dim*X* = 2.\nBeyond the fundamental domain\n-----------------------------\nTheorem [thm:movable] is enough to understand completely the birational geometry of any hyperpolygon space *X**n*(*θ*) for *θ* ∈ *F*. For completeness, we explain how the Namikawa Weyl group allows us to extend this understanding to any hyperpolygon space.\nLet *n* ≥ 5. The Namikawa Weyl group is generated by reflections in the supporting hyperplanes *L*1, …, *L**n* of the cone *F* from, so it is Z2*n* in this case. Note that this action permutes the GIT chambers in Θ, and it is explicitly realised on the hyperpolygon space $X\\_n(\\theta)=\\mf{M}\\_{0}({\\boldsymbol{v}},\\theta)$ by Nakajima’s reflection functors since $({\\boldsymbol{v}},e\\_i) = 0$ for all *i* > 0. As explained in and references therein, the reflection *s**i*: Θ → Θ at each external vertex 1 ≤ *i* ≤ *n* can be lifted to a reflection isomorphism $S\\_i \\colon \\Hyp\\_n(\\theta) \\to \\Hyp\\_n(s\\_i(\\theta))$ of varieties over *Y* provided *θ**i* ≠ 0. In particular, for any $\\theta \\in \\Theta^{\\reg}$ there is a Poisson isomorphism $S\\_i \\colon \\Hyp\\_n(\\theta) {{\\;\\stackrel{\\_\\sim}{\\longrightarrow}\\;}}\\Hyp\\_n(s\\_i(\\theta))$. The reflections *s*1, …, *s**n* generate the Namikawa Weyl group Z2*n*, so we obtain the following.\n[lem:reflection] Let *n* ≥ 5. Let *C*, *C*′ ⊂ Θ be chambers and let *θ* ∈ *C*, *θ*′ ∈ *C*′. Then *X**n*(*θ*) ≅ *X**n*(*θ*′) as schemes over *Y* if and only if there exists *w* ∈ Z2*n* such that *w*(*C*) = *C*′. In particular, if *C*, *C*ʹ are separated by a wall contained in some *L**i*, then *X**n*(*θ*) ≅ *X**n*(*θ*′) for *θ* ∈ *C* and *θ*′ ∈ *C*′.\nThe proof of the analogous result from is more involved because one generator of the Namikawa Weyl group in that case is not a Nakajima reflection.\nIn the proof of Theorem [thm:movable], we established that *X**n*(*θ*) undergoes a Mukai flop as *θ* crosses any interior wall of *F*. This result is due originally to Godinho–Mandini , but their approach is quite different. They identify *X**n*(*θ*) with a moduli space of stable, rank two, holomorphically trivial parabolic Higgs bundles over P1 with fixed determinant and trace-free Higgs field, in which case they are able to reduce to previous work on variation of GIT quotient by Thaddeus . In fact, Lemma [lem:reflection] implies more: every wall crossing in Θ induces either a Mukai flop (for walls in some *H**I*) or an isomorphism of hyperpolygon spaces (for walls in some *L**i*).\nCounting crepant resolutions\n----------------------------\nThe result of Theorem [thm:movable] allows us to count the number of projective crepant resolutions of the singularity *X**n*(0) by counting the number of GIT chambers in the fundamental domain *F* for the action of the Namikawa Weyl group on Θ.\nThe number of projective crepant resolutions of *X**n*(0) is equal to 1 when *n* = 4, it is 81 when *n* = 5, and it is 1684 when *n* = 6.\nWe remark that this is the only proof in this article that relies on computer calculation.\nWhen *n* = 4, the characteristic polynomial of the arrangement $\\mc{A}$ is *q*4 − 12*q*3 + 50*q*2 − 84*q* + 45. Evaluating this polynomial at − 1 gives 192, so $\\Theta^{\\reg}$ contains 192 chambers. In this case *X*4(0) is the Kleinian singularity of type *D*4. Therefore, the Namikawa Weyl group is the Weyl group of type *D*4, which has order 23 ⋅ 4! = 192. Any chamber provides a fundamental domain for this action, and indeed the minimal resolution of the Kleinian singularity is unique. Note that *F* contains 12 chambers in this case.\nWhen *n* = 5, the characteristic polynomial of $\\mc{A}$ is *q*5 − 21*q*4 + 170*q*3 − 650*q*2 + 1125*q* − 625. Evaluating this polynomial at − 1 gives − 2592, so $\\Theta^{\\reg}$ contains 2592 chambers. Taking into account the action of the Namikawa Weyl group Z25 implies that *F* contains 2592/32 = 81 chambers.\nFinally, when *n* = 6, a brutal computer calculation using Singular shows that characteristic polynomial of the arrangement $\\mc{A}$ is *q*6 − 38*q*5 + 607*q*4 − 5100*q*3 + 22935*q*2 − 48750*q* + 30345. Evaluating this polynomial at − 1 gives 107776, so $\\Theta^{\\reg}$ comprises 107776 chambers. The Namikawa Weyl group is Z26 in this case, so *F* contains 107776/64 = 1684 chambers.\n[rem:adk] Alastair King subsequently observed that the sequence 12, 81, 1684, ... recording the number of chambers in *F* for *n* ≥ 4 coincides with the sequence counting the number of positive self-dual threshold functions of *n* variables; this sequence for *n* ≥ 1 appears in. The known terms in this sequence for *n* ≥ 4 are:\n| *n* | 4 | 5 | 6 | 7 | 8 | 9 |\n| --- | --- | --- | --- | --- | --- | --- |\n| chamber count | 12 | 81 | 1,684 | 122,921 | 33,207,256 | 34,448,225,389. |\nFor *n* ≥ 5, these numbers count the projective crepant resolutions of *X**n*(0) by Theorem [thm:movable]. For *n* = 4, the (Namikawa) Weyl group is the group generated by reflections in all hyperplanes of $\\mc{A}$, whereas the group generated by reflections in the supporting hyperplanes of *F* is a subgroup of index 12 (see Remark [rem:surfaceCase]).\nThe finite quotient singularity\n===============================\nFrom now on we fix *n* = 5. To prove Theorem [t:isom], we first establish some information about the group *G*, expanding on and. We then identify C4/*G* with the affine hyperpolygon space *X*5(0) by combining the analysis from with our description of the semi-invariant ring from Proposition [p:gens-repsl2]. This section is logically independent from the results of Section [sec:theresofhyperpoly].\nFacts about *G*\n---------------\nAs in the introduction, set *V* = C4 and consider $G = Q\\_8 \\times\\_{{\\mathbb{Z}}\\_2} D\\_8 < \\Sp(V)$. Recall that there is a two-to-one covering $\\Sp\\_4 {\\twoheadrightarrow}\\operatorname{SO}\\_5$, which can be explicitly realised as follows. The representation ∧ 2*V*\\* decomposes as a sum of the invariant part C ⋅ *ω* (for *ω* the symplectic form) and a five-dimensional irreducible representation *W*. The form *ω*− 1 ⊗ *ω*− 1 defines a nondegenerate symmetric bilinear form on ∧ 2*V*\\* which restricts to one on *W*. We then obtain a natural map \n$$\\gamma\\colon \\Sp(V) \\to \\operatorname{SO}(W)$$ \n with kernel { ± 1}, which is surjective for dimension reasons. This isogeny realises the equality of root systems C2 = B2. The centre of *G* is Z2 × Z2Z2 = { ± 1}, which is also equal to [*G*, *G*]. Thus, the image *γ*(*G*) is isomorphic to the abelianisation Ab(*G*) ≅ Z24. This subgroup of SO(*W*) has a simultaneous orthogonal eigenbasis, so that it is identified under this choice of basis with the subgroup of diagonal matrices.\n[p:G-facts] There is an orthonormal basis of *W*, unique up to scaling and reordering, such that *G* is the preimage under *γ* of the group of diagonal matrices Z24 < SO(*W*). The conjugation action of the permutation group *S*5 < O(*W*) lifts uniquely to a conjugation action *S*5 → Aut(*G*), whose composition with $\\operatorname{Aut}(G) {\\twoheadrightarrow}\\operatorname{Out}(G)$ is an isomorphism *S*5 ≅ Out(*G*). In particular, we have a semidirect product decomposition $\\operatorname{Aut}(G) \\cong \\operatorname{Inn}(G) \\rtimes \\operatorname{Out}(G) \\cong {\\mathbb{Z}}\\_2^4 \\rtimes S\\_5$.\nThe first statement follows from the spectral theorem for orthogonal matrices. For the second, we have O(*W*) = { ± Id} ⋅ SO(*W*), giving a two-to-one covering $\\mu\\_4 \\cdot \\Sp(V) \\to \\operatorname{O}(W)$. The preimage of *S*5 in $\\mu\\_4 \\cdot \\Sp(V)$, call it $\\widetilde S\\_5$, acts on *G* by conjugation, with kernel ± Id (note that *μ*4 is not a subgroup of $\\widetilde S\\_5$, since ± Id is not a subgroup of *S*5). Therefore this action descends to the quotient *S*5. Since these induce all (nontrivial) outer automorphisms of Z24, the composition *S*5 → Aut(*G*) → Out(*G*) is injective. Since Out(*G*) ≅ *S*5 by, this is actually an isomorphism, and the first map gives a homomorphic section of the quotient Aut(*G*) → Out(*G*). This establishes the last fact.\nSince there is a unique irreducible representation of *G* of dimension greater than one (see ), every subgroup of $\\Sp\\_4$ isomorphic to *G* is actually conjugate to it. Similarly, by the spectral theorem as in the beginning of the proof of Proposition [p:G-facts], every subgroup of SO5 isomorphic to Z24 is conjugate to the diagonal subgroup. So *G* is also the unique subgroup of $\\Sp\\_4$, up to conjugation, whose image in SO5 is isomorphic to Z24.\nThe quotient *V*/*G*\n--------------------\nAs observed in, the coordinate ring of *V*/*G* can be written as \n$${\\mathbb{C}}[V]^G\\cong\\big({\\mathbb{C}}[V]^{[G,G]}\\big)^{\\operatorname{Ab}(G)},$$ \n where C[*V*][*G*, *G*] = C[*V*]{ ± Id} is the algebra of even-degree polynomial functions on *V*; this is generated by $\\Sym^2 V^\\*$. The $\\Sp(V)$ action here descends under the two-to-one cover to an action of SO(*W*). As pointed out in *loc. cit.* , we have an isomorphism \n$$\\iota\\colon \\Sym^2 V^\\* \\longrightarrow \\wedge^2 W$$ \n of irreducible SO(*W*)-representations that must be unique up to scaling.\n[rem:iota-compute] The isomorphism *ι* can be computed explicitly as follows: recall that *W* ⊆ ∧ 2*V*\\* and hence ∧ 2*W* ⊂ ∧ 2( ∧ 2*V*\\*). We can contract with the symplectic form *ω*− 1 ∈ ∧ 2*V* and obtain an element of *V*\\* ⊗ *V*\\*. Since *W* is the kernel of the contraction ∧ 2*V*\\* → C, the image of the contraction is symmetric, giving a map $\\wedge^2 W \\to \\Sym^2 V^\\*$. It is clear that this is compatible with the action of the symplectic group. Now *ι* is the inverse of this map (up to scaling).\nPutting this together, the orthonormal basis *w*1, …, *w*5 of *W* as in Proposition [p:G-facts] (unique up to scaling and reordering) gives C-algebra generators *ψ**i*, *j* : = *ι*− 1(*w**i* ∧ *w**j*) of C[*V*][*G*, *G*], where 1 ≤ *i* < *j* ≤ 5.\n[p:rels-qt] The ten elements *ψ**i*, *j* in C[*V*][*G*, *G*] satisfy the equations\n \n$$\\begin{gathered}\n \\psi\\_{i,j} \\psi\\_{k,\\ell} - \\psi\\_{i,k} \\psi\\_{j,\\ell} + \\psi\\_{i,\\ell} \\psi\\_{j, k} = 0, \\\\\n \\sum\\_i \\psi\\_{i,j} \\psi\\_{i,k} = 0.\n \\end{gathered}$$ \nThe subring C[*V*]*G* is spanned by products of generators in which every index occurs an even number of times.\nThis result was established by and with some computer assistance, at least up to a choice of signs in the relations and in each *ψ**i*, *j*.\nThe relations can be checked by an explicit computation using the description of *ι* from Remark [rem:iota-compute]; note by symmetry that it suffices to establish the first identity for a particular choice of distinct *i*, *j*, *k*, ℓ (the identity is trivial if any two indices are equal), and the second for a particular choice of *j* ≠ *k* and a particular choice of *j* = *k*.\nFor the convenience of the reader, let us explain how to derive the formulas for *ψ**i*, *j*. First one must find the aforementioned orthonormal basis of *W*. Actually, any orthonormal basis will do, since by conjugating *G* by an element of $\\Sp(V)$, the image Z24 of *G* under $\\Sp(V) \\twoheadrightarrow \\operatorname{SO}(W)$ is the group of diagonal matrices in any chosen orthonormal basis. To write one, let *v**i* be a basis of *V* and *x**i* the dual basis of *V*\\*. Let *ω*− 1 = *v*1 ∧ *v*3 + *v*2 ∧ *v*4 on *V*. This induces a nondegenerate symmetric bilinear form on *V*\\* ⊗ *V*\\*, which restricts to one on *W*. Up to overall scaling an orthonormal basis is, for $i := \\sqrt{-1}$: *w*1 = *x*1 ∧ *x*3 − *x*2 ∧ *x*4, *w*2 = *x*1 ∧ *x*2 + *x*3 ∧ *x*4, *w*3 = *i*(*x*1 ∧ *x*2 − *x*3 ∧ *x*4), *w*4 = *x*1 ∧ *x*4 + *x*2 ∧ *x*3, *w*5 = *i*(*x*1 ∧ *x*4 − *x*2 ∧ *x*3). Now the explicit map of Remark [rem:iota-compute] immediately yields the following:\n| | | | | | | |\n| --- | --- | --- | --- | --- | --- | --- |\n| *ψ*1, 2 | = | 2(*x*1*x*2 + *x*3*x*4) | | *ψ*2, 4 | = | *x*12 − *x*22 + *x*32 − *x*42 |\n| *ψ*1, 3 | = | 2*i*(*x*1*x*2 − *x*3*x*4) | | *ψ*2, 5 | = | *i*(*x*12 + *x*22 − *x*32 − *x*42) |\n| *ψ*1, 4 | = | 2(*x*1*x*4 − *x*2*x*3) | | *ψ*3, 4 | = | *i*(*x*12 − *x*22 − *x*32 + *x*42) |\n| *ψ*1, 5 | = | 2*i*(*x*1*x*4 + *x*2*x*3) | | *ψ*3, 5 | = | − *x*12 − *x*22 − *x*32 − *x*42 |\n| *ψ*2, 3 | = | − 2*i*(*x*1*x*3 + *x*2*x*4) | | *ψ*4, 5 | = | 2*i*( − *x*1*x*3 + *x*2*x*4). |\nFor example, *ψ*1, 2 is the contraction of *w*1 ∧ *w*2 with *ω*: sum over all ways of contracting the symplectic form *ω* to one component of *w*1 and one of *w*2 and multiply the other two components with the correct sign, yielding *x*1*x*2 + *x*3*x*4 + *x*2*x*1 + *x*4*x*3 = 2(*x*1*x*2 + *x*3*x*4). (Note that the above table corrects, which almost matches ours after reordering and applying an overall scaling, but some coefficients in *op. cit.* are incorrect.)\nThe algebra C[*V*]*G* is then identified with the subalgebra of C[*V*][*G*, *G*] generated by those products of weight vectors (for $\\Z\\_2^4$) in $\\Sym^2 V^\\* \\cong \\wedge^2 W$ for which the weights sum to zero. Explicitly, we may identify $\\operatorname{\\mathrm{Hom}}({\\mathbb{Z}}\\_2^4, {\\mathbb{C}}^{\\times})$ with Z24 so that the weights on *W* are *e*1, *e*2, *e*3, *e*4, − *e*1 − *e*2 − *e*3 − *e*4, with *e**i* ∈ Z24 the standard coordinate vectors (we have − *e*1 − *e*2 − *e*3 − *e*4 = *e*1 + *e*2 + *e*3 + *e*4, but writing in the above way resembles the reflection representation of $\\SL\\_5$). Then the weights on $\\wedge^2 W \\cong \\Sym^2 V^\\*$ are *e**i* + *e**j*, *e**i* − *e*1 − *e*2 − *e*3 − *e*4 for *i* < *j*. Therefore, the products of generators *ψ**i*, *j* having trivial weights are those in which the parity of the number of occurrences of every index *i* equals the parity of the number of occurrences of the index 5. But since there must be a total even number of indices occurring, this means that actually every index must occur an even number of times.\nThe isomorphism\n---------------\nWe are now in a position to prove Theorem [t:isom].\n[thm:4foldsing] The assignment *φ**a**i*, *a**j*, *φ**a**i*, *a**j*\\*, *φ**a**i*\\*, *a**j*, *φ**a**i*\\*, *a**j*\\* ↦ *ψ**i*, *j* extends to a surjective C-algebra homomorphism ${\\mathbb{C}}[\\mu^{-1}(0)]^{\\SL\\_2} \\to {\\mathbb{C}}[V]^{[G,G]}$ that restricts to an isomorphism ${\\mathbb{C}}[\\mu^{-1}(0)]^{\\operatorname{\\mathrm{GL}}\\_{{\\boldsymbol{v}}}} \\to {\\mathbb{C}}[V]^G$. In particular, $X\\_5(0)\\cong \\mathfrak{M}\\_0({\\boldsymbol{v}},0)\\cong V/G$.\nSince *λ* = 0, Proposition [p:gens-repsl2] says that the equations\n \n*φ**a*, *b**φ**c*, *d* − *φ**a*, *c**φ**b*, *d* + *φ**a*, *d**φ**b*, *c* = 0 \n \n∑*i**φ**a**i*, *b**φ**a**i*\\*, *c* = 0 \nhold in ${\\mathbb{C}}[\\mu^{-1}(0)]^{\\SL\\_2}$, so the surjective C-algebra homomorphism is well-defined by Proposition [p:rels-qt]. The restriction of this map to ${\\mathbb{C}}[\\mu^{-1}(0)]^{\\operatorname{\\mathrm{GL}}\\_{{\\boldsymbol{v}}}}$ is well-defined and maps onto C[*V*]*G* by these same propositions, giving a closed immersion $V/G {\\hookrightarrow}{\\mathfrak{M}}\\_{0}({\\boldsymbol{v}},0)$. But these are irreducible varieties of the same dimension, so this closed immersion is an isomorphism.\nAffine Hyperpolygon spaces are not quotient singularities more generally\n========================================================================\nWe have seen that the hyperpolygon space $\\Hyp\\_n(0)$ is a finite symplectic quotient singularity for *n* = 4 or 5. It is therefore natural to ask if we can identify $\\Hyp\\_n(0)$ for *n* > 5 with a symplectic quotient singularity. In this section, we establish the following result.\n[t:not-quotient-sing] For *n* > 5, the affine hyperpolygon space $\\Hyp\\_n(0)$ is not isomorphic to *V*/*G* for any finite group $G \\subset \\Sp(V)$.\nThe movable cone of a product\n-----------------------------\nThe proof of Theorem [t:not-quotient-sing] requires preparatory results. The following is presumably well-known, but we were unable to find the statement in the literature; for the statement under the additional assumption that *X*1 is rational, see.\n[lem:productpicfgnew] Let *X*1, *X*2 be smooth (connected) quasi-projective varieties with $\\Pic(X\\_1)$ finitely generated. Then, $\\Pic (X\\_1 \\times X\\_2) \\cong \\Pic(X\\_1) \\oplus \\Pic(X\\_2)$.\nWe choose a projective compactification $\\overline{X}\\_i$ of *X**i*. Since any quasi-projective variety over C admits a projective resolution of singularities by Hironaka’s Theorem, we may assume that each $\\overline{X}\\_i$ is smooth. If $E\\_{1}, \\ds, E\\_k$ (resp. $F\\_1, \\ds, F\\_{\\ell}$) are the irreducible components of codimension one in $\\overline{X}\\_1 \\smallsetminus X\\_1$ (resp. in $\\overline{X}\\_2 \\smallsetminus X\\_2$) then standard exact sequence on divisor class groups gives \n$$\\bigoplus\\_{i = 1}^k \\Z [\\mc{O}(E\\_i)] \\longrightarrow \\Pic(\\overline{X}\\_1) \\longrightarrow \\Pic(X\\_1) \\longrightarrow 0,$$ \n and similarly for *X*2. In particular, $\\Pic(\\overline{X}\\_1)$ is finitely generated. Since $\\overline{X}\\_1$ is projective over C, we may consider $\\Pic(\\overline{X}\\_1)$ as a group scheme; see. Since it is a discrete group, the connected component $\\Pic^0(\\overline{X}\\_1)$ of the identity is trivial.\nNow assume $\\mc{M}$ is a line bundle on $\\overline{X}\\_1 \\times \\overline{X}\\_2$ and there exists $y\\_0 \\in \\overline{X}\\_2$ with $\\mc{M}\\_{y\\_0}$ (the restriction of $\\mc{M}$ to $\\overline{X}\\_1 \\times \\{ x\\_2 \\}$) trivial. Then implies that every $\\mc{M}\\_{y}$, for $y \\in \\overline{X}\\_2$, is trivial since $\\overline{X}\\_2$ is connected. We deduce, more generally, that if $\\mc{L}$ is a line bundle on $\\overline{X}\\_1 \\times \\overline{X}\\_2$ then $\\mc{L}\\_{y\\_1} \\cong \\mc{L}\\_{y\\_2}$ for all $y\\_i \\in \\overline{X}\\_2$. Set $\\mc{K} = \\mc{L}\\_{y\\_1}$ and consider the projections $\\pi\\_i \\colon \\overline{X}\\_1 \\times \\overline{X}\\_2 \\to \\overline{X}\\_i$. Since *π*2 is a flat projective morphism, says that there exists a line bundle $\\mc{N}$ on $\\overline{X}\\_2$ such that $\\mc{L} \\cong \\pi\\_1^\\* \\mc{K} \\otimes \\pi^\\*\\_2 \\mc{N}$. That is, $\\mc{L} \\cong \\mc{K} \\boxtimes \\mc{N}$. Thus, the canonical map $\\Pic(\\overline{X}\\_1) \\oplus \\Pic(\\overline{X}\\_2) \\to \\Pic(\\overline{X}\\_1 \\times \\overline{X}\\_2)$ is surjective. The fact that it is injective is immediate: if $\\mc{K} \\boxtimes \\mc{N} \\cong \\mc{O}$ then pulling back along $i \\colon \\overline{X}\\_1 \\times \\{ y\\_0 \\} \\to \\overline{X}\\_1 \\times \\overline{X}\\_2$ implies that $\\mc{K} \\cong \\mc{O}$ and similarly for $\\mc{N}$. We deduce that $\\Pic(\\overline{X}\\_1) \\oplus \\Pic(\\overline{X}\\_2) \\cong \\Pic(\\overline{X}\\_1 \\times \\overline{X}\\_2)$.\nFinally, the corresponding result for *X*1 × *X*2 follows from the fact that the rows in the commutative diagram \n$$\\begin{tikzcd}\n\\bigoplus\\_{i = 1}^k \\Z [\\mc{O}(E\\_i)] \\oplus \\bigoplus\\_{j = 1}^{\\ell} \\Z [\\mc{O}(F\\_j)] \\ar[r] \\ar[d,equal] & \\Pic(\\overline{X}\\_1 \\times \\overline{X}\\_2) \\ar[r] & \\Pic(X\\_1 \\times X\\_2) \\ar[r] & 0 \\\\\n\\bigoplus\\_{i = 1}^k \\Z [\\mc{O}(E\\_i)] \\oplus \\bigoplus\\_{j = 1}^{\\ell} \\Z [\\mc{O}(F\\_j)] \\ar[r] & \\Pic(\\overline{X}\\_1) \\oplus \\Pic(\\overline{X}\\_2) \\ar[r] \\ar[u,\"\\wr\"'] & \\Pic(X\\_1) \\oplus \\Pic(X\\_2) \\ar[r] \\ar[u] & 0, \n\\end{tikzcd}$$ \n are exact.\nLet *Y* be a conical symplectic singularity (see Section [sec:sympleaves]). Suppose there exists a projective, crepant resolution of singularities *f*: *X* → *Y*. Recall from Hu–Keel (see Grab and Ohta for the relative case) that *X* is a *relative Mori Dream Space over* *Y* if\n1. $\\Pic(X/Y)\\_{\\mathbb{Q}}\\cong N^1(X/Y);$\n2. the relative nef cone $\\Nef(X/Y)$ is generated by finitely many semiample line bundles; and\n3. there exists *k* ≥ 0 and Q-factorial varieties *X* = *X*0, *X*1, …, *X**k*, each projective over *Y*, as well as birational maps $\\psi\\_i\\colon X\\dashrightarrow X\\_i$ over *Y* for 0 ≤ *i* ≤ *k* that are isomorphisms in codimension-one, such that\n \n$$\\label{eqn:movconedecomp}\n\\Mov(X/Y) = \\bigcup\\_{0\\leq i\\leq k} \\psi\\_i^\\*\\;\\Nef(X\\_i/Y),$$ \nwhere each cone in this description is generated by finitely many semiample line bundles.\nIn particular, the decomposition describes $\\Mov(X/Y)$ as a finite polyhedral fan. The following result was first observed in the current context by Andreatta–Wiśniewski in dimension four, and by Namikawa in general.\n[lem:BCHM] The resolution *X* is a relative Mori Dream Space over *Y*. In particular, the polyhedral decomposition depends only on *Y*, not on the choice of the projective crepant resolution.\nObserve that any divisor *D* on *X* is effective: the coherent sheaf *f*\\*(O*X*(*D*)) has nonvanishing global sections as *Y* is affine and *f* is birational. In particular, − *A* is effective for any *f*-ample divisor *A*. The same is true for Δ : = − *ε**A* for any rational number *ε* > 0. Letting *ε* be sufficiently small, we see that (*X*, Δ) is klt, so it is divisorially log terminal by definition (cf. ). Since − Δ is *f*-ample by construction, gives that the Cox ring of *X* is finitely generated which, by (see also Grab or Ohta ) implies that *X* is a relative Mori Dream Space over *Y*.\nFinally, let *X* → *Y* and *X*′ → *Y* be projective crepant resolutions and let $\\psi \\colon X \\dashrightarrow X^\\prime$ be the resulting rational map over *Y*. Then *ψ* is an isomorphism in codimension one, so we may identify the relative Néron–Severi spaces of *X* and *X*′ over *Y* via strict transform along *ψ*. This, in turn, identifies $\\mr{Mov}(X/Y)$ and $\\mr{Mov}(X^\\prime/Y)$ as fans; see.\n[rem:BCHM] The proof of Lemma [lem:BCHM] relies on, because in order to understand the decomposition of $\\Mov(X/Y)$ beyond the nef cone cone *X* over *Y*, one must know that flops exist in arbitrary dimension. For projective resolutions constructed by VGIT, an alternative approach is given in.\nNamikawa proves that the fan decomposition from Lemma [lem:BCHM] is given by intersecting the movable cone with the complement of a hyperplane arrangement $\\mc{D}\\_Y \\subset N^1(X/Y)$. By Lemma [lem:BCHM], the fan decomposition depends only on *Y* and hence so does the hyperplane arrangement. Namikawa introduces the action of a group *W**Y* on *N*1(*X*/*Y*), depending only on *Y* and, moreover, the movable cone is a fundamental domain for this action.\nWe call $\\mc{D}\\_Y$ the Namikawa hyperplane arrangement of *Y* and *W**Y* the Namikawa Weyl group of *Y*.\nIt follows that in the case *Y* = *X**n*(0), the arrangement $\\mc{D}\\_Y$ equals A and the Namikawa Weyl group is Z2*n* as we found earlier.\nNow let *Y*1, *Y*2 be conical symplectic singularities. We assume that there exist projective crepant resolutions *f**i*: *X**i* → *Y**i* for *i* = 1, 2. Define *X* : = *X*1 × *X*2 and *Y* : = *Y*1 × *Y*2, and consider the projective, crepant resolution *f* = *f*1 × *f*2: *X* → *Y*. In particular, we may apply Lemma [lem:BCHM] to each of *X*1, *X*2 and *X* over *Y*1, *Y*2 and *Y* respectively.\n[prop:movableproduct] The Picard group of *X**i* is finitely generated for *i* = 1, 2, and moreover, the isomorphism of Lemma [lem:productpicfgnew] induces an isomorphism of relative Néron–Severi spaces\n \n*N*1(*X*/*Y*) ≅ *N*1(*X*1/*Y*1) × *N*1(*X*2/*Y*2). \nThis, in turn, induces an identification of finite polyhedral fans\n \n$$\\label{eqn:Movproduct}\n\\mr{Mov}(X/Y)\\cong \\mr{Mov}(X\\_1/Y\\_1)\\times \\mr{Mov}(X\\_2/Y\\_2).$$ \nIn particular, the Namikawa hyperplane arrangement of *Y* is equal to the product of the Namikawa hyperplane arrangements of *Y*1 and *Y*2.\nTo prove that $\\Pic(X\\_i)$ for *i* = 1, 2 is finitely generated, it suffices to prove that the Z-linear map $\\Pic(X\\_i)\\to N^1(X\\_i/Y\\_i)$ sending a line bundle to its numerical class is injective. Let $L\\in \\Pic(X\\_i)$ satisfy deg(*L*|*C*) = 0 for each curve *C* contracted by *f**i*. Consider the pair (*X**i*, Δ) constructed in the proof of Lemma [lem:BCHM], where − Δ is *f**i*-ample. Since *K**Y**i* = 0 and *f**i* is crepant, we have *K**X**i* = 0 and hence − (*K**X**i* + Δ) = − Δ is *f**i*-ample. The relative version of Kleiman’s ampleness criterion now gives (*K**X**i* + Δ).*z* < 0 for all $z\\in \\overline{NE}(X\\_i/Y\\_i)\\setminus \\{0\\}$, in which case the relative version of the Contraction Theorem gives that *L* ≅ (*f**i*)\\*(*M*) for some ample line bundle *M* on *Y**i*. However, the Picard group of *Y**i* is trivial because *Y**i* is a conical singularity, so *L* ≅ O*X**i*. This shows that the map $\\Pic(X\\_i)\\to N^1(X\\_i/Y\\_i)$ is injective as required.\nWe may now apply Lemma [lem:productpicfgnew] to obtain $\\Pic (X) \\cong \\Pic(X\\_1) \\oplus \\Pic(X\\_2)$. For the statement, it suffices to show that if a line bundle *L* has degree zero on all proper curves in *X*1 and *X*2, then it also has degree zero on every proper (connected) curve *C* in *X*1 × *X*2. But the intersection number *L*.*C* = deg(*L*|*C*) depends only on the class of *C* in *H*2(*X*, Q) = *H*2(*X*1, Q) ⊕ *H*2(*X*2, Q). Since this class can be written in terms of classes of curves in *X*1 and *X*2, we obtain deg(*L*|*C*) = 0.\nFor the statement, recall that any relative Mori Dream Space *X* over *Y* can be constructed by GIT from its Cox ring R(*X*). The Cox ring of *X* is the tensor product R(*X*) ≅ R(*X*1) ⊗ CR(*X*2) of the Cox rings of *X*1 and *X*2, equipped with the canonical grading by $\\Pic(X\\_1)\\oplus \\Pic(X\\_2)$; see. In particular, *X* = *X*1 × *X*2 is a GIT quotient for the action of the quasitorus $\\operatorname{\\mathrm{Spec}}{\\mathbb{C}}[\\Pic(X\\_1)]\\times \\operatorname{\\mathrm{Spec}}{\\mathbb{C}}[\\Pic(X\\_2)]$ on $\\operatorname{\\mathrm{Spec}}\\mathcal{R}(X\\_1)\\times \\operatorname{\\mathrm{Spec}}\\mathcal{R}(X\\_2)$ determined by the product group action, so the GIT chamber decomposition for the $\\Pic(X)$-grading of R(*X*) is the product of the GIT chamber decompositions for the $\\Pic(X\\_i)$-gradings of R(*X**i*) for *i* = 1, 2.\nWithin this product chamber decomposition, we need to check that the chambers whose union makes up $\\mr{Mov}(X/Y)$ is precisely the the union of the product of chambers making up $\\mr{Mov}(X\\_1/Y\\_1)$ and $\\mr{Mov}(X\\_2/Y\\_2)$. For all three of these GIT constructions, the movable cone is a subfan of the GIT fan characterised by the $\\Pic$-degrees of a system of homogeneous generators of the relevant Cox ring . A system of homogeneous generators of R(*X*) is given by taking the product {*g**i**h**j* ∣ *i* ∈ *I*, *j* ∈ *J*} over systems of homogeneous generators {*g**i* ∣ *i* ∈ *I*} and {*h**j* ∣ *j* ∈ *J*} of R(*X*1) and R(*X*2) respectively. It follows that the cone supporting the movable cone of *X*1 × *X*2 over *Y*1 × *Y*2 coincides with the product of the cones supporting the movable cones of *X*1 over *Y*1 and *X*2 over *Y*2 respectively.\nFinally, the movable cone contains the ample cone which is of full dimension for a projective morphism over an affine base. The statement about the Namikawa hyperplane arrangements now follows from the statement about the movable cones. This completes the proof.\nResolutions of quotient singularities\n-------------------------------------\nBelow, *V*1, *V*2 and *V* will denote finite-dimensional symplectic vector spaces.\n[lem:nontrivaiNamarragement] Let $G \\subset \\Sp(V)$ be a non-trivial finite group such that *Y* = *V*/*G* admits a projective crepant resolution *f*: *X* → *Y*. The Namikawa hyperplane arrangement in *N*1(*X*/*Y*) is non-trivial.\nObserve in this case that *Y* is singular by results of Shephard–Todd and Chevelley because *G* is not a complex reflection group; indeed it contains contains no complex reflections.\nSince *V*/*G* is singular, *f* is not an isomorphism. Let O*X*(*D*) be *f*-ample. The divisor *D* is numerically positive against all (necessarily proper) curves contracted by *f*. In particular, [*D*] ≠ 0 in *N*1(*X*/*Y*), so *N*1(*X*/*Y*) ≠ 0. Since *f* is not an isomorphism, 0 ∈ *N*1(*X*/*Y*) cannot be *f*-ample. Thus, there is a hyperplane containing 0; in fact, the Namikawa arrangement is central so it lies on all hyperplanes.\nA hyperplane arrangement is reducible if it is a product of non-trivial hyperplane arrangements. More formally, A is *reducible* if if there exist nonempty subarrangements A1 and A2 such that A = A1 ∪ A2 is a disjoint union, and after a linear change of coordinates, the linear polynomials defining the hyperplanes in A1 and in A2 share no common variable.\n[lem:GproductNamarrangement] Let $G\\_i \\subset \\Sp(V\\_i)$ for *i* = 1, 2 be finite groups, both non-trivial. If *Y* = *V*1/*G*1 × *V*2/*G*2 admits a projective crepant resolution *X*′ → *Y*, then the Namikawa hyperplane arrangement in *N*1(*X*′/*Y*) is reducible.\nApplying to the action of *G* = *G*1 × *G*2 on *V* = *V*1 × *V*2 implies that each *Y**i* = *V**i*/*G**i* admits a projective crepant resolution *X**i* → *Y**i*. Thus, *X*1 × *X*2 → *Y* is also a projective crepant resolution of *Y*. As described in the final paragraph of the proof of Lemma [lem:BCHM], we may identify $\\Mov(X^\\prime/Y)$ with $\\Mov(X\\_1\\times X\\_2/Y)$. The result follows from Proposition [prop:movableproduct].\n[lem:hyperirreducible] The hyperplane arrangement $\\mc{A}$ in Θ is irreducible.\nAssume that Θ = Θ1 × Θ2 and $\\mc{A} = \\mc{A}\\_1 \\cup \\mc{A}\\_2$ for $\\mc{A}\\_i$ an arrangement in Θ*i*. Then, for each $H \\in \\mc{A}$, $H \\in \\mc{A}\\_1$ means that Θ2 ⊂ *H* etc. Since the coordinate hyperplanes *L**i* belong to $\\mc{A}$, this forces $\\Theta\\_1 = (\\theta\\_i = 0 : i \\notin J)$ and Θ2 = (*θ**j* = 0 : *j* ∈ *J*) for some *J* ⊂ [1, *n*]. But then the *H**I* do not contain either Θ1 or Θ2 unless *J* ≠ ∅, [1, *n*]. Thus, $\\mc{A}$ is irreducible.\nIf $G \\subset \\Sp(V)$ is a finite group then the pair (*V*, *G*) is said to be *symplectically reducible* if *V* = *V*1 × *V*2 is a proper decomposition into symplectic *G*-submodules. Otherwise, (*V*, *G*) is said to be *symplectically irreducible*; see for more on this notion. Recall that *g* ∈ *G* is a *symplectic reflection* if it fixes a linear subspace of codimension two, and *G* is a *symplectic reflection group if it is generated by symplectice reflections.*\n[cor:hyperpolyquotiso] If, for some *n* ≥ 4, there is an isomorphism $\\Hyp\\_n(0) \\cong V/G$ with $G \\subset \\Sp(V)$ finite, then (*V*, *G*) is a symplectically irreducible symplectic reflection group.\nChoose *θ* ∈ Θ generic. By Corollary [cor:symplecticreshyperpoly], the morphism $X = \\Hyp\\_n(\\theta) \\to \\Hyp\\_n(0) = Y$ is a projective crepant resolution. Moreover, under the identification $L\\_F \\colon \\Theta {{\\;\\stackrel{\\_\\sim}{\\longrightarrow}\\;}}N^1(X/Y)$ of Theorem [thm:resolutionsofstar], the arrangement $\\mc{A}$ in Θ is identified with the Namikawa arrangement. Therefore, Lemma [lem:hyperirreducible] implies that the latter is irreducible.\nIf *X* ≅ *V*/*G*, then *G* is a symplectic reflection group by Verbitsky’s Theorem because *X* admits a projective crepant resolution. In this case, if *V* is symplectically reducible then *V* = *V*1 × *V*2 and *G* = *G*1 × *G*2, with *G**i* acting on *V**i*. Moreover, *G**i* ≠ 1 because *X**n*(0) has a zero-dimensional leaf by Lemma [lem:sympleavesH0] and symplectic leaves of a symplectic singularity are preserved under isomorphism (they are the strata of the singular locus stratification ). Lemma [lem:GproductNamarrangement] then implies that the Namikawa arrangement in *N*1(*X*/*Y*) is reducible. This contradicts the conclusion of the previous paragraph. Thus, (*V*, *G*) is symplectically irreducible.\n[Proof of Theorem [t:not-quotient-sing]] Assume that $\\Hyp\\_n(0) \\cong V/G$. By Corollary [cor:hyperpolyquotiso], (*V*, *G*) is symplectically irreducible and *V*/*G* admits a projective crepant resolution. We also know from Proposition [prop:beyondF] that its associated Namikawa Weyl group is $\\Z\\_2^n$. Based on the the paragraph after Question 9.5 of and, either $G = \\s\\_n \\wr K$ is a wreath product group (with $K \\subset \\SL(2)$) or potentially *G* is one of the (complex) primitive symplectic reflection groups *W*(*S*1), *W*(*R*) or *W*(*U*) acting on a 6, 8 or 10 dimensional space, respectively (in the notation of ).\nIf $G = \\s\\_n \\wr K$ for *n* > 1 and *W**K* is the (irreducible) Weyl group of type ADE associated to *K* via the McKay correspondence, then by the Namikawa Weyl group of a projective symplectic resolution *X* of *V*/*G* is $\\Z\\_2 \\times W\\_K$. Then $\\Z\\_2 \\times W\\_K \\cong \\Z\\_2^n$, but the only abelian Weyl group of type *A**D**E* is of type *A*1, where $W\\_K\\cong \\Z\\_2$ and hence *n* = 2. But we assumed *n* > 5.\nTherefore, we must rule out *G* = *W*(*S*1), *W*(*R*) or *W*(*U*). It was shown recently by computer computations in that these quotient singularities do not admit projective symplectic resolutions, but to avoid relying on computer calculations, we do not use this result here. To rule these cases out directly, note that if $\\Hyp\\_n(0)$ is isomorphic to *V*/*G* then $\\Hyp\\_n(\\theta) \\to V/G$ is a projective symplectic resolution for *θ* generic. On the one hand, as in the proof of Proposition [prop:dimN1], $\\dim H^2(\\Hyp\\_n(\\theta),{\\mathbb{C}}) = n$. On the other, by Lemma [lem:H2Wexceptional] below, $\\dim H^2(\\Hyp\\_n(\\theta),{\\mathbb{C}}) = 1$; a contradiction.\nIn the statement, and proof, of the following lemma, we use freely the notation from where the irreducible symplectic reflection groups with primitive complexification are classified.\n[lem:H2Wexceptional] Let *G* equal one of the (complex) primitive symplectic reflection groups *W*(*S*1), *W*(*R*) or *W*(*U*) and assume that *V*/*G* admits a projective symplectic resolution *X* → *V*/*G*. Then *H*2(*X*, Q) is one-dimensional.\nBy, dim*H*2(*X*, Q) equals the number of “junior” conjugacy classes in *G* which, in light of, can be defined to be the conjugacy classes of symplectic reflections. As noted in, each symplectic reflection in *G* (for our choice of groups) is of the form *s**a*, *ξ* for some (*a*, *ξ*) ∈ Σ, where Σ is the root system associated to *G*, as in. Then, \n*ξ* ∈ *H**a* : = {*λ* ∈ *S**U*(2) ∣ *s**a*, *λ* ∈ *G*}. \n Inspecting shows that $H\\_a = \\Z\\_2$, and hence *ξ* = − 1 for our three groups. It is explained in the proof of that the roots *a* in Σ form a single *G*-orbit. We deduce that all symplectic reflections are conjugate.\nHyperkähler metrics\n===================\nRecall that *X**n*(*θ*) carries a complete hyperkähler metric for any generic choice of *θ*. Recently, there has been much interest in the asymptotic geometry of such spaces. In this final (discursive) section, we note the implications of our main results for asymptotic geometry of *X**n*(*θ*). The resolution of singularities *X**n*(*θ*) → *X**n*(0) identifies the singularity *X**n*(0) with the tangent cone at infinity of the hyperkähler manifold *X**n*(*θ*). The full description of the asymptotic geometry of *X**n*(*θ*) requires precise estimates on the decay of the true hyperkähler metric on *X**n*(*θ*) to the (singular) cone metric on *X**n*(0).\nMetrics: the case *n* = 4\n-------------------------\nIt is known that *X*4(*θ*) is *asymptotically locally Euclidean* (ALE). In general, for a smooth, noncompact hyperkähler manifold of real dimension 4*k*, this means that the asymptotic geometry is modelled on the singular space C2*k*/*G* for some finite subgroup *G* in $\\Sp(k) = \\Sp(2k,{\\mathbb{C}}) \\cap \\mathrm{U}(2k)$, whose action on the unit sphere *S*4*k* − 1 ⊂ C2*k* is free. In particular, the orbifold cone C2/*G* has an isolated singularity at the origin. For *k* = 2, the hyperkähler ALE-spaces were classified by Kronheimer to be the crepant resolutions of the Kleinian singularities C2/*G* for the finite subgroups $G < \\mathrm{SU}(2)=\\Sp(1)$. The hyperpolygon space *X*4(*θ*) corresponds to the (affine) *D*4 case, that is, to the quaternion group *Q*8 < SU(2), and hence *X*4(0) is the orbifold cone C2/*Q*8 which has an isolated singularity at the origin.\nMetrics: the case *n* = 5\n-------------------------\nAs a further generalisation of ALE, a smooth noncompact hyperkähler manifold is called *quasi-asymptotically locally Euclidean* (QALE) if its asymptotic geometry is modelled on C2*k*/*G* for a finite subgroup *G* in $\\Sp(k)$ that does not necessarily act freely on the unit sphere in C2*k*; here C2*k*/*G* has singular strata given by non-isolated quotient singularities. Both ALE and QALE metrics have maximal, that is, Euclidean volume growth. Regarding larger values of *n*, Theorem [t:isom] shows that the hyperpolygon space *X*5(0) is the orbifold cone C4/*G* for the group *G* = *Q*8 × Z2*D*8 in $\\Sp(2)=\\Sp(4,{\\mathbb{C}}) \\cap \\mathrm{U}(4)$. This provides significant evidence for the following conjecture:\n[conj:QALE] For generic *θ*, the space *X*5(*θ*) is quasi-asymptotically locally Euclidean.\nThere are two reasons that make this conjecture plausible. First, by work of Joyce, there exists a unique hyperkähler QALE metric on the crepant resolution of a symplectic quotient singularity in each QALE Kähler class (with regards to the complex structure *I*). However, it is *a priori* not at all clear that the natural metric on *X*5(*θ*)—that is, the one given by hyperkähler reduction —can also be obtained by the construction of Joyce. To apply Joyce’s result one would first have to construct a QALE metric in the given Kähler class. While this may presumably be carried out by an inductive procedure, applying the uniqueness portion of Joyce’s theorem would require specific knowledge about the asymptotics of the competing hyperkähler metric; see for a discussion in the context of hyperkähler metrics for the planar Hilbert scheme of points.\nSecond, one may compare the metric geometry of the moduli space of (parabolic) Higgs bundles on an *n*-punctured P1 to that of *X**n*(*θ*), given the embedding of the former into the latter as described in. In particular, one of the moment map conditions that defines *X**n*(*θ*) as a hyperkähler quotient is an explicit linearisation of the relation that defines the character variety of an *n*-punctured sphere. The Higgs bundle moduli space, which is one of the avatars of the character variety under nonabelian Hodge theory, comes with certain expectations about the decay of its hyperkähler metric to a certain “semiflat” model metric on a torus bundle over a half-dimensional base as one goes to infinity. This behaviour is broadly classed as “ALG” — see, for example,. We note that ALG is not an acronym; rather, it is obtained by iteration from ALE, with ALF as the intermediary. In general, this entails a much slower volume growth. (In complex dimension 2, the volume growth for ALG is quadratic and is explicitly verified for the *n* = 4 parabolic Higgs moduli space in.) The realisation of *X**n*(*θ*) as a linear version of this geometry suggests, for arbitrary *n*, a more rapid, namely fully Euclidean volume growth in contrast to its Higgs bundle counterpart. For more on this point of view, see.\nMetrics: the general case\n-------------------------\nFor *n* ≥ 6, Theorem [t:not-quotient-sing] shows that *X**n*(0) is not an orbifold cone of the form C2*k*/*G* for a finite subgroup *G* in $\\Sp(k)=\\Sp(2k),{\\mathbb{C}}) \\cap \\mathrm{U}(2k)$, where *k* = *n* − 3. As a consequence, we obtain a proof of Corollary [cor:QALEnot] which we restate here for convenience:\n[cor:restated1.6] For *n* > 5, *X**n*(*θ*) is not quasi-asymptotically locally Euclidean for any *θ*.\nIf the hyperkähler metric on *X**n*(*θ*) given by hyperkähler reduction were QALE, then the tangent cone at infinity *X**n*(0)—that is, the limit of the sequence of homothetic rescalings (*X**n*(*θ*), *λ*2*g*) as *λ* → 0—would be the Riemannian orbifold C2*k*/*G*. This contradicts Theorem [t:not-quotient-sing].\nIt is natural to ask whether *X**n*(*θ*) is *quasi-asymptotically conical* (QAC) in the sense of Degeratu–Mazzeo . This notion generalises Q"},"data_id":{"kind":"string","value":"arxiv_0002032"}}},{"rowIdx":494,"cells":{"content":{"kind":"string","value":"Electrostatic stability of insulating surfaces: Theory and applications\n=======================================================================\nWe analyze the electrostatic stability of insulating surfaces in the framework of the bulk modern theory of polarization. We show that heuristic arguments based on a fully ionic limit find formal justification at the microscopic level, even in solids where the bonding has a mixed ionic/covalent character. Based on these arguments, we propose simple criteria to construct arbitrary non-polar terminations of a given bulk crystal. We illustrate our ideas by performing model calculations of several LaAlO3 and SrTiO3 surfaces. We find, in the case of ideal LaAlO3 surfaces, that cleavage along a higher-index (*n*10) direction is energetically favorable compared to the polar (100) orientation. In the presence of external adsorbates or defects the picture can change dramatically, as we demonstrate in the case of H2O/LaAlO3(100).\nIntroduction\n============\nA uniformly charged plane separating two semi-infinite regions of space yields a divergent electrostatic energy; for this reason, *polar* surfaces or interfaces cannot exist. Yet, thanks to recent progress in epitaxial growth techniques, nominally polar terminations of insulating crystals are now routinely prepared and characterized within well-controlled experimental conditions. This is possible because, in practice, there are several mechanisms available for a polar surface to neutralize the problematic excess charge, and possibly become thermodynamically stable. These include adsorption of foreign gas-phase species, changes in the surface stoichiometry, ionic and/or electronic reconstructions, or local metallization via accumulation of intrinsic free carriers; each of the above can, in principle, prevent the “polar catastrophe” by restoring the correct charge balance at the surface.\nUnderstanding and controlling these compensation mechanisms is a subject of great importance for many areas of fundamental science and technology. For example, surface polarity is of great interest for catalysis, gas sensing and energy applications , as adsorption and/or redox of gas-phase species are known to be strongly influenced by the electrostatic environment . In the context of perovskite-structure thin films and heterostructures, control of surface charge/polarity is currently investigated as a route towards the development of novel field-effect devices (e.g. in the LaAlO3/SrTiO3 system ), or ferroelectric memories based on the tunneling electroresistance effect . Central to rationalizing all these phenomena is the intimate relationship between surface charge and bulk polarization in crystalline insulators, which was formally established by Vanderbilt and King-Smith in 1993. \nFrom the point of view of the theoretical analysis, it is crucial to establish an unambiguous criterion to classify a given surface as “polar-compensated” (i.e. an originally polar surface that was neutralized via one of the aforementioned mechanisms) or intrinsically non-polar. This is not just a matter of nomenclature, but has very concrete practical relevance: non-polar terminations generally tend to be more stable, as extrinsic (i.e. not originated from the primitive building blocks of the insulating bulk crystal) sources of compensating charge tend to have a high energy cost. Furthermore, if a given surface is polar, one needs to know precisely how much external charge is needed to neutralize it; this greatly facilitates the theoretical analysis by restricting the number of possible candidate structures. We shall see in the following that, while the energetics is a genuine surface property, an exact answer to the latter question can be given already at the bulk level. Many authors have already addressed this issue in the past – we shall briefly mention hereafter the approaches that are most directly relevant to our work.\nTasker modeled a given ionic crystal as a lattice of point charges, corresponding to the nominal valence of the ions. Based on an abrupt truncation of this lattice, a given surface is then classified as polar on non-polar, depending on the behavior of the electrostatic energy. In particular, in the former (polar) case, the bulk repeated unit cell carries a finite dipole moment; this produces a diverging electrostatic potential unless compensated by an equal and opposite external surface charge density. This model, despite its simplicity, turned out to be surprisingly effective, and was able to correctly predict, at least at the qualitative level, the polar or non-polar nature of the vast majority of insulating surfaces. However, at the quantitative level this model has clear limitations. Many oxides and semiconductors display a marked covalent character, and the bulk polarization departs significantly from the value that can be inferred from atomic positions and nominal valence charges. Hence the need for a more accurate treatment.\nTo address these issues, and adopt a more realistic description of the charge density of the solid, Goniakowski *et al.* proposed a different criterion for classifying surfaces as polar or non-polar. At the heart of the strategy of Ref. is the concept of “dipole-free” unit cell. Given a certain plane orientation, one can demonstrate that it is always possible to choose a dipole-free repeated unit along the normal to that plane; then, the remainder charge that is left at the surface (once the bulk units have been removed) determines the polar or non-polar character of the termination. This criterion, however, is not free from ambiguities: for the same surface termination there may exist more than one possible choice of the dipole-free bulk unit cell, which might lead to opposite conclusions about the polar or non-polar character of the surface (see section [sec:dipole-free] for a detailed discussion). Also, identifying the dipole-free unit cell might be cumbersome in the case of higher index surfaces, where the structural complexity of the larger cell could complicate this type of analysis. Finally, the intuitive appeal of Tasker’s model is apparently lost in the strategy of Ref. : one needs to look at the ground-state charge density (e.g. as provided by a first-principles calculation) before drawing a conclusion.\nThere are two further issues that are common to both methods. First, it is universally agreed that the surface polarity is a property of the actual lattice termination. This means that, for a given material and surface plane orientation, there might be polar and non-polar terminations, depending on the surface stoichiometry. However, there is no established recipe to unambiguously decide, given a compound crystal and a surface orientation, whether a stoichiometric 1 × 1 non-polar termination is allowed at all. Furthermore, it is not clear how to construct, in general, a non-polar candidate structure without relying on a heuristic counting of the layer charges. Second, it was correctly recognized by both Tasker and Goniakowski that the issue of surface polarity is directly related to the bulk polarization of the material. However, neither model traces a formal link to the modern theory of polarization in periodic insulators , where the macroscopic ${\\bf P}$ is a multivalued vector field, written in terms of the *phases* of the wavefunctions. Only the total charge density (modulus of the wavefunctions) is considered in the model of Ref. , while explicit electronic orbitals are not addressed by Tasker’s approach. Recent theoretical works have indeed highlighted the importance of the formal Berry-phase polarization in discussing polarity at surfaces and interfaces , but a general formulation of the problem, based on the formalism established in Ref. , is still missing.\nHere we show that a Wannier function representation together with the “interface theorem” of Ref. , provide a very natural framework for addressing the above issues. Wannier functions were already shown to be a very useful tool, in layered superlattices , for partitioning the polarization of a crystal into the contribution of individual charge-neutral units. Most importantly, Wannier functions are intimately linked to the modern theory of polarization in solids , and therefore appear to be the most appropriate ingredient to discuss the issue of surface polarity, where the basic question concerns the existence of a finite dipole moment perpendicular to the surface plane. We shall provide, based on this description, precise criteria to establish whether truncating a bulk crystal along a given crystallographic orientation can yield a non-polar surface. We shall demonstrate that answering this question involves only an analysis of the bulk, and that our scheme naturally leads to candidate structures that can be used as a starting point for the subsequent determination of the thermodynamic ground state. To demonstrate our arguments, we focus on the surfaces of LaAlO3 and SrTiO3, two prototypical perovskite materials that have been at the center of the attention in the past few years as their polar (100) interface exhibits numerous peculiar properties.\nThis work is organized as follows. In section [sec:theory] we introduce our definition of polar surface and its formal relationship to the theory of bulk polarization. We also establish a direct link to Tasker’s model and we compare it to the “dipole-free” cell approach. In section [sec:results] we apply this formalism to a variety of systems, including non-polar LaAlO3(*n*10) and SrTiO3(111) surfaces. We also discuss electronic/ionic compensation mechanisms of polar LaAlO3(100). In section [sec:discussion] we briefly address some related topics, including the case of ferroelectric surfaces, and possible extensions to covalent semiconductors. Finally, in section [sec:conclusions] we present a brief summary and the conclusions.\nTheory\n======\nDefinition of polar surface\n---------------------------\nIn full generality, for the surface of a crystalline insulator to be electrostatically stable, it must have a vanishing density of *physical* surface charge, $\\sigma\\_{\\rm surf}=0$. In order to introduce the notion of surface polarity, it is useful to separate $\\sigma\\_{\\rm surf}$ into two distinct contributions, and rewrite the stability condition as\n \n$$\\sigma\\_{\\rm ext} + {\\bf P}\\_{\\rm bulk} \\cdot {\\hat{n}} = 0.\n\\label{eq:neutral}$$ \nHere ${\\bf P}\\_{\\rm bulk}$ is the bulk polarization, *n̂* is the normal to the surface plane, and $\\sigma\\_{\\rm ext}$ is a surface density of “external” compensating charges, which encompasses all contributions that cannot conveniently be described as “bulk-like” in nature (we include the latter in ${\\bf P}\\_{\\rm bulk}$). $\\sigma\\_{\\rm ext}$ typically includes free charges (e.g. in the form of a confined electron gas) and/or bound charges (either in the form of surface adsorbates, vacancies, non-stoichiometric reconstructions, or non-isoelectronic substitutions).\nWe define a given surface as non-polar if the stability criterion Eq. ([eq:neutral]) can be satisfied in the absence of external charges $\\sigma\\_{\\rm ext}$, which implies\n \n$${\\bf P}\\_{\\rm bulk} \\cdot {\\hat{n}} = 0.\n\\label{eq:nonpolar}$$ \nThis equation, at first sight, looks inconsistent with the current understanding of the surface polarity problem. It is now widely accepted that the polar or non-polar attribute is a property of the *termination*, not only of the material and surface plane orientation, contrary to what Eq. ([eq:nonpolar]) seems to suggest. We shall see in the following section that the choice of the termination is only apparently absent from Eq. ([eq:nonpolar]). It is implicitly included through the intrinsically multivalued nature of ${\\bf P}\\_{\\rm bulk}$, which is a well-established aspect of the modern theory of polarization in bulk insulators. \nThe bulk polarization\n---------------------\n### As a Berry phase\nWe consider a crystalline insulator described by three primitive translation vectors ${\\bf a}\\_{1,...,3}$ and a basis of *N* atoms located at positions ${\\bf R}\\_\\alpha$, with *α* = 1, ..., *N*. The “formal” bulk polarization is usually defined as\n \n$${\\bf P}\\_{\\rm bulk} = \\frac{1}{\\Omega} \\Big( \\sum\\_{\\alpha=1}^N {\\bf R}\\_\\alpha Z\\_\\alpha -\n2e \\sum\\_{i=1}^3 \\frac{\\phi\\_{\\rm el}^{(i)} {\\bf a}\\_i}{2\\pi} \\Big).\n\\label{eq:berry}$$ \nHere *Z**α* is the charge of the ionic core *α*, *e* is the (positive) electron charge and $\\phi\\_{\\rm el}^{(i)}$ is the Berry phase along the reciprocal-space vector *i*; for simplicity we assume spin pairing, hence the factor of 2 in the electronic contribution.\nIt is important to note that ${\\bf P}\\_{\\rm bulk}$, as defined in Eq. ([eq:berry]) is only defined modulo a “quantum of polarization”; in other words, it is not a single-valued but a multi-valued function of the electronic and structural degrees of freedom. This indeterminacy concerns both the ionic and the electronic parts in Eq. ([eq:berry]). On one hand, one has the freedom to choose any of the periodically repeated images of each atomic specie, and thus change ${\\bf R}\\_\\alpha$ by an arbitrary translation vector of the type $\\Delta {\\bf R} = n\\_1 {\\bf a}\\_1 + n\\_2 {\\bf a}\\_2 + n\\_3 {\\bf a}\\_3$. On the other hand, $\\phi\\_{\\rm el}^{(i)}$ are phases of complex numbers, and therefore only defined modulo 2*π*.\nIn the following sections we shall use the equivalent formulation of ${\\bf P}\\_{\\rm bulk}$ in terms of Wannier functions to illustrate the relationship between the multivaluedness of ${\\bf P}\\_{\\rm bulk}$ and the termination of the crystal lattice.\n### From the Wannier functions\nWe shall explicitly assume, from now on, a single-particle picture, in terms of a Kohn-Sham set of orbitals, and a conventional (as opposed to topological) insulating state. Within these assumptions, it is possible to express the electronic ground state of the bulk solid in terms of a set of maximally-localized Wannier functions , which are exponentially localized in direct space . Based on this representation, Eq. ([eq:berry]) can be rewritten as\n \n$${\\bf P}\\_{\\rm bulk} = \\frac{1}{\\Omega} \\Big( \\sum\\_{\\alpha=1}^N {\\bf R}\\_\\alpha Z\\_\\alpha -\n2e \\sum\\_{j=1}^{N\\_{\\rm el}/2} \\langle {\\bf r} \\rangle\\_j \\Big),\n\\label{eq:wannier}$$ \nwhere $N\\_{\\rm el}$ is the total number of electrons in the primitive cell, and $\\langle {\\bf r} \\rangle\\_j$ is the location of the center of the *j*-th Wannier function.\nAlternatively, we can think in terms of a charge density distribution that consists in the basis of atomic point charges together with the Wannier densities,\n \n$$\\rho\\_{\\rm cell} ({\\bf r}) = \\sum\\_{\\alpha=1}^N Z\\_\\alpha \\delta({\\bf r}-{\\bf R}\\_\\alpha)\n- 2e \\sum\\_{j=1}^{N\\_{\\rm el}/2} |w\\_j({\\bf r})|^2,\n\\label{eq:rhocell}$$ \nwhere $w\\_j({\\bf r})$ is the *j*-th Wannier function of the primitive cell. By construction, the sum of all the periodic images of $\\rho\\_{\\rm cell}$ “tiles” the total charge density of the extended solid. Then, by combining Eq. ([eq:wannier]) and Eq. ([eq:rhocell]) one immediately obtains the intuitive connection to the Clausius-Mossotti formula,\n \n$${\\bf P}\\_{\\rm bulk} = \\frac{\\bf d}{\\Omega},$$ \nwhere ${\\bf d}$ is the dipole moment of $\\rho\\_{\\rm cell}$.\nThis formulation provides a transparent way to partition the total charge density into individual primitive units, whose dipole moment correctly yields the formal value of ${\\bf P}\\_{\\rm bulk}$. In doing so, the *phase* indeterminacy of the electronic contribution to the polarization discussed in the previous Section has been reduced to a *lattice* indeterminacy, in all respects analogous to that characterizing the ionic contribution. In other words, all the complications related to the quantum-mechanical nature of the electrons have been mapped into a system of classical point charges, where the atoms and the electrons are formally treated on the same footing. In the following section we discuss how this Wannier representation can be further partitioned into smaller units that retain the chemical information about the formal oxidation state of each ion, which is central to the notion of “polar surface”.\nFormal ionic charges\n--------------------\nThe location of the Wannier functions generally reflects the bonding properties of the material – in ionic solids they will cluster around the atoms, while in covalent materials they will tend to occupy the bond centers. We shall assume that the solid has at least a certain degree of ionic character, so it is possible to “assign” each Wannier function to a given atom without ambiguities; this is certainly true in most known oxide materials. (With some caution the ideas developed here can be conveniently adapted to any crystalline insulator; we shall briefly discuss the example of purely covalent semiconductors in Sec. [sec:discussion].) We then combine each ion *α* with the Wannier orbitals *j* that “belong” to it and define a set of compound charge distributions that we call “Wannier ions” (WI),\n \n$$\\rho\\_{\\rm WI}^{(\\alpha)} ({\\bf r}) = Z\\_\\alpha \\delta({\\bf r})\n- 2e \\sum\\_{j \\in \\alpha} |w\\_j({\\bf r + {\\bf R}\\_\\alpha})|^2 \\Big).\n\\label{eq:rhowi0}$$ \n(We operated a translation so that the nucleus sits in the origin.) As the Wannier function locations usually agree remarkably well with chemical intuition, each of these *N* charge distributions will carry a monopole *Q**α* corresponding to the “nominal” charge of the ion (e.g. − 2*e* for O, + 2*e* for Sr, + 4*e* for Ti). In addition to their net charge, the WI are non-spherical and generally carry non-zero dipole moments ${\\bf d}\\_\\alpha$. (Higher multipoles are also present, but are not directly relevant for the present discussion.) $\\rho\\_{\\rm cell}({\\bf r})$ can now be rewritten in terms of the WI densities,\n \n$$\\rho\\_{\\rm cell} ({\\bf r}) = \\sum\\_\\alpha \\rho\\_{\\rm WI}^{(\\alpha)} ({\\bf r}-{\\bf R}\\_\\alpha),\n\\label{eq:rhowi}$$ \nwhich is equivalent to Eq. ([eq:rhocell]) except that here we use precautions to keep the basic WI units intact. It follows that the dipole moment of $\\rho\\_{\\rm cell}({\\bf r})$ can be written in terms of two contributions,\n \n$${\\bf d} = {\\bf d}\\_{\\rm PC} + {\\bf d}\\_{\\rm WI}.\n\\label{eqd}$$ \nThe first term is the dipole moment of a system of point charges located at positions ${\\bf R}\\_\\alpha$,\n \n$${\\bf d}\\_{\\rm PC}=\\sum\\_\\alpha {\\bf R}\\_\\alpha Q\\_\\alpha.$$ \nThe second term is the sum of the individual dipole moments of the WI,\n \n$${\\bf d}\\_{\\rm WI}=\\sum\\_\\alpha {\\bf d}\\_\\alpha,$$ \nIt is possible to show that ${\\bf d}\\_{\\rm WI}$ is a single-valued, gauge-invariant quantity; this number contains all the non-trivial electronic contributions to the polarization that are due to the deformation of the ionic orbitals in the crystalline environment. The gauge invariance of ${\\bf d}\\_{\\rm WI}$ might be surprising at first sight, as the individual dipole moments ${\\bf d}\\_\\alpha$ are manifestly gauge-dependent, i.e. they depend on the specific algorithm used to localize the Wannier functions. This arbitrariness cancels out when all ${\\bf d}\\_\\alpha$ are summed up, as long as the assignment of each Wannier function to a specific lattice site remains unambiguous. This is equivalent to stating that a given choice of ${\\bf R}\\_\\alpha$ uniquely determines the branch choice of the electronic polarization, which is a reasonable assumption in ionic materials where the nature of the valence wavefunction has typically a marked atomic character.\nWith this new decomposition of ${\\bf d}$, we have overcome an important drawback of Eq. ([eq:wannier]) and Eq. ([eq:berry]): in the latter two equations the decomposition of ${\\bf P}\\_{\\rm bulk}$ into ionic and electronic contributions is physically meaningless – only the sum of the two terms is well defined. Here both ${\\bf d}\\_{\\rm PC}$ and ${\\bf d}\\_{\\rm WI}$ are formally meaningful objects. All the indeterminacy in the definition of ${\\bf d}$ has been recast into the term ${\\bf d}\\_{\\rm PC}$, which has an intuitive interpretation as the dipole moment of a system of classical ions, each of them carrying its formal valence charge. The term which contains the quantum-mechanical information about the electronic polarization effects is now ${\\bf d}\\_{\\rm WI}$, which is now a single-valued quantity; this has also a simple physical interpretation as the total dipole moment of the electronic clouds of the WI.\nBefore closing this part, it is useful to point out a direct relationship between our formalism and the “layer polarizations” (LP), *p**l*, introduced in Refs. and . If one is interested in layered perovskites with stacking axis along the (001) direction, it is useful to consider a decomposition of the charge density of the ABO3 cell into individual AO and BO2 layers. In the framework of the present work this implies grouping together the WI that belong to a given oxide layer. In particular, the total charge density of each layer *l* can be then written as a sum of all WI densities that belong to layer *l*,\n \n$$\\rho\\_l({\\bf r}) = \\sum\\_{\\alpha \\in l} \\rho\\_{\\rm WI}^{(\\alpha)} ({\\bf r}-{\\bf R}\\_\\alpha).\n\\label{eq:rhol}$$ \nIn II-IV perovskites the layers are formally charge-neutral . $\\rho\\_l({\\bf r})$ then carries a well-defined dipole moment, which is directly related to the LP,\n \n$$p\\_l = \\int \\bar{\\rho}\\_l(z) z dz = \\frac{1}{S} \\sum\\_{\\alpha \\in l} \n({\\bf d}\\_\\alpha + {\\bf R}\\_\\alpha Q\\_\\alpha) \\cdot \\hat z.\n\\label{eq:pl}$$ \n(The bar indicates in-plane averaging, and the integral is carried out along the stacking axis; *S* is the cell cross-section.) We shall illustrate this layer-by-layer decomposition of the total charge density with practical examples in section [sec:dipole-free].\nCrystal termination as a bulk property\n--------------------------------------\nWe shall illustrate in this section that the multivaluedness of the term ${\\bf d}\\_{\\rm PC}$ can be formally related to the surface termination of a semi-infinite crystal. This fact is not new, and was rigorously established within the modern theory of polarization . Here we discuss the implications of the “interface theorem” for the electrostatics of polar surfaces.\nFollowing Goniakowski *et al.* , we define a *frozen bulk termination* as a surface that is obtained by piling up “bulk unit cells” without any further electronic or ionic relaxation. For the time being, we shall limit our discussion of surface polarity to this (somewhat unrealistic) type of surface, that we further specify hereafter; we shall make the link to more realistic surface models in the next section. In contrast with Goniakowski *et al.* , here we define our “bulk unit cell” as a charge density distribution that results from a superposition of bulk WI, as in Eq. ([eq:rhowi]). Then, we construct the charge density distribution of the semi-infinite surface system as a superposition of $\\rho\\_{\\rm cell}({\\bf r})$,\n \n$$\\rho({\\bf r}) = \\sum\\_{{\\bf R} \\cdot \\hat n \\leq 0} \\rho\\_{\\rm cell}({\\bf r-R}),\n\\label{eq:rhosurf}$$ \nwhere ${\\bf R} = n\\_1 {\\bf a}\\_1 + n\\_2 {\\bf a}\\_2 + n\\_3 {\\bf a}\\_3$ is a real-space translation vector, and again ${\\hat n}$ is the normal to the surface plane. This way of defining a frozen bulk termination has two crucial advantages: (i) The choice of using the compound WI as our “elementary particles” naturally ensures that every ion in the surface system will have exactly the same formal oxidation state as in the bulk. This is a central point in the definition of a polar surface – if we allowed for fractional orbital occupations no surface would be polar. (ii) The choice of cleaving the *Bravais* lattice, rather than the *crystal* lattice is particularly advantageous, as it naturally preserves the bulk stoichiometry everywhere in the system (if we allowed for stoichiometry changes of reconstructions again the notion of polar surface would be inconsistent).\nIt can be easily verified that several types of unreconstructed surfaces can be generated by using Eq. ([eq:rhosurf]), simply by changing the definition of $\\rho\\_{\\rm cell}({\\bf r})$. In particular, we have the freedom to construct $\\rho\\_{\\rm cell}({\\bf r})$ in many different ways, simply by shifting each WI in the basis by an arbitrary Bravais lattice vector $\\Delta {\\bf R}$. Thus, the choice of the basis vectors ${\\bf r}\\_i$ uniquely determines the surface structure, according to Eq. ([eq:rhowi]) and Eq. ([eq:rhosurf]). (Of course, different choices of ${\\bf r}\\_i$ can lead to the same termination; the “$\\{{\\bf r}\\_i\\} \\rightarrow $ termination” relationship is a many-to-one function.) On the other hand, we have shown in the previous section that the choice of ${\\bf r}\\_i$ uniquely determines the value of ${\\bf P}\\_{\\rm bulk}$, out of the infinite possibilities allowed precisely by the arbitrariness in the choice of the basis vectors. This formally establishes the relationship between ${\\bf P}\\_{\\rm bulk}$ and the termination of the lattice. By construction, the *physical* net charge that lies at the surface of a frozen bulk termination as defined above is simply $\\sigma\\_{\\rm surf} = {\\bf P}\\_{\\rm bulk} \\cdot \\hat n$, where ${\\bf P}\\_{\\rm bulk}$ is the dipole moment (per unit volume) of an appropriate bulk unit cell, i.e. one that tiles the semi-infinite solid according to Eq. ([eq:rhosurf]).\nWithin these assumptions, we define a frozen bulk termination polar if the bulk building block used to construct it has a net dipole perpendicular to the surface plane; we define it non-polar otherwise. The problem of determining whether a surface is polar or not is, therefore, reduced to the problem of calculating the dipole moment of a bulk unit cell made of WI. This, in turn, can be directly related to the result of a Berry-phase calculation in the bulk crystal, which can be routinely performed with most publicly available codes. In other words, the termination itself can be understood as a *bulk* property.\nFrozen and relaxed surfaces\n---------------------------\nIt might appear artificial to consider surfaces that are constructed by stacking electronic orbitals corresponding to bulk Wannier functions. At a real surface, electronic states always depart from their bulk counterparts because of the peculiar chemical and electrostatic environment produced by the truncation of the crystal. Furthermore, also the ionic lattice undergoes nontrivial structural relaxations in the surface layers, in response to the perturbation of the bonding network. A central point of our formalism is that both (electronic and ionic) surface relaxation effects are essentially irrelevant in the context of deciding whether a given surface is polar on non-polar. As a matter of fact, either type of relaxation only affects the surface dipole moment, and not the surface charge density. Thus, genuine surface properties (e.g. the alignment between the bulk bands and the vacuum levels, or the surface energy) certainly depend on these mechanisms, but the polarity (which depends only on the physical surface charge) won’t be affected. This formally establishes the surface polarity as a property that can be completely understood at the bulk level – note that the termination dependence can also be understood as a bulk property as specified in the previous section. Then, all mechanisms that alter the surface *charge* (either in the form of a local composition change or as a modification of the formal oxidation state of the surface ions) are unambiguously understood as external compensation effects, and enter the definition of $\\sigma\\_{\\rm ext}$.\nConstruction of arbitrary non-polar terminations\n------------------------------------------------\nSo far we have addressed the question of deciding whether a given surface, of a certain orientation and termination, is polar or non-polar. One could wonder now, for a given bulk compound, (i) whether non-polar terminations can be constructed at all; (ii) if yes along which surface plane orientation; finally, it would be helpful to (iii) identify candidate non-polar surface structures based only on bulk information. In this section we shall illustrate how this is done within the present definition of surface polarity.\nEssentially, the question (i) boils down to finding all possible values of ${\\bf P}\\_{\\rm bulk}$. This is, within the modern theory of polarization, a periodic lattice of points. The difference between two arbitrary values of ${\\bf P}$ is a multiple of a real-space primitive translation vector,\n \n$${\\bf P}'\\_{\\rm bulk} - {\\bf P}\\_{\\rm bulk} = \\frac{Q\\_0}{\\Omega} (i {\\bf a}\\_1 +\nj {\\bf a}\\_2 + k {\\bf a}\\_3).$$ \nHere *Q*0 = *n**e* is an integer *n* times the electron charge *e*. *n*, which determines the resolution of the ${\\bf P}\\_{\\rm bulk}$ mesh depends on the convention of how the Wannier functions and the ion cores are grouped together. In particular, the constraint adopted here of assigning each Wannier function to a specific ionic site generally restricts the lattice of possible values of ${\\bf P}\\_{\\rm bulk}$ to a subset of those allowed by Eq. ([eq:wannier]) and Eq. ([eq:berry]). This can be understood by observing that the new elementary building blocks of the lattice are the “compound objects” WI, rather than single electrons or ions.\nAnswering question (ii) consists in finding the intersections between the infinite lattice of ${\\bf P}\\_{\\rm bulk}$ values and a given surface plane, which is a straightforward geometrical problem.\nAnswering question (iii) then is easy by recalling the direct relationship between a given value of ${\\bf P}\\_{\\rm bulk}$ and the dipole moment of a well defined bulk unit. More specifically, once a value (or a subset of values) of ${\\bf P}\\_{\\rm bulk}$ is found for which ${\\bf P}\\_{\\rm bulk} \\cdot \\hat n = 0$, models of the non-polar surface can be readily built by stacking [using Eq. ([eq:rhosurf])] bulk unit cells that correspond to those same values of ${\\bf P}\\_{\\rm bulk}$. We shall present several practical examples of this strategy in Sec. [sec:results].\nRelationship to previous approaches\n-----------------------------------\n### Tasker model\n \nEq. ([eqd]) constitutes the rigorous link between Tasker’s model and the modern theory of polarization in periodic insulators. Within our formalism, the total excess charge at a frozen bulk termination can be exactly written as\n \n$$\\sigma\\_{\\rm surf} = {\\bf P}\\_{\\rm bulk} \\cdot \\hat n = \\frac{ (\\sum\\_\\alpha {\\bf R}\\_\\alpha Q\\_\\alpha + \n{\\bf d}\\_{\\rm WI}) \\cdot \\hat{n}}{\\Omega},\n\\label{eqtasker}$$ \nwhere the sum is extended over all atoms in the semi-infinite crystal. The only difference between Tasker’s model and Eq. [eqtasker] is the additional, purely electronic contribution ${\\bf d}\\_{\\rm WI}$, which comes from the polarization of the Wannier ions in the crystalline environment. This contribution vanishes in all solids that are characterized by a center of symmetry; in these materials the discussion of the surface polarity problem in terms of nominal charges is therefore rigorous and exact. Even in materials where ${\\bf d}\\_{\\rm WI} \\neq 0$, neglecting this term is usually not crucial to assessing the polar or non-polar nature of a given surface. However, considering the WI contribution is essential for a quantitative estimation of *σ* (which is the excess charge that needs to be compensated); this is especially true in ferroelectric materials, which generally have a large anomalous contribution to ${\\bf P}$. Thus, our formalism provides a formal justification to Tasker’s model, and completes it by introducing an additional well-defined electronic dipolar contribution, ${\\bf d}\\_{\\rm WI}$.\nIn addition to this, our strategy has important practical advantages. Tasker’s approach involves a direct calculation of Eq.([eqtasker]) by means of an infinite lattice sum, whose convergence is ensured by using Ewald summation techniques. This procedure might be cumbersome in practice, and it requires a specialized computer code to perform the calculation. Our strategy greatly simplifies the problem, by reducing it to the calculation of the dipole moment of a small set of point charges. This can be done with paper and pencil in few minutes for a surface of arbitrary orientation, provided that one knows ${\\bf d}\\_{\\rm WI}$. This vanishes in many cases of practical interest – whenever it doesn’t vanish, only a single bulk Berry-phase calculation is needed to evaluate this contribution. Moreover, our strategy allows one to easily answer a number of physical questions that were difficult to address within Tasker’s approach, e.g. those discussed in the previous section.\n### Other approaches\nIn order to fully appreciate the advantages of our formalism, it is useful to compare it, in a practical case, with the alternative notion of “dipole-free unit cell” proposed by Goniakowski *et al.* For illustrative purposes, we consider the (100) surfaces of two prototypical perovskite materials, LaAlO3 and SrTiO3, in their cubic high-symmetry phase. We shall use either the arguments developed in the previous sections, based solely on bulk properties of the respective materials, or the theory of Ref. , in order to assess the polar or non-polar character of these surfaces.\nIn Fig. [fig1] we plot, along the (001)-oriented *z* axis, the *x**y*-planar average of the total valence charge density. (The left panels refer to LaAlO3, the right ones to SrTiO3.) The upper panels show a possible decomposition of the electronic charge that leads to a dipole-free unit cell, which we construct as follows. First, we count the total charge of the ionic cores of the individual oxide layers. With the pseudopotentials used in this work, these are LaO(+17), AlO2(+15), SrO(+16) and TiO2(+24). Next, we decompose the total valence charge by cutting it with abrupt (001) planes located in the interstitial regions. The location of those planes is chosen as to (i) respect the inversion symmetry of the crystal, and to (ii) assign to each layer an electron density that exactly cancels the positive core charge of that layer. The resulting electron charge assigned to the AO layers is highlighted with a dark shading (light for the BO2 layers). By construction, the “unit cell” obtained by combining two adjacent layers (evidenced by the arrow and dashed lines in the figure) has zero dipole moment in both LaAlO3 and SrTiO3. Hence, this construction fails at detecting any fundamental difference between LaAlO3 and SrTiO3: both are predicted to have non-polar (001) surfaces. Of course, this prediction relies on a completely arbitrary partition of the total electronic charge density. There are many other ways to do it. For example, if one chooses a different location of the cut planes (e.g. at the mid-point distance between the atomic planes), or yet a more sophisticated prescription (e.g. based on the Bader analysis), one generally gets a non-vanishing layer charge in both LaAlO3 and SrTiO3. From this perspective, one would have to conclude that the (001) surfaces of both materials are polar. The main point that we want to stress here is that, if we base our analysis solely on the total electronic density [as we have done in Fig. [fig1](a-b)], (i) the choice between one partitioning scheme and the other is arbitrary; (ii) any statement about the surface polarity inferred from such a partitioning is ambiguous; and (iii) such an analysis cannot be linked in any ways to the bulk polarization of the material (the latter cannot be defined, even in principle, in terms of the total charge density of a periodic crystal).\nIn Fig. [fig1](c-d) we demonstrate how the Wannier-based decomposition of the valence density solves this problem. The thin solid lines show, as above, the ground-state electronic charge densities *ρ̄*(*z*) (again, the bar symbol on *ρ* indicates that an in-plane averaging was performed). The total Wannier densities of each layer, defined as *ρ̄**l* in Eq. ([eq:rhol]), are shown as thick lines (solid for the AO layers, dashed for the BO2 layers). Note that we show the electron density as positive, and we omit the bare pseudopotential charges from the plots. (These are a lattice of Dirac delta functions, centered at the oxide layer locations.) As the Wannier functions are discrete objects, the total electronic charges are integer numbers. Most importantly, the Wannier functions carry some crucial information (that is absent in the total valence density) on how the localized bound charges are organized in the insulating state of each compound. It turns out that (summing up the contributions from the cores) the LaO and AlO2 layers have a total charge of +1 and -1, respectively, while the SrO and TiO2 layers result charge-neutral, in perfect agreement with the naive assumption of perfect ionicity. Thus, the Wannier decomposition correctly identifies LaAlO3(001) as polar and SrTiO3 as non-polar, in agreement with Tasker’s classification.\nTo corroborate our arguments, a further consideration is in order. One could be tempted to criticize our reasoning by observing that Wannier functions are by no means uniquely defined starting from a given set of Bloch orbitals. In Fig. [fig1](c-d) we have chosen a maximally localized representation, but there is nothing really fundamental behind this choice. Is the identification of a surface as polar or non-polar robust against this arbitrariness? The answer is yes. The formal proof of this statement was derived in 1993 , several years before the maximally localized Wannier functions were first introduced. Our choice of a maximally localized representation is motivated by its intuitive relationship to elementary chemical concepts (e.g. formal valence), but the reader should keep in mind that this is just a convenient way of expressing a concept that has solid mathematical grounds. In particular, the exact value of the surface charge can be computed at the bulk level, regardless of the degree of ionicity of the material. \nApplication to perovskite surfaces\n==================================\nLaAlO3 and SrTiO3: bulk properties\n----------------------------------\nWe now use two prototypical perovskite materials, LaAlO3 and SrTiO3, to illustrate our strategy in practice. This choice of materials is motivated by the recent discovery of a conducting electron gas at their polar (100) interface. This has generated a lively excitement in the research community and a renewed interest in the theoretical foundations of the surface/interface polarity problem. \nWe shall address questions (i-iii) raised in Sec. [sec:questions]. For simplicity, in both materials we consider only surfaces of the type (0ij). This means that only the projection of ${\\bf P}\\_{\\rm bulk}$ on the *y**z* plane is relevant, and our procedure can be conveniently represented on 2D graphs. Our strategy is general, and this choice was made only to simplify the notation and the graphical representations. We also consider both bulk compounds within their high symmetry cubic phase. (Both materials are characterized by zone-boundary distortions, related to rotations and tilts of the oxygen octahedral network; however, as these distortions are non-polar in nature, they are irrelevant for the present discussion.)\nRelationship between ${\\bf P}\\_{\\rm bulk}$ and the dipole moment of the bulk unit cell. Red balls represent O ions, with charge $Q\\_{\\rm O} = -2e$. Gold balls represent the A-site cation, either Sr ($Q\\_{\\rm Sr} = +2e$) or La ($Q\\_{\\rm La} = +2e$); green balls are the B-site cation, either Ti ($Q\\_{\\rm Ti} = +4e$) or Al ($Q\\_{\\rm Al} = +3e$). The sketches (a-d) represent the projection of the atomic positions onto a (100)-oriented plane. On the top and the bottom are shown the values of ${\\bf P}\\_{\\rm bulk}$ (in units of *e*/*a*02, where *a*0 is the lattice parameter in either material) resulting in LaAlO3 and SrTiO3, respectively, from each cell arrangement. [fig:dipoles]\nLattices of allowed values of ${\\bf P}\\_{\\rm bulk}$ in LaAlO3 (left) and SrTiO3 (right). In the left panel we show four possible surface plane orientations. The polar orientations (dashed red lines) have no intersections with the ${\\bf P}\\_{\\rm bulk}$ lattice. The contrary is true for the non-polar orientation (solid black lines). [fig:platt]\nTo start with (question i), we need to find the lattice of “allowed” values of ${\\bf P}\\_{\\rm bulk}$ in either material. Recalling that both compounds are characterized by a center of symmetry, it follows that ${\\bf d}\\_{\\rm WI}=0$, and ${\\bf P}\\_{\\rm bulk}$ is exactly determined by the formal valence charges of the participating ions, all sitting in their high-symmetry lattice sites. Now, the formal ionic charges are La(+3), Al(+3) in LaAlO3, Sr(+2), Ti(+4) in SrTiO3; oxygens in either compound have a formal charge of (-2). In Fig. [fig:dipoles] we show how different choices of the crystal basis of five atoms lead to different dipole moments per unit cell, and hence to a different formal polarization. If we could take all (infinite) combinations, we would obtain an infinite lattice of points, which is isomorphic with the real-space Bravais lattice of the cubic crystal. The 2D projection of the lattice of ${\\bf P}\\_{\\rm bulk}$ in either compound is shown in Fig. [fig:platt]. Even if the two compounds are isostructural (recall that we consider both LAO and STO in their cubic phase), there are two important differences in their formal polarization lattice. First, ${\\bf P}\\_{\\rm STO}$ is centered in the origin, while ${\\bf P}\\_{\\rm LAO}$ is centered in (1/2, 1/2, 1/2). Second, ${\\bf P}\\_{\\rm STO}$ has a coarser mesh than ${\\bf P}\\_{\\rm LAO}$ – the spacings are doubled because the constituent point charges are all even in the former. Note that both ${\\bf P}\\_{\\rm bulk}$ lattices are centrosymmetric, consistent with the absence of a spontaneous polarization in either material. \nTo answer question (ii), we need to find all possible intersections between a surface plane and the allowed values of ${\\bf P}\\_{\\rm bulk}$; the projection of a few representative surface planes are plotted in Fig. [fig:platt](a). As it can be readily appreciated from the diagram, the aforementioned qualitative differences between the respective ${\\bf P}\\_{\\rm bulk}$ lattices of STO and LAO have important consequences on the electrostatics of the surfaces. In particular, in STO the origin belongs to the allowed values of ${\\bf P}\\_{\\rm STO}$, and any surface plane intersects the origin by construction; therefore, a non-polar surface of any possible orientation can be readily constructed [we shall illustrate the case of STO(111) in Sec. [sec:sto111]]. Conversely, in LAO only specific plane orientations intersect the ${\\bf P}\\_{\\rm LAO}$ lattice [note that the (100) orientation is correctly classified as polar]. In the following Section we shall consider a subset of these (infinite) possibilities, i.e. the vicinal (01*n*) surfaces, where *n* is an arbitrary odd integer number. We shall focus on the lowest-index cases with *n* = 1, 3, 5.\nVicinal LaAlO3 surfaces\n-----------------------\n \n \nWe first construct preliminary models for the (01*n*) surfaces with *n* = 1, 3, 5. These are obtained by using slab geometries, within the supercell method. First we choose a unit cell with the appropriate translational periodicity in plane, and enough room along the out-of-plane direction to accommodate both the slab and a vacuum region (slab and vacuum thicknesses are treated as convergence parameters). Second, we tile the slab region with repeated copies of a well-defined primitive basis of atoms, which is chosen in a such a way that its dipole moment lies exactly parallel to the surface plane. (This implies that the choice of the basis depends on the surface orientation.) This procedure leads to the slab models sketched in Fig. [fig:models].\nc|ccc Surface type & \n& (011) & (013) & (015) \nA & 1.93 & 2.23 & 2.15 \nB & 1.93 & 1.85 & 1.83 \nCleavage & 3.86 & 4.08 & 3.98\nThe first observation is that all these surface models [except maybe the (011) case] present alternating LaO-type and AlO2-type terraces, and these terraces tend to grow wider and wider for increasing *n*. Note that [again, with the only exception of the (011) case], the construction described above produces, in fact, *two* inequivalent surface structures for each orientation. In other words, the models of Fig. [fig:models] do not enjoy inversion symmetry. We shall refer to these two surfaces as “type A” and “type B”, where type A presents LaO-type step edges and type B has AlO-type edges. Remarkably, it is easy to realize that one can change from A-type to B-type simply by displacing an oxygen atom from one step edge to the neighboring one. This way, starting from the “mixed” AB-type slabs Fig. [fig:models] one can readily construct pure AA or BB slabs. One can verify that the resulting AA and BB models do enjoy inversion symmetry. Since going from A to B preserves the bulk stoichiometry, this allows for a rigorous definition of the surface energy for all individual surface structures.\nIn practice, in the simulations we use a slab thickness of approximately 4-5 LAO cells in each case, which is more than sufficient to obtain a well-converged value of the surface energy. The surface energy is defined as\n \n$$E\\_{\\rm surf} = \\frac{1}{2S} (E\\_{\\rm slab} - N E\\_{\\rm bulk}),$$ \nwhere $E\\_{\\rm slab}$ and $E\\_{\\rm bulk}$ are the relaxed total energy of the slab supercell and of LAO bulk, *N* is the total number of LAO units in the slab model, and $S=a\\_0^2\\sqrt{1+n^2} $ is the surface area in each case (*a*0 is the lattice parameter of cubic LAO; the factor of 2 takes into account the fact that a slab has two surfaces). In Table [tab1] we report the results. Comparing these values with previous literature studies is difficult, as studies of vicinal perovskite surfaces are scarce. Only the lowest-index (011) surface type has been investigated to some extent, although we weren’t able to find data specific to LAO. Concerning other perovskite materials, Eglitis and Vanderbilt reported an energy of 1.52 J/m2 for an isostructural O-terminated SrTiO3(011) surface structure. The value we obtain for LAO, 1.93 J/m2, is somewhat larger but otherwise of comparable magnitude. Note that in the study of Ref. a different (hybrid) functional was used – LDA might well overestimate surface energy values due to the well-known overbinding issues.\nIt is interesting to note that in the case of B-type surfaces the energy decreases slightly for increasing index *n*. We ascribe this behavior to the lower steps-to-terraces ratio in the (013) and (015) surfaces (undercoordinated step sites are likely to be less favorable). We consider it unlikely, however, that this energy be further reduced for *n* > 5. Increasing *n* would lead to larger and larger terraces that are locally charged [either of the LaO(+) or AlO2(-) type], and the electrostatic cost (roughly linear in *n*) would eventually dominate over the step energy (proportional to 1/*n*) in a way that bears many analogies to Kittel’s theory of domain walls. Still, the increased stability of the vicinal (013) and (015) surfaces [compared to the (011) orientation] suggests that these geometries could be, in principle, fabricated under appropriate experimental conditions. The simultaneous presence AO and BO2 domains appears promising for applications, e.g. in selective self-assembly of functional nanostructures, as it was recently shown in the case of SrTiO3 \nTotal density of states of the LAO(013) slab models. Surface A (black curve), B (red dashed) and bulk (shaded area) are shown. [fig:ldos] \nThe above considerations on the energetics haven’t answered an important question yet: how can we verify that these surfaces are indeed non-polar, consistent with our predictions? A useful indication comes from the density of states. If a surface is polar, then there is a need for compensation via additional charge carriers (either electron or holes) that deplete or populate the energy bands of the crystal in proximity of the problematic termination. This typically results in a metallic surface. Conversely, if the surface is non-polar, the bulk-derived Wannier functions alone are sufficient to ensure electrostatic stability, and therefore the system can remain *insulating*. In Fig. [fig:ldos] we show the total density of states extracted from a (013) (A- or B-type) slab model, compared with the bulk LAO density of states. In all cases there is a wide gap separating the unoccupied from the occupied states. This fact, together with the inversion symmetry and perfect bulk stoichiometry of the slabs, directly demonstrates that the surfaces are non-polar, and that every atom contributes with a total number of electrons that exactly corresponds to its formal ionic valence. Similar considerations apply to the (110) and (510) surface models (not shown).\nIt is important to stress that, contrary to a common misconception, all these surfaces are perfectly *stoichiometric* by construction, and they are *non-reconstructed* as they have the highest possible translational symmetry that is allowed by each plane orientation. It is often assumed that the only “legitimate” structures that can be named frozen bulk terminations are those that are obtained upon cleavage of the *crystal* lattice, i.e. preserving the integrity of the bulk-like atomic planes. This is, however, just a convention that has nothing fundamental to it. We believe it is more practical to truncate the *Bravais* lattice instead. This automatically preserves stoichiometry and translational symmetry, and dramatically simplifies the description of surface electrostatics.\nAs a final remark, it is fairly easy to realize that all the (01*n*) surface models presented in this section are non-polar for *any* non-ferroelectric perovskite material (or for a ferroelectric one in its high-temperature symmetric phase). This can be simply understood by observing that, by replacing the cations in each bulk primitive basis (see Fig. [fig:models]) with those of a different charge family (i.e. I-V or II-IV), the dipole moment changes its magnitude but not its direction. Also, all the surface models enjoy inversion symmetry and have ideal bulk stoichiometry – the absence of a dipole moment normal to the surface plane is therefore automatically guaranteed. Therefore, many of the considerations made here in relationship to the specific LAO case are actually completely general, and apply to all cubic perovskite compounds.\nNon-polar SrTiO3(111) surfaces\n------------------------------\nTo further illustrate our arguments, we move now to the case of SrTiO3. According to our definition of polar surface, and by observing that the simplest choice of the SrTiO3 bulk unit has zero dipole moment, one would conclude that in SrTiO3 *any* surface orientation is “non-polar”. To illustrate this point we shall consider here the (111) surface, which has been classified as polar by most authors. Note that this surface is indeed polar if one insists on terminating the crystal lattice with either a Ti or a SrO3 layer. Our prescription of cleaving the Bravais lattice and tiling it with well-defined bulk-like formula units is less restrictive, and allows for non-polar terminations as we shall see in the following.\n Primitive bulk SrTiO3 unit cells for the model A (left) and B (right) (111) slabs. Sr atoms (large green circle) lie at the corners of the cube; O atoms (small red circles) lie at the face-centered sites; Ti atoms lie at the center of the cube. Both choices of the primitive unit have zero dipole moment along any direction. [fig111cell] \nWe build two inequivalent stoichiometric slab models (that we call A and B) by stacking the primitive building blocks schematically shown in Fig. [fig111cell]. It is easy to verify that both choices of the primitive cell have zero dipole moment. (Again, to compute the dipole moment we use the *formal* charges. This is substantiated by the Wannier-based decomposition described in section [sec:theory], which provides the formal link to the theory of polarization in bulk solids. ) The primitive translation vectors of the supercell are (in units of the bulk equilibrium lattice parameter *a*0 = 7.275 a.u.) ${\\bf a}\\_1 = (\\sqrt{1/2},\\sqrt{3/2},0)$, ${\\bf a}\\_2 = (\\sqrt{1/2},-\\sqrt{3/2},0)$ and ${\\bf a}\\_3 = (0,0,10)$; the out-of-plane spacing ${\\bf a}\\_3$ was chosen in order to include a sufficiently thick vacuum region separating the repeated images of the 10-layer slabs. As the slabs do not enjoy inversion symmetry (there are a total of four inequivalent surfaces in our simulations), we apply a dipole correction in the vacuum layer to avoid unphysical macroscopic fields in the bulk region of the SrTiO3 films. We use a regular (8 × 8 × 1) Γ-centered *k*-point mesh to sample the surface Brillouin zone, and we fully relax our structures within the symmetry constraints allowed by the surface composition. Note that the A-type slab preserves the point group of the bulk (111) orientation, while the B-type slab has a lower symmetry due to the presence of an incomplete oxygen plane on one side.\n \nIn Fig. [fig:stoslab] we show the relaxed structures of the two slab models described above (the primitive unit of the supercell was repeated three times in both in-plane directions to obtain a clearer view of the structure). Henceforth we shall indicate A1, A2, B1 and B2 the four inequivalent surfaces, where A and B refer to the specific slab model, and 1/2 refer to the top/bottom surface, respectively. A1 has a TiO3-type termination (i.e. an ideal Ti-type surface where the Sr atom has been removed from the topmost SrO3 layer), and correspondingly A2 contains a Sr-type termination, where undercoordinated Sr atoms protrude from the underlying oxygen group. B1 has a supplementary O atom accommodated on top of an ideal Ti-terminated surface, and this atom forms a tetrahedron surrounding the topmost Ti atom. B2 has an O vacancy in the terminating SrO2 layer; model B can be therefore obtained from model A simply by displacing a neutral SrO unit from the 2 (bottom) to the 1 (top) surface. Most of these surfaces were already considered in Ref. , and indicated as “small unit-cell reconstructions” of SrTiO3(111). We note that surface reconstructions are typically associated with a reduction in the translational symmetry group, which is not the case for any of these models. Therefore, we rather regard these as primitive, stoichiometric bulk terminations. Whatever is the nomenclature, the authors of Ref. correctly recognized the formal charge neutrality of these “valence-compensated” terminations.\n \nSimilarly to the LaAlO3 case, we analyze the electronic properties of these surface models to verify their insulating character. We plot in Fig. [figldossto] the local density of states (LDOS) integrated on spheres of radius 3.0 bohr surrounding the Ti atoms. In the main panels we show the average Ti LDOS in the middle of the slab (gray shaded areas), which we take as our bulk-like SrTiO3 reference curve. We also show the LDOS corresponding to the outermost Ti atom at the top (red dashed curve) and bottom (solid blue curve) surfaces. At A1 the gap is smaller than in the bulk, as a narrow band of Ti-derived unoccupied orbitals splits from the conduction band. The band gap narrowing is rather extreme at A2, where a highly dispersive surface state makes the gap as small as 0.1 eV at the Γ point (presumably this free-electron-like state is originated from the *s* and *p* state of the protruding Sr ions). To better illustrate this, we show a blowup of the LDOS in the inset. Here we also plot (thin black curve) the LDOS of the outermost Sr atom, where the surface state has its maximum weight. The nearly flat DOS (the wiggles are caused by the finite *k* resolution) between 0 and 1.5-2 eV, typical of a parabolic band in 2D, is clear. The B slab presents, overall, an energy gap which is much closer to the bulk value; this suggests that the system is electronically more stable than in A. Both at B1 and B2 the gap reduction is caused by valence-band derived surface states; these are reminiscent of the states that are found at some BO2-terminated (100) perovskite surfaces. Conversely, no conduction band-derived states are present. Especially interesting are the sharp peaks appearing at B1; these are derived from the atomic-like orbitals of the outermost O atom. To illustrate this point, we plot in the inset the LDOS on the terminating O, which lies at the vertex of the surface tetrahedron surrounding Ti; for comparison, we also show the LDOS of a bulk-like oxygen far from the surfaces. The 2*s*- and the 2*p*- derived features of the surface O appear extremely sharp and atomic-like, in contrast with the substantial broadening in the SrTiO3 bulk caused by band dispersion. A tetrahedral coordination might look unusual for Ti, which tends to adopt octahedral coordination in most (if not all) stable bulk oxide phases. Nevertheless, at the SrTiO3 surface, analogous tetrahedral units were recently shown both experimentally and theoretically to be energetically favorable, even in the case of the (110) orientation where there exist alternative (1 × 1) structures with relatively low energy. \nFinally, we shall comment on the relative energy of these structures. Unlike the (*n*10) models discussed in the previous section, here it is not possible to construct a stoichiometric *and* symmetric slab; therefore, we can only calculate a *cleavage* energy for A and B, $E\\_{\\rm cl}$. Resolving this value into the contributions of the top and bottom terminations would require further considerations about the chemical potential of Sr, Ti and O; this goes beyond the scopes of the present work. We find $E\\_{\\rm cl}$(A)= 6.27 J/m2 and $E\\_{\\rm cl}$(B)= 3.94 J/m2. These values are both larger than the previously reported cleavage energies along the (100) or (110) directions. Especially the A model has a high energy cost, consistent with the “open” nature of the low-coordinated surface sites, and with the relatively unfavorable electronic configuration discussed in the previous section. Model B, on the other hand, has a cleavage energy that is significantly smaller, and (on average) reasonably close to typical (110) surface energies. It should be kept in mind that the cleavage energy might be unequally distributed between the B1 and B2 surfaces – we cannot exclude that one of the two might be quite stable in a wide range of thermodynamic conditions. $E\\_{\\rm cl}$(B) can be directly compared to the values reported in Ref. , where it appears to correspond to the sum of the surface energies of model 3 and 4. The authors of Ref. did not use LDA but a variety of different density functionals, with values ranging from 4.94 eV (PBE) to 6.41 eV (TPSSh) per surface cell. Our LDA value of 6.31 eV per surface cell compares favorably with the highest value reported there, consistent with the systematic tendency of LDA towards overbinding.\nIt is interesting to compare our calculated $E\\_{\\rm cl}$(A)=10.0 eV/cell to the energy associated with the “textbook” cleavage, i.e. that leaving atomically flat, metallic and polar Ti / SrO3 terminations. Assuming that our LDA values are comparable to the TPSSh results of Marks *et al.*, we can infer the Ti / SrO3 cleavage energy by summing up the TPSSh surface energies of model 1 and 2 in the aforementioned work; this yields a value of 12.3 eV / cell. As surprising as it may sound, our non-polar cleavage model A, with the severely undercoordinated Sr atoms protruding from surface A2, is still about 2 eV /cell lower in energy than the atomically flat cleavage model. This fact highlights the importance of achieving electrostatic stability from bulk-like building blocks, without invoking external compensation mechanisms (such as hole or electron doping as in the case of Ti / SrO3).\nPolarity compensation of LaAlO3(100)\n------------------------------------\n (a) and (c): Total DOS of the (100) LaAlO3 slabs, highlighting the population of the states involved in the compensation of the surface polarity (shaded area, indicated with an arrow). (b) and (d): Plane-averaged density of compensating charge, related to the shaded portion of the DOS in (a) and (c). Top (a-b) and bottom (c-d) panels refer to the LaO- and AlO2-terminated slabs, respectively. [figmetal] \nSo far we restricted our analysis to ideal non-polar surfaces, i.e. systems where only bulk-like building blocks are present. To complete our discussion we now consider a prototypical polar surface, LaAlO3(100), and illustrate how our arguments apply to the analysis of selected compensation mechanisms, where we introduce extrinsic sources of compensating charge $\\sigma\\_{\\rm ext}$.\n### Via metallic carriers\nFirst, we consider the clean (1 × 1) LaO- and AlO2-terminated (100) surfaces. Due to the built-in dipole of the bulk unit cell that we must use to construct these terminations, there is an excess charge of + 0.5*e* and − 0.5*e* per surface unit cell, respectively. If we don’t relax the translational symmetry, the only possible compensation comes from metallic carriers, either in the form of conduction band electrons or valence holes. By performing two separate calculations of symmetrically terminated 6.5-unit cell thick slabs, we indeed obtain metallic surfaces. In Fig. [figmetal] we plot the total density of states for both slabs, where the Fermi level clearly crosses either the valence band (AlO2 termination) or the conduction band (LaO termination).\nAn interesting feature of the DOS of Fig. [figmetal](a-c) is that, in both cases, a clear gap persists in the spectrum. This means that, in spite of the partial metallization, the conduction and valence bands preserve their respective identities. This observation implies that we can rigorously separate what we consider “bound charges” (which are all of bulk origin here, as we don’t introduce extrinsic species in the supercell) from “external compensating charge”, following the prescriptions of Ref. and . The former, which we take as the total charge density of the (completely filled) valence-band manifold, are implicitly included in the definition of ${\\bf P}\\_{\\rm bulk}$; the latter can be either a positive external density of valence-band holes (ext,h) or a negative density of conduction-band electrons (ext,e),\n \n$$\\begin{aligned}\n\\rho\\_{\\rm ext,e}(z) & = & \\int\\_{E\\_{\\rm mid-gap}}^{E\\_{\\rm F}} \\tilde{\\rho}(E,z) dE \\\\\n\\label{eqrhoe}\n\\rho\\_{\\rm ext,h}(z) & = & -\\int\\_{E\\_{\\rm F}}^{E\\_{\\rm mid-gap}} \\tilde{\\rho}(E,z) dE.\n\\label{eqrhoh}\\end{aligned}$$ \nHere *ρ̃*(*E*, *z*) is the planar-averaged and energy-smeared local density of states defined in Ref. , and $E\\_{\\rm F}$ is the Fermi level. The electronic states that contribute to the integrated charge densities $\\rho\\_{\\rm ext,e}$ and $\\rho\\_{\\rm ext,h}$ are evidenced as shaded areas in the DOS plot of Fig. [figmetal]. (Note that the DOS is the volume integral of *ρ̃*(*E*, *z*).) In Fig. [figmetal](b-d) we plot the compensating surface densities $\\rho\\_{\\rm ext,e}$ and $\\rho\\_{\\rm ext,h}$. Both appear localized to the surface region, although they display a relatively slow decay into bulk LaAlO3, and amount (within machine precision) to a total of exactly plus or minus half an electron per side. This demonstrates the full consistency [in the sense of Eq. ([eq:neutral])] between the “external charge” defined in Eq. ([eqrhoe]) and ([eqrhoh]), and the prediction of excess bound charge coming from the analysis of ${\\bf P}\\_{\\rm bulk}$. Note that, in the case of the LaO-terminated slab, part of the charge spills out into the vacuum region. This is a consequence of the vacuum level being very close to the conduction band edge, which is populated by the compensating electrons. Conversely, only O(2*p*)-derived states contribute to $\\rho\\_{\\rm ext,h}$. By combining the total energies of the reference slabs and subtracting an appropriate number of bulk reference units, we obtain a relaxed cleavage energy of 4.53 eV per surface cell (5.13 J/m2). This is larger than the cleavage energies we computed in Sec. [sec:vicinal] for the primitive non-polar (*n*10) surface models.\n### Via external bound charges\n Relaxed structure of the LaAlO3(100) slab with LaO (top) and AlO2 (bottom) terminations, compensated with OH(-) and H(+) groups respectively. [figh2o] \nWe shall now consider a different compensation mechanism, where instead of metallic carriers the surface acquires bound charge via adsorption of external species. As a matter of fact, this is of concrete relevance for the interpretation of a vast number of physical phenomena. A real surface is always in contact with an atmosphere where various gas-phase species are present, and a number of exchange/adsorption/redox processes are usually thermodynamically accessible. On a more general basis, there are many situations where the surface layer differs, either compositionally and chemically, from the bulk of the crystal. Consider, for instance, the Sr-decorated Si(100) surface that is used to promote coherent epitaxial growth of SrTiO3. Given their technological and fundamental importance, it is useful here to provide some examples, without the pretention to be exhaustive, of how our arguments can be translated to address those situations.\nIn the specific case of LaAlO3, H atoms adsorbed at the surface of a film deposited on a SrTiO3 substrate were found to significantly alter the electrical boundary conditions, e.g. by reducing or enhancing the residual internal electric field in LaAlO3 and by influencing the free carrier concentration at the interface. Experimentally, a humid atmosphere was demonstrated to be necessary to stabilize conducting paths at the buried interface . In turn, these “writing” and “erasing” processes appear to be mediated by charged surface adsorbates . All in all, there is growing evidence that OH and H species are crucial to explain many outstanding phenomena experimentally observed at LaAlO3 surfaces and thin films. Hence the motivation for studying H2O-based compensation mechanisms, where the surface retains the insulating character of the bulk.\nAdding external species to a (1 × 1) surface won’t make it insulating, as the excess charge is half an electron per cell (a periodic array of external species can provide only integer multiples of *e*). Doubling the surface unit cell leads to exactly plus or minus one electron, that now allows for an insulating state. We compensate this excess charge with a split water molecule (H adsorbed on the “negative” AlO2 side, OH on the “positive” LaO side) per $\\sqrt{2} \\times \\sqrt{2}$ surface cell. We use a stoichiometric LaAlO3 slab with a thickness of four unit cells, a *c*(2 × 2) in-plane translational symmetry and we relax the structure without imposing any symmetry constraint. As usual, we use a vacuum dipole correction to ensure the correct cancellation of the macroscopic electric fields due to the asymmetry of the slab. At equilibrium, the structure appears as in Fig. [figh2o]. Note the tilted position of the H atoms on the AlO2 side, consistent with the geometry found in Ref. for the H-LaAlO3/SrTiO3 system. The OH groups on the LaO side lie in a bridge site between two surface La atoms, thus occupying a natural lattice site for O. (An analogous location of the OH group was found on the SrO-terminated SrTiO3 surface decorated with dissociated water .) The system has a large insulating gap, almost equal to the bulk value, suggesting that this configuration might be fairly stable. We can estimate the energetics by considering a “wet cleavage” experiment where two LaAlO3 (001) surface are created and at the same time one free H2O molecule is split between the two terminations,\n \n$$E\\_{\\rm cl} = E\\_{\\rm slab} - 8 E\\_{\\rm bulk} - E\\_{\\rm H\\_2O}.$$ \nHere $E\\_{\\rm slab}$ is the energy of the supercell described above with two LaAlO3 cells per surface unit and a thickness of four unit cells; $E\\_{\\rm bulk}$ is the bulk energy, calculated by including the antiferrodistortive tilt of the O octahedra, which now are allowed by symmetry; $E\\_{\\rm H\\_2O}$ is the energy of a free water molecule, calculated by using a cubic box of approximately 10 Å lateral size. The resulting cleavage energy per surface area is 2.50 J/m2, which is the lowest value calculated in this work. This result suggests that adsorption of OH and H groups, which are ubiquitous in most experimental setups, is a very likely candidate to stabilize the LaAlO3 surface polarity. A study of LaAlO3(100) compensation via point defects was also recently reported in Ref. .\nAs a final remark, note that bound compensating charges, unlike the metallic carriers mediating electronic compensation, come in discrete units of *e*. Therefore, in cases where bound-charge compensation occur, it is most appropriate to “count” the external charges per unit area, which must satisfy the relationship\n \n$${\\bf P}\\_{\\rm bulk} \\cdot \\hat{n} = -\\frac{Q}{S}.\n\\label{qs}$$ \nHere *Q* = *n**e* (with *n* integer) is the formal oxidation state of the external defect or adsorbate, and *S* is the surface area per defect. Note that there exist defects (e.g. transition metal cations) that are stable in several oxidation states; of course, the actual *Q* that occurs in the situation of interest must be used in Eq. ([qs]). In case of doubt, the Wannier-based analysis of Sec. [sec:theory] can be used to assess the formal oxidation state of a given defect.\nDiscussion\n==========\nHere we shall put our results in the context of the current state-of-the-art in the field, especially regarding the fundamental theoretical understanding of the electrostatic stability of insulator surfaces.\nInsulating nature of the surface\n--------------------------------\nAs we mentioned several times when discussing our applications, it is likely that a non-polar (in the sense specified in this work) primitive surface will have a well-defined surface band gap. Here we shall further specify this point, to prevent dangerous generalizations.\nIt is certainly true that a polar surface with all the atoms in their bulk oxidation state cannot exist. If we insist on keeping the local stoichiometry fixed, some of the atoms must change their valence in order to avoid a diverging electrostatic energy. In many cases this produces partially filled electronic bands and a metallic surface. It is not difficult to imagine cases, however, where the surface atoms may change their oxidation state while preserving a gap in the spectrum. This would happen, for example, whenever the excess/defect charge amounts to an integer number of electrons, and there are ions in the lattice that have multiple stable oxidation states, e.g. most transition metals. Oxygen might also, in principle, change its formal valence from -2 to -1 to compensate a net surface charge – such a mechanism, stabilized via the formation of a peroxo bond, was reported in the case of a SrTiO3 surface by Bottin *et al.* . Therefore, a polar surface does not necessarily lead to a metallic surface. There are several other compensation mechanisms available (often accompanied by a reduction in the translational periodicity) that leave the surface insulating even without changes in the stoichiometry. We stress that in this latter case, however, the formal oxidation state of some atoms *must* change.\nAlso the statement that non-polar surfaces are insulating is far from universal. Indeed, by replicating a sufficiently pathological choice for the bulk primitive unit, one might end up with a surface that has a very awkward bonding configuration. This might produce a dramatic departure from the bulk bonding environment, and in such cases it is well possible that one or more surface bands may close the band gap. An example of how this may happen is provided by our A2-type SrTiO3 surface, where the band gap is reduced to a tiny value of about 0.1 eV. Gap closure would also occur for our (*n*10) LaAlO3 surface models for a large enough *n*; at some point the electrostatic energy of the large LaO- and AlO2-type terraces would become too large and eventually the gap would close. Note that surface relaxation usually helps stabilizing the truncated bonding network; in cases where our arguments would predict an insulating and non-polar surface, it is not infrequent to observe that a sizeable band gap opens only after full atomic relaxation.\nIn conclusion, the relationship between electrostatic stability and insulating nature is certainly not a rigorous one. It is nonetheless a useful guideline, in the sense that if the surface bonding environment is not too pathological and the solid has a marked ionic character one usually expects a non-polar surface to be insulating. Note that the insulating/metallic nature of a formally non-polar termination is more an issue of chemistry than of electrostatics – it boils down to the chemical driving force for the atoms to preserve their bulk-like oxidation state. If the surface becomes metallic this will happen through a *local* rearrangement of the electron cloud; the total surface charge won’t change.\nCovalency arguments and “weak polarity”\n---------------------------------------\nThe (001) surfaces of II-IV perovskites such as SrTiO3, BaTiO3, etc. are classified as non-polar within our definitions, consistent with Tasker’s assumption of formal ionic valence. Goniakowski, Finocchi and Noguera challenged Tasker’s classification by invoking covalent bonding effects, which would produce a smearing of the electron cloud. This, in turn, would produce a distribution of the charge between the oxide layers that differs from their formal ionic charges. According to this interpretation, SrTiO3(001) was classified as “weakly polar”.\nThe disagreement between the two interpretations is rooted in the way the electronic charge density is partitioned into individual building blocks at the bulk level. Choosing Mulliken or Bader populations inevitably leads to a fractional charge per ion that is typically smaller than the (integer) formal oxidation state, and the individual SrO and TiO2 layers appear charged. While these ideas have certainly some merit, there are severe drawbacks as well. The most important one is that, by insisting on a partition of the electron cloud based exclusively on the total charge density $\\rho({\\bf r})$, one thwarts any further attempt at linking the discussion of surface electrostatics to the theory of polarization in bulk solids. Powerful and rigorous results of the modern theory of polarization, e.g. the “interface theorem”, are inconsistent with a description of surface polarity in terms of Mulliken, Bader or even Born effective charges. (The fact that in II-IV perovskites the individual AO and BO2 layer do not satisfy the acoustic sume rule separately is sometimes taken as a further argument in support of the weak polarity concept.) The theory of bulk polarization implies a *wavefunction*-based partition of $\\rho({\\bf r})$. The natural tool in that sense are the spatially localized Wannier function as we discussed in section [sec:theory]. Covalency effects are irrelevant in this context, except that they are implicitly accounted for through the location of the ground-state Wannier centers and their spatial spread. Using Wannier functions might appear unnatural and complicated at first sight, but eventually they really lead to drastic simplifications and to an intuitive physical picture. In particular, if one wants to recover the intuitive classical formula ${\\bf P} \\cdot \\hat{n} = \\sigma$, there is simply no other way around.\nWe believe that full consistency between the theory of bulk polarization and the theory of surface polarity is a must. Therefore, we caution against the use of concepts such as weak polarity or covalent charges as they are are inconsistent with the former.\nOther oxide surfaces\n--------------------\n### Ferroelectric perovskites\nAs the concept of surface polarity is intimately linked to the polarization of the bulk solid, ${\\bf P}\\_{\\rm bulk}$, it is particularly insightful to discuss cases where ${\\bf P}\\_{\\rm bulk}$ has a non-trivial behavior, as in ferroelectric perovskite materials. Consider a (100)-oriented slab of BaTiO3, with the spontaneous polarization vector, ${\\bf P}\\_{\\rm S}$ oriented along the normal to the surface. Imagine that we have a stoichiometric slab with ideal BaO and TiO2 terminations, and a monodomain state with perfect 1 × 1 periodicity; assume also that ${\\bf P}$ points towards the TiO2-type surface.\nBoth surfaces are polar, as ${\\bf P}\\_{\\rm bulk} \\cdot \\hat n = P\\_{\\rm S} \\neq 0$. Contrary to the LaAlO3 example, however, here ${\\bf P}\\_{\\rm bulk} \\cdot \\hat n$ is not a simple fraction (plus or minus one half) of the polarization quantum *e*/*S*. Here $P\\_{\\rm S} = pe/S$, where *p* is a real number of the order of 0.25-0.35 (depending on the in-plane strain imposed to the film). Therefore, it might be technically difficult in a calculation to construct a commensurate supercell where ${\\bf P}\\_{\\rm S}$ is accurately compensated by an appropriate coverage of charged adsorbates or defects (unless *p* happens by accident to be exactly equal to a rational number with a small denominator). A possible trick to circumvent this difficulty is using the so-called “virtual crystal approximation” (VCA) . Here a fractionally charged pseudopotential is introduced at the surface to reproduce the effect of a disordered array of defects with the appropriate coverage. This way, the surface can be made insulating and charge-neutral at a low computational cost; the price to pay is that the VCA does not lend itself easily to the calculation of surface-specific properties, e.g. the energetics of a given compensation mechanism.\nA second important example is that of I-V ferroelectric perovskites, e.g. KNbO3. Here the ferroelectric contribution to the polarization, $P\\_{\\rm S}$, adds up to the “compositional” built-in cell dipole, which is *P*0 = ± *e*/2*S* (as in LaAlO3, the layers are formally charged, although here AO layers are negative and BO2 are positive). Note that, if $|P\\_{\\rm S}|$ were (again, by accident) equal to half a quantum of polarization, both NbO2- and KO-terminated (100) surfaces would be non-polar (i.e. $P\\_{\\rm S}$ would cancel out *P*0), provided that the spontaneous polarization points in the correct direction. This would be away from the surface for the *p*-type NbO2 termination, and towards the surface for the KO termination.\nFinally, it is worthwile mentioning the case of BiFeO3. This material appears complicated at first sight, because of the tilted polarization axis [${\\bf P}\\_{\\rm S}$ is oriented along the (111) direction] and the compositional layer charges of ± 1 (both Fe and Bi are formally 3+ ions). However, within the formalism established in this work, predicting the excess surface charge density that will be present at a given ideal termination becomes trivially simple. For example, at the FeO2 (100) termination we have [just like in the case of the AlO2-terminated LaAlO3(100) surface] an excess built-in charge of − *e*/2*S* ∼ − 0.5 C/m2. To this value we need to add the projection of ${\\bf P}\\_{\\rm S}$ along the (100) axis, which amounts to approximately the same value. Therefore, if ${\\bf P}\\_{\\rm S}$ points towards the FeO2-type (100) termination, this surface will be in practice only very weakly charged or even neutral. It goes without saying that the composition of the surface “pins” the out-of-plane component of the polarization to a fixed value that cannot be switched (unless the surface composition itself is changed, see Ref. ).\n### ZnO\nThe use of ZnO in many technological areas, as well as the recent progress in fabricating tailored nanostructures and functional surfaces with this material, have generated a widespread interest in the fundamental properties of its polar (0001) surface. . Several possible compensation mechanisms involving, e.g. metallic free carriers , hydroxylation/protonation or stoichiometry changes have been proposed over the years. In spite of this activity, the question of exactly *how much* excess charge is present at the polar Zn- or O- terminated surfaces is still a source of confusion.\nFor instance, there is a common belief that, starting from an ideal unreconstructed termination, removal of 1/4 of the surface ions will lead to perfect compensation of the polarity. This would be true if ZnO crystallized in zincblende phase. However, bulk ZnO is wurtzite-type, which means that on top of the compositional (zincblende-like) dipole it has also a non-trivial spontaneous $P\\_{\\rm S}$. This $P\\_{\\rm S}$ is of course not switchable, unlike the ferroelectric materials discussed in the previous section, but it does need to be taken into account when computing the surface charge. First-principles calculations of $P\\_{\\rm S}$ have reported relatively small values (compared to a hypothetical zincblende reference structure) of $P\\_{\\rm S}$ in bulk ZnO, of the order of 0.02-0.07 C/m2 . This implies that the necessary correction to the zincblende-like excess charge of 0.5*e* per surface cell is of the order of 0.01-0.03 electrons. Even if this correction is not large, one should keep in mind that, in a hypothetical free-slab calculation of ZnO where 1/4 of the O (and Zn) surface ions have been removed, after full relaxation there will be a non-zero residual macroscopic electric field in the slab of approximately $\\mathcal{E}\\_{\\rm slab} = P\\_{\\rm S} / \n(\\epsilon\\_0 \\epsilon\\_{\\rm r})$. Here *ε*0 is the vacuum permittivity and $\\epsilon\\_{\\rm r}$ is the static dielectric constant of ZnO (including piezoelectric effects). In fact, this observation was used to calculate the spontaneous polarization of wurtzite BeO several years before the modern theory of polarization was developed. \nSemiconductor surfaces\n----------------------\nWhile our arguments apply most naturally to ionic materials, where the assignment of the localized Wannier charges to a given atom is unambiguous, with some care they can be easily adapted to covalently bonded insulators. The main difficulty is that in semiconductors (e.g. Si) the maximally-localized Wannier functions tend to occupy bond-centered sites, and are shared between two atoms – assigning a given Wannier function to either atom that participate to the bond is then entirely arbitrary. Nevertheless, one can usually establish a reasonable convention for partitioning the bulk solid into well-defined units. For instance, in Si one could assign four spin-up Wannier functions (their centers would form a tetrahedron around the nucleus) to one atom and four spin-down Wannier functions to the other atom in the basis. (It might appear somewhat artificial to use such a spin-split basis; however, for the present discussion, the information about the spin is irrelevant, only the charge density of the Wannier functions really matters.) Then, this decomposition yields a basis of two WIs that individually retain the full symmetry of the lattice, are charge-neutral and have zero dipole moment. Primitive Si surfaces are then predicted to be non-polar, but chemically they will be highly reactive because of the singly occupied “dangling bonds”; this picture is consistent with the widely accepted understanding of Si surfaces. It is easy to see that by saturating these bonds with H one always obtains a non-polar and chemically stable surface (H does not add a net charge density as it contributes one electron and one proton to each dangling orbital). Alternatively, one could supply one Sr atom every two dangling bonds; this stabilization mechanism is important for growth of perovskite oxide films on Si substrates. Interestingly, in the case of the Sr-decorated surface, further oxidation does not change the surface charge count, as additional O atoms achieve a closed-shell configuration by incorporating the electron pairs already present in the saturated dangling bonds. This is a system where oxygen adsorption does not change the surface charge, in striking contrast with typical ferroelectric surfaces. \nOf course, one could prefer to use other conventions, e.g. assign two doubly occupied Wannier functions to each Si atom. This way the Si(100) or (111) surfaces would be understood as “polar”, and they *are* indeed polar if one insists on counting electrons two-by-two (the dangling bonds would need to be either empty or saturated, without the necessary countercharge to balance the electrostatics). This means that the concept of polar surface becomes somewhat ill-defined if the solid has no ionic character whatsoever. Note that the formalism developed in Ref. , on which the present work heavily relies, provides always a rigorous means of calculating the surface charge from bulk properties, regardless of the (ionic or non-ionic) nature of the insulator, and independently of the convention that one uses to “assign” the bound electron charges to a given lattice site. A more extensive treatment of the covalent case can be found in Ref. , and was recently discussed also in Ref. .\nInterfaces\n----------\nIn this work we decided to focus on surfaces, which is a special case of interface between two materials (one of them is vacuum). Whenever the second material is another crystalline insulator, the same arguments apply, but the “electrostatic phase diagram” can be substantially richer. The simplest case is that of two materials that have the same crystal structure, and we assume coherent epitaxy, i.e. both semi-infinite regions have the same in-plane periodicity and the same crystallographic orientation of the atomic planes. However, there might be more complex cases – for example, the participating materials have different bulk structures, or they are not oriented along the same crystallographic direction. In any case, the electrostatics is always governed by the intuitive classical formula,\n \n$$({\\bf P}\\_2-{\\bf P}\\_1) \\cdot \\hat n = \\sigma\\_{\\rm ext}.$$ \nHere ${\\bf P}\\_{1,2}$ is the polarization in either material, calculated by choosing a certain basis for the primitive basis of atoms and Wannier functions; $\\sigma\\_{\\rm ext}$ is the “remainder” interface charge, that is left behind once one removes all the bulk-like primitive units on either side; $\\hat n$ is the normal to the surface plane. As in the case of surfaces, we define an interface non-polar if, for an ideal termination of both materials with the maximum allowed translational symmetry one has $\\sigma\\_{\\rm ext}=0$.\nConclusions\n===========\nIn summary, we have revisited the concept of polar surface within the context of the modern theory of bulk polarization. Our definition, which is consistent with the bound (and discrete) nature of electrons in the insulating state of matter, puts Tasker’s classification on a firmer theoretical grounds, and corroborates it at the microscopic level. We further complete Tasker’s formalism with an additional term, which comes from the polarization of the electron cloud in solids that spontaneously break space inversion symmetry. Our calculations of non-polar LaAlO3(*n*10) and SrTiO3(111) surfaces, and of compensation mechanisms at LaAlO3(100), demonstrate that our formalism provides a convenient way of describing the net surface charge in terms of bulk polarization and external sources (either “bound” or “free”). We have also illustrated some practical analysis tools that can be used to monitor the equilibrium distribution of compensating charge in a calculation. We hope that these techniques will be helpful for future first-principles studies, and more generally as a conceptual basis to rationalize the many interesting phenomena occurring at the surfaces of insulating materials.\nAcknowledgments\n===============\nI am indebted to D. Vanderbilt for a critical read of the manuscript and many illuminating discussions. This work was supported by DGI-Spain through Grants No. MAT2010-18113 and No. CSD2007-00041, and by the European Union through the project EC-FP7, Grant No. NMP3-SL-2009-228989 “OxIDes”. I thankfully acknowledge the computer resources, technical expertise and assistance provided by the Red Española de Supercomputación (RES) and by the Supercomputing Center of Galicia (CESGA).\n\nElectrostatic stability of insulating surfaces: Theory and applications\n=======================================================================\nWe analyze the electrostatic stability of insulating surfaces in the framework of the bulk modern theory of polarization. We show that heuristic arguments based on a fully ionic limit find formal justification at the microscopic level, even in solids where the bonding has a mixed ionic/covalent character. Based on these arguments, we propose simple criteria to construct arbitrary non-polar terminations of a given bulk crystal. We illustrate our ideas by performing model calculations of several LaAlO3 and SrTiO3 surfaces. We find, in the case of ideal LaAlO3 surfaces, that cleavage along a higher-index (*n*10) direction is energetically favorable compared to the polar (100) orientation. In the presence of external adsorbates or defects the picture can change dramatically, as we demonstrate in the case of H2O/LaAlO3(100).\nIntroduction\n============\nA uniformly charged plane separating two semi-infinite regions of space yields a divergent electrostatic energy; for this reason, *polar* surfaces or interfaces cannot exist. Yet, thanks to recent progress in epitaxial growth techniques, nominally polar terminations of insulating crystals are now routinely prepared and characterized within well-controlled experimental conditions. This is possible because, in practice, there are several mechanisms available for a polar surface to neutralize the problematic excess charge, and possibly become thermodynamically stable. These include adsorption of foreign gas-phase species, changes in the surface stoichiometry, ionic and/or electronic reconstructions, or local metallization via accumulation of intrinsic free carriers; each of the above can, in principle, prevent the “polar catastrophe” by restoring the correct charge balance at the surface.\nUnderstanding and controlling these compensation mechanisms is a subject of great importance for many areas of fundamental science and technology. For example, surface polarity is of great interest for catalysis, gas sensing and energy applications , as adsorption and/or redox of gas-phase species are known to be strongly influenced by the electrostatic environment . In the context of perovskite-structure thin films and heterostructures, control of surface charge/polarity is currently investigated as a route towards the development of novel field-effect devices (e.g. in the LaAlO3/SrTiO3 system ), or ferroelectric memories based on the tunneling electroresistance effect . Central to rationalizing all these phenomena is the intimate relationship between surface charge and bulk polarization in crystalline insulators, which was formally established by Vanderbilt and King-Smith in 1993. \nFrom the point of view of the theoretical analysis, it is crucial to establish an unambiguous criterion to classify a given surface as “polar-compensated” (i.e. an originally polar surface that was neutralized via one of the aforementioned mechanisms) or intrinsically non-polar. This is not just a matter of nomenclature, but has very concrete practical relevance: non-polar terminations generally tend to be more stable, as extrinsic (i.e. not originated from the primitive building blocks of the insulating bulk crystal) sources of compensating charge tend to have a high energy cost. Furthermore, if a given surface is polar, one needs to know precisely how much external charge is needed to neutralize it; this greatly facilitates the theoretical analysis by restricting the number of possible candidate structures. We shall see in the following that, while the energetics is a genuine surface property, an exact answer to the latter question can be given already at the bulk level. Many authors have already addressed this issue in the past – we shall briefly mention hereafter the approaches that are most directly relevant to our work.\nTasker modeled a given ionic crystal as a lattice of point charges, corresponding to the nominal valence of the ions. Based on an abrupt truncation of this lattice, a given surface is then classified as polar on non-polar, depending on the behavior of the electrostatic energy. In particular, in the former (polar) case, the bulk repeated unit cell carries a finite dipole moment; this produces a diverging electrostatic potential unless compensated by an equal and opposite external surface charge density. This model, despite its simplicity, turned out to be surprisingly effective, and was able to correctly predict, at least at the qualitative level, the polar or non-polar nature of the vast majority of insulating surfaces. However, at the quantitative level this model has clear limitations. Many oxides and semiconductors display a marked covalent character, and the bulk polarization departs significantly from the value that can be inferred from atomic positions and nominal valence charges. Hence the need for a more accurate treatment.\nTo address these issues, and adopt a more realistic description of the charge density of the solid, Goniakowski *et al.* proposed a different criterion for classifying surfaces as polar or non-polar. At the heart of the strategy of Ref. is the concept of “dipole-free” unit cell. Given a certain plane orientation, one can demonstrate that it is always possible to choose a dipole-free repeated unit along the normal to that plane; then, the remainder charge that is left at the surface (once the bulk units have been removed) determines the polar or non-polar character of the termination. This criterion, however, is not free from ambiguities: for the same surface termination there may exist more than one possible choice of the dipole-free bulk unit cell, which might lead to opposite conclusions about the polar or non-polar character of the surface (see section [sec:dipole-free] for a detailed discussion). Also, identifying the dipole-free unit cell might be cumbersome in the case of higher index surfaces, where the structural complexity of the larger cell could complicate this type of analysis. Finally, the intuitive appeal of Tasker’s model is apparently lost in the strategy of Ref. : one needs to look at the ground-state charge density (e.g. as provided by a first-principles calculation) before drawing a conclusion.\nThere are two further issues that are common to both methods. First, it is universally agreed that the surface polarity is a property of the actual lattice termination. This means that, for a given material and surface plane orientation, there might be polar and non-polar terminations, depending on the surface stoichiometry. However, there is no established recipe to unambiguously decide, given a compound crystal and a surface orientation, whether a stoichiometric 1 × 1 non-polar termination is allowed at all. Furthermore, it is not clear how to construct, in general, a non-polar candidate structure without relying on a heuristic counting of the layer charges. Second, it was correctly recognized by both Tasker and Goniakowski that the issue of surface polarity is directly related to the bulk polarization of the material. However, neither model traces a formal link to the modern theory of polarization in periodic insulators , where the macroscopic ${\\bf P}$ is a multivalued vector field, written in terms of the *phases* of the wavefunctions. Only the total charge density (modulus of the wavefunctions) is considered in the model of Ref. , while explicit electronic orbitals are not addressed by Tasker’s approach. Recent theoretical works have indeed highlighted the importance of the formal Berry-phase polarization in discussing polarity at surfaces and interfaces , but a general formulation of the problem, based on the formalism established in Ref. , is still missing.\nHere we show that a Wannier function representation together with the “interface theorem” of Ref. , provide a very natural framework for addressing the above issues. Wannier functions were already shown to be a very useful tool, in layered superlattices , for partitioning the polarization of a crystal into the contribution of individual charge-neutral units. Most importantly, Wannier functions are intimately linked to the modern theory of polarization in solids , and therefore appear to be the most appropriate ingredient to discuss the issue of surface polarity, where the basic question concerns the existence of a finite dipole moment perpendicular to the surface plane. We shall provide, based on this description, precise criteria to establish whether truncating a bulk crystal along a given crystallographic orientation can yield a non-polar surface. We shall demonstrate that answering this question involves only an analysis of the bulk, and that our scheme naturally leads to candidate structures that can be used as a starting point for the subsequent determination of the thermodynamic ground state. To demonstrate our arguments, we focus on the surfaces of LaAlO3 and SrTiO3, two prototypical perovskite materials that have been at the center of the attention in the past few years as their polar (100) interface exhibits numerous peculiar properties.\nThis work is organized as follows. In section [sec:theory] we introduce our definition of polar surface and its formal relationship to the theory of bulk polarization. We also establish a direct link to Tasker’s model and we compare it to the “dipole-free” cell approach. In section [sec:results] we apply this formalism to a variety of systems, including non-polar LaAlO3(*n*10) and SrTiO3(111) surfaces. We also discuss electronic/ionic compensation mechanisms of polar LaAlO3(100). In section [sec:discussion] we briefly address some related topics, including the case of ferroelectric surfaces, and possible extensions to covalent semiconductors. Finally, in section [sec:conclusions] we present a brief summary and the conclusions.\nTheory\n======\nDefinition of polar surface\n---------------------------\nIn full generality, for the surface of a crystalline insulator to be electrostatically stable, it must have a vanishing density of *physical* surface charge, $\\sigma\\_{\\rm surf}=0$. In order to introduce the notion of surface polarity, it is useful to separate $\\sigma\\_{\\rm surf}$ into two distinct contributions, and rewrite the stability condition as\n \n$$\\sigma\\_{\\rm ext} + {\\bf P}\\_{\\rm bulk} \\cdot {\\hat{n}} = 0.\n\\label{eq:neutral}$$ \nHere ${\\bf P}\\_{\\rm bulk}$ is the bulk polarization, *n̂* is the normal to the surface plane, and $\\sigma\\_{\\rm ext}$ is a surface density of “external” compensating charges, which encompasses all contributions that cannot conveniently be described as “bulk-like” in nature (we include the latter in ${\\bf P}\\_{\\rm bulk}$). $\\sigma\\_{\\rm ext}$ typically includes free charges (e.g. in the form of a confined electron gas) and/or bound charges (either in the form of surface adsorbates, vacancies, non-stoichiometric reconstructions, or non-isoelectronic substitutions).\nWe define a given surface as non-polar if the stability criterion Eq. ([eq:neutral]) can be satisfied in the absence of external charges $\\sigma\\_{\\rm ext}$, which implies\n \n$${\\bf P}\\_{\\rm bulk} \\cdot {\\hat{n}} = 0.\n\\label{eq:nonpolar}$$ \nThis equation, at first sight, looks inconsistent with the current understanding of the surface polarity problem. It is now widely accepted that the polar or non-polar attribute is a property of the *termination*, not only of the material and surface plane orientation, contrary to what Eq. ([eq:nonpolar]) seems to suggest. We shall see in the following section that the choice of the termination is only apparently absent from Eq. ([eq:nonpolar]). It is implicitly included through the intrinsically multivalued nature of ${\\bf P}\\_{\\rm bulk}$, which is a well-established aspect of the modern theory of polarization in bulk insulators. \nThe bulk polarization\n---------------------\n### As a Berry phase\nWe consider a crystalline insulator described by three primitive translation vectors ${\\bf a}\\_{1,...,3}$ and a basis of *N* atoms located at positions ${\\bf R}\\_\\alpha$, with *α* = 1, ..., *N*. The “formal” bulk polarization is usually defined as\n \n$${\\bf P}\\_{\\rm bulk} = \\frac{1}{\\Omega} \\Big( \\sum\\_{\\alpha=1}^N {\\bf R}\\_\\alpha Z\\_\\alpha -\n2e \\sum\\_{i=1}^3 \\frac{\\phi\\_{\\rm el}^{(i)} {\\bf a}\\_i}{2\\pi} \\Big).\n\\label{eq:berry}$$ \nHere *Z**α* is the charge of the ionic core *α*, *e* is the (positive) electron charge and $\\phi\\_{\\rm el}^{(i)}$ is the Berry phase along the reciprocal-space vector *i*; for simplicity we assume spin pairing, hence the factor of 2 in the electronic contribution.\nIt is important to note that ${\\bf P}\\_{\\rm bulk}$, as defined in Eq. ([eq:berry]) is only defined modulo a “quantum of polarization”; in other words, it is not a single-valued but a multi-valued function of the electronic and structural degrees of freedom. This indeterminacy concerns both the ionic and the electronic parts in Eq. ([eq:berry]). On one hand, one has the freedom to choose any of the periodically repeated images of each atomic specie, and thus change ${\\bf R}\\_\\alpha$ by an arbitrary translation vector of the type $\\Delta {\\bf R} = n\\_1 {\\bf a}\\_1 + n\\_2 {\\bf a}\\_2 + n\\_3 {\\bf a}\\_3$. On the other hand, $\\phi\\_{\\rm el}^{(i)}$ are phases of complex numbers, and therefore only defined modulo 2*π*.\nIn the following sections we shall use the equivalent formulation of ${\\bf P}\\_{\\rm bulk}$ in terms of Wannier functions to illustrate the relationship between the multivaluedness of ${\\bf P}\\_{\\rm bulk}$ and the termination of the crystal lattice.\n### From the Wannier functions\nWe shall explicitly assume, from now on, a single-particle picture, in terms of a Kohn-Sham set of orbitals, and a conventional (as opposed to topological) insulating state. Within these assumptions, it is possible to express the electronic ground state of the bulk solid in terms of a set of maximally-localized Wannier functions , which are exponentially localized in direct space . Based on this representation, Eq. ([eq:berry]) can be rewritten as\n \n$${\\bf P}\\_{\\rm bulk} = \\frac{1}{\\Omega} \\Big( \\sum\\_{\\alpha=1}^N {\\bf R}\\_\\alpha Z\\_\\alpha -\n2e \\sum\\_{j=1}^{N\\_{\\rm el}/2} \\langle {\\bf r} \\rangle\\_j \\Big),\n\\label{eq:wannier}$$ \nwhere $N\\_{\\rm el}$ is the total number of electrons in the primitive cell, and $\\langle {\\bf r} \\rangle\\_j$ is the location of the center of the *j*-th Wannier function.\nAlternatively, we can think in terms of a charge density distribution that consists in the basis of atomic point charges together with the Wannier densities,\n \n$$\\rho\\_{\\rm cell} ({\\bf r}) = \\sum\\_{\\alpha=1}^N Z\\_\\alpha \\delta({\\bf r}-{\\bf R}\\_\\alpha)\n- 2e \\sum\\_{j=1}^{N\\_{\\rm el}/2} |w\\_j({\\bf r})|^2,\n\\label{eq:rhocell}$$ \nwhere $w\\_j({\\bf r})$ is the *j*-th Wannier function of the primitive cell. By construction, the sum of all the periodic images of $\\rho\\_{\\rm cell}$ “tiles” the total charge density of the extended solid. Then, by combining Eq. ([eq:wannier]) and Eq. ([eq:rhocell]) one immediately obtains the intuitive connection to the Clausius-Mossotti formula,\n \n$${\\bf P}\\_{\\rm bulk} = \\frac{\\bf d}{\\Omega},$$ \nwhere ${\\bf d}$ is the dipole moment of $\\rho\\_{\\rm cell}$.\nThis formulation provides a transparent way to partition the total charge density into individual primitive units, whose dipole moment correctly yields the formal value of ${\\bf P}\\_{\\rm bulk}$. In doing so, the *phase* indeterminacy of the electronic contribution to the polarization discussed in the previous Section has been reduced to a *lattice* indeterminacy, in all respects analogous to that characterizing the ionic contribution. In other words, all the complications related to the quantum-mechanical nature of the electrons have been mapped into a system of classical point charges, where the atoms and the electrons are formally treated on the same footing. In the following section we discuss how this Wannier representation can be further partitioned into smaller units that retain the chemical information about the formal oxidation state of each ion, which is central to the notion of “polar surface”.\nFormal ionic charges\n--------------------\nThe location of the Wannier functions generally reflects the bonding properties of the material – in ionic solids they will cluster around the atoms, while in covalent materials they will tend to occupy the bond centers. We shall assume that the solid has at least a certain degree of ionic character, so it is possible to “assign” each Wannier function to a given atom without ambiguities; this is certainly true in most known oxide materials. (With some caution the ideas developed here can be conveniently adapted to any crystalline insulator; we shall briefly discuss the example of purely covalent semiconductors in Sec. [sec:discussion].) We then combine each ion *α* with the Wannier orbitals *j* that “belong” to it and define a set of compound charge distributions that we call “Wannier ions” (WI),\n \n$$\\rho\\_{\\rm WI}^{(\\alpha)} ({\\bf r}) = Z\\_\\alpha \\delta({\\bf r})\n- 2e \\sum\\_{j \\in \\alpha} |w\\_j({\\bf r + {\\bf R}\\_\\alpha})|^2 \\Big).\n\\label{eq:rhowi0}$$ \n(We operated a translation so that the nucleus sits in the origin.) As the Wannier function locations usually agree remarkably well with chemical intuition, each of these *N* charge distributions will carry a monopole *Q**α* corresponding to the “nominal” charge of the ion (e.g. − 2*e* for O, + 2*e* for Sr, + 4*e* for Ti). In addition to their net charge, the WI are non-spherical and generally carry non-zero dipole moments ${\\bf d}\\_\\alpha$. (Higher multipoles are also present, but are not directly relevant for the present discussion.) $\\rho\\_{\\rm cell}({\\bf r})$ can now be rewritten in terms of the WI densities,\n \n$$\\rho\\_{\\rm cell} ({\\bf r}) = \\sum\\_\\alpha \\rho\\_{\\rm WI}^{(\\alpha)} ({\\bf r}-{\\bf R}\\_\\alpha),\n\\label{eq:rhowi}$$ \nwhich is equivalent to Eq. ([eq:rhocell]) except that here we use precautions to keep the basic WI units intact. It follows that the dipole moment of $\\rho\\_{\\rm cell}({\\bf r})$ can be written in terms of two contributions,\n \n$${\\bf d} = {\\bf d}\\_{\\rm PC} + {\\bf d}\\_{\\rm WI}.\n\\label{eqd}$$ \nThe first term is the dipole moment of a system of point charges located at positions ${\\bf R}\\_\\alpha$,\n \n$${\\bf d}\\_{\\rm PC}=\\sum\\_\\alpha {\\bf R}\\_\\alpha Q\\_\\alpha.$$ \nThe second term is the sum of the individual dipole moments of the WI,\n \n$${\\bf d}\\_{\\rm WI}=\\sum\\_\\alpha {\\bf d}\\_\\alpha,$$ \nIt is possible to show that ${\\bf d}\\_{\\rm WI}$ is a single-valued, gauge-invariant quantity; this number contains all the non-trivial electronic contributions to the polarization that are due to the deformation of the ionic orbitals in the crystalline environment. The gauge invariance of ${\\bf d}\\_{\\rm WI}$ might be surprising at first sight, as the individual dipole moments ${\\bf d}\\_\\alpha$ are manifestly gauge-dependent, i.e. they depend on the specific algorithm used to localize the Wannier functions. This arbitrariness cancels out when all ${\\bf d}\\_\\alpha$ are summed up, as long as the assignment of each Wannier function to a specific lattice site remains unambiguous. This is equivalent to stating that a given choice of ${\\bf R}\\_\\alpha$ uniquely determines the branch choice of the electronic polarization, which is a reasonable assumption in ionic materials where the nature of the valence wavefunction has typically a marked atomic character.\nWith this new decomposition of ${\\bf d}$, we have overcome an important drawback of Eq. ([eq:wannier]) and Eq. ([eq:berry]): in the latter two equations the decomposition of ${\\bf P}\\_{\\rm bulk}$ into ionic and electronic contributions is physically meaningless – only the sum of the two terms is well defined. Here both ${\\bf d}\\_{\\rm PC}$ and ${\\bf d}\\_{\\rm WI}$ are formally meaningful objects. All the indeterminacy in the definition of ${\\bf d}$ has been recast into the term ${\\bf d}\\_{\\rm PC}$, which has an intuitive interpretation as the dipole moment of a system of classical ions, each of them carrying its formal valence charge. The term which contains the quantum-mechanical information about the electronic polarization effects is now ${\\bf d}\\_{\\rm WI}$, which is now a single-valued quantity; this has also a simple physical interpretation as the total dipole moment of the electronic clouds of the WI.\nBefore closing this part, it is useful to point out a direct relationship between our formalism and the “layer polarizations” (LP), *p**l*, introduced in Refs. and . If one is interested in layered perovskites with stacking axis along the (001) direction, it is useful to consider a decomposition of the charge density of the ABO3 cell into individual AO and BO2 layers. In the framework of the present work this implies grouping together the WI that belong to a given oxide layer. In particular, the total charge density of each layer *l* can be then written as a sum of all WI densities that belong to layer *l*,\n \n$$\\rho\\_l({\\bf r}) = \\sum\\_{\\alpha \\in l} \\rho\\_{\\rm WI}^{(\\alpha)} ({\\bf r}-{\\bf R}\\_\\alpha).\n\\label{eq:rhol}$$ \nIn II-IV perovskites the layers are formally charge-neutral . $\\rho\\_l({\\bf r})$ then carries a well-defined dipole moment, which is directly related to the LP,\n \n$$p\\_l = \\int \\bar{\\rho}\\_l(z) z dz = \\frac{1}{S} \\sum\\_{\\alpha \\in l} \n({\\bf d}\\_\\alpha + {\\bf R}\\_\\alpha Q\\_\\alpha) \\cdot \\hat z.\n\\label{eq:pl}$$ \n(The bar indicates in-plane averaging, and the integral is carried out along the stacking axis; *S* is the cell cross-section.) We shall illustrate this layer-by-layer decomposition of the total charge density with practical examples in section [sec:dipole-free].\nCrystal termination as a bulk property\n--------------------------------------\nWe shall illustrate in this section that the multivaluedness of the term ${\\bf d}\\_{\\rm PC}$ can be formally related to the surface termination of a semi-infinite crystal. This fact is not new, and was rigorously established within the modern theory of polarization . Here we discuss the implications of the “interface theorem” for the electrostatics of polar surfaces.\nFollowing Goniakowski *et al.* , we define a *frozen bulk termination* as a surface that is obtained by piling up “bulk unit cells” without any further electronic or ionic relaxation. For the time being, we shall limit our discussion of surface polarity to this (somewhat unrealistic) type of surface, that we further specify hereafter; we shall make the link to more realistic surface models in the next section. In contrast with Goniakowski *et al.* , here we define our “bulk unit cell” as a charge density distribution that results from a superposition of bulk WI, as in Eq. ([eq:rhowi]). Then, we construct the charge density distribution of the semi-infinite surface system as a superposition of $\\rho\\_{\\rm cell}({\\bf r})$,\n \n$$\\rho({\\bf r}) = \\sum\\_{{\\bf R} \\cdot \\hat n \\leq 0} \\rho\\_{\\rm cell}({\\bf r-R}),\n\\label{eq:rhosurf}$$ \nwhere ${\\bf R} = n\\_1 {\\bf a}\\_1 + n\\_2 {\\bf a}\\_2 + n\\_3 {\\bf a}\\_3$ is a real-space translation vector, and again ${\\hat n}$ is the normal to the surface plane. This way of defining a frozen bulk termination has two crucial advantages: (i) The choice of using the compound WI as our “elementary particles” naturally ensures that every ion in the surface system will have exactly the same formal oxidation state as in the bulk. This is a central point in the definition of a polar surface – if we allowed for fractional orbital occupations no surface would be polar. (ii) The choice of cleaving the *Bravais* lattice, rather than the *crystal* lattice is particularly advantageous, as it naturally preserves the bulk stoichiometry everywhere in the system (if we allowed for stoichiometry changes of reconstructions again the notion of polar surface would be inconsistent).\nIt can be easily verified that several types of unreconstructed surfaces can be generated by using Eq. ([eq:rhosurf]), simply by changing the definition of $\\rho\\_{\\rm cell}({\\bf r})$. In particular, we have the freedom to construct $\\rho\\_{\\rm cell}({\\bf r})$ in many different ways, simply by shifting each WI in the basis by an arbitrary Bravais lattice vector $\\Delta {\\bf R}$. Thus, the choice of the basis vectors ${\\bf r}\\_i$ uniquely determines the surface structure, according to Eq. ([eq:rhowi]) and Eq. ([eq:rhosurf]). (Of course, different choices of ${\\bf r}\\_i$ can lead to the same termination; the “$\\{{\\bf r}\\_i\\} \\rightarrow $ termination” relationship is a many-to-one function.) On the other hand, we have shown in the previous section that the choice of ${\\bf r}\\_i$ uniquely determines the value of ${\\bf P}\\_{\\rm bulk}$, out of the infinite possibilities allowed precisely by the arbitrariness in the choice of the basis vectors. This formally establishes the relationship between ${\\bf P}\\_{\\rm bulk}$ and the termination of the lattice. By construction, the *physical* net charge that lies at the surface of a frozen bulk termination as defined above is simply $\\sigma\\_{\\rm surf} = {\\bf P}\\_{\\rm bulk} \\cdot \\hat n$, where ${\\bf P}\\_{\\rm bulk}$ is the dipole moment (per unit volume) of an appropriate bulk unit cell, i.e. one that tiles the semi-infinite solid according to Eq. ([eq:rhosurf]).\nWithin these assumptions, we define a frozen bulk termination polar if the bulk building block used to construct it has a net dipole perpendicular to the surface plane; we define it non-polar otherwise. The problem of determining whether a surface is polar or not is, therefore, reduced to the problem of calculating the dipole moment of a bulk unit cell made of WI. This, in turn, can be directly related to the result of a Berry-phase calculation in the bulk crystal, which can be routinely performed with most publicly available codes. In other words, the termination itself can be understood as a *bulk* property.\nFrozen and relaxed surfaces\n---------------------------\nIt might appear artificial to consider surfaces that are constructed by stacking electronic orbitals corresponding to bulk Wannier functions. At a real surface, electronic states always depart from their bulk counterparts because of the peculiar chemical and electrostatic environment produced by the truncation of the crystal. Furthermore, also the ionic lattice undergoes nontrivial structural relaxations in the surface layers, in response to the perturbation of the bonding network. A central point of our formalism is that both (electronic and ionic) surface relaxation effects are essentially irrelevant in the context of deciding whether a given surface is polar on non-polar. As a matter of fact, either type of relaxation only affects the surface dipole moment, and not the surface charge density. Thus, genuine surface properties (e.g. the alignment between the bulk bands and the vacuum levels, or the surface energy) certainly depend on these mechanisms, but the polarity (which depends only on the physical surface charge) won’t be affected. This formally establishes the surface polarity as a property that can be completely understood at the bulk level – note that the termination dependence can also be understood as a bulk property as specified in the previous section. Then, all mechanisms that alter the surface *charge* (either in the form of a local composition change or as a modification of the formal oxidation state of the surface ions) are unambiguously understood as external compensation effects, and enter the definition of $\\sigma\\_{\\rm ext}$.\nConstruction of arbitrary non-polar terminations\n------------------------------------------------\nSo far we have addressed the question of deciding whether a given surface, of a certain orientation and termination, is polar or non-polar. One could wonder now, for a given bulk compound, (i) whether non-polar terminations can be constructed at all; (ii) if yes along which surface plane orientation; finally, it would be helpful to (iii) identify candidate non-polar surface structures based only on bulk information. In this section we shall illustrate how this is done within the present definition of surface polarity.\nEssentially, the question (i) boils down to finding all possible values of ${\\bf P}\\_{\\rm bulk}$. This is, within the modern theory of polarization, a periodic lattice of points. The difference between two arbitrary values of ${\\bf P}$ is a multiple of a real-space primitive translation vector,\n \n$${\\bf P}'\\_{\\rm bulk} - {\\bf P}\\_{\\rm bulk} = \\frac{Q\\_0}{\\Omega} (i {\\bf a}\\_1 +\nj {\\bf a}\\_2 + k {\\bf a}\\_3).$$ \nHere *Q*0 = *n**e* is an integer *n* times the electron charge *e*. *n*, which determines the resolution of the ${\\bf P}\\_{\\rm bulk}$ mesh depends on the convention of how the Wannier functions and the ion cores are grouped together. In particular, the constraint adopted here of assigning each Wannier function to a specific ionic site generally restricts the lattice of possible values of ${\\bf P}\\_{\\rm bulk}$ to a subset of those allowed by Eq. ([eq:wannier]) and Eq. ([eq:berry]). This can be understood by observing that the new elementary building blocks of the lattice are the “compound objects” WI, rather than single electrons or ions.\nAnswering question (ii) consists in finding the intersections between the infinite lattice of ${\\bf P}\\_{\\rm bulk}$ values and a given surface plane, which is a straightforward geometrical problem.\nAnswering question (iii) then is easy by recalling the direct relationship between a given value of ${\\bf P}\\_{\\rm bulk}$ and the dipole moment of a well defined bulk unit. More specifically, once a value (or a subset of values) of ${\\bf P}\\_{\\rm bulk}$ is found for which ${\\bf P}\\_{\\rm bulk} \\cdot \\hat n = 0$, models of the non-polar surface can be readily built by stacking [using Eq. ([eq:rhosurf])] bulk unit cells that correspond to those same values of ${\\bf P}\\_{\\rm bulk}$. We shall present several practical examples of this strategy in Sec. [sec:results].\nRelationship to previous approaches\n-----------------------------------\n### Tasker model\n \nEq. ([eqd]) constitutes the rigorous link between Tasker’s model and the modern theory of polarization in periodic insulators. Within our formalism, the total excess charge at a frozen bulk termination can be exactly written as\n \n$$\\sigma\\_{\\rm surf} = {\\bf P}\\_{\\rm bulk} \\cdot \\hat n = \\frac{ (\\sum\\_\\alpha {\\bf R}\\_\\alpha Q\\_\\alpha + \n{\\bf d}\\_{\\rm WI}) \\cdot \\hat{n}}{\\Omega},\n\\label{eqtasker}$$ \nwhere the sum is extended over all atoms in the semi-infinite crystal. The only difference between Tasker’s model and Eq. [eqtasker] is the additional, purely electronic contribution ${\\bf d}\\_{\\rm WI}$, which comes from the polarization of the Wannier ions in the crystalline environment. This contribution vanishes in all solids that are characterized by a center of symmetry; in these materials the discussion of the surface polarity problem in terms of nominal charges is therefore rigorous and exact. Even in materials where ${\\bf d}\\_{\\rm WI} \\neq 0$, neglecting this term is usually not crucial to assessing the polar or non-polar nature of a given surface. However, considering the WI contribution is essential for a quantitative estimation of *σ* (which is the excess charge that needs to be compensated); this is especially true in ferroelectric materials, which generally have a large anomalous contribution to ${\\bf P}$. Thus, our formalism provides a formal justification to Tasker’s model, and completes it by introducing an additional well-defined electronic dipolar contribution, ${\\bf d}\\_{\\rm WI}$.\nIn addition to this, our strategy has important practical advantages. Tasker’s approach involves a direct calculation of Eq.([eqtasker]) by means of an infinite lattice sum, whose convergence is ensured by using Ewald summation techniques. This procedure might be cumbersome in practice, and it requires a specialized computer code to perform the calculation. Our strategy greatly simplifies the problem, by reducing it to the calculation of the dipole moment of a small set of point charges. This can be done with paper and pencil in few minutes for a surface of arbitrary orientation, provided that one knows ${\\bf d}\\_{\\rm WI}$. This vanishes in many cases of practical interest – whenever it doesn’t vanish, only a single bulk Berry-phase calculation is needed to evaluate this contribution. Moreover, our strategy allows one to easily answer a number of physical questions that were difficult to address within Tasker’s approach, e.g. those discussed in the previous section.\n### Other approaches\nIn order to fully appreciate the advantages of our formalism, it is useful to compare it, in a practical case, with the alternative notion of “dipole-free unit cell” proposed by Goniakowski *et al.* For illustrative purposes, we consider the (100) surfaces of two prototypical perovskite materials, LaAlO3 and SrTiO3, in their cubic high-symmetry phase. We shall use either the arguments developed in the previous sections, based solely on bulk properties of the respective materials, or the theory of Ref. , in order to assess the polar or non-polar character of these surfaces.\nIn Fig. [fig1] we plot, along the (001)-oriented *z* axis, the *x**y*-planar average of the total valence charge density. (The left panels refer to LaAlO3, the right ones to SrTiO3.) The upper panels show a possible decomposition of the electronic charge that leads to a dipole-free unit cell, which we construct as follows. First, we count the total charge of the ionic cores of the individual oxide layers. With the pseudopotentials used in this work, these are LaO(+17), AlO2(+15), SrO(+16) and TiO2(+24). Next, we decompose the total valence charge by cutting it with abrupt (001) planes located in the interstitial regions. The location of those planes is chosen as to (i) respect the inversion symmetry of the crystal, and to (ii) assign to each layer an electron density that exactly cancels the positive core charge of that layer. The resulting electron charge assigned to the AO layers is highlighted with a dark shading (light for the BO2 layers). By construction, the “unit cell” obtained by combining two adjacent layers (evidenced by the arrow and dashed lines in the figure) has zero dipole moment in both LaAlO3 and SrTiO3. Hence, this construction fails at detecting any fundamental difference between LaAlO3 and SrTiO3: both are predicted to have non-polar (001) surfaces. Of course, this prediction relies on a completely arbitrary partition of the total electronic charge density. There are many other ways to do it. For example, if one chooses a different location of the cut planes (e.g. at the mid-point distance between the atomic planes), or yet a more sophisticated prescription (e.g. based on the Bader analysis), one generally gets a non-vanishing layer charge in both LaAlO3 and SrTiO3. From this perspective, one would have to conclude that the (001) surfaces of both materials are polar. The main point that we want to stress here is that, if we base our analysis solely on the total electronic density [as we have done in Fig. [fig1](a-b)], (i) the choice between one partitioning scheme and the other is arbitrary; (ii) any statement about the surface polarity inferred from such a partitioning is ambiguous; and (iii) such an analysis cannot be linked in any ways to the bulk polarization of the material (the latter cannot be defined, even in principle, in terms of the total charge density of a periodic crystal).\nIn Fig. [fig1](c-d) we demonstrate how the Wannier-based decomposition of the valence density solves this problem. The thin solid lines show, as above, the ground-state electronic charge densities *ρ̄*(*z*) (again, the bar symbol on *ρ* indicates that an in-plane averaging was performed). The total Wannier densities of each layer, defined as *ρ̄**l* in Eq. ([eq:rhol]), are shown as thick lines (solid for the AO layers, dashed for the BO2 layers). Note that we show the electron density as positive, and we omit the bare pseudopotential charges from the plots. (These are a lattice of Dirac delta functions, centered at the oxide layer locations.) As the Wannier functions are discrete objects, the total electronic charges are integer numbers. Most importantly, the Wannier functions carry some crucial information (that is absent in the total valence density) on how the localized bound charges are organized in the insulating state of each compound. It turns out that (summing up the contributions from the cores) the LaO and AlO2 layers have a total charge of +1 and -1, respectively, while the SrO and TiO2 layers result charge-neutral, in perfect agreement with the naive assumption of perfect ionicity. Thus, the Wannier decomposition correctly identifies LaAlO3(001) as polar and SrTiO3 as non-polar, in agreement with Tasker’s classification.\nTo corroborate our arguments, a further consideration is in order. One could be tempted to criticize our reasoning by observing that Wannier functions are by no means uniquely defined starting from a given set of Bloch orbitals. In Fig. [fig1](c-d) we have chosen a maximally localized representation, but there is nothing really fundamental behind this choice. Is the identification of a surface as polar or non-polar robust against this arbitrariness? The answer is yes. The formal proof of this statement was derived in 1993 , several years before the maximally localized Wannier functions were first introduced. Our choice of a maximally localized representation is motivated by its intuitive relationship to elementary chemical concepts (e.g. formal valence), but the reader should keep in mind that this is just a convenient way of expressing a concept that has solid mathematical grounds. In particular, the exact value of the surface charge can be computed at the bulk level, regardless of the degree of ionicity of the material. \nApplication to perovskite surfaces\n==================================\nLaAlO3 and SrTiO3: bulk properties\n----------------------------------\nWe now use two prototypical perovskite materials, LaAlO3 and SrTiO3, to illustrate our strategy in practice. This choice of materials is motivated by the recent discovery of a conducting electron gas at their polar (100) interface. This has generated a lively excitement in the research community and a renewed interest in the theoretical foundations of the surface/interface polarity problem. \nWe shall address questions (i-iii) raised in Sec. [sec:questions]. For simplicity, in both materials we consider only surfaces of the type (0ij). This means that only the projection of ${\\bf P}\\_{\\rm bulk}$ on the *y**z* plane is relevant, and our procedure can be conveniently represented on 2D graphs. Our strategy is general, and this choice was made only to simplify the notation and the graphical representations. We also consider both bulk compounds within their high symmetry cubic phase. (Both materials are characterized by zone-boundary distortions, related to rotations and tilts of the oxygen octahedral network; however, as these distortions are non-polar in nature, they are irrelevant for the present discussion.)\nRelationship between ${\\bf P}\\_{\\rm bulk}$ and the dipole moment of the bulk unit cell. Red balls represent O ions, with charge $Q\\_{\\rm O} = -2e$. Gold balls represent the A-site cation, either Sr ($Q\\_{\\rm Sr} = +2e$) or La ($Q\\_{\\rm La} = +2e$); green balls are the B-site cation, either Ti ($Q\\_{\\rm Ti} = +4e$) or Al ($Q\\_{\\rm Al} = +3e$). The sketches (a-d) represent the projection of the atomic positions onto a (100)-oriented plane. On the top and the bottom are shown the values of ${\\bf P}\\_{\\rm bulk}$ (in units of *e*/*a*02, where *a*0 is the lattice parameter in either material) resulting in LaAlO3 and SrTiO3, respectively, from each cell arrangement. [fig:dipoles]\nLattices of allowed values of ${\\bf P}\\_{\\rm bulk}$ in LaAlO3 (left) and SrTiO3 (right). In the left panel we show four possible surface plane orientations. The polar orientations (dashed red lines) have no intersections with the ${\\bf P}\\_{\\rm bulk}$ lattice. The contrary is true for the non-polar orientation (solid black lines). [fig:platt]\nTo start with (question i), we need to find the lattice of “allowed” values of ${\\bf P}\\_{\\rm bulk}$ in either material. Recalling that both compounds are characterized by a center of symmetry, it follows that ${\\bf d}\\_{\\rm WI}=0$, and ${\\bf P}\\_{\\rm bulk}$ is exactly determined by the formal valence charges of the participating ions, all sitting in their high-symmetry lattice sites. Now, the formal ionic charges are La(+3), Al(+3) in LaAlO3, Sr(+2), Ti(+4) in SrTiO3; oxygens in either compound have a formal charge of (-2). In Fig. [fig:dipoles] we show how different choices of the crystal basis of five atoms lead to different dipole moments per unit cell, and hence to a different formal polarization. If we could take all (infinite) combinations, we would obtain an infinite lattice of points, which is isomorphic with the real-space Bravais lattice of the cubic crystal. The 2D projection of the lattice of ${\\bf P}\\_{\\rm bulk}$ in either compound is shown in Fig. [fig:platt]. Even if the two compounds are isostructural (recall that we consider both LAO and STO in their cubic phase), there are two important differences in their formal polarization lattice. First, ${\\bf P}\\_{\\rm STO}$ is centered in the origin, while ${\\bf P}\\_{\\rm LAO}$ is centered in (1/2, 1/2, 1/2). Second, ${\\bf P}\\_{\\rm STO}$ has a coarser mesh than ${\\bf P}\\_{\\rm LAO}$ – the spacings are doubled because the constituent point charges are all even in the former. Note that both ${\\bf P}\\_{\\rm bulk}$ lattices are centrosymmetric, consistent with the absence of a spontaneous polarization in either material. \nTo answer question (ii), we need to find all possible intersections between a surface plane and the allowed values of ${\\bf P}\\_{\\rm bulk}$; the projection of a few representative surface planes are plotted in Fig. [fig:platt](a). As it can be readily appreciated from the diagram, the aforementioned qualitative differences between the respective ${\\bf P}\\_{\\rm bulk}$ lattices of STO and LAO have important consequences on the electrostatics of the surfaces. In particular, in STO the origin belongs to the allowed values of ${\\bf P}\\_{\\rm STO}$, and any surface plane intersects the origin by construction; therefore, a non-polar surface of any possible orientation can be readily constructed [we shall illustrate the case of STO(111) in Sec. [sec:sto111]]. Conversely, in LAO only specific plane orientations intersect the ${\\bf P}\\_{\\rm LAO}$ lattice [note that the (100) orientation is correctly classified as polar]. In the following Section we shall consider a subset of these (infinite) possibilities, i.e. the vicinal (01*n*) surfaces, where *n* is an arbitrary odd integer number. We shall focus on the lowest-index cases with *n* = 1, 3, 5.\nVicinal LaAlO3 surfaces\n-----------------------\n \n \nWe first construct preliminary models for the (01*n*) surfaces with *n* = 1, 3, 5. These are obtained by using slab geometries, within the supercell method. First we choose a unit cell with the appropriate translational periodicity in plane, and enough room along the out-of-plane direction to accommodate both the slab and a vacuum region (slab and vacuum thicknesses are treated as convergence parameters). Second, we tile the slab region with repeated copies of a well-defined primitive basis of atoms, which is chosen in a such a way that its dipole moment lies exactly parallel to the surface plane. (This implies that the choice of the basis depends on the surface orientation.) This procedure leads to the slab models sketched in Fig. [fig:models].\nc|ccc Surface type & \n& (011) & (013) & (015) \nA & 1.93 & 2.23 & 2.15 \nB & 1.93 & 1.85 & 1.83 \nCleavage & 3.86 & 4.08 & 3.98\nThe first observation is that all these surface models [except maybe the (011) case] present alternating LaO-type and AlO2-type terraces, and these terraces tend to grow wider and wider for increasing *n*. Note that [again, with the only exception of the (011) case], the construction described above produces, in fact, *two* inequivalent surface structures for each orientation. In other words, the models of Fig. [fig:models] do not enjoy inversion symmetry. We shall refer to these two surfaces as “type A” and “type B”, where type A presents LaO-type step edges and type B has AlO-type edges. Remarkably, it is easy to realize that one can change from A-type to B-type simply by displacing an oxygen atom from one step edge to the neighboring one. This way, starting from the “mixed” AB-type slabs Fig. [fig:models] one can readily construct pure AA or BB slabs. One can verify that the resulting AA and BB models do enjoy inversion symmetry. Since going from A to B preserves the bulk stoichiometry, this allows for a rigorous definition of the surface energy for all individual surface structures.\nIn practice, in the simulations we use a slab thickness of approximately 4-5 LAO cells in each case, which is more than sufficient to obtain a well-converged value of the surface energy. The surface energy is defined as\n \n$$E\\_{\\rm surf} = \\frac{1}{2S} (E\\_{\\rm slab} - N E\\_{\\rm bulk}),$$ \nwhere $E\\_{\\rm slab}$ and $E\\_{\\rm bulk}$ are the relaxed total energy of the slab supercell and of LAO bulk, *N* is the total number of LAO units in the slab model, and $S=a\\_0^2\\sqrt{1+n^2} $ is the surface area in each case (*a*0 is the lattice parameter of cubic LAO; the factor of 2 takes into account the fact that a slab has two surfaces). In Table [tab1] we report the results. Comparing these values with previous literature studies is difficult, as studies of vicinal perovskite surfaces are scarce. Only the lowest-index (011) surface type has been investigated to some extent, although we weren’t able to find data specific to LAO. Concerning other perovskite materials, Eglitis and Vanderbilt reported an energy of 1.52 J/m2 for an isostructural O-terminated SrTiO3(011) surface structure. The value we obtain for LAO, 1.93 J/m2, is somewhat larger but otherwise of comparable magnitude. Note that in the study of Ref. a different (hybrid) functional was used – LDA might well overestimate surface energy values due to the well-known overbinding issues.\nIt is interesting to note that in the case of B-type surfaces the energy decreases slightly for increasing index *n*. We ascribe this behavior to the lower steps-to-terraces ratio in the (013) and (015) surfaces (undercoordinated step sites are likely to be less favorable). We consider it unlikely, however, that this energy be further reduced for *n* > 5. Increasing *n* would lead to larger and larger terraces that are locally charged [either of the LaO(+) or AlO2(-) type], and the electrostatic cost (roughly linear in *n*) would eventually dominate over the step energy (proportional to 1/*n*) in a way that bears many analogies to Kittel’s theory of domain walls. Still, the increased stability of the vicinal (013) and (015) surfaces [compared to the (011) orientation] suggests that these geometries could be, in principle, fabricated under appropriate experimental conditions. The simultaneous presence AO and BO2 domains appears promising for applications, e.g. in selective self-assembly of functional nanostructures, as it was recently shown in the case of SrTiO3 \nTotal density of states of the LAO(013) slab models. Surface A (black curve), B (red dashed) and bulk (shaded area) are shown. [fig:ldos] \nThe above considerations on the energetics haven’t answered an important question yet: how can we verify that these surfaces are indeed non-polar, consistent with our predictions? A useful indication comes from the density of states. If a surface is polar, then there is a need for compensation via additional charge carriers (either electron or holes) that deplete or populate the energy bands of the crystal in proximity of the problematic termination. This typically results in a metallic surface. Conversely, if the surface is non-polar, the bulk-derived Wannier functions alone are sufficient to ensure electrostatic stability, and therefore the system can remain *insulating*. In Fig. [fig:ldos] we show the total density of states extracted from a (013) (A- or B-type) slab model, compared with the bulk LAO density of states. In all cases there is a wide gap separating the unoccupied from the occupied states. This fact, together with the inversion symmetry and perfect bulk stoichiometry of the slabs, directly demonstrates that the surfaces are non-polar, and that every atom contributes with a total number of electrons that exactly corresponds to its formal ionic valence. Similar considerations apply to the (110) and (510) surface models (not shown).\nIt is important to stress that, contrary to a common misconception, all these surfaces are perfectly *stoichiometric* by construction, and they are *non-reconstructed* as they have the highest possible translational symmetry that is allowed by each plane orientation. It is often assumed that the only “legitimate” structures that can be named frozen bulk terminations are those that are obtained upon cleavage of the *crystal* lattice, i.e. preserving the integrity of the bulk-like atomic planes. This is, however, just a convention that has nothing fundamental to it. We believe it is more practical to truncate the *Bravais* lattice instead. This automatically preserves stoichiometry and translational symmetry, and dramatically simplifies the description of surface electrostatics.\nAs a final remark, it is fairly easy to realize that all the (01*n*) surface models presented in this section are non-polar for *any* non-ferroelectric perovskite material (or for a ferroelectric one in its high-temperature symmetric phase). This can be simply understood by observing that, by replacing the cations in each bulk primitive basis (see Fig. [fig:models]) with those of a different charge family (i.e. I-V or II-IV), the dipole moment changes its magnitude but not its direction. Also, all the surface models enjoy inversion symmetry and have ideal bulk stoichiometry – the absence of a dipole moment normal to the surface plane is therefore automatically guaranteed. Therefore, many of the considerations made here in relationship to the specific LAO case are actually completely general, and apply to all cubic perovskite compounds.\nNon-polar SrTiO3(111) surfaces\n------------------------------\nTo further illustrate our arguments, we move now to the case of SrTiO3. According to our definition of polar surface, and by observing that the simplest choice of the SrTiO3 bulk unit has zero dipole moment, one would conclude that in SrTiO3 *any* surface orientation is “non-polar”. To illustrate this point we shall consider here the (111) surface, which has been classified as polar by most authors. Note that this surface is indeed polar if one insists on terminating the crystal lattice with either a Ti or a SrO3 layer. Our prescription of cleaving the Bravais lattice and tiling it with well-defined bulk-like formula units is less restrictive, and allows for non-polar terminations as we shall see in the following.\n Primitive bulk SrTiO3 unit cells for the model A (left) and B (right) (111) slabs. Sr atoms (large green circle) lie at the corners of the cube; O atoms (small red circles) lie at the face-centered sites; Ti atoms lie at the center of the cube. Both choices of the primitive unit have zero dipole moment along any direction. [fig111cell] \nWe build two inequivalent stoichiometric slab models (that we call A and B) by stacking the primitive building blocks schematically shown in Fig. [fig111cell]. It is easy to verify that both choices of the primitive cell have zero dipole moment. (Again, to compute the dipole moment we use the *formal* charges. This is substantiated by the Wannier-based decomposition described in section [sec:theory], which provides the formal link to the theory of polarization in bulk solids. ) The primitive translation vectors of the supercell are (in units of the bulk equilibrium lattice parameter"},"data_id":{"kind":"string","value":"arxiv_0002037"}}},{"rowIdx":495,"cells":{"content":{"kind":"string","value":"Bayes correlated equilibria and no-regret dynamics\n==================================================\nThis paper explores equilibrium concepts for Bayesian games, which are fundamental models of games with incomplete information. We aim at three desirable properties of equilibria. First, equilibria can be naturally realized by introducing a mediator into games. Second, an equilibrium can be computed efficiently in a distributed fashion. Third, any equilibrium in that class approximately maximizes social welfare, as measured by the price of anarchy, for a broad class of games. These three properties allow players to compute an equilibrium and realize it via a mediator, thereby settling into a stable state with approximately optimal social welfare. Our main result is the existence of an equilibrium concept that satisfies these three properties.\nToward this goal, we characterize various (non-equivalent) extensions of correlated equilibria, collectively known as Bayes correlated equilibria. In particular, we focus on communication equilibria (also known as coordination mechanisms), which can be realized by a mediator who gathers each player’s private information and then sends correlated recommendations to the players. We show that if each player minimizes a variant of regret called untruthful swap regret in repeated play of Bayesian games, the empirical distribution of these dynamics converges to a communication equilibrium. We present an efficient algorithm for minimizing untruthful swap regret with a sublinear upper bound, which we prove to be tight up to a multiplicative constant. As a result, by simulating the dynamics with our algorithm, we can efficiently compute an approximate communication equilibrium. Furthermore, we extend existing lower bounds on the price of anarchy based on the smoothness arguments from Bayes Nash equilibria to equilibria obtained by the proposed dynamics.\nIntroduction\n============\nIn this paper, we study equilibrium concepts for *Bayesian games*,[2](#fn2) in which players do not share complete information about the game. Since introduced by, Bayesian games have been studied as one of the most significant models in game theory.\nThe best-known equilibrium concept for Bayesian games must be a *Bayes Nash equilibrium*, which is a natural generalization of a (mixed) Nash equilibrium in games with complete information. However, this equilibrium concept has a crucial issue concerning efficient computability. Even for a special case with complete information, computing an approximate Nash equilibrium is PPAD-complete and requires exponentially many queries in the number of players. Even for two-player Bayesian games with a constant number of actions, computing an approximate Bayes Nash equilibrium is PPAD-complete. This is why players cannot be expected to reach a Bayes Nash equilibrium in practical situations.\nA possible remedy for this issue is to introduce a *mediator* to the game. For games with complete information, if a mediator can recommend (possibly correlated) actions to players, a broader class of equilibria called *correlated equilibria* is a natural equilibrium concept. Correlated equilibria allow players’ actions to be correlated via the mediator’s recommendation, while their actions must be independently decided in a Nash equilibrium. An example of correlated equilibria in our society is realized by traffic signals, which randomly recommend cars in one direction to “proceed” and cars in the other direction to “stop.” As we can see from this example, players can coordinate and achieve a better outcome in a correlated equilibrium.\nIn contrast to Nash equilibria, correlated equilibria can be efficiently computed even for multi-player non-zero-sum games. A standard approach to computing a correlated equilibrium is to simulate a variant of *no-regret dynamics*, in which players repeat the same game many times and independently decide their own action in each round. If each player decides their action according to an algorithm that exhibits no internal regret or swap regret, the dynamics are known to converge to a correlated equilibrium.[3](#fn3) An advantage of this approach is that the dynamics are *uncoupled*, that is, they can be simulated by independent players in a distributed fashion. Furthermore, the existence of such dynamics can explain why people automatically reach an equilibrium in our society. No-regret dynamics in Bayesian games are known to converge to a *Bayes coarse correlated equilibrium*[4](#fn4). However, this equilibrium concept is usually unnatural because it is assumed that once observing the recommendation by the mediator, each player must follow it, even if it is more beneficial to deviate. The convergence of a variant of no-regret dynamics to more natural equilibrium concepts is still unknown.\nIn addition to efficient computability, another desirable property of equilibrium concepts is that any equilibrium in that class approximately maximizes social welfare. As the famous example of the prisoners’ dilemma illustrates, players can fall into a bad equilibrium by trying to maximize their own payoffs. To measure how bad equilibria could be, proposed the *price of anarchy* (POA), which is defined as the ratio of the social welfare in the worst equilibrium to the maximum social welfare. An argument using *smoothness* is a versatile framework for bounding the POA for a broad class of games. For games with complete information, proved that smoothness of games implies a lower bound on the POA of correlated equilibria,[5](#fn5) and various classes of games satisfy smoothness. later developed a variant of smoothness for mechanism design and provided lower bounds on the POA of correlated equilibria for a large class of mechanisms. For Bayesian games, and derived bounds on the POA of Bayes Nash equilibria by using the smoothness arguments, but it remains to be seen whether these bounds can be extended to a broader class of equilibria. There are several exceptions that considered Bayes coarse correlated equilibria, but their applications are limited (see for more detailed discussions).\n[line width=1.5pt] (origin) at (0,0); ([yshift=+1.45]origin) [rounded corners=0.1] – ([xshift=+0.6,yshift=+1.0]origin) – ([xshift=+0.6]origin) – ([xshift=-0.6]origin) – ([xshift=-0.6,yshift=+1.0]origin) – cycle; (sf) at (origin) [anchor=south, rectangle, draw, rounded corners=0.1, minimum width=, minimum height=] ; ([xshift=0.9]origin) – ([xshift=0.9,yshift=+1.7]origin) to node[anchor=north, align=right] Communication \nequilibria ([xshift=+0.3,yshift=+1.7]origin) – ([xshift=-0.6,yshift=+1.0]origin) – ([xshift=-0.6]origin) – ([xshift=0.9]origin) – cycle; ([xshift=-0.9]origin) – ([xshift=-0.9,yshift=+1.7]origin) to node[anchor=north, align=left] (anfce) Agent-normal-form \ncorrelated equilibria ([xshift=-0.3,yshift=+1.7]origin) – ([xshift=+0.6,yshift=+1.0]origin) – ([xshift=+0.6]origin) – cycle; ([xshift=-1.0]origin) – ([xshift=-1.0,yshift=+2.0]origin) to node[anchor=north, align=left] Bayesian solutions ([xshift=+1.0,yshift=+2.0]origin) – ([xshift=+1.0]origin) – cycle; at ([yshift=0.8]sf.south) Strategic-form correlated equilibria (SFCEs); at (sf.south) [draw, rounded corners=0.05, minimum height=0.5, minimum width=0.8, anchor=south, align=center]Bayes Nash equilibria (BNEs); at (anfce.south west) (ANFCEs);\n[fig:relation]\nIf there exists an equilibrium concept that satisfies these desirable properties, we can compute an equilibrium efficiently in a distributed fashion and then realize it by introducing a mediator, which enables players to settle into an equilibrium with high social welfare. Thus, the following question regarding Bayesian games naturally arises.\n*Is there any equilibrium concept for Bayesian games that can be realized by a mediator, efficiently computable via uncoupled dynamics, and guaranteed to have POA bounds for various games?*\nTo answer this question, we investigate the concept of *communication equilibria*, which can be realized by introducing a mediator to Bayesian games. In Bayesian games, each player has private information represented by a *type* generated from a commonly known prior distribution. To deal with this private information, a mediator in a communication equilibrium communicates with players bidirectionally. A mediator first gathers the types from players and then sends recommendations to each player based on the received types. This is an equilibrium in the sense that players have no incentive to deviate from reporting the type truthfully or following the action recommendation. Note that originally called communication equilibria *coordination mechanisms* with an emphasis on their relations to Bayesian incentive-compatible mechanisms. Although these two concepts are similar in that truthful type reporting is incentive-compatible, a mediator in communication equilibria is required to recommend incentive-compatible actions to players, while a mechanism determines an outcome in a top-down fashion. A communication equilibrium can be interpreted as a solution concept combining truthful type reporting of Bayesian incentive-compatible mechanisms and action recommendations of correlated equilibria. These two incentive constraints complicate the structure of this concept and make its computation challenging.\nOur central result is an efficient algorithm for computing a communication equilibrium. To derive this algorithm, we first show that if each player minimizes a variant of regret, which we call *untruthful swap regret*, the dynamics converge to a communication equilibrium. We then develop an efficient algorithm for minimizing untruthful swap regret with a sublinear upper bound. We also prove a lower bound that shows the tightness of the upper bound. As a byproduct, we characterize various extensions of correlated equilibria, collectively called *Bayes correlated equilibria* (). This reveals an additional property called *strategy representability*, which other extensions have but communication equilibria lack. The limit of dynamics minimizing untruthful swap regret also satisfies strategy representability. Furthermore, we show that most existing POA bounds of Bayes Nash equilibria based on the smoothness arguments can be directly extended to communication equilibria with strategy representability. Thus, the concept of communication equilibria with strategy representability satisfies the desirable properties.\nPreliminaries\n-------------\n#### Basic notations\nThe sets of reals and integers are denoted by $\\bbR$ and $\\bbZ$, respectively. For any $a,b \\in \\bbR$, we define $[a,b] = \\{ x \\in \\bbR \\mid a \\le x \\le b\\}$. For any positive integer $n \\in \\bbZ$, we define $[n] = \\{i \\in \\bbZ \\mid 1 \\le i \\le n\\} = \\{ 1,2,\\dots,n \\}$. For any finite set *A*, we denote the probability simplex by Δ(*A*) = {*π* ∈ [0, 1]*A* ∣ ∑*a* ∈ *A**π*(*a*) = 1}. For any distribution *π* ∈ Δ(*A*), we write *a* ∼ *π* when *a* ∈ *A* is generated from the distribution *π*. By abuse of notation, we write *a* ∼ *A* when *a* is generated from the uniform distribution over *A*. To specify a random variable that a probability or an expected value is concerned with, we write Pr*a* ∼ *π*( ⋅ ) or ${\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi} \\left[ \\cdot \\right]$. If the distribution is obvious, we simply write Pr*a*( ⋅ ) or ${\\mathop{\\mathbb{E}}}\\_{a} \\left[ \\cdot \\right]$. For vectors $x,y \\in \\bbR^n$, the inner product is written as ⟨*x*, *y*⟩ = ∑*i* = 1*n**x*(*i*)*y*(*i*). Similarly, for matrices $P,Q \\in \\bbR^{n \\times m}$, the matrix inner product is written as ⟨*P*, *Q*⟩ = ∑*i* = 1*n*∑*j* = 1*m**P*(*i*, *j*)*Q*(*i*, *j*).\n#### Bayesian games\nLet $n \\in \\bbZ$ be the number of players and *N* = [*n*] a set of players. Let *A**i* be a finite set of actions for player *i* ∈ *N* and *A* = *A*1 × … × *A**n* the set of action profiles of all players. Let Θ*i* be a set of finite possible types for each player *i* ∈ *N* and Θ = Θ1 × … × Θ*n* the set of type profiles of all players. Let *v**i*: Θ × *A* → [0, 1] be a utility function that maps a type profile and action profile to a payoff value for player *i*.[6](#fn6) Since the size of Θ and *A* is exponential in *n*, we assume that each *v**i* is given by an oracle that returns the value of *v**i*(*θ*; *a*) for any *θ* ∈ Θ and *a* ∈ *A*. For each player *i* ∈ *N*, subscript − *i* represents the other players *N* \\ {*i*}, i.e., *a*− *i* = (*a**j*)*j* ∈ *N* \\ {*i*} and *θ*− *i* = (*θ**j*)*j* ∈ *N* \\ {*i*}.\nIn this paper, we distinguish between the two words “strategies” and “actions.” While actions are defined as above, strategies are decisions that determine an action for every type. Formally, a strategy *s**i*: Θ*i* → *A**i* of player *i* ∈ *N* determines action *s**i*(*θ**i*) ∈ *A**i* to be selected given their own type *θ**i* ∈ Θ*i*. For each player *i* ∈ *N*, let *S**i* = *A**i*Θ*i* be the set of all strategies. We write a strategy profile as *s* = (*s*1, …, *s**n*), and let *S* = *S*1 × … × *S**n* be the set of all strategy profiles. For notational simplicity, we write *s*(*θ*) = (*s**j*(*θ**j*))*j* ∈ *N* and *s*− *i*(*θ*− *i*) = (*s**j*(*θ**j*))*j* ∈ *N* \\ {*i*} for each *s* ∈ *S*, *θ* ∈ Θ, and *i* ∈ *N*.\nIn Bayesian games, we assume that the type profile is generated from the probability distribution *ρ* ∈ Δ(Θ) that every player knows in advance. Since the size of Θ is exponential in *n* in general, the distribution *ρ* cannot be expressed in polynomial size. In most parts of this paper, we assume that it is possible to efficiently sample *θ**i* from the marginal distribution of *ρ*, which is denoted by *ρ**i*, and *θ*− *i* from the distribution *ρ* conditioned on any *θ**i* ∈ Θ*i*, which is denoted by *ρ*∣*θ**i*. This assumption holds for a special case where *ρ* is a product distribution, i.e., *ρ*(*θ*) = ∏*i* ∈ *N**ρ**i*(*θ**i*) holds for each *θ* ∈ Θ. For POA bounds in, we assume that *ρ* is a product distribution.\nA Bayesian game proceeds in the following steps: (1) First, a type profile *θ* is generated from *ρ* ∈ Δ(Θ), and each player *i* ∈ *N* is notified of their own type *θ**i* ∈ Θ*i*. (2) Each player then decides their action *a**i* ∈ *A**i* without knowing the other players’ types *θ*− *i*. (3) Finally, each player obtains the payoff *v**i*(*θ*; *a*) depending on all players’ actions and types.\nThere are two widely-used interpretations of Bayesian games: the strategic form (also called the random-vector model or the normal form) and the agent normal form (also called the posterior-lottery model, the Selten model, or the population interpretation). Both interpretations reduce a Bayesian game to a normal-form game with complete information as follows.\n* In the strategic form of a Bayesian game, the strategies *S**i* are considered as a decision space for player *i* ∈ *N*. Each player *i* ∈ *N* obtains the expected payoff ∑*θ* ∈ Θ*ρ*(*θ*)*v**i*(*θ*; *s*(*θ*)) for strategy profile *s* ∈ *S*.\n* In the agent normal form, we hypothetically consider the same player with different types as different players, that is, the set of players are {(*i*, *θ**i*) ∣ *i* ∈ *N*, *θ**i* ∈ Θ*i*}, which we denote by *N*ʹ. Each player (*i*, *θ**i*) ∈ *N*ʹ selects an action from *A**i*. A type profile *θ* ∈ Θ is randomly generated from *ρ*, and then only one player (*i*, *θ**i*) becomes active for each *i* ∈ *N*. Only the active players can obtain non-zero payoffs, which are determined only by the active players’ actions. The inactive players always obtain payoff 0 and do not affect the other players’ payoffs. If we denote player (*i*, *θ**i*)’s action by *a**i*, *θ**i*, the expected payoff for player (*i*, *θ**i*) is ∑*θ*ʹ ∈ Θ: *θ*ʹ*i* = *θ**i**ρ*(*θ*ʹ)*v**i*(*θ*ʹ; (*a**j*, *θ*ʹ*j*)*j* ∈ *N*).\n#### Correlated equilibria\nA normal-form game with complete information is specified by players *N* = {1, 2, …, *n*}, action profiles *A* = ∏*i* ∈ *N**A**i*, and a utility function *v**i*: *A* → [0, 1] for each player *i* ∈ *N*. In this game, a distribution *π* ∈ Δ(*A*) is an *ε*-approximate correlated equilibrium if\n \n$${\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi} \\left[ v\\_i(a) \\right]\n\\ge \n{\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi} \\left[ v\\_i(\\phi(a\\_i),a\\_{-i}) \\right] - \\epsilon.$$ \nholds for any *i* ∈ *N* and *ϕ*: *A**i* → *A**i*, where *ε* ∈ [0, 1] represents an additive error. In an *ε*-approximate correlated equilibrium, each player has gain at most *ε* in expectation by taking an action *ϕ*(*a**i*) ∈ *A**i* instead of the recommended action *a**i* ∈ *A**i*.\nSuppose that the same players play this game *T* times and decide a randomized action in each round. Let *π**i**t* ∈ Δ(*A**i*) be the decision by each player *i* ∈ *N* for each round *t* ∈ [*T*]. Then the payoff obtained by each player *i* ∈ *N* in each round *t* ∈ [*T*] is ${\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi^t} \\left[ v\\_i(a) \\right]$, where *π**t* ∈ Δ(*A*) is the product distribution that independently generates *a**j* ∼ *π**j**t* for each *j* ∈ *N*. This payoff can be expressed as ${\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t} \\left[ u\\_i^t(a\\_i) \\right]$ if we set $u\\_i^t(a\\_i) = {\\mathop{\\mathbb{E}}}\\_{a\\_{-i} \\sim \\pi\\_{-i}^t} \\left[ v\\_i(a) \\right]$ for each action *a**i* ∈ *A**i*, where *π*− *i**t* ∈ Δ(*A*− *i*) is the product distribution that independently generates *a**j* ∼ *π**j**t* for each *j* ∈ *N* \\ {*i*}. Thus, the online decision-making problem that each player *i* ∈ *N* faces can be regarded as an online learning problem with decision space *A**i* and rewards (*u**i**t*)*t* = 1*T*. For this problem, swap regret is defined as\n \n$${R\\_{\\mathsf{S},i}}= \\max\\_{\\phi \\colon A\\_i \\to A\\_i} \\sum\\_{t=1}^T {\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t} \\left[ u\\_i^t(\\phi(a\\_i)) \\right] - \\sum\\_{t=1}^T {\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t} \\left[ u\\_i^t(a\\_i) \\right].$$ \nThis quantity expresses regret for how much the player would have gained by playing *ϕ*(*a**i*) instead of each *a**i* ∈ *A**i*. proved that if each player minimizes their swap regret, the empirical distribution of the dynamics for *T* rounds is a (max*i* ∈ *N**R*S, *i*/*T*)-approximate correlated equilibrium. also proposed an efficient algorithm that achieves $O(\\sqrt{T |A\\_i| \\log |A\\_i|})$ swap regret.\nOverview of our results\n-----------------------\n[tab:bce]\n| | |\n| --- | --- |\n| Class | Characterization |\n| (lr)1-2 Strategic-form correlated equilibria | Incentive constraints w.r.t. strategies & SR |\n| Agent-normal-form correlated equilibria | Incentive constraints w.r.t. actions & SR |\n| Communication equilibria | Incentive constraints w.r.t. types and actions |\n| Bayesian solutions | Incentive constraints w.r.t. actions |\n#### Bayes correlated equilibria and no-regret dynamics\nOur first goal is to define untruthful swap regret and prove that dynamics minimizing it converge to a communication equilibrium with strategy representability. In the process leading to this, we define approximate versions of Bayes correlated equilibria and consider extensions of no-swap-regret dynamics.\nTo discuss the convergence of dynamics, we need to characterize each class as a set of distributions (). For this purpose, we first clarify the difference between Δ(*S*) and Δ(*A*)Θ by considering two corresponding scenarios. In the first scenario, a mediator generates *s* ∼ *σ* ∈ Δ(*S*) without observing type profile *θ* ∼ *ρ*, and then each player *i* ∈ *N* takes action *s**i*(*θ**i*). In the second scenario, a mediator generates *a* ∼ *π*(*θ*) ∈ Δ(*A*) based on type profile *θ* ∈ Θ. Thus, given a type profile distribution *ρ* ∈ Δ(Θ), both distributions *σ* ∈ Δ(*S*) and *π* ∈ Δ(*A*)Θ determine a distribution over action profiles *A* for each *θ* ∈ Θ. However, not all distributions obtained from Δ(*A*)Θ can be realized by Δ(*S*). There exists a simple example *π* ∈ Δ(*A*)Θ that cannot be represented by any distribution *σ* ∈ Δ(*S*) (see and ). Strategic-form correlated equilibria (SFCEs) and agent-normal-form correlated equilibria (ANFCEs) are defined as a subset of Δ(*S*), while communication equilibria and Bayesian solutions are defined as a subset of Δ(*A*)Θ. To compare these classes, we consider the set of distributions in Δ(*A*)Θ that can be realized by SFCEs or ANFCEs, and we call *π* ∈ Δ(*A*)Θ strategy-representable if there exists some *σ* ∈ Δ(*S*) that yields the same distribution over action profiles for each *θ* ∈ Θ ().\nSFCEs and ANFCEs are defined as correlated equilibria of the strategic form and agent normal form, respectively. Their incentive constraints are natural extensions of those for correlated equilibria. In an SFCE, each player *i* ∈ *N* cannot gain by selecting strategy *ϕ*SF(*s**i*) ∈ *S**i* instead of the recommended strategy *s**i* ∈ *S**i* according to any *ϕ*SF: *S**i* → *S**i*, while in an ANFCE, by selecting action *ϕ*(*θ**i*, *a**i*) ∈ *A**i* instead of the recommended action *a**i* ∈ *A**i* when the type is *θ**i* ∈ Θ*i* according to any *ϕ*: Θ*i* × *A**i* → *A**i*. No-swap-regret dynamics converging to a correlated equilibrium can be directly extended to these classes. Note that this naive extension does not immediately imply efficient computability of an SFCE because it requires swap regret minimization with decision space *S**i*, whose size is exponential in ∣Θ*i*∣. See for SFCEs and for ANFCEs.\nOur main focus is the notion of communication equilibria. The incentive constraints for communication equilibria are specified by two different kinds of deviations: reporting an untruthful type *ψ*(*θ**i*) ∈ Θ*i* instead of the true type *θ**i* ∈ Θ*i* according to *ψ*: Θ*i* → Θ*i* and taking an action instead of the recommended action according to *ϕ*: Θ*i* × *A**i* → *A**i*. Formally, we say *π* ∈ Δ(*A*)Θ is an *ε*-approximate communication equilibrium if the incentive constraint\n \n$$\\tag{$\\mathrm{IC}\\_{\\mathrm{Com}}$}\n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi(\\theta)} \\left[ v\\_i(\\theta; a) \\right] \\right]\n\\ge\n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi(\\psi(\\theta\\_i),\\theta\\_{-i})} \\left[ v\\_i(\\theta; \\phi(\\theta\\_i, a\\_i), a\\_{-i}) \\right] \\right] - \\epsilon.$$ \nholds for each *i* ∈ *N*, *ψ*: Θ*i* → Θ*i*, and *ϕ*: Θ*i* × *A**i* → *A**i*. As it allows untruthful type reporting, this constraint is stronger than that for ANFCEs. On the other hand, ANFCEs, defined on Δ(*S*), do not include communication equilibria, defined on Δ(*A*)Θ.\nTo deal with these two kinds of deviations, we introduce a new variant of regret called untruthful swap regret. In repeated play of Bayesian games, each player faces an online learning problem with rewards determined by a stochastic type. In each round *t* ∈ [*T*] of this problem, each player *i* ∈ *N* decides a distribution *π**i**t* ∈ Δ(*A**i*)Θ*i* that determines a (randomized) action for each type and receives reward[7](#fn7)\n \n$$u\\_i^t(\\theta\\_i, a\\_i) = {\\mathop{\\mathbb{E}}}\\_{\\theta\\_{-i} \\sim \\rho|\\theta\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{a\\_{-i} \\sim \\pi^t\\_{-i}(\\theta\\_{-i})} \\left[ v\\_i( \\theta; a) \\right] \\right]$$ \ndepending on their realized type *θ**i* ∼ *ρ**i* and action *a**i* ∼ *π**i**t*(*θ**i*), where *π*− *i**t*(*θ*− *i*) is the product distribution that independently generates *a**j* ∼ *π**j**t*(*θ**j*) for each *j* ∈ *N* \\ {*i*}. For this online learning problem, the untruthful swap regret *R*US, *i**T* for player *i* ∈ *N* is defined as\n \n$${R\\_{\\mathsf{US},i}}^T = \\max\\_{\\psi \\colon \\Theta\\_i \\to \\Theta\\_i} \\max\\_{\\phi \\colon \\Theta\\_i \\times A\\_i \\to A\\_i} \\sum\\_{t=1}^T {\\mathop{\\mathbb{E}}}\\_{\\theta\\_i \\sim \\rho\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t(\\psi(\\theta\\_i))} \\left[ u\\_i^t(\\theta\\_i, \\phi(\\theta\\_i, a\\_i)) \\right] - {\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t(\\theta\\_i)} \\left[ u\\_i^t(\\theta\\_i, a\\_i) \\right] \\right].$$ \nNote that this definition coincides with swap regret when ∣Θ∣ = 1. We show that if every player achieves sublinear untruthful swap regret, then the dynamics converge to a communication equilibrium. Furthermore, the empirical distribution of uncoupled dynamics is always strategy-representable ().\n[Informal version of ][thm:dynamics-com-intro] The empirical distribution of the dynamics for *T* rounds is a max*i* ∈ *N**R*US, *i*/*T*-approximate communication equilibrium with strategy representability, where *R*US, *i* is the untruthful swap regret for each player *i* ∈ *N*.\nNote that the set of all communication equilibria with strategy representability equals the intersection of communication equilibria and ANFCEs (). We present several other results: we clarify the relations between these classes as well as Bayesian solutions and Bayes Nash equilibria (see ), we slightly improve an upper bound on strategy swap regret from the direct application of the existing swap regret minimization algorithm (), and we provide various definitions of Bayes coarse correlated equilibria for comparison ().\n#### Algorithm for minimizing untruthful swap regret\nTo approximately compute a communication equilibria with strategy representability, we propose an efficient algorithm for minimizing untruthful swap regret. We construct our algorithm in the following three steps to obtain an upper bound of the optimal order.\nIn the first step (), the problem of minimizing untruthful swap regret is interpreted as an instance of *Φ-regret minimization*. To our knowledge, the definition of Φ-regret was first provided by when the decision space is a probability simplex and then extended to a general decision space by. For a set of transformations Φ, the Φ-regret measures how the algorithm gains by transforming their decisions with an optimal *ϕ* ∈ Φ in hindsight. To interpret untruthful swap regret as Φ-regret, we change the decision space from the set of distributions for each type Δ(*A**i*)Θ*i* to the set of vectors $\\calX$ defined by\n \n$$\\calX = \\left\\{ x \\in [0,1]^{\\Theta\\_i \\times A\\_i} \\;\\middle|\\; \\sum\\_{a\\_i \\in A\\_i} x(\\theta\\_i,a\\_i) = 1 ~ (\\forall \\theta\\_i \\in \\Theta\\_i) \\right\\},$$ \nwhere we set *x**t*(*θ**i*, *a**i*) = *π**i**t*(*θ**i*, *a**i*) for each *t* ∈ [*T*], *θ**i* ∈ Θ*i*, and *a**i* ∈ *A**i*. Then the space of all transformations expressed by *ψ*: Θ*i* → Θ*i* and *ϕ*: Θ*i* × *A**i* → *A**i* can be written as $\\calQ$ defined by\n \n$$\\calQ = \\left\\{ Q \\in [0,1]^{(\\Theta\\_i \\times A\\_i) \\times (\\Theta\\_i \\times A\\_i)} \n\\;\\middle|\\;\n\\begin{array}{l}\n\\text{there exists some $W \\in [0,1]^{\\Theta\\_i \\times \\Theta\\_i}$ such that}\\\\\n\\sum\\_{\\theta'\\_i \\in \\Theta\\_i} W(\\theta\\_i,\\theta'\\_i) = 1 ~ \\text{($\\forall \\theta\\_i \\in \\Theta\\_i$) and}\\\\\n\\sum\\_{a\\_i \\in A\\_i} Q((\\theta\\_i,a\\_i),(\\theta'\\_i,a'\\_i)) = W(\\theta\\_i,\\theta'\\_i) ~ \\text{($\\forall \\theta\\_i, \\theta'\\_i \\in \\Theta\\_i, a'\\_i \\in A\\_i$)}\n\\end{array}\n\\right\\},$$ \nwhere *W* represents a convex combinations of all possible *ψ* and each block matrix *Q**θ**i*, *ψ*(*θ**i*) represents a convex combination of all possible *ϕ*(*θ**i*, ⋅ ). Then we obtain\n \n$${R\\_{\\mathsf{US},i}}= \\max\\_{Q \\in \\calQ} \\sum\\_{t=1}^T \\langle Q x^t, \\bar{u}^t \\rangle - \\sum\\_{t=1}^T \\langle x^t, \\bar{u}^t \\rangle,$$ \nwhere *ū**t* is the reward vector weighted by prior probabilities *ρ**i*, i.e., *ū**t*(*θ**i*, *a**i*) = *ρ**i*(*θ**i*)*u**i**t*(*θ**i*, *a**i*) for each *θ**i* ∈ Θ*i* and *a**i* ∈ *A**i* ().\nIn the second step (), we apply an efficient reduction framework that reduces Φ-regret minimization to online linear optimization with decision space Φ, which was proposed by. This framework internally uses fixed point computation of each $Q \\in \\calQ$. We show that the fixed point $x \\in \\calX$ such that *Q**x* = *x* can be obtained by using eigenvector computation. Now our problem is reduced to online linear optimization with decision space $\\calQ$. Since for online linear optimization with a polytope constraint, there exist efficient algorithms with sublinear external regret (e.g., Follow-the-Perturbed-Leader or Component Hedge ), this problem is already tractable. However, they do not provide an upper bound of the optimal order. We thus need the following third step.\nIn the third step (), we decompose online linear optimization with decision space $\\calQ$ into small external regret minimization problems. Since $\\calQ$ can be regarded as a product of block stochastic matrices and a stochastic matrix in some sense, it can be decomposed into ∣Θ*i*∣2∣*A**i*∣ + ∣Θ*i*∣ probability simplices. Then we can apply algorithms for external regret minimization with a probability simplex. The final algorithm is described in, whose regret upper bound is the following.\n[Informal version of ] There exists an efficiently computable algorithm with $O(\\sqrt{T \\max\\{\\log|\\Theta\\_i|, |A\\_i| \\log |A\\_i|\\}} + |A\\_i| \\log |A\\_i|)$ untruthful swap regret.\nAs a corollary, we obtain an algorithm that computes an approximate communication equilibrium with strategy representability in polynomial time with high probability (). This theorem also answers the problem of minimizing *linear swap regret* proposed by. See and for more details on linear swap regret.\n#### Lower bound for untruthful swap regret\nFor swap regret, already proved an $\\Omega(\\sqrt{ T |A\\_i| \\log |A\\_i|})$ lower bound, which can apply to untruthful swap regret. To show the tightness of the leading term in our upper bound, it is sufficient to prove an $\\Omega(\\sqrt{T \\log|\\Theta\\_i|})$ lower bound.\nThe proof idea for the swap regret lower bounds can be summarized as follows. A problem instance is constructed such that the reward for each action *a**i* ∈ *A**i* in each round *t* ∈ [*T*] is independently chosen from the uniform distribution over {0, 1}. We can assume that the algorithm selects most of the actions *A**i* for at least Ω(*T*/∣*A**i*∣) rounds (This can be guaranteed by “blocking” operations, which set the rewards of actions selected more than a certain number of times to 0 for the remaining rounds). For the rounds when the algorithm selects such an action *a**i* ∈ *A**i*, the gap between the total expected rewards for *a**i* and an optimal action *ϕ*(*a**i*) ∈ *A**i* is $\\Omega \\left(\\sqrt{\\frac{T}{|A\\_i|} \\log |A\\_i|} \\right)$ due to an anti-concentration bound for binomial distributions. The swap regret is at least the sum of $\\Omega \\left(\\sqrt{\\frac{T}{|A\\_i|} \\log |A\\_i|} \\right)$ for these actions, hence $\\Omega(\\sqrt{T |A\\_i| \\log |A\\_i|})$.\nBased on a similar idea, for untruthful swap regret, we can easily prove a lower bound if the algorithm’s decisions are guaranteed to be sufficiently different for different types. For example, we consider a problem instance with two actions *A**i* = {*α*0, *α*1}, in which for each type *θ**i* ∈ Θ*i* and round *t* ∈ [*T*], the reward for *α*0 is chosen from {0, 1} uniformly at random, and the reward for *α*1 is decided by flipping the corresponding reward for *α*0. Then the total expected reward for any algorithm is *T*/2. Assume that the algorithm’s decisions are completely different for different types. For example, we assume that for each type *θ**i* ∈ Θ*i* and round *t* ∈ [*T*], the algorithm chooses *α*0 or *α*1 uniformly at random. Recall that the competitor of untruthful swap regret can apply any type swap *ψ*: Θ*i* → Θ*i*. Since for each pair of *θ**i* and *ψ*(*θ**i*) ∈ Θ*i*, the expected total reward obtained by applying the decision for *ψ*(*θ**i*) to the rewards for *θ**i* follows a binomial distribution, the optimal reward is $T/2 + \\Omega(\\sqrt{T \\log|\\Theta\\_i|})$ again from an anti-concentration bound, which leads to a desired lower bound.\nHowever, it is not easy to guarantee that the algorithm’s decisions are sufficiently different for different types. For any payoffs, if the algorithm’s decisions are completely the same for all types, *ψ* does not yield any untruthful swap regret. We thus need to consider a combination of *ψ* and *ϕ*. For example, if there is a type *θ**i* ∈ Θ*i* for which the payoff for *α*0 is always better than *α*1, the algorithm suffers untruthful swap regret due to *ϕ*(*θ**i*, *α*0) = *ϕ*(*θ**i*, *α*1) = *α*0 unless always choosing *α*0 (as formally stated in ).\n[line width=0.001] (block) [minimum width=0.6, minimum height=0.2] ; (random) at (block.south) [anchor=south, minimum width=0.6, minimum height=0.1, fill=gray, draw=gray, text=white] ; (branch) at (block.north) [anchor=north, minimum width=0.6, minimum height=0.1, draw=gray] ; at (branch.north west) [anchor=north west, minimum width=0.15, minimum height=0.05, fill=black] ; at ([xshift=0.15]branch.north west) [anchor=north west, minimum width=0.15, minimum height=0.025, inner sep=0, fill=black] ; at ([xshift=0.15,yshift=-0.05]branch.north west) [anchor=north west, minimum width=0.15, minimum height=0.025, inner sep=0, fill=black] ; in 0,1,...,3 at ([xshift=0.30,yshift=-0.025\\*]branch.north west) [anchor=north west, minimum width=0.15, minimum height=0.0125, inner sep=0, fill=black] ; in 0,1,...,7 at ([xshift=0.45,yshift=-0.0125\\*]branch.north west) [anchor=north west, minimum width=0.15, minimum height=0.00625, inner sep=0, fill=black] ; in 0.00625,0.0125,...,0.09375 ([yshift=-]branch.north west) – ([yshift=-]branch.north east); in -0.29375,-0.28750,...,0.29375 ([xshift=]branch.north) – ([xshift=]branch.south); at (random) independently from the uniform distribution over {0, 1}; at ([xshift=-0.225]branch.north) [anchor=south] 1st block; at ([xshift=-0.075]branch.north) [anchor=south] 2nd block; at ([xshift=+0.075]branch.north) [anchor=south] ⋯; at ([xshift=+0.225]branch.north) [anchor=south] *B*th block; (par) at (branch.east) [anchor=west, inner sep=0, scale=1.5] $\\Biggr\\}$; at (par.east) [anchor=west, inner sep=0] randomly shuffled; at (branch.west) [anchor=east] Θʹ*i*; at (random.west) [anchor=east] Θʺ*i*;\n[fig:hard-instance]\nBased on this observation, we create a problem instance with *A**i* = {*α*0, *α*1} and the uniform distribution *ρ**i* (). We assume ∣Θ*i*∣ = 2*B* + 1 for some $B \\in \\bbZ$ for simplicity. The set of types Θ*i* is partitioned into Θʹ*i* and Θʺ*i* of equal size. The set of rounds [*T*] is partitioned into *B* blocks of equal length. The rewards for Θʺ*i* are randomly determined as in the problem instance described above. Each *θ*ʹ*i* ∈ Θʹ*i* is associated with a binary sequence of length *B* by a random bijection *ζ*: Θʹ*i* → {0, 1}*B*, and the *b*th bit of this sequence indicates which of *α*0 or *α*1 has reward 1 for *θ*ʹ*i* in the *b*th block.\nThe rewards for Θʹ*i* are designed so that the algorithm must use different decisions for different types. Let *θ**i*0, *θ**i*1 ∈ Θʹ*i* be the types for which *α*0’s and *α*1’s rewards are always 1 (the completely black row and completely white row), respectively. As mentioned above, *ϕ*(*θ**i*, *α*0) = *ϕ*(*θ**i*, *α*1) = *α*0 forces the algorithm to almost always select *α*0 for the type *θ**i*0. Similarly, the algorithm almost always selects *α*1 for the type *θ**i*1 ∈ Θʹ*i*. Since the rewards for Θʹ*i* are randomly shuffled, the algorithm does not know which are *θ**i*0 and *θ**i*1 before reaching the final block. In the first block, since the algorithm does not know which type in the black rows is *θ**i*0 and which type in the white rows is *θ**i*1, the algorithm must almost always choose *α*0 for all the black cells and *α*1 for all the white cells.\nWe are tempted to expect that for the second and later blocks, the algorithm also almost always selects *α*0 for the black cells and *α*1 for the white cells. Unfortunately, there is a counterexample to this expectation. Suppose that the algorithm uses the same action for each contiguous *two* blocks. For example, if the first, second, third, and fourth blocks are black, white, white, and black, respectively, the algorithm uses *α*0 for the first two blocks and *α*1 for the next two blocks. Then the algorithm successfully selects *α*0 for *θ**i*0 and *α*1 for *θ**i*1 in almost all rounds. Furthermore, even though the algorithm obtains the total reward only *T*/2 for some type, this is optimal among all decisions that select the same action for each contiguous two blocks, and then the algorithm suffers no untruthful swap regret.[8](#fn8) We prove that this example is tight, that is, if the untruthful swap regret is small, then the difference of the algorithm’s decisions between the “adjacent” black and white rows must be at least *T*/2 approximately (see for a formal statement). By using this fact, we show that the algorithm’s decisions must be significantly different at each branching point with some constant probability ().\nIf the algorithm always selects *α*0 for the black cells and *α*1 for the white cells, by choosing a better action separately for *θ*ʺ*i* ∈ Θʺ*i* in each block, we can show that the algorithm suffers $B \\cdot \\Omega(\\sqrt{T/B}) = \\Omega(\\sqrt{T \\log |\\Theta\\_i|})$ untruthful swap regret. Since we can control the algorithm’s decisions only for a constant fraction of all rounds, we need to carefully determine *ψ* so that the algorithm cannot manipulate the untruthful swap regret () and trace the difference of cumulative rewards by using martingale analysis (). These analyses lead to the following desired lower bound ().\n[Informal version of ] For any algorithm, there exists a problem instance such that the untruthful swap regret is $\\Omega(\\sqrt{T \\log|\\Theta\\_i|})$.\n#### Smoothness and price of anarchy\nGiven an equilibrium class Π ⊆ Δ(*A*)Θ, the price of anarchy (POA) of a Bayesian game is defined as\n \n$${{\\mathsf{POA}}}\\_\\Pi = \n\\frac{\\displaystyle\\inf\\_{\\pi \\in \\Pi}{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi(\\theta)} \\left[ {v\\_{\\mathsf{SW}}}(\\theta; a) \\right] \\right]}{\\displaystyle{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ \\max\\_{a \\in A} {v\\_{\\mathsf{SW}}}(\\theta; a) \\right]},$$ \nwhere *v*SW: Θ × *A* → [0, 1] is the social welfare function. For analyzing the POA, we make two assumptions: *ρ* is a product distribution and the value of *v**i*(*θ*; *a*) does not depend on *θ*− *i* for each *i* ∈ *N*. These two assumptions are natural when a type represents a preference or attribute of each player. They were assumed in existing studies.\nOur analysis is based on smoothness, which is a standard assumption for bounding the POA of games with complete information. Intuitively, smoothness is a condition that for any action profile, there exists some deviation that significantly increases the social welfare toward the optimum. By assuming (*λ*, *μ*)-smoothness, proved that the POA of coarse correlated equilibria is bounded below by *λ*/(1 + *μ*). Furthermore, showed that a broad class of games satisfies smoothness.\nThe smoothness argument cannot be extended directly to Bayesian games, even for Bayes Nash equilibria. Note that the POA of Bayesian games does not coincide with the POA of the strategic form or agent normal form. This is because the denominator takes an optimal action profile for each *θ* ∈ Θ, which can be better than the expected social welfare for the optimal strategy profile $\\max\\_{s \\in S} {\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {v\\_{\\mathsf{SW}}}(\\theta; s(\\theta)) \\right]$. Therefore, we cannot directly apply the smoothness analysis of the strategic form or agent normal form. Existing studies analyze the POA of Bayes Nash equilibria by making an additional assumption. We will show that their results can be extended from Bayes Nash equilibria to communication equilibria with strategy representability.\nIn, we consider the case where the social welfare is the sum of all players’ payoffs, which is often assumed from the viewpoint of utilitarianism. We extend the proof given by from Bayes Nash equilibria to communication equilibria with strategy representability and obtain a lower bound of *λ*/(1 + *μ*). We make an assumption called *(*λ*, *μ*)-conditional smoothness*, which considers deviations of each player depending on the current action. Since stronger deviations can be used, conditional smoothness is a weaker assumption than (unconditional) smoothness. This assumption seems to correspond to correlated equilibria, and actually used it to bound the POA of correlated equilibria for games with complete information. However, since our current proof uses the property of communication equilibria with strategy representability also for another purpose, even if we assume (unconditional) smoothness, it cannot extend to Bayes coarse correlated equilibria.\nIn, we consider Bayesian games derived from Bayesian mechanisms. defined (*λ*, *μ*)-conditional smoothness[9](#fn9) for mechanisms, which is satisfied by the simultaneous and sequential composition of first-price, second-price, all-pay, and many other auctions. We extend their lower bound of *λ*/max{1, *μ*} to communication equilibria with strategy representability.\nIn, we consider the case where the set of actions for each player changes depending on their type. For example, in a routing game, each player selects their route out of all paths from an origin to a destination, but if the origin-destination pair is determined by their type, then the set of possible actions changes depending on the type. A standard assumption for this setting is *(*λ*, *μ*)-universal smoothness*,[10](#fn10) which was satisfied by routing games and Bayesian effort games. Under the conditional version of this assumption, we prove a lower bound of *λ*/(1 + *μ*) on the POA of communication equilibria with strategy representability. We also briefly discuss how we can extend the definition of communication equilibria, the dynamics, and the untruthful swap regret minimization algorithm to this setting.\nFurther related work\n--------------------\n#### Computational studies on correlated equilibria\nThere exist many studies for computing a correlated equilibrium in games with *complete* information. A pioneering study on this topic was given by, but the time complexity of the proposed algorithm is polynomial in the number of action profiles ∣*A*∣, which is exponential in our setting. There are mainly two approaches to designing efficient algorithms. One approach is based on no-regret dynamics. No-internal-regret dynamics converging to correlated equilibria were developed by and originally in the economics community. It was later applied to efficiently computing correlated equilibria and extended to stronger regret notions. Another approach is *Ellipsoid Against Hope* proposed by (and modified by ), which computes an exact correlated equilibrium of succinctly represented games by the ellipsoid method. On the hardness side, proved that there is no deterministic or exact algorithm for computing a correlated equilibrium with polynomial query complexity.\n#### Computational studies on Bayes correlated equilibria\nWe can find definitions of Bayes correlated equilibria in several computer science papers, but their varieties have not been considered. In Appendix D.2.2 of their paper, mentioned a class of correlated equilibria of two-player Bayesian games, in which *a single player* cannot benefit from any type swap *or* action swap but not their combination. Their definition is somewhat similar to communication equilibria, but the incentive constraints are weak. For Bayesian first-price auctions, considered a class of Bayes correlated equilibria, which is equivalent to Bayesian solutions. Both of them did not provide any algorithm for computing an equilibrium in these classes.\nSome studies mentioned Bayes coarse correlated equilibria. considered no-external-regret dynamics in the agent normal form of Bayesian games and their convergence to an agent-normal-form coarse correlated equilibrium (ANFCCE). Since communication equilibria with strategy representability are included in ANFCCEs, our proposed dynamics converge to a narrower equilibrium concept. The definition of Bayes coarse correlated equilibria used by and is similar to ANFCCEs, but it is defined on Δ(*A*)Θ, not Δ(*S*). We briefly discuss definitions of Bayes coarse correlated equilibria in.\n#### Bayesian persuasion\nBayesian persuasion was proposed as a framework for information design by. As discussed by, Bayesian persuasion is optimization over *universal Bayesian solutions*, which is a variant of Bayes correlated equilibria for the setting where a mediator has more information than players. The set of universal Bayesian solutions is the largest among the variants described in this paper as shown by. Computing a universal Bayesian solution is easier than the other classes considered in this paper, but computing an *optimal* equilibrium is rather difficult. Even for games with complete information, the problem of computing an optimal correlated equilibrium is in general harder than finding a correlated equilibrium. Most of the algorithmic studies of Bayesian persuasion make some assumption to alleviate complexity of equilibria: there is only one player or players’ actions do not affect other players’ payoffs (no-externalities assumption). There are only a small number of exceptions, which deal with specific games such as atomic singleton congestion games.\n#### Correlated equilibria in extensive-form games with imperfect information\n extended the notion of communication equilibria to extensive-form games, in which the mediator receives signals from players and then sends signals to players at the beginning of each node of the game tree. Bayesian games can be regarded as a special case of extensive-form games with imperfect information by regarding types as an action of a chance node. Note that the size of the game tree obtained by this reduction is exponential: the number of actions of this chance node is ∣Θ∣, and even in the case of complete information, the size of the game tree is at least the number of action profiles ∣*A*∣. There is a large amount of literature on computing extensions of correlated equilibrium to extensive-form games with imperfect information, but they consider algorithms running in polynomial time in the size of the game tree, which is exponential in our formulation. mentioned how to design efficiently computable dynamics converging to a communication equilibrium as an open problem, which we partially resolve.\n#### Linear swap regret and polytope swap regret\n considered repeated play of Bayesian games with two players: an optimizer and a learner who follows an online learning algorithm. Their goal is to determine which concept of regret corresponds to the optimizer’s benefit compared with the Stackelberg value. Although their goal is different from ours, they proposed similar variants of regret: *linear swap regret* and *polytope swap regret*. For any polytope, linear swap regret is defined by the competitor that can use *any* valid linear transformation[11](#fn11) for swapping the algorithm’s decision. Note that already proposed the same concept for general convex compact sets as *linear regret*. Since untruthful swap regret is defined by a specific class of valid linear transformations represented by $\\calQ$, it is easier to minimize than linear swap regret. proposed an algorithm for minimizing linear swap regret, but they did not prove any rigorous regret bound, and this algorithm was not proved to be efficient. In, we prove that linear swap regret is equivalent to untruthful swap regret when the decision space is $\\calX$ defined in, and then our algorithm can be used for minimizing linear swap regret.\nPolytope swap regret is defined by the competitor that can use any swap of the vertices of the polytope, while the vertex decomposition of the algorithm’s decision in each round can be optimally decided by the learner in hindsight. Since our strategy swap regret requires the algorithm to submit vertex decomposition in each round irrevocably, it is more difficult to minimize than polytope swap regret.\n#### Price of anarchy of Bayesian games\nThe definition of the POA for games with complete information was provided by and then extended from Nash equilibria to (coarse) correlated equilibria by. Most existing studies for Bayesian games focus on the POA of Bayes Nash equilibria. An exception is a study by, which extended the smoothness framework for mechanisms developed by to ANFCCEs. However, its applications are not clear due to the following reason. showed that if a Bayesian mechanism satisfies (unconditional) smoothness, the POA of Bayes coarse correlated equilibria is bounded. showed that a wide class of mechanisms satisfies *conditional* smoothness, which assumes that the deviation can depend on the current action (see for a formal definition). By carefully reading proofs of conditional smoothness of mechanisms presented by, we can see that some mechanisms introduced in their paper satisfy (unconditional) smoothness, but not all. We prove that conditional smoothness is sufficient for extending the POA bounds to communication equilibria with strategy representability in.\nAs mentioned above, and considered a variant of ANFCCEs defined on Δ(*A*)Θ. proved POA bounds of this class for generalized second-price auctions. For first-price auctions, considered the price of stability, which is defined to be the ratio of the *optimal* social welfare achieved by equilibria to the maximum social welfare.\nIn, we assume that the players’ types are independent, i.e., the prior distribution *ρ* is a product distribution. called the POA for the correlated case *cPOA* (correlated POA) and showed a lower bound if the deviation of each player *i* ∈ *N* can be determined only by *θ**i* in the definition of smoothness.\nOrganization of this paper\n--------------------------\nThe remaining part of this paper is organized as follows.\n* provides various definitions and characterizations of approximate Bayes correlated equilibria and then shows the convergence of variants of no-regret dynamics. defines strategy representability, which is used for the characterizations. specifies the online learning setting that each player is faced with in dynamics. provides a definition of approximate SFCEs and shows that dynamics minimizing a variant of regret called strategy swap regret converge to them. provides a definition of approximate ANFCEs and shows that dynamics minimizing a variant of regret called type-wise swap regret converge to them., the main part of this section, provides a definition of approximate communication equilibria and shows that dynamics minimizing untruthful swap regret converge to communication equilibria with strategy representability. shows several inclusion relations between the classes of Bayes correlated equilibria and Bayes Nash equilibria.\n* derives an efficient algorithm for minimizing untruthful swap regret in three steps, each of which is presented in. applies this algorithm to computing a communication equilibrium with strategy representability.\n* proves a lower bound on untruthful swap regret. The proof consists of four steps, each of which is presented in.\n* shows lower bounds on the price of anarchy for each of three cases: assumes that the social welfare is the sum of payoffs, assumes that the game is derived from a Bayesian mechanism with a quasilinear utility function, and assumes a stronger version of smoothness called universal smoothness.\nBayes correlated equilibria and no-regret dynamics\n==================================================\nIn this section, we review various classes of Bayes correlated equilibria surveyed by and propose variants of no-regret dynamics converging to them. The highlight of this section is the definition of untruthful swap regret and the convergence of dynamics minimizing it to communication equilibria with strategy representability (). As a byproduct, we characterize each class of Bayes correlated equilibria, propose dynamics converging to SFCEs and ANFCEs, and show relations among these classes.\nStrategy representability\n-------------------------\nTo classify Bayes correlated equilibria in terms of distributions, we need to compare different kinds of distributions. As we will see, SFCEs and ANFCEs are defined as distributions over strategy profiles, that is, *σ* ∈ Δ(*S*). On the other hand, communication equilibria and Bayesian solutions are defined as distributions over action profiles for each type profile, which we call *type-wise distributions*. A type-wise distribution *π* ∈ Δ(*A*)Θ determines a distribution *π*(*θ*) ∈ Δ(*A*) for each type *θ* ∈ Θ. We write *π*(*θ*; *a*) instead of *π*(*θ*)(*a*) for readability.\nTo compare these distributions, we adopt the approach of transforming *σ* ∈ Δ(*S*) into *π* ∈ Δ(*A*)Θ by considering the marginal distribution. Suppose that the mediator samples *s* ∼ *σ* independently of *θ* ∼ *ρ* and then recommends *s**i*(*θ**i*) to each player *i* ∈ *N*. Then, for each *σ* ∈ Δ(*S*), we can define its corresponding distribution *π* ∈ Δ(*A*)Θ by\n \n*π*(*θ*; *a*) = Pr*s* ∼ *σ*(*s*(*θ*) = *a*) \nfor each *θ* ∈ Θ and *a* ∈ *A*. This operation defines a function *η*: Δ(*S*) → Δ(*A*)Θ.\nWe define the strategy representability as the property that *π* ∈ *η*(Δ(*S*)). This property implies that the mediator can realize *π* by sending a randomized recommendation of *s* ∈ *S* without observing the type profile *θ* ∈ Θ.\n[Strategy representability][def:sr] We call a type-wise distribution *π* ∈ Δ(*A*)Θ *strategy-representable* if there exists *σ* ∈ Δ(*S*) such that *π*(*θ*; *a*) = Pr*s* ∼ *σ*(*s*(*θ*) = *a*) for each *θ* ∈ Θ and *a* ∈ *A*.\nAs we show below, not all type-wise distributions Δ(*A*)Θ are strategy-representable. That is, if we generate *s* ∼ *σ* independently of *θ* ∼ *ρ*, we cannot realize all distributions in Δ(*A*)Θ.\n[line width=1pt]\nat (-20pt,55pt)*θ*1; at (-20pt,10pt)*θ*ʹ1;\nat (-10pt,65pt)*a*1; at (-10pt,45pt)*a*ʹ1; at (-10pt,20pt)*a*1; at (-10pt,00pt)*a*ʹ1;\nat (10pt,85pt)*θ*2; at (55pt,85pt)*θ*ʹ2;\nat (00pt,75pt)*a*2; at (20pt,75pt)*a*ʹ2; at (45pt,75pt)*a*2; at (65pt,75pt)*a*ʹ2;\nat (00pt,45pt)0; at (00pt,65pt)1/2; at (20pt,45pt)1/2; at (20pt,65pt)0;\nat (45pt,45pt)0; at (45pt,65pt)1/2; at (65pt,45pt)1/2; at (65pt,65pt)0;\nat (00pt,00pt)0; at (00pt,20pt)1/2; at (20pt,00pt)1/2; at (20pt,20pt)0;\nat (45pt,00pt)1/2; at (45pt,20pt)0; at (65pt,00pt)0; at (65pt,20pt)1/2;\n[fig:non-representable]\n[ex:sr] We give an example *π* ∈ Δ(*A*)Θ that cannot be represented by any *σ* ∈ Δ(*S*). Suppose that *A*1 = {*a*1, *a*ʹ1}, *A*2 = {*a*2, *a*ʹ2}, Θ1 = {*θ*1, *θ*ʹ1}, and Θ2 = {*θ*2, *θ*ʹ2}. Let *ρ* be the uniform distribution on Θ1 × Θ2. As illustrated in, we define *π* ∈ Δ(*A*)Θ by $\\pi(\\theta\\_1,\\theta\\_2; a\\_1,a\\_2)\n= \\pi(\\theta\\_1,\\theta\\_2; a'\\_1,a'\\_2)\n= \\pi(\\theta\\_1,\\theta'\\_2; a\\_1,a\\_2)\n= \\pi(\\theta\\_1,\\theta'\\_2; a'\\_1,a'\\_2)\n= \\pi(\\theta'\\_1,\\theta\\_2; a\\_1,a\\_2)\n= \\pi(\\theta'\\_1,\\theta\\_2; a'\\_1,a'\\_2)\n= \\pi(\\theta'\\_1,\\theta'\\_2; a\\_1,a'\\_2)\n= \\pi(\\theta'\\_1,\\theta'\\_2; a'\\_1,a\\_2)\n= \\frac{1}{2}$. We show that *π* is not strategy-representable. If there exists *σ* ∈ Δ(*S*) that satisfies *π*(*θ*; *a*) = Pr*s* ∼ *σ*(*s*(*θ*) = *a*) for any *θ* ∈ Θ and *a* ∈ *A*, then $\\Pr(s\\_1(\\theta\\_1) = a\\_1, ~ s\\_2(\\theta\\_2) = a\\_2) = \\Pr(s\\_1(\\theta\\_1) = a'\\_1, ~ s\\_2(\\theta\\_2) = a'\\_2) = \\frac{1}{2}$. This implies that the event *s*1(*θ*1) = *a*1 coincides with the event *s*2(*θ*2) = *a*2, and *s*1(*θ*1) = *a*ʹ1 coincides with *s*2(*θ*2) = *a*ʹ2. Similarly, from $\\Pr(s\\_1(\\theta\\_1) = a\\_1, ~ s\\_2(\\theta'\\_2) = a\\_2) = \\Pr(s\\_1(\\theta\\_1) = a'\\_1, ~ s\\_2(\\theta'\\_2) = a'\\_2) = \\frac{1}{2}$, we can see that *s*1(*θ*1) = *a*1 coincides with *s*2(*θ*ʹ2) = *a*2, and *s*1(*θ*1) = *a*ʹ1 coincides with *s*2(*θ*ʹ2) = *a*ʹ2. Therefore, *s*2(*θ*2) = *s*2(*θ*ʹ2) holds with probability 1. By symmetry, we can see that *s*1(*θ*1) = *s*1(*θ*ʹ1) holds with probability 1. Hence, the distributions of (*s*1(*θ*1), *s*2(*θ*2)) and (*s*1(*θ*ʹ1), *s*2(*θ*ʹ2)) must be identical, but they are not, which leads to a contradiction. This implies that *π* is not strategy-representable.\n did not formally define the strategy representability but instead claimed that a property called *conditional independence property* is important to classify Bayes correlated equilibria. As claimed, the conditional independence property holds for any strategy-representable distribution. Although it was not stated explicitly, the conditional independence property is not equivalent to the strategy representability (see ).[12](#fn12) As discussed below, the strategy representability is the exact difference between several classes of Bayes correlated equilibria.\nWe say *π* ∈ Δ(*A*)Θ is a *type-wise product distribution* if there exists some *π**i*, *θ**i* ∈ Δ(*A**i*) for each *i* ∈ *N* and *θ**i* ∈ Θ*i* such that *π*(*θ*; *a*) = ∏*i* ∈ *N**π**i*, *θ**i*(*a**i*) for every *θ* ∈ Θ and *a* ∈ *A*. Let Πprod ⊆ Δ(*A*)Θ be the set of all type-wise product distributions. As we show below, a finite mixture of type-wise product distributions is always strategy-representable. We will use this fact to prove that the average of finite rounds of uncoupled dynamics in the agent normal form is strategy-representable since the distribution in each round is type-wise product. Furthermore, the inverse also holds.\n[prop:sr] A distribution *π* ∈ Δ(*A*)Θ is strategy-representable if and only if *π* is a finite mixture of type-wise product distributions, that is, there exist a positive integer *T*, *σ*ʹ ∈ Δ([*T*]), and *π**t* ∈ ΠProd for each *t* ∈ [*T*] such that *π*(*θ*; *a*) = ∑*t* ∈ [*T*]*σ*ʹ(*t*)*π**t*(*θ*; *a*) for each *θ* ∈ Θ and *a* ∈ *A*.\nAssume *π* is strategy-representable, that is, there exists *σ* ∈ Δ(*S*) such that *π*(*θ*; *a*) = Pr*s* ∼ *σ*(*s*(*θ*) = *a*) for each *θ* ∈ Θ and *a* ∈ *A*. Let *σ*ʹ = *σ* and define *π**s* ∈ Δ(*A*)Θ by *π**s*(*θ*; *a*) = **1**{*s*(*θ*) = *a*}. We can check *π**s* ∈ ΠProd by setting *π**i*, *θ**i*(*a**i*) = **1**{*s**i*(*θ**i*) = *a**i*}. Then\n \n*π*(*θ*; *a*) = Pr*s* ∼ *σ*(*s*(*θ*) = *a*) = ∑*s* ∈ *S**σ*(*s*)**1**{*s*(*θ*) = *a*} = ∑*s* ∈ *S**σ*ʹ(*s*)*π**s*(*θ*; *a*) \nfor each *θ* ∈ Θ and *a* ∈ *A*.\nAssume there exists a positive integer *T*, *σ*ʹ ∈ Δ(*T*), and *π**t* ∈ ΠProd for each *t* ∈ [*T*] such that *π*(*θ*; *a*) = ∑*t* ∈ [*T*]*σ*ʹ(*t*)*π**t*(*θ*; *a*). Since *π**t* ∈ ΠProd for each *t* ∈ [*T*], there exists *π**i*, *θ**i**t* ∈ Δ(*A**i*) for each *i* ∈ *N* and *θ**i* ∈ Θ*i* such that *π**t*(*θ*; *a*) = ∏*i* ∈ *N**π**i*, *θ**i**t*(*a**i*) for each *θ* ∈ Θ and *a* ∈ *A*. We define *σ**t* ∈ Δ(*S*) by *σ**t*(*s*) = ∏*i* ∈ *N*∏*θ**i* ∈ Θ*i**π**i*, *θ**i**t*(*s**i*(*θ**i*)) for each *s* ∈ *S*. Then for each *θ* ∈ Θ and *a* ∈ *A*, it holds that\n \n$$\\Pr\\_{s \\sim \\sigma^t} (s(\\theta) = a)\n= \\sum\\_{\\substack{s \\in S \\colon \\\\ s(\\theta) = a}} \\prod\\_{i \\in N} \\prod\\_{\\theta\\_i \\in \\Theta\\_i} \\pi^t\\_{i,\\theta\\_i}(s\\_i(\\theta\\_i))\n= \\prod\\_{i \\in N} \\pi^t\\_{i,\\theta\\_i}(a\\_i)\n= \\pi^t(\\theta;a).$$ \nDefine *σ* ∈ Δ(*S*) by *σ*(*s*) = ∑*t* ∈ [*T*]*σ*ʹ(*t*)*σ**t*(*s*) for every *s* ∈ *S*. Then\n \nPr*s* ∼ *σ*(*s*(*θ*) = *a*) = ∑*t* ∈ [*T*]*σ*ʹ(*t*)Pr*s* ∼ *σ**t*(*s*(*θ*) = *a*) = ∑*t* ∈ [*T*]*σ*ʹ(*t*)*π**t*(*θ*; *a*) = *π*(*θ*; *a*) \nfor each *θ* ∈ Θ and *a* ∈ *A*, which implies that *π* is strategy-representable.\nOnline learning with stochastic types\n-------------------------------------\nTo discuss dynamics converging to Bayes correlated equilibria, we define *online learning with stochastic types*, which is an online decision-making problem that each player is faced with in repeated play of Bayesian games.\nThis problem is defined as an extension of an ordinary online learning setting. Recall that in each round of online learning, a learner with a set of finite actions *A**i* decides a randomized action *a**i* ∈ *A**i* and then obtains payoff *u**i**t*(*a**i*) according to a reward vector *u**i**t* ∈ [0, 1]*A**i*. In online learning with stochastic types, in addition to actions *A**i*, the learner is associated with a set of finite types Θ*i* and a prior distribution *ρ**i* ∈ Δ(Θ*i*). In each round *t* ∈ [*T*], the learner decides a strategy distribution *σ**i**t* ∈ Δ(*S**i*). This distribution randomly determines a strategy *s**i* ∈ *A**i*Θ*i* that assigns an action *a**i* ∈ *A**i* to each type *θ**i* ∈ Θ*i*. Then the learner observes a reward vector *u**i**t* ∈ [0, 1]Θ*i* × *A**i* that assigns a reward value for each type *θ**i* ∈ Θ*i* and action *a**i* ∈ *A**i*. The learner’s action *s**i*(*θ**i*) ∈ *A**i* is randomly determined by strategy *s**i* ∼ *σ**i**t* and type *θ**i* ∼ *ρ**i*. The expected reward for randomized strategy *σ**i**t* ∈ Δ(*A**i*Θ*i*) is therefore ${\\mathop{\\mathbb{E}}}\\_{\\theta\\_i \\sim \\rho\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{s\\_i \\sim \\sigma\\_i^t} \\left[ u\\_i^t(\\theta\\_i, s\\_i(\\theta\\_i)) \\right] \\right]$.\nSuppose that players are repeatedly involved in the same Bayesian game and decide their own randomized strategies in each round. Then the online decision-making problem that each player is faced with can be formulated as online learning with stochastic types. In each round *t* ∈ [*T*], each player *i* ∈ *N* decides a randomized strategy *σ**i**t* ∈ Δ(*S**i*) and then obtains the expected payoff ${\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma^t} \\left[ v\\_i(\\theta;s(\\theta)) \\right] \\right]$, where *σ**t* is the product distribution that independently generates *s**j* ∼ *σ**j**t* for each *j* ∈ *N*. We define the reward vector *u**i**t* ∈ [0, 1]Θ*i* × *A**i* by\n \n$$\\label{eq:reward}\nu\\_i^t(\\theta\\_i, a\\_i) = {\\mathop{\\mathbb{E}}}\\_{\\theta\\_{-i} \\sim \\rho|\\theta\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{s\\_{-i} \\sim \\sigma^t\\_{-i}} \\left[ v\\_i( \\theta; a\\_i, s\\_{-i}(\\theta\\_{-i})) \\right] \\right]$$ \nfor each type *θ**i* ∈ Θ*i* and action *a**i* ∈ *A**i*, where *σ*− *i**t* is the product distribution that independently generates *s**j* ∼ *σ**j**t* for each *j* ∈ *N* \\ {*i*}. By using this reward vector, the expected payoff for player *i* ∈ *N* can be expressed as\n \n$$\\begin{aligned}\n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma^t} \\left[ v\\_i(\\theta;s(\\theta)) \\right] \\right]\n&=\n{\\mathop{\\mathbb{E}}}\\_{\\theta\\_i \\sim \\rho\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{s\\_i \\sim \\sigma\\_i^t} \\left[ {\\mathop{\\mathbb{E}}}\\_{\\theta\\_{-i} \\sim \\rho|\\theta\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{s\\_{-i} \\sim \\sigma^t\\_{-i}} \\left[ v\\_i( \\theta; s\\_i(\\theta\\_i), s\\_{-i}(\\theta\\_{-i})) \\right] \\right] \\right] \\right]\\\\\n&=\n{\\mathop{\\mathbb{E}}}\\_{\\theta\\_i \\sim \\rho\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{s\\_i \\sim \\sigma\\_i^t} \\left[ u\\_i^t(\\theta\\_i, s\\_i(\\theta\\_i)) \\right] \\right].\\end{aligned}$$ \nThus, the problem for player *i* ∈ *N* in repeated play of Bayesian games is online learning with stochastic types specified by actions *A**i*, types Θ*i*, prior distribution *ρ**i* ∈ Δ(Θ*i*), and reward vector *u**i**t* ∈ [0, 1]Θ*i* × *A**i* for each *t* ∈ [*T*] as defined above. Note that the decision space is *S**i*, but the rewards for them can be succinctly represented by *u**i**t* ∈ [0, 1]Θ*i* × *A**i*. If the goal is to minimize external regret, this problem can be solved separately for each type. By considering various extensions of swap regret to this setting, we will define dynamics converging to each class of Bayes correlated equilibria.\nFor ANFCEs () and communication equilibria (), we restrict the decision *σ**i**t* ∈ Δ(*S**i*) of each player *i* ∈ *N* to a product distribution, i.e., there exists some *π**i**t* ∈ Δ(*A**i*)Θ*i* such that *σ**i**t*(*s**i*) = ∏*θ**i* ∈ Θ*i**π**i**t*(*θ**i*; *s**i*(*θ**i*)) for each *s**i* ∈ *S**i*. Then the definition of reward vector *u**i**t* ∈ [0, 1]Θ*i* × *A**i* is\n \n$$\\label{eq:reward-pi}\nu\\_i^t(\\theta\\_i, a\\_i) = {\\mathop{\\mathbb{E}}}\\_{\\theta\\_{-i} \\sim \\rho|\\theta\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{a\\_{-i} \\sim \\pi^t\\_{-i}(\\theta\\_{-i})} \\left[ v\\_i( \\theta; a) \\right] \\right]$$ \nfor each type *θ**i* ∈ Θ*i* and action *a**i* ∈ *A**i*, where *π*− *i**t*(*θ*− *i*) is the product distribution that independently generates *a**j* ∼ *π**j**t*(*θ**j*) for each *j* ∈ *N* \\ {*i*}. While a strategy distribution *σ**i**t* ∈ Δ(*S**i*) requires space exponential in ∣Θ*i*∣, a product distribution can be succinctly represented by *π**i**t* ∈ Δ(*A**i*)Θ*i* in polynomial space. For ANFCEs and communication equilibria, we will define type-wise swap regret and untruthful swap regret, for which a product distribution is sufficient. On the other hand, for SFCEs, we define strategy swap regret, for which we need to consider a strategy distribution ().\nStrategic-form correlated equilibria\n------------------------------------\n*Strategic-form correlated equilibria* (SFCEs; also known as normal-form correlated equilibria) are defined as correlated equilibria of the strategic form of a Bayesian game. Recall that in the strategic form, the set of strategies *S**i* = *A**i*Θ*i* is considered as a decision space for each player *i* ∈ *N*. This class is defined as a set of joint distributions *σ* ∈ Δ(*S*) on strategy profiles in which each player *i* ∈ *N* cannot gain by deviating from recommended strategy *s**i* ∈ *S**i* in expectation. This deviation is represented by a swap *ϕ*SF: *S**i* → *S**i* that maps recommended strategy *s**i* ∈ *S**i* to another strategy *ϕ*SF(*s**i*). Here we define an approximate version in which each player can gain at most *ε* in expectation by deviating from the recommendation.\n[*ε*-Approximate strategic-form correlated equilibria (*ε*-SFCEs)] For any *ε* ≥ 0, a distribution *σ* ∈ Δ(*S*) is an *ε*-approximate strategic-form correlated equilibrium if for any *i* ∈ *N* and any *ϕ*SF: *S**i* → *S**i*, it holds that\n \n$$\\label{eq:ic-sf}\n\\tag{$\\mathrm{IC}\\_{\\mathrm{SF}}$}\n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta; s(\\theta)) \\right] \\right]\n\\ge \n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta; ({\\phi\\_{\\mathsf{SF}}}(s\\_i))(\\theta\\_i),s\\_{-i}(\\theta\\_{-i})) \\right] \\right] - \\epsilon.$$ \nLet ΣSF*ε* ⊆ Δ(*S*) be the set of all *ε*-SFCEs.\nThe notion of SFCE is a natural equilibrium concept for the following scenario. A mediator samples *s* ∈ *S* according to a distribution *σ* ∈ Δ(*S*) and then informs each player *i* ∈ *N* of *s**i*. Independently, a type profile *θ* is sampled from *ρ* and each player *i* ∈ *N* observes their own type *θ**i*. Then, each player *i* ∈ *N* decides their action *s**i*(*θ**i*) ∈ *A**i*. If every player cannot gain by deviating from action *s**i*(*θ**i*) in expectation, the distribution *σ* is an SFCE.\nThe role of this mediator can be interpreted as a correlation device. A correlation device privately sends (possibly correlated) signals to each player according to some distribution. An SFCE can be defined as a Nash equilibrium of an extended game equipped with a correlation device. A correlation device can send any signal in general, but from the revelation principle, we can assume that each signal corresponds to a recommendation of each strategy without loss of generality.\nSince SFCEs are correlated equilibria of the strategic form of Bayesian games, no-swap-regret dynamics converging to a correlated equilibrium can be directly extended to dynamics converging to an SFCE. Recall that the decision space for each player *i* ∈ *N* in the strategic form is *S**i*. Then each player *i* ∈ *N* decides a strategy distribution *σ**i**t* ∈ Δ(*S**i*) in each round *t* ∈ [*T*] and obtains the expected payoff ${\\mathop{\\mathbb{E}}}\\_{\\theta\\_i \\sim \\rho\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{s\\_i \\sim \\sigma\\_i^t} \\left[ u\\_i^t(\\theta\\_i, s\\_i(\\theta\\_i)) \\right] \\right]$ according to reward vector *u**i**t* ∈ [0, 1]Θ*i* × *A**i* defined as, which is the setting of online learning with stochastic types. We consider an extension of swap regret to this online learning problem with decision space *S**i*, which we call *strategy swap regret*.\n[Strategy swap regret] For online learning with stochastic types specified by actions *A**i*, types Θ*i*, prior distribution *ρ**i*, and reward vector *u**i**t* ∈ [0, 1]Θ*i* × *A**i* for every time round *t* ∈ [*T*], strategy swap regret is defined as\n \n$${R\\_{\\mathsf{SS},i}}^T = \\max\\_{{\\phi\\_{\\mathsf{SF}}}\\colon S\\_i \\to S\\_i} \\sum\\_{t=1}^T {\\mathop{\\mathbb{E}}}\\_{\\theta\\_i \\sim \\rho\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{s\\_i \\sim \\sigma\\_i^t} \\left[ u\\_i^t(\\theta\\_i, ({\\phi\\_{\\mathsf{SF}}}(s\\_i))(\\theta\\_i)) - u\\_i^t(\\theta\\_i, s\\_i(\\theta\\_i)) \\right] \\right].$$ \n[t] [alg:sf-dynamics] For each player *i* ∈ *N*, let $\\calA\\_{i}$ be a subroutine that minimizes strategy swap regret for online learning with stochastic types\nTheorem 3 of relates each player’s swap regret with the convergence to a correlated equilibrium. By directly applying it to the strategic form of Bayesian games, we can show that if the strategy swap regret of each player *i* ∈ *N* grows sublinear in *T*, we can guarantee the convergence of the dynamics to an SFCE. This is claimed in the following proposition formally.\n[prop:dynamics-sf] Let *σ**i**t* ∈ Δ(*S**i*) be the randomized strategy of each player *i* ∈ *N* for each round *t* ∈ [*T*] in. Then, the empirical distribution *σ* ∈ Δ(*S*) defined by $\\sigma(s) = \\frac{1}{T} \\sum\\_{t=1}^T \\prod\\_{i \\in N} \\sigma\\_i^t(s\\_i)$ for each *s* ∈ *S* is a $\\frac{\\max\\_{i \\in N} {R\\_{\\mathsf{SS},i}}^T}{T}$-SFCE, where *R*SS, *i**T* is the strategy swap regret for each subroutine $\\calA\\_i$.\nFrom the definition of *ε*-SFCEs, it is sufficient to prove\n \n$${\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta; ({\\phi\\_{\\mathsf{SF}}}(s\\_i))(\\theta\\_i),s\\_{-i}(\\theta\\_{-i})) \\right] \\right]\n-\n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta; s(\\theta)) \\right] \\right]\n\\le \\frac{\\max\\_{j \\in N} {R\\_{\\mathsf{SS},j}}^T}{T}$$ \nfor each *i* ∈ *N* and *ϕ*SF: *S**i* → *S**i*. Let *σ**t* ∈ Δ(*S*− *i*) be the product distribution that independently generates *s**j* ∼ *σ**j**t* for each *j* ∈ *N* and *σ*− *i**t* ∈ Δ(*S*) the product distribution that independently generates *s**j* ∼ *σ**j**t* for each *j* ∈ *N* \\ {*i*}. Then the left-hand side can be bounded as\n \n$$\\begin{aligned}\n&{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta; ({\\phi\\_{\\mathsf{SF}}}(s\\_i))(\\theta\\_i),s\\_{-i}(\\theta\\_{-i})) \\right] \\right]\n- \n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta; s(\\theta)) \\right] \\right]\\\\\n&= \n{\\mathop{\\mathbb{E}}}\\_{t \\sim [T]} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma^t} \\left[ {\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ v\\_i(\\theta; ({\\phi\\_{\\mathsf{SF}}}(s\\_i))(\\theta\\_i),s\\_{-i}(\\theta\\_{-i})) - v\\_i(\\theta; s(\\theta)) \\right] \\right] \\right] \\tag{from the definition of $\\sigma$} \\\\\n&= \n\\frac{1}{T} \\sum\\_{t=1}^T {\\mathop{\\mathbb{E}}}\\_{s\\_i \\sim \\sigma\\_i^t} \\left[ {\\mathop{\\mathbb{E}}}\\_{s\\_{-i} \\sim \\sigma\\_{-i}^t} \\left[ {\\mathop{\\mathbb{E}}}\\_{\\theta\\_i \\sim \\rho\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{\\theta\\_{-i} \\sim \\rho|\\theta\\_i} \\left[ v\\_i(\\theta; ({\\phi\\_{\\mathsf{SF}}}(s\\_i))(\\theta\\_i), s\\_{-i}(\\theta\\_{-i}))\n- \nv\\_i(\\theta; s(\\theta))\\right] \\right] \\right]\\right] \\tag{since $\\sigma^t$ is the product distribution}\\\\\n&= \n\\frac{1}{T} \\sum\\_{t=1}^T {\\mathop{\\mathbb{E}}}\\_{s\\_i \\sim \\sigma\\_i^t} \\left[ {\\mathop{\\mathbb{E}}}\\_{\\theta\\_{i} \\sim \\rho\\_{i}} \\left[ u\\_i^t(\\theta\\_i, ({\\phi\\_{\\mathsf{SF}}}(s\\_i))(\\theta\\_i))\n- \nu\\_i^t(\\theta\\_i, s\\_i(\\theta\\_i))\\right] \\right] \\tag{from the definition of $u\\_i^t$}\\\\\n&\\le \\frac{{R\\_{\\mathsf{SS},i}}^T}{T} \\\\\n&\\le \\frac{\\max\\_{j \\in N} {R\\_{\\mathsf{SS},j}}^T}{T},\\end{aligned}$$ \nwhich completes the proof.\nWe provide a detailed description of the dynamics in. We should note that this proposition does not imply the existence of a polynomial-time algorithm for computing an approximate SFCE. This is because the size of the decision space *S**i* for each player *i* ∈ *N* is exponential. In, we provide an algorithm that has an $O\\left(\\sqrt{T |A\\_i|^{|\\Theta\\_i|} \\log |A\\_i|} + |A\\_i|^{|\\Theta\\_i|} \\log |A\\_i| \\right)$ upper bound on strategy swap regret, which is slightly better than a direct application of the swap regret minimization algorithm proposed by but still exponential in ∣Θ*i*∣. Computing an *ε*-SFCE by simulating the dynamics with this algorithm requires time exponential in ∣Θ*i*∣. Whether there exists an algorithm that computes an *ε*-SFCE in time polynomial in *n*, ∣*A**i*∣, ∣Θ*i*∣, and 1/*ε* with an oracle for utility functions (*v**i*)*i* = 1*n* is an open problem left for future work.\nWe define approximate SFCEs based on swap regret, but it is also possible to define them based on internal regret. Note that we have two different definitions of approximate correlated equilibria that are based on internal regret or swap regret.[13](#fn13) These definitions coincide when the error is 0, but different when the error is non-zero. In this paper, we adopt the extension based on swap regret, but it is also possible to use the extension based on internal regret, in which *ϕ*SF: *S**i* → *S**i* in the definition is restricted to a swap for a single strategy:\n \n$${\\phi\\_{\\mathsf{SF}}}(s\\_i) = \n\\begin{cases}\ns''\\_i & \\text{if $s\\_i = s'\\_i$}\\\\\ns\\_i & \\text{otherwise}\n\\end{cases}$$ \nfor some *s*ʹ*i*, *s*ʺ*i* ∈ *S**i*. However, the problem of computing this version of approximate SFCEs is rather easy. It is because if we set *σ* ∈ Δ(*S*) to the uniform distribution over *S*, then the error is always at most 1/(min*i* ∈ *N*∣*S**i*∣). Hence, for *ε* ≥ 1/(min*i* ∈ *N*∣*S**i*∣), computing it can be done easily. On the other hand, for *ε* ≤ 1/(min*i* ∈ *N*∣*S**i*∣), if we assume ∣*S**i*∣ has the same order for every *i* ∈ *N*, then we can straightforwardly run an internal regret minimization algorithm with decision space *S**i* in time polynomial in 1/*ε* = min*i* ∈ *N*∣*S**i*∣.\nTo compare SFCEs with other classes defined on Δ(*A*)Θ, we define the set of distributions obtained by mapping SFCEs to Δ(*A*)Θ by *η*: Δ(*S*) → Δ(*A*)Θ (see for the definition of *η*). Formally, it is defined as ΠSF*ε* = {*η*(*σ*) ∈ Δ(*A*)Θ ∣ *σ* ∈ ΣSF*ε*}. From the definition, every *π* ∈ ΠSF*ε* is strategy-representable.\nAs we will see, ΠSF*ε* is a subset of approximate communication equilibria. Informally, computing an *ε*-SFCE is harder than computing an *ε*-approximate communication equilibrium. If an *ε*-SFCE *σ* ∈ Δ(*S*) can be obtained, then we can compute its corresponding distribution *η*(*σ*) ∈ Δ(*A*)Θ, which is also an *ε*-approximate communication equilibrium. Note that this reduction is informal because distributions in Δ(*S*) and Δ(*A*)Θ have an exponentially large support in general, and the computational complexity of computing *η*(*σ*) depends on the succinct representation of *σ*.\nWhether *σ* ∈ Δ(*S*) is an SFCE cannot be determined by its corresponding *η*(*σ*) ∈ Δ(*A*)Θ. As in the following example, even if *σ*, *σ*ʹ ∈ Δ(*S*) correspond to the same *π* ∈ Δ(*A*)Θ, it is possible that *σ* is an SFCE, but *σ*ʹ is not.\nSuppose Θ1 = {*θ*1, *θ*ʹ1} and Θ2 = {*θ*2} with *A*1 = {*a*1, *a*ʹ1} and *A*2 = {*a*2, *a*ʹ2}. The prior distribution *ρ* ∈ Δ(Θ) is the uniform distribution over Θ. Let *v*1(*θ̃*; *a*1, *a*2) = *v*1(*θ̃*; *a*ʹ1, *a*ʹ2) = 1 and *v*1(*θ̃*; *a*ʹ1, *a*2) = *v*1(*θ̃*; *a*1, *a*ʹ2) = 0 for any *θ̃* ∈ Θ, while the second player’s payoff is always 0, i.e., *v*2 ≡ 0. We consider two distinct distributions *σ*, *σ*ʹ ∈ Δ(*S*) with the same corresponding *π* ∈ Δ(*A*)Θ. The first one *σ* is the uniform distribution over *S*. The second one *σ*ʹ is the uniform distribution over *s*1, *s*2, *s*3, *s*4 defined as\n \n$$\\begin{aligned}\n&s^1\\_1(\\theta\\_1) = a\\_1, ~s^1\\_1(\\theta'\\_1) = a\\_1, ~s^1\\_2(\\theta\\_2) = a\\_2, \\\\\n&s^2\\_1(\\theta\\_1) = a'\\_1, ~s^2\\_1(\\theta'\\_1) = a'\\_1, ~s^2\\_2(\\theta\\_2) = a\\_2, \\\\\n&s^3\\_1(\\theta\\_1) = a\\_1, ~s^3\\_1(\\theta'\\_1) = a'\\_1, ~s^3\\_2(\\theta\\_2) = a'\\_2, \\\\\n&s^4\\_1(\\theta\\_1) = a'\\_1, ~s^4\\_1(\\theta'\\_1) = a\\_1, ~s^4\\_2(\\theta\\_2) = a'\\_2.\\end{aligned}$$ \nBoth *π*(*θ*1, *θ*2) and *π*(*θ*ʹ1, *θ*2) are the uniform distribution over *A*1 × *A*2. The expected payoff for each type of *θ*1 and *θ*ʹ1 is $\\frac{1}{2}$.\nFirst, we show that *σ* is an SFCE. Since *v*2 ≡ 0, the second player does not have any incentive to deviate. Since *σ* is the uniform distribution over *S*, when *s*1 ∈ *S*1 is recommended, the posterior distribution of *s*2 is the uniform distribution over *S*2. Hence, the expected payoff of each action of *A*1 is $\\frac{1}{2}$, and the first player does not have any incentive to deviate.\nNext, we show that *σ*ʹ is not an SFCE. Since *s*11, *s*12, *s*13, *s*14 are all distinct, the first player can infer the recommendation *s*2(*θ*2) for the second player. The first player can increase the payoff to 1 by taking the same action as the second player. Therefore, *σ*ʹ is not an SFCE.\nAgent-normal-form correlated equilibria\n---------------------------------------\nAgent-normal-form correlated equilibria (ANFCEs) are defined as correlated equilibria of the agent normal form of a Bayesian game. Recall that in the agent normal form, the same player with different types is hypothetically considered as distinct players. In an ANFCE, the mediator recommends possibly correlated actions to all hypothetical players (*i*, *θ**i*) ∈ *N*ʹ. Since an action profile for the agent normal form assigns some *a**i* ∈ *A**i* to every *i* ∈ *N* and *θ**i* ∈ Θ*i*, it can be identified with a strategy profile *s* ∈ *S*. Hence, we can define ANFCEs as the set of distributions over *S*. The only difference from SFCEs is the information that each player can use for deviations. While each player *i* ∈ *N* can decide their action according to the recommended actions for all types Θ*i* in SFCEs, each player *i* ∈ *N* can use only the recommended action *s**i*(*θ**i*) for their realized type *θ**i* ∈ Θ*i* in ANFCEs. Here we define an approximate version of ANFCEs with additive error *ε*.\n[*ε*-Approximate agent-normal-form correlated equilibria (*ε*-ANFCEs)] For any *ε* ≥ 0, a distribution *σ* ∈ Δ(*S*) is an *ε*-approximate agent-normal-form correlated equilibrium if for any *i* ∈ *N* and *ϕ*: Θ*i* × *A**i* → *A**i*, it holds that\n \n$$\\label{eq:ic-anf}\n\\tag{$\\mathrm{IC}\\_{\\mathrm{ANF}}$}\n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta; s(\\theta)) \\right] \\right]\n\\ge \n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta; \\phi(\\theta\\_i, s\\_i(\\theta\\_i)), s\\_{-i}(\\theta\\_{-i})) \\right] \\right] - \\epsilon.$$ \nLet ΣANF*ε* ⊆ Δ(*S*) be the set of all *ε*-ANFCEs.\nIn contrast to SFCEs, each player *i* ∈ *N* in ANFCEs is informed of only the action for the realized type *s**i*(*θ**i*), not the full strategy *s**i*. In an SFCE, each player can use recommendations for unrealized types, which differentiates SFCEs from ANFCEs. In an ANFCE, each player *i* ∈ *N* with type *θ**i* ∈ Θ*i* can use only *s**i*(*θ**i*) to decide deviation *ϕ*(*θ**i*, *s**i*(*θ**i*)). This difference makes ΣANF*ε* broader than ΣSF*ε*.\nThe incentive constraint in our definition is slightly stronger than the incentive constraint for each hypothetical player (*i*, *θ*ʹ*i*) ∈ *N*ʹ, which can be written as\n \n$${\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathbf{1}}\\_{\\{\\theta\\_i=\\theta'\\_i\\}} {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta; s(\\theta)) \\right] \\right]\n\\ge \n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathbf{1}}\\_{\\{\\theta\\_i=\\theta'\\_i\\}} {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta; \\phi\\_{\\theta'\\_i}(s\\_i(\\theta\\_i)), s\\_{-i}(\\theta\\_{-i})) \\right] \\right] - \\epsilon$$ \nfor any *ϕ**θ*ʹ*i*: *A**i* → *A**i*. This can be checked by letting *ϕ*(*θ*ʹ*i*, ⋅ ) = *ϕ**θ*ʹ*i* and *ϕ*(*θ*ʺ*i*, ⋅ ) be the identity map for each *θ*ʺ*i* ∈ Θ*i* \\ {*θ*ʹ*i*} in. Note that this difference is only for the error parameter, and these two incentive constraints are equivalent for *ε* = 0.\nSince the setting of the agent normal form is hypothetical, it is difficult to consider a natural scenario for ANFCEs. In an ANFCE, the mediator determines a strategy profile *s* ∈ *S* according to *σ* ∈ Δ(*S*) but recommends to each player only *s**i*(*θ**i*). To realize it, the mediator must know the type profile *θ* ∈ Θ in advance. However, if it is the case, then it is possible to realize any distribution *π*(*θ*) ∈ Δ(*A*) for each *θ* ∈ Θ, for which the notion of Bayesian solutions introduced later is more appropriate.\nBy extending no-swap-regret-dynamics to the agent normal form, we can obtain uncoupled dynamics converging to an ANFCE. In these dynamics, for every time round *t* ∈ [*T*], each hypothetical player (*i*, *θ**i*) ∈ *N*ʹ decides a distribution *π**i**t*(*θ**i*) ∈ Δ(*A**i*) over their actions. Hence, swap regret for each hypothetical player (*i*, *θ*ʹ*i*) ∈ *N*ʹ considers an action swap *ϕ**θ*ʹ*i*: *A**i* → *A**i*. As with the incentive constraints discussed above, we consider the sum of swap regrets for all hypothetical players corresponding to *i* ∈ *N*, which we call *type-wise swap regret*.\n[Type-wise swap regret] For online learning with stochastic types specified by actions *A**i*, types Θ*i*, prior distribution *ρ**i*, and reward vector *u**i**t* ∈ [0, 1]Θ*i* × *A**i* for each round *t* ∈ [*T*], type-wise swap regret is defined as\n \n$${R\\_{\\mathsf{TS},i}}^T = \\max\\_{\\phi \\colon \\Theta\\_i \\times A\\_i \\to A\\_i} \\sum\\_{t=1}^T {\\mathop{\\mathbb{E}}}\\_{\\theta\\_i \\sim \\rho\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t(\\theta\\_i)} \\left[ u\\_i^t(\\theta\\_i, \\phi(\\theta\\_i, a\\_i)) - u\\_i^t(\\theta\\_i, a\\_i) \\right] \\right].$$ \n[t] [alg:anf-dynamics] For each *i* ∈ *N*, let $\\calA\\_i$ be a subroutine that minimizes type-wise swap regret (ANFCEs) or untruthful swap regret (communication equilibria with strategy representability) for online learning with stochastic types\nAs with SFCEs, by directly applying Theorem 3 of to the agent normal form of Bayesian games, we obtain the following claim.\n[prop:dynamics-anf] Let *π**i**t* ∈ Δ(*A**i*)Θ*i* be the type-wise distribution of each player *i* ∈ *N* for each round *t* ∈ [*T*] in. Then, the empirical distribution *σ* ∈ Δ(*S*), which is defined by $\\sigma(s) = \\frac{1}{T} \\sum\\_{t=1}^T \\prod\\_{i \\in N} \\prod\\_{\\theta\\_i \\in \\Theta\\_i} \\pi\\_{i}^t(\\theta\\_i; s\\_i(\\theta\\_i))$ for each *s* ∈ *S*, is a $\\frac{\\max\\_{i \\in N} {R\\_{\\mathsf{TS},i}}^T}{T}$-ANFCE, where *R*TS, *i**T* is the type-wise swap regret for each subroutine $\\calA\\_i$.\nFrom the definition of *ε*-ANFCEs, it is sufficient to prove\n \n$${\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta; \\phi(\\theta\\_i, s\\_i(\\theta\\_i)), s\\_{-i}(\\theta\\_{-i})) \\right] \\right] - {\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta; s(\\theta)) \\right] \\right]\n\\le \\frac{\\max\\_{j \\in N} {R\\_{\\mathsf{TS},j}}^T}{T}$$ \nfor each *i* ∈ *N* and *ϕ*: Θ*i* × *A**i* → *A**i*. Let *σ**t* ∈ Δ(*S*) be the product distribution that independently generates *s**j*(*θ**j*) ∼ *π**j**t*(*θ**j*) for each *j* ∈ *N* and *θ**j* ∈ Θ*j*. For each *θ*− *i* ∈ Θ− *i*, let *π*− *i**t*(*θ*− *i*) ∈ Δ(*A*− *i*) be the product distribution that independently generates *a**j* ∼ *π**j**t*(*θ**j*) for each *j* ∈ *N* \\ {*i*}. Then the left-hand side can be bounded as\n \n$$\\begin{aligned}\n&{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[v\\_i(\\theta; \\phi(\\theta\\_i, s\\_i(\\theta\\_i)), s\\_{-i}(\\theta\\_{-i})) \\right] \\right] - {\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta; s(\\theta)) \\right] \\right]\\\\\n&= \n{\\mathop{\\mathbb{E}}}\\_{t \\sim [T]} \\left[ {\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma^t} \\left[ v\\_i(\\theta; \\phi(\\theta\\_i, s\\_i(\\theta\\_i)), s\\_{-i}(\\theta\\_{-i})) - v\\_i(\\theta; s(\\theta)) \\right] \\right] \\right] \\tag{from the definition of $\\sigma$} \\\\\n&= \n{\\mathop{\\mathbb{E}}}\\_{t \\sim [T]} \\left[ {\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t(\\theta\\_i)} \\left[ {\\mathop{\\mathbb{E}}}\\_{a\\_{-i} \\sim \\pi\\_{-i}^t(\\theta\\_{-i})} \\left[ v\\_i(\\theta; \\phi(\\theta\\_i, a\\_i), a\\_{-i}) - v\\_i(\\theta; a) \\right] \\right] \\right] \\right] \\tag{from the definition of $\\sigma^t$} \\\\\n&=\n\\frac{1}{T} \\sum\\_{t=1}^T {\\mathop{\\mathbb{E}}}\\_{\\theta\\_i \\sim \\rho\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t(\\theta\\_i)} \\left[ {\\mathop{\\mathbb{E}}}\\_{\\theta\\_{-i} \\sim \\rho|\\theta\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{a\\_{-i} \\sim \\pi\\_{-i}^t(\\theta\\_{-i})} \\left[ v\\_i(\\theta; \\phi(\\theta\\_i,a\\_i),a\\_{-i})\n- \nv\\_i(\\theta; a)\\right] \\right] \\right] \\right]\\\\\n&=\n\\frac{1}{T} \\sum\\_{t=1}^T {\\mathop{\\mathbb{E}}}\\_{\\theta\\_i \\sim \\rho\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t(\\theta\\_i)} \\left[ u\\_i^t(\\theta\\_i, \\phi(\\theta\\_i, a\\_i)) - u\\_i^t(\\theta\\_i, a\\_i) \\right] \\right] \\tag{from the definition of $u\\_i^t$}\\\\\n&\\le \\frac{{R\\_{\\mathsf{TS},i}}^T}{T} \\\\\n&\\le \\frac{\\max\\_{j \\in N} {R\\_{\\mathsf{TS},j}}^T}{T},\\end{aligned}$$ \nwhich completes the proof.\nA detailed description of the dynamics can be found in. We can obtain an algorithm for minimizing type-wise swap regret by running the swap regret minimization algorithm proposed by separately for each (*i*, *θ**i*) ∈ *N*ʹ. Since the rewards for each (*i*, *θ**i*) ∈ *N*ʹ are always bounded above by *ρ**i*(*θ**i*), an upper bound of $O(\\sqrt{T |A\\_i| \\log |A\\_i|})$ on swap regret directly applies to type-wise swap regret.\nAs with *ε*-SFCEs, we can define its counterpart ΠANF*ε* ⊆ Δ(*A*)Θ by ΠANF*ε* = *η*(ΣANF*ε*) (see for the definition of *η*). Since the incentive constraints for ANFCEs only rely on the action profile *s*(*θ*) instead of the strategy profile *s*, we can characterize ΠANF*ε* as follows.\n[prop:anf-pi] For any type-wise distribution *π* ∈ Δ(*A*)Θ, the following are equivalent:\n* *π* ∈ ΠANF*ε*.\n* *π* is strategy-representable, and for any *i* ∈ *N* and any *ϕ*: Θ*i* × *A**i* → *A**i*, it holds that\n \n$$\\label{eq:ic-anf-pi}\n\\tag{$\\mathrm{IC}'\\_{\\mathrm{ANF}}$}\n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi(\\theta)} \\left[ v\\_i(\\theta; a) \\right] \\right]\n\\ge\n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi(\\theta)} \\left[ v\\_i(\\theta; \\phi(\\theta\\_i, a\\_i),a\\_{-i}) \\right] \\right] - \\epsilon.$$\nFirst, we prove (ii) assuming (i). Suppose *π* ∈ ΠANF*ε*. Since ΠANF*ε* = *η*(ΣANF*ε*), there exists *σ* ∈ ΣANF*ε* such that *η*(*σ*) = *π*. From the definition of ΣANF*ε*, the incentive constraint holds for every *i* ∈ *N* and *ϕ*: Θ*i* × *A**i* → *A**i*. Since *π* = *η*(*σ*), when *s* is generated from *σ*, the distribution of *s*(*θ*) equals *π*(*θ*). Then implies.\nNext, we prove (i) assuming (ii). Suppose *π* is strategy-representable and satisfies the incentive constraint for every *i* ∈ *N* and *ϕ*: Θ*i* × *A**i* → *A**i*. From the strategy representability, there exists *σ* ∈ Δ(*S*) such that *π* = *η*(*σ*). As discussed above, when *s* is generated from *σ*, the distribution of *s*(*θ*) equals *π*(*θ*). Then implies.\nCommunication equilibria\n------------------------\nA communication equilibria is defined as a type-wise distribution *π* ∈ Δ(*A*)Θ that is incentive-compatible in the following scenario. First, type profile *θ* ∈ Θ is jointly generated from prior distribution *ρ*, and then each player observes their own type *θ**i* ∈ Θ*i*. Next, the players privately report their own types to the mediator. Then the mediator generates an action profile *a* ∈ *A* from *π*(*θ*) and privately recommends *a**i* to each player *i* ∈ *N*. If each player *i* ∈ *N* has no incentive to deviate from reporting the true type *θ**i* and following the recommendation *a**i*, then this type-wise distribution *π* ∈ Δ(*A*)Θ is called a communication equilibrium.\nFormally, the incentive constraints for each player *i* ∈ *N* are defined for the following two mappings. The first one *ψ*: Θ*i* → Θ*i* specifies a type *ψ*(*θ**i*) to be reported when the true type is *θ**i*. As a result, the distribution *π*(*ψ*(*θ**i*), *θ*− *i*) that generates the recommended action profile *a* ∈ *A* may be different from the original one *π*(*θ*). The second one *ϕ*: Θ*i* × *A**i* → *A**i* specifies an action *ϕ*(*θ**i*, *a**i*) when *a**i* is recommended by the mediator and the true type is *θ**i*. If each player cannot gain by using any combination of *ψ* and *ϕ*, a distribution *π* is a communication equilibrium. Here we define its approximate version.\n[*ε*-Approximate communication equilibria] For any *ε* ≥ 0, a type-wise distribution *π* ∈ Δ(*A*)Θ is an *ε*-approximate communication equilibrium if for any *i* ∈ *N*, *ψ*: Θ*i* → Θ*i*, and *ϕ*: Θ*i* × *A**i* → *A**i*, it holds that\n \n$$\\label{eq:ic-com}\n\\tag{$\\mathrm{IC}\\_{\\mathrm{Com}}$}\n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi(\\theta)} \\left[ v\\_i(\\theta; a) \\right] \\right]\n\\ge\n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi(\\psi(\\theta\\_i),\\theta\\_{-i})} \\left[ v\\_i(\\theta; \\phi(\\theta\\_i, a\\_i), a\\_{-i}) \\right] \\right] - \\epsilon.$$ \nLet ΠCom*ε* ⊆ Δ(*A*)Θ be the set of all *ε*-approximate communication equilibria.\nA communication equilibrium can be defined as a Nash equilibrium for the extended game equipped with a *communication device*, which is a device that privately receives a signal from each player and then privately sends a stochastic (possibly correlated) signal to each player based on the received signals. In general, the inputs and outputs for the communication device may be any signals different from type profiles Θ and action profiles *A*. By the revelation principle, it is sufficient to consider *direct* (or *canonical*) ones, in which the sets of inputs and outputs coincide with Θ and *A*. In this paper, we consider direct communication equilibria only.\nThere are two differences between communication equilibria and ANFCEs. First, in an ANFCE, the mediator must generate a strategy profile *s* ∈ *S* before observing the type profile *θ* ∈ Θ, while in a communication equilibrium, the mediator can generate an action profile *a* ∈ *A* after observing the type profile. Since there exists a type-wise distribution *π* that cannot be expressed by any strategy distribution (), ANFCEs are more restrictive than communication equilibria in this aspect. Second, in an ANFCE, the players must inform the mediator of their true types, while in a communication equilibrium, the players can untruthfully report their types. If the mediator can validate the reported types in some way, ANFCEs are more natural, but communication equilibria are more natural otherwise.\nTo introduce dynamics converging to a communication equilibrium, we present the definition of untruthful swap regret. In addition to *ϕ*: Θ*i* × *A**i* → *A**i* that represents the deviation from the recommended action, the competitor can use *ψ*: Θ*i* → Θ*i* that represents untruthful type reporting.\n[Untruthful swap regret][def:untruthful] For online learning with stochastic types specified by actions *A**i*, types Θ*i*, prior distribution *ρ**i*, and reward vector *u**i**t* ∈ [0, 1]Θ*i* × *A**i* for every round *t* ∈ [*T*], untruthful swap regret is defined as\n \n$${R\\_{\\mathsf{US},i}}^T = \\max\\_{\\psi \\colon \\Theta\\_i \\to \\Theta\\_i} \\max\\_{\\phi \\colon \\Theta\\_i \\times A\\_i \\to A\\_i} \\sum\\_{t=1}^T {\\mathop{\\mathbb{E}}}\\_{\\theta\\_i \\sim \\rho\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t(\\psi(\\theta\\_i))} \\left[ u\\_i^t(\\theta\\_i, \\phi(\\theta\\_i, a\\_i)) \\right] - {\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t(\\theta\\_i)} \\left[ u\\_i^t(\\theta\\_i, a\\_i) \\right] \\right].$$ \nIf we restrict *ψ* to the identity map, untruthful swap regret is reduced to type-wise swap regret. Then untruthful swap regret is no less than type-wise swap regret.\nNow we consider dynamics in which each player *i* ∈ *N* minimizes untruthful swap regret *R*US, *i*. In the following theorem, we show that these dynamics converge to a communication equilibrium. Moreover, the empirical distribution is always strategy-representable from.\n[thm:dynamics-com] Let *π**i**t* ∈ Δ(*A**i*)Θ*i* be the type-wise distribution of each player *i* ∈ *N* for each round *t* ∈ [*T*] in. Then, the empirical distribution *π* ∈ Δ(*A*)Θ, which is defined by $\\pi(\\theta;a) = \\frac{1}{T} \\sum\\_{t=1}^T \\prod\\_{i \\in N} \\pi\\_{i}^t(\\theta\\_i;a\\_i)$ for each *θ* ∈ Θ and *a* ∈ *A*, is a $\\frac{\\max\\_{i \\in N} {R\\_{\\mathsf{US},i}}^T}{T}$-approximate communication equilibrium that satisfies strategy representability, where *R*US, *i**T* is the untruthful swap regret for each subroutine $\\calA\\_i$.\nFrom, *π* is strategy-representable. From the definition of *ε*-approximate communication equilibria, it is sufficient to prove\n \n$${\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi(\\psi(\\theta\\_i),\\theta\\_{-i})} \\left[ v\\_i(\\theta; \\phi(\\theta\\_i, a\\_i), a\\_{-i}) \\right] \\right]\n-\n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi(\\theta)} \\left[ v\\_i(\\theta; a) \\right] \\right]\n\\le \\frac{\\max\\_{j \\in N} {R\\_{\\mathsf{US},j}}^T}{T}$$ \nfor each *i* ∈ *N*, *ψ*: Θ*i* → Θ*i*, and *ϕ*: Θ*i* × *A**i* → *A**i*. For each *t* ∈ [*T*] and *θ* ∈ Θ, let *π**t*(*θ*) ∈ Δ(*A*) be the product distribution that independently generates *a**j* ∼ *π**j**t*(*θ**j*) for each *j* ∈ *N*. Similarly, let *π*− *i**t*(*θ*) ∈ Δ(*A*− *i*) be the product distribution that independently generates *a**j* ∼ *π**j**t*(*θ**j*) for each *j* ∈ *N* \\ {*i*}. The left-hand side can be bounded as\n \n$$\\begin{aligned}\n&{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi(\\psi(\\theta\\_i), \\theta\\_{-i})} \\left[ v\\_i(\\theta; \\phi(\\theta\\_i, a\\_i), a\\_{-i}) \\right] \\right] - {\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi(\\theta)} \\left[ v\\_i(\\theta; a) \\right] \\right]\\\\\n&=\n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{t \\sim [T]} \\left[ {\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi^t(\\psi(\\theta\\_i), \\theta\\_{-i})} \\left[ v\\_i(\\theta; \\phi(\\theta\\_i, a\\_i), a\\_{-i}) \\right] - {\\mathop{\\mathbb{E}}}\\_{a \\sim \\pi^t(\\theta)} \\left[ v\\_i(\\theta; a) \\right] \\right] \\right]\\\\\n&=\n\\frac{1}{T} \\sum\\_{t=1}^T {\\mathop{\\mathbb{E}}}\\_{\\theta\\_i \\sim \\rho\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{\\theta\\_{-i} \\sim \\rho|\\theta\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{a\\_{-i} \\sim \\pi\\_{-i}^t(\\theta\\_{-i})} \\left[ {\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t(\\psi(\\theta\\_i))} \\left[ v\\_i(\\theta; \\phi(\\theta\\_i,a\\_i),a\\_{-i}) \\right]\n- \n{\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t(\\theta\\_i)} \\left[ v\\_i(\\theta; a)\\right] \\right] \\right] \\right] \\tag{since $\\pi^t(\\theta)$ is a product distribution}\\\\\n&=\n\\frac{1}{T} \\sum\\_{t=1}^T {\\mathop{\\mathbb{E}}}\\_{\\theta\\_i \\sim \\rho\\_i} \\left[ {\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t(\\psi(\\theta\\_i))} \\left[ u\\_i^t(\\theta\\_i, \\phi(\\theta\\_i, a\\_i)) \\right] - {\\mathop{\\mathbb{E}}}\\_{a\\_i \\sim \\pi\\_i^t(\\theta\\_i)} \\left[ u\\_i^t(\\theta\\_i, a\\_i) \\right] \\right] \\tag{from the definition of $u\\_i^t$}\\\\\n&\\le\n\\frac{{R\\_{\\mathsf{US},i}}^T}{T}\\\\\n&\\le \\frac{\\max\\_{j \\in N} {R\\_{\\mathsf{US},j}}^T}{T},\\end{aligned}$$ \nwhich completes the proof.\nWe present characterizations of communication equilibria with strategy representability, to which the proposed dynamics converge.\n[prop:com-sr] For any type-wise distribution *π* ∈ Δ(*A*)Θ, the following are equivalent:\n* *π* ∈ ΠCom*ε* and *π* is strategy-representable.\n* *π* ∈ ΠCom*ε* ∩ ΠANF*ε*.\n* There exists some *σ* ∈ Δ(*S*) such that *η*(*σ*) = *π* and\n \n$$\\label{eq:ic-comanf}\n\\tag{$\\mathrm{IC}\\_{\\mathrm{ComSR}}$}\n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta\\_i; s(\\theta)) \\right] \\right]\n\\ge \n{\\mathop{\\mathbb{E}}}\\_{\\theta \\sim \\rho} \\left[ {\\mathop{\\mathbb{E}}}\\_{s \\sim \\sigma} \\left[ v\\_i(\\theta\\_i; \\phi(\\theta\\_i, s\\_i(\\psi(\\theta\\_i))), s\\_{-i}(\\theta\\_{-i})) \\right] \\right] - \\epsilon.$$ \nholds for any *i* ∈ *N*, *ψ*: Θ*i* → Θ*i*, and *ϕ*: Θ*i* × *A**i* → *A**i*.\nFirst, we prove (ii) assuming (i). Suppose *π* ∈ ΠCom*ε* and *π* is strategy-representable. Since is a weaker condition than, *π* satisfies for any *ϕ*: Θ*i* × *A**i* → *A**i*. We thus obtain *π* ∈ ΠANF*ε* from.\nNext, we prove (iii) assuming (ii). Suppose *π* ∈ ΠCom*ε* ∩ ΠANF*ε*. Since ΠANF*ε* = *η*(ΣANF*ε*) from the definition, *π* ∈ ΠANF*ε* implies that there exists *σ* ∈ ΣANF*ε* such that *η*(*σ*) = *π*. Moreover, since *π* ∈ ΠCom*ε*, holds for any *i* ∈ *N*, *ψ*: Θ*i* → Θ*i*, and *ϕ*: Θ*i* × *A**i* → *A**i*. Since *η* is defined by *π*(*θ*; *a*) = Pr*s* ∼ *σ*(*s"},"data_id":{"kind":"string","value":"arxiv_0002050"}}},{"rowIdx":496,"cells":{"content":{"kind":"string","value":" a straight vortex has zero interaction with itself. On the other hand a curved vortex acts on itself with a divergent force (made finite by cutting off the integral at a distance on the order of the core radius). This force can be described by the ``local induction approximation\", Eq. ([eq:LIA]). This force points toward the center of curvature of the vortex, therefore encouraging it to shrink. To see that this is the same as the force (Eq. [eq:shrinkforce]) which drives the vortex towards thinner regions where the kinetic energy is less,\nTo see this, note that if the opposing boundaries S, S' are not at a constant distance, the vortex must curve in order to connect them, because it meets both boundaries at right angles. The resulting curvature of the vortex, *κ*, is equal to $\\frac{\\nabla D}{D}$ (see Fig. [fig:crevice]), so that the force per\n![[fig:crevice]A figure for reference in determining the curvature of a vortex. If the upper and lower surface \\mathcal{S} and \\mathcal{S'} are nearly but not quite parallel, the curvature of a vortex connecting them can be estimated simply. Extend the tangents at the end-points X,Y of the vortex until they intersect at the center O of curvature of the vortex (which happens because the vortex is perpendicular to \\mathcal{S,S'}). The radius of curvature, OX, can then be found by using Eq. ([eq:ezekiel]) and noticing that OX=XX'\\tan\\beta and \\sin\\beta\\approx 1. ](trigdiag)[fig:crevice]A figure for reference in determining the curvature of a vortex. If the upper and lower surface S and S' are nearly but not quite parallel, the curvature of a vortex connecting them can be estimated simply. Extend the tangents at the end-points *X*, *Y* of the vortex until they intersect at the center *O* of curvature of the vortex (which happens because the vortex is perpendicular to S,S'). The radius of curvature, *O**X*, can then be found by using Eq. ([eq:ezekiel]) and noticing that *O**X* = *X**X*ʹtan*β* and sin*β* ≈ 1. \nunit length given by Eq. ([eq:LIA]), multiplied by *D*, agrees with the force derived by the energy method, Eq. ([eq:shrinkforce]). On the other hand, for films of constant thickness, the surfaces S, S' can be connected by straight vortices. According to the Biot-Savart law a straight vortex has zero interaction with itself. In addition, there is still an interaction energy with its image, which is where the geometric force, the third term in Eq. [eq:shrinkforce], comes from. When the helium has boundaries, a distribution of ``image vorticity\" beyond the boundaries can be supplied to simulate the effects of the boundary conditions. The integral of the Coulomb interaction with the distribution of surface curvature (Eq. ([eq:curvature-defect]) is reminiscent of the long-range integral of the Biot-Savart interaction with the distribution of vorticity beyond the surface.\nThe mathematical principle connecting these different points of view is the following (see Fig. [fig:railroad]): The constancy of *d* is equivalent to the possibility of drawing straight lines from any point on S to a point S', which are perpendicular to *both* surfaces. The equivalence is derived by differentiating *d*(**x**)2 = ∣**x** − **x'**∣2; if the two surfaces can be connected by uncurved vortices then 2*d**δ**d* = 2(**x** − **x'**) ⋅ (*δ***x** − *δ***x'**) = 2(**x'** − **x**) ⋅ *δ***x'**, since the line connecting **x** to **x'** is perpendicular to S. In other words,\n \n*δ**d* = *d*cos*β* \nwhere *β* is illustrated in Fig. [fig:railroad]. Hence the local induction force is equivalent to a vortex energy per unit length which is roughly equal to $\\frac{\\rho\\_3\\hbar^2}{m^2}\\ln\\frac{L\\_{th}}{a}+\\epsilon\\_{c3}$. Hence if the vortex connecting the points**x** and **x'** is a straight line normal to both surfaces (so that there is no local induction force), then *d* is constant (so that there is no variation in the logarithmically divergent energy around the core to compete with the geometrical force).\n[sec:unevenexptl]Vortex Depinning\n---------------------------------\nBecause of the divergence of the force described by Eq. ([eq:relic]) in the limit *a* → 0, it will be hard to see effects of the geometric force without films of very nearly uniform thickness. In an experiment is described in which a vortex extending along a wire in a helium-filled tube leaves the wire more easily when the wire is connected to a bump on the bottom of the tube than when the bottom is flat. In this case, any geometric repulsion from the bump should be negligible in comparison to the extra energy associated with the stretching of the vortex. (See Fig. [fig:last].) Geometric energies should be of order $R\\rho\\_3\\frac{\\hbar^2}{m^2}$ where *R* is the radius of the tube and approximately the height of the bump, but the vortex has to stretch an additional length on the order of *R* in order to leave the bump. Therefore, the extra kinetic energy from Eq. ([eq:linetension]), $R\\rho\\_3\\frac{\\hbar^2}{m^2}\\ln\\frac{R}{a}$, probably overwhelms the geometric effects. Thus, the energy barrier for detachment of the vortex line should increase when the wire is attached to a bump.\n![[fig:last] Sketch of the experimental geometry from Ref.. This experiment may give some indication about the difference in energy between a vortex stretching from A to B and one stretching from C to D. A wire is placed along AB and a vortex is formed around it. Ref. found that this vortex depins more easily than a vortex in a tube with a flat bottom. This suggests that the energy of a vortex (when there is no wire) would decrease by moving from AB to CD, but estimates seem to rule out this explanation.](last)[fig:last] Sketch of the experimental geometry from Ref.. This experiment may give some indication about the difference in energy between a vortex stretching from *A* to *B* and one stretching from *C* to *D*. A wire is placed along *A**B* and a vortex is formed around it. Ref. found that this vortex depins more easily than a vortex in a tube with a flat bottom. This suggests that the energy of a vortex (when there is no wire) would decrease by moving from *A**B* to *C**D*, but estimates seem to rule out this explanation.\nThis suggests that depinning is not simply caused by thermally activated barrier crossings (which would occur at the rate proportional to *e*− Δ*E*/*T*). In fact, for the length scale of the bump in the experiment (a few millimeters), the additional energy barrier due to the stretching of the vortex line is millions of degrees kelvin at the temperature where the vortex depins! The depinning may instead depend on ``remanent vorticity\" in the form of extra pinned vortices stretching from the top of the bump to the cylinder’s walls. This would decrease the energy barrier because the vortex could leave the wire by attaching itself to some of the vortices that already exist.\nIn a more nearly two-dimensional geometry, Eq. [eq:shrinkforce] shows that observing the equilibrium positions of vortices in a film of varying thickness could shed light on the value of the core size and core-energy and whether they are fixed functions of temperature as most models imply. The structure of vortex cores is still not well understood, and there are several alternative models. Experiments on the geometric force in contexts where the two contributions to Eq. ([eq:shrinkforce]) are comparable could give information about the effective core radius *ã*. By allowing the film thickness to vary by about $\\epsilon = \\frac{\\Delta D}{D}\\sim\\frac{1}{\\ln\\frac{r\\_0}{a}}$, see Eq.([eq:epsilonmanners]), one ensures that the vortex core size has a decisive effect on the equilibrium positions of the vortices (comparable to the effects of the long range forces).\nWhereas the local induction approximation does not tell the whole story for vortices in bulk because the long-range portions of the Biot-Savart interaction depend on the complicated tangling of the vortex line, the self interaction of a vortex connecting nearly parallel surfaces is independent of the configuration of the vortex as long as it is reasonably straight. The force on a short vortex is accurately described by just the local induction aproximation together with the geometrical potential, which describes all the long-range forces. If these two forces are comparable (i.e., $\\epsilon\\sim\\frac{1}{\\ln\\frac{r\\_0}{a}}$), then the competition of these forces determines the vortex-equilibrium positions. If pinning is not important and the vortices can be observed, this would help to measure the quantity *ã* the only quantity which can be determined based on macroscopic properties of vortices.\n[sec:complex]Complex Surface Morphologies\n=========================================\nUp to this point, our discussion has been confined to rotationally symmetric surfaces and slightly deformed surfaces for which the electrostatic analogy and perturbation theory can be successfully employed to determine the geometric potential. To investigate geometric effects that arise for strong deformations and for surfaces with the topology of a sphere, we adopt a more versatile geometric approach based on the method of conformal mapping, often employed in the study of complicated boundary problems in electromagnetism and fluid mechanics. This approach also sheds light on the physical origin of the geometric potential. A concrete goal is to solve for the energetics (and the associated flows) of topological defects on a complicated substrate *T*, the target surface, whose metric tensor we denote by *g**T**a**b*. This is accomplished by means of a conformal map *C* that transforms the target surface into a reference surface *R*, with metric tensor *g**R**a**b*. The computational advantages result from choosing the conformal map so that *R* is a simple surface (e.g. an infinite flat plane, a flat disk, or a regular sphere) that preserves the topology of the target surface. Figure [fig:splat] represents a complicated planar domain denoted by *T* which can be mapped conformally onto a simple annulus labeled by *R*. We will introduce all the basic concepts in the context of this simple planar problem before turning to the conformal mapping between target and reference surfaces which is represented schematically in Fig. [fig:emental]. Such mappings can always be found in principle.\nThe conformal transformation will map the original positions of the defects on *T*, denoted by **u**, onto a new set of coordinates on *R* denoted by $\\mathbf{\\mathcal{U}}=C(\\mathbf{u})$. In what follows, capital calligraphic fonts always indicate coordinates on the reference surface. For sufficiently small objects near a point **u** the map will act as a similarity transformation; that is, an infinitesimal length, $ds\\_T =\\sqrt{ g\\_{Tab} du^{a} du^{b}}$, will be rescaled by a scale factor *e**ω*(**u**) which is independent of the orientation of the length on *T*:\n \n*d**s**R* = *e**ω*(**u**)*d**s**T*, \nwhere $ds\\_R=\\sqrt{g\\_{RAB}d\\mathcal{U}^Ad\\mathcal{U}^B}$. This result in turn implies a simple relation between the metric tensors of the two surfaces:\n \n*g**R**A**B* = *e*2*ω*(**u**)*g**T**A**B*, \nwhere we have assumed for simplicity that the coordinates used on the target surface are chosen so that corresponding points on the two surfaces have the same coordinates U*A*.\nWe will demonstrate that, once the geometric quantity *ω*(**u**) is calculated, the geometric potential of an isolated vortex interacting with the curvature is automatically determined. For multiple vortices, the energy consists of single-vortex terms and vortex-vortex interactions. On a deformed sphere or plane, the geometric potential reads\n \n*E*1(**u\\_i**) = − *π**n**i*2*K**ω*(**u***i*), \nwhere *K* is the stiffness parameter defined in Eq. ([eq:stiffness]). For a deformed disk, there are boundary interactions not included in Eq. ([eq:conformalself-energy]). (We will not consider multiply connected surfaces here, but the single particle energy on a multiply connected surface has additional contributions which cannot be described by a local Poisson equation.) The interaction energy is\n \n$$E\\_2(\\mathbf{u}\\_i,\\mathbf{u}\\_j)=-2\\pi n\\_i n\\_j K\\ln\\frac{\\mathcal{D}\\_{ij}}{a},\n\\label{eq:conformalpair-energy}$$ \nwhere D*i**j* is the distance between the two *image* points on the reference surface. When the reference surface is an undeformed sphere (the other possibilities are a plane or disk), D*i**j* is the distance between the points along a chord rather than a great circle. We will show below that on a deformed plane *ω* is equal to *U**G*, but from now on we will use *ω* instead; the two functions are conceptually different, and are not even equal on a deformed sphere.\nEquation ([eq:conformalself-energy]) is derived by the method of conformal mapping in section [sec:map] and its computational efficiency is illustrated in section [sec:bubble] where the geometric potential of a vortex is evaluated on an Enneper disk, a minimal surface that naturally arises in the context of soap films, but whose geometry is distorted enough compared to flat space that it cannot be analyzed with perturbation theory. Changing the geometry of the substrate has interesting effects not only on the one-body geometric potential but also on the two-body interaction between vortices. In section [sec:bumps] we use conformal methods to show how a periodic lattice of bumps can cause the vortex interaction to become anisotropic. In section [sec:zucchini], we demonstrate that the quantization of circulation leads to an extremely long-range force on an elongated surface with the topology of a sphere. The interaction energy is no longer logarithmic as in Eq.([eq:conformalpair-energy]), but now grows linearly with the distance between the two vortices. Indeed the charge neutrality constraint imposed by the compact topology of a sphere, blurs the distinction between geometric potential and vortex-interaction drawn in Equations ([eq:conformalself-energy]) and ([eq:conformalpair-energy]). This is most easily seen by bypassing vortex energetics on the reference surface (which is after all an auxiliary concept) and opting for a more direct restatement of the problem in terms of Green’s functions on the actual target surface coated by the helium layer. The interaction energy now reads\n \n*E*2ʹ(**u***i*, **u***j*) = 4*π*2*K**n**i**n**j*Γ(**u***i*, **u***j*), \nand the single-particle energy takes the form of a self-energy\n \n*E*1ʹ(**u***i*) = − *π**n**i*2*K**U**G*(**u***i*) = *π**n**i*2*K*∬*G*(**u'**)Γ(**u\\_i**, **u'**)*d*2**u'**, \nwhere Γ is a Green’s function for the surface that generalizes the logarithmic potential familiar from two dimensional electrostatics. For a deformed plane the two descriptions of the interaction energy are equivalent since the Green’s function on a deformed plane can be obtained by conformal mapping,\n \n$$\\Gamma(\\mathbf{u}\\_i,\\mathbf{u}\\_j)=-\\frac{1}{2\\pi}\\ln\\frac{\\mathcal{D}\\_{ij}}{a}.\n\\label{eq:greentransf}$$ \nWe will see that the expressions for the single-particle energies are also equivalent. In contrast, for a deformed sphere, we show in section [sec:noodles] and appendix [app:CG] that the two formulations do not agree term by term ( *E*1ʹ ≠ *E*1 and *E*2ʹ ≠ *E*2), although the *combined* effect of one-particle and interaction terms is the same (up to an additive constant). Both self-energies and interaction energies include effects of the geometry and explicit formulas are provided on an azimuthally symmetric deformed sphere. Finally, in section [sec:geomineq], we present a discussion of some general upper bounds to which the strength of geometric forces is subjected (even in the regime of strong deformations) which are useful in experimental estimates and which illustrate a major difference between electrostatic and geometric forces: The former can always be increased by piling-up physical charges but the latter are generated by the Gaussian curvature that can grow only at the price of ``warping\" the underlying geometry of space. Too much warping either leads to self-intersection of the surface or a dilution of the long-range force.\n[sec:map]Using conformal mapping\n--------------------------------\nWe start by proving a simple relationship between the total energies (including self- and interaction parts) *E**T* and *E**R* of two corresponding vortex-configurations on the target and reference surfaces respectively:\n \n*E**T* = *E**R* − *π**K*∑*i* = 1*N**ω*(**u***i*). \nThe right-hand side of Eq. ([eq:anom]) can be calculated for the reference surface and then subsequently decomposed into single-vortex and vortex-vortex interactions; several examples are worked out in detail in Sections [sec:bubble] and [sec:bumps].\n![[fig:splat] The flow in a wiggily annular region (the target substrate T), obtained by conformally mapping the flow from a circular annulus (the reference substrate R). Since every pair of radial spokes in the first picture comprises the same energy, this is true in the conformal image as well. The small region in the constriction on the right manages this by compensating for its small area by having a high flow speed.](splatmap2)[fig:splat] The flow in a wiggily annular region (the target substrate *T*), obtained by conformally mapping the flow from a circular annulus (the reference substrate *R*). Since every pair of radial spokes in the first picture comprises the same energy, this is true in the conformal image as well. The small region in the constriction on the right manages this by compensating for its small area by having a high flow speed.\nThe general approach is best illustrated by considering the planar flow in the complicated annular container shown in Fig. [fig:splat]. which can be tackled by conformally mapping it to a simpler circular annulus.\nThe flow in the reference annulus is clearly circular, and it has the same $\\frac{1}{r}$ dependence as for a vortex. A crucial property of conformal transformation allows us to transplant this understanding of the reference flow to the target annulus. This property concerns the following mapping of the stream function *χ* (see Eq. ([eq:chidef])) from the reference surface to the target surface:\n \n$$\\chi\\_T(\\mathbf{u})=\\chi\\_R(\\mathbf{\\mathcal{U}})\n\\label{eq:mapping}$$ \nOnce the coordinate change $\\mathbf{\\mathcal{U}}=C(\\mathbf{u})$ is found, the prescription to determine *χ**T* provided in Eq.([eq:mapping]) guarantees that the corresponding flow will be irrotational and incompressible, as required. Visually, all we are doing is taking the streamlines on the reference surface, which are level curves of *χ**R*, and mapping them by *C*− 1 to the target surface. We can informally state this first property of conformal maps as follows:\nNote that any multiple of the mapped stream function, *α**χ**R*(*C*(**u**)), corresponds to an irrotational and incompressible flow but in this case the rates of flow in the target and reference substrate are different. Only the choice *α* = 1 ensures that both flows have the same number *n* of circulation quanta around the hole (or around each vortex if some are present.) This follows from another basic property of conformal maps:\nThis property follows from the fact that the contribution to the circulation from each element of the contour, **v\\cdot dl**, is conformally invariant. The infinitesimal length **dl** and the velocity **v** scale inversely to each other under conformal transformations and the angle between them is preserved by the map. To understand why, note that flow lines are compressed together (stretched apart) when they are mapped to the target space if the conformal parameter *ω* is greater (less) than zero. As a result, the velocity increases (decreases) by the same factor as distances are decreased (increased). This heuristic argument is confirmed by noting that the velocity is given by the *c**o**v**a**r**i**a**n**t* curl of the stream function, see Equations ([eq:chidef]) and ([eq:mapping]). By definition, the covariant curl carries a multiplicative factor of *g*− 1/2 = *e**ω*, hence\n \n*v**T* = *e**ω**v**R*. \nOn the other hand, ∣**dl***T*∣ = *e*− *ω*∣**dl***R*∣ rescales in the opposite way, as indicated in Eq. ([eq:confdef]).\nThe problem of finding the *energy* in the deformed annulus can now be reduced to a simple-rotationally symmetric problem by appealing to a third property of the conformal mapping:\nThe proof of this statement relies on the previous discussion: the kinetic energy contained in an element of the area of the surface *d**A* is $\\frac{1}{2}\\rho\\_s \\mathbf{v}^2dA$. By Eq. ([eq:mapvelocity]), **v**2 scales as *e*2*ω* whereas *d**A* scales by *e*− 2*ω* making the energy conformally invariant. Figure [fig:splat] illustrates pictorially that the energy density in the original flow on the target surface is smoothed out and simplified, its variations being replaced by variations in the conformal scale factor.\nWe now return to the energetics of flows containing point vortices. The starting point of our analysis follows from the defining property of a conformal map, namely that a conformal image of a small figure has the same shape as the original figure, while a larger shape becomes distorted (consider Greenland, which has an elongated shape, but appears to round out at the top in a Mercator projection, which is itself a conformal map). To quantify the size limits, note that if a shape has size *l*, *ω* changes by about *l*∇*ω* across the shape. Thus, as long as\n \n$$l<<\\frac{1}{|\\nabla\\omega|},\n\\label{eq:columbus}$$ \nthe mapping rescales the shape uniformly. The right-hand side is ordinarily of order *L*, the curvature scale of the surface. As a result we can conclude that\nOn a deformed substrate with a flow induced by vortices, the flow speeds will increase or decrease not just depending on the distance to the vortex, but also depending on the shape of the surface. For example, the vortex on top of the bump in the example of Sec. [subsec:anomalous] has a flow that decays more slowly with distance than in flat space. Also, for a vortex well off to the side of a bump, if the bump’s height *h* is larger than its width 2*r*0, it turns out that the flow pattern penetrates only up to an elevation of about *r*0 up the side of the bump.\n![[fig:emental] Comparing the energy of T and R by splitting T up into portions I and O and using different maps to map them to R. Left, a target surface, which has the topology of an infinite plane and is distorted by a three dimensional lump. Right, the reference plane (shown in the plane of the page). I consists of the interior of radius l disks in T, and is mapped rigidly to the heavily demarcated disks in R. O consists of the gray portion of T, and is mapped conformally to the gray portion of R. I and O contain the same kinetic energy as their images in R, but the images do not fit together perfectly. Thus the difference in energy between the flows in T and in R is determined by calculating how much energy is contained in the annuli which are either mismatched or covered twice.](emental)[fig:emental] Comparing the energy of *T* and *R* by splitting *T* up into portions *I* and *O* and using different maps to map them to *R*. Left, a target surface, which has the topology of an infinite plane and is distorted by a three dimensional lump. Right, the reference plane (shown in the plane of the page). *I* consists of the interior of radius *l* disks in *T*, and is mapped rigidly to the heavily demarcated disks in *R*. *O* consists of the gray portion of *T*, and is mapped conformally to the gray portion of *R*. *I* and *O* contain the same kinetic energy as their images in *R*, but the images do not fit together perfectly. Thus the difference in energy between the flows in *T* and in *R* is determined by calculating how much energy is contained in the annuli which are either mismatched or covered twice.\nThe method of conformal mapping elucidates these geometrical rearrangements of the flow pattern. To find the flow pattern around the vortices at positions **u***i*, we find the flow pattern around vortices at the corresponding positions $\\mathbf{\\mathcal{U}}\\_i$ on the reference surface and then map these streamlines onto the target surface by Eq. ([eq:mapping]). The energies are *not* equal in this case, in spite of property 3. Property 3 does not apply to a region containing vortex cores, because we would have to suppose the area of the cores on the reference surface to be greater by *e*2*ω* and the energy in the cores to be smaller by a factor of *e*2*ω*, in order for the conformal relation Eq. $\\frac{1}{2}\\rho\\_s v\\_T^2dA\\_T=\n\\frac{1}{2}\\rho\\_s v\\_R^2dA\\_R$ to continue to hold. In contrast, the core radius is fixed by the short-distance correlations of the helium atoms and the core energy is related to the interaction energy of the atoms.\nThe vortex cores are not significantly affected by the curvature of the substrate; moreover, the whole flow pattern in the vicinity of the core is nearly independent of the location of a vortex. We observe that each vortex has a ``dominion,\" a region where the flows are forced by the presence of the vortex to be\n \n$$v=n\\frac{\\hbar}{mr}+\\delta v\n\\label{eq:alsouseful}$$ \nThe leading term has the same form as one expects for a vortex in a rotationally symmetric situation, and the effects of geometry are accounted for by *δ**v* ; by dimensional analysis, this error is of the order of $\\frac{\\hbar}{mL}$ where *L* is the radius of curvature of the substrate (or possibly the distance to another vortex or to the boundary, whichever is shortest). Therefore we can introduce any length *l* < < *L* and note that *l* is then a distance below which the effects of curvature do not have a significant effect (compared to the diverging velocity field). The geometry correction gives a contribution to the energy within this radius that is also small, as seen by integrating the kinetic energy over the annulus between *a* and *l* (using Eq. ([eq:alsouseful])):\n \n$$\\pi K\\ln\\frac{l}{a}+ \\epsilon\\_c + O(K)\\frac{l^2}{L^2}\n\\label{eq:dominionenergy}$$ \nwhere *a* is the core radius and *ε**c* the core energy. The error term is *quadratic* in $\\frac{l}{L}$ because the integral over the cross-term from squaring Eq. ([eq:alsouseful]) cancels.\nWhen we wish to find the kinetic energy of the superflow, these near-vortex regions are thus the simplest to account for, as their energy is nearly independent of their positions relative features such as bumps on the substrate as long as\n \n*l* < < *L*. \nSince the target and reference configurations have the same number *N* of vortices, the energies contained within radius *l* of the vortices are the same:\n \n$$E^T\\_{<l}=E^R\\_{<l}=\\pi NK\\ln\\frac{l}{\\tilde{a}}\n\\label{eq:inert}$$ \nIn order to find the forces on a set of vortices, we need to account for all the energy of the vortices in regions away from the vortices where the flow *has* been affected by the curvature. Let us imagine cutting the target surface up into an inner region *I* (the union of the radius *l* disks around each vortex) and an outer region *O* (consisting of everything else) as illustrated in Fig. [fig:emental]. We can map *I* to the reference surface by simply translating each of the disks so that they surround the vortices on *R*. The modifications to the flow are all in *O*; for example, streamlines there are deformed from their circular shape. These irregularities can be removed (or at least simplified) by using the conformal mapping to map *O* to *R*, just as in the case of the annulus illustrated above. The inhomogeneity of the conformal map compensates for the irregularity of the flow. Some portions of *O* are expanded and some are contracted, but its circular boundaries are small enough (see Eqs. ([eq:useful]), ([eq:columbus])) that they are simply rescaled into circles of different radii (property 4):\n \n*l**i* = *e**ω*(**u***i*)*l* \nNow we have mapped *I* and *O* from the target surface to regions on the reference surface which contain the same energy. But these images of the regions on *R* do not fit together. The conformal map on *O* stretches or contracts each hole in it, to circles of radii *e**ω*(**u***i*)*l*. These stretched edges (corresponding to the solid black circle in Fig. [fig:emental]) do not fit together with the images of *O*, which have been moved rigidly from the target surface. The energies are related by\n \n*E**T* = *E*< *l**T* + *E*> *l**T* = *E*< *l**R* + *E*> *l**i**R* \nWe must correct for the gaps and overlaps between the two image regions on *R* in order to relate the last expression to *E**R*. If *ω*(**u***i*) > 0, there is an annular gap near vortex *i*; using Eq. ([eq:alsouseful]) (since this gap is part of the flow controlled by this vortex):\n \n$$\\begin{aligned}\n\\Delta E\\_i&=&\n-\\frac{1}{2}\\rho\\_s\\int\\_{l}^{l\\_i}2\\pi rdr\\frac{n\\_i^2\\hbar^2}{m^2r^2}\\\\\n&=&-\\pi Kn\\_i^2\\omega(\\mathbf{u}\\_i)\\end{aligned}$$ \nSumming all these contributions gives our desired result, Eq. ([eq:anom]).\nWe emphasize the energy difference is not produced within the cores, or anywhere near the vortices. In fact, the fact that the energies on *T* and *R* differ is a result of assuming that there is *no change* in the flow within a macroscopic distance *l* of the vortex. The scale *l* ≪ *L* only has to be small compared to the geometry of the system and has no relation to the atomic structure of the core. On the other hand, taking *l* as small as possible has an elegant consequence: Eq. ([eq:anom]) actually has an error which is $O\\left(K\\left(\\frac{a}{L}\\right)^2\\right)$, smaller than $O\\left(K\\left(\\frac{l}{L}\\right)^2\\right)$ as predicted at first. Taking smaller values of *l* gives a more accurate result, since the conformal mapping (which does not suffer from the error in Eq. ([eq:dominionenergy])) is used to calculate the energy of a larger portion of the flow pattern.\nNow Eq. ([eq:anom]) shows that the position-dependent scale factor, *ω*(**u**), plays the role of a single-particle energy. Additionally, this single particle energy can be regarded as the ``geometric potential,\" since it turns out to be related to the curvature in a way analogous to how the electrostatic potential is related to the charge. The function *ω* depends on the shape of the boundaries and on the curvature of the surface. A varying scale factor is necessary to map between surfaces with different distributions of curvature (such as planes with and without bumps). (A constant scale factor only rescales the curvature.) The curvature therefore depends on the *variation* of *ω*. In fact it can be shown that\n \n$$\\begin{aligned}\n{1}\nG\\_T(u,v)=e^{2\\omega(u,v)}&G\\_R(U(u,v),V(u,v))\\nonumber\\\\\n&+\\frac{1}{\\sqrt{det(g\\_{T,cd})}}\n\\partial\\_{a}\\sqrt{det(g\\_{T,cd})}g\\_T^{ab}\\partial\\_{b}\\omega.\n\\label{eq:Greenland}\\end{aligned}$$ \nThe second term is the Laplacian of *ω* as a function on the target surface[4](#fn4). The correspondence with electrostatic potentials, with *G**T*(*u*, *v*) as a source, is clear if the reference surface is a plane or disk and *G**R* = 0. Then Eq. ([eq:Greenland]) reduces to Eq. ([eq:geompdeq]).\nFor a deformed sphere or plane, single particle potentials come entirely from the second term in Eq. ([eq:anom]). The reference energy, corresponding to vortices on a sphere or plane cannot favor one position over another because of the homogeneity of these reference surfaces. The first term, *E**R*, leads to the vortex-vortex interactions which depend only on the separation of the vortices *on the reference surface*, again by symmetry; this energy is\n \n$$E\\_R=4\\pi^2 K\\sum\\_{i<j}n\\_in\\_j\\Gamma(\\mathbf{\\mathcal{U}\\_i},\\mathbf{\\mathcal{U}\\_j})\n\\label{eq:refenergy}$$ \nwhere Γ depends on the reference surface:\n \n$$\\Gamma\\_{plane}(\\mathbf{\\mathcal{X}},\\mathbf{\\mathcal{Y}})=\n-\\frac{1}{2\\pi} \\ln\\frac{|\\mathbf{\\mathcal{X}}-\\mathbf{\\mathcal{Y}}|}{a}$$ \nand\n \n$$\\begin{aligned}\n\\Gamma\\_{sphere}(\\mathbf{\\mathcal{X}},\\mathbf{\\mathcal{Y}})&=&\n-\\frac{1}{2\\pi} \\ln\\frac{2R\\sin\\frac{\\gamma}{2}}{a}\\\\\n&=&-\\frac{1}{2\\pi}\\ln\\frac{|\\mathbf{\\mathcal{X}}-\\mathbf{\\mathcal{Y}}|}{a}\n\\label{eq:sphereen}\\end{aligned}$$ \nwhere *γ* is the angle between the two points. ($2R\\sin\\frac{\\gamma}{2}$ is the *chordal* distance between the points, not the geodesic distance along the surface, as one might have guessed for the natural generalization of a Green’s function to curved space.) The detailed derivation of the second formulation of the vortex interaction energies in terms of the Green’s functions on the deformed surface (see Eqs. ([eq:greenpair-energy]) and ([eq:greenself-energy])) is contained in Appendix [app:CG].\n[sec:bubble]Vortices on a ``Soap Film\" Surface\n----------------------------------------------\nThere are experimental and theoretical motivations for studying substrates shaped as minimal surfaces. An example of a minimal surface is easy to make by dipping a loop of wire in soap; the spanning soap film tries to minimize its area. Vortices can be studied on a helium film coating a *solid* substrate whose surface has the shape of such a film. Such surfaces are characterized by a vanishing mean curvature, $H({\\bf x})$, so the contribution of the surface tension 2*γ**H*(**x**) to the thickness variation equation, Eq. ([eq:wet-8]) is drastically reduced. From the mathematical point of view, there is a widely known parametrization due to Weierstrass which readily leads to an *exact* expression for the geometric potential of a vortex on such a surface.\nWeierstrass’s formulae, which give a minimal surface for each choice of an analytic function *R*(*ζ*), read:\n \n$$\\begin{aligned}\nx(\\zeta)&=&Re\\int\\_0^\\zeta {R(\\zeta')(\\zeta'^2-1)d\\zeta'}\\nonumber\\\\\ny(\\zeta)&=&Im\\int\\_0^\\zeta {R(\\zeta')(\\zeta'^2+1)d\\zeta'}\\nonumber\\\\\nz(\\zeta)&=&Re\\int\\_0^\\zeta {R(\\zeta')2\\zeta'd\\zeta'}\n\\label{eq:weierstrass}\\end{aligned}$$ \nThe correspondence between this parametric surface and the complex variable *ζ* = X + *i*Y is a conformal map, and the conformal factor can be expressed in terms of *R*(*ζ*). Therefore, the analysis of vortices on such a surface is not difficult at all when the X, Y-plane is used as the reference surface.\nAs an example, let *R*(*ζ*) be equal to *L**ζ* where *L* controls the size of the target surface. Then the surface produced is given in parametric form by\n \n$$\\begin{aligned}\nx&=&L\\left[\\frac{\\mathcal{X}^3}{3}-\\mathcal{X}\\mathcal{Y}^2\n-\\mathcal{X}\\right]\\nonumber\\\\\ny&=&L\\left[\\frac{-\\mathcal{Y}^3}{3}+\\mathcal{Y}\\mathcal{X}^2+\\mathcal{Y}\\right]\n\\nonumber\\\\\nz&=&L(\\mathcal{X}^2-\\mathcal{Y}^2)\n\\label{eq:enneper}\\end{aligned}$$ \n![[fig:escaping?]Vortex and its streamlines on an ``Enneper Disk\".](ennepescape2)[fig:escaping?]Vortex and its streamlines on an ``Enneper Disk\".\nWe consider a superfluid film coating only a circle of radius *A* about the origin of the X,Y plane because the complete surface has self-intersections. This surface can be called the Enneper disk and is illustrated in Fig. [fig:escaping?]. The figure illustrates that the left and right hand sides of the saddle fold over it and would pass through each other if allowed to extend further while the front and the back would eventually intersect each other underneath the saddle. The former pair of intersection curves correspond to the two branches of the hyperbola X2 = 3(Y2 + 1). When the reference surface is curved into the Enneper surface, the *X* axis bends upward so that the branches map to the same intersection curve in the *y**z* plane. (The other intersection curves are obtained by exchanging X and Y.) Since the points where these hyperbolae are closest to the origin are $(\\pm\\sqrt{3},0),(0,\\pm\\sqrt{3})$, a non-self intersecting portion of the Enneper surface results as long as $A<\\sqrt{3}$.\nNow we explicitly calculate how a single vortex interacts with the curvature of such a surface by using Eq. ([eq:anom]). (We will use conformal mapping instead of Eq. ([eq:curvature-defect]) since the latter equation does not hold on a surface with the topology of a disk, as *ω* does not satisfy the Dirichlet boundary conditions which are implied by such an expression.) The metric obtained from ([eq:enneper]) is given by\n \n*d**x*2 + *d**y*2 + *d**z*2 = *L*2(*d*X2 + *d*Y2)(1 + X2 + Y2)2. \n(Surprisingly, this *metric* is rotationally symmetric. This implies that the surface may be slid along itself without stretching, but with changing amounts of bending.) Hence\n \n*ω**E**n**n**e**p**e**r* = − ln*L*(1 + R2), \nwhere R2 = X2 + Y2. According to Eq. ([eq:anom]), this indicates that the vortex should be attracted to the middle of the surface, but of course this force competes with the boundary interaction $K\\pi \\ln\\frac{A^2-\\mathcal{R}^2}{aA}$ which tries to pull the vortex to the edge. This expression for the boundary interaction is obtained from the familiar formula for the energy of a vortex interacting with its image in a flat reference disk. The total energy is then\n \n$$E=K[\\pi\\ln\\frac{L}{aA}+ \\pi\\ln (A^2-\\mathcal{R}^2)(1+\\mathcal{R}^2)].$$ \nAs long as *A* > 1, the central point of the saddle is a local minimum and this condition is compatible with the requirement $A<\\sqrt{3}$ for non-self-intersecting disks. Fig. [fig:escaping?] shows the flow lines of a vortex forced by the geometric interactions towards the center of an Enneper surface with *A* = 1.5.\nIn general, conformal mapping allows us to express the energy of a single vortex on a deformed surface with a boundary in the form:\n \n$$E=\\pi K[\\ln\\frac{A^2-\\mathcal{R}(\\mathbf{u})^2}{aA}-\\omega(\\mathbf{u})],\n\\label{eq:netenergy}$$ \nwhere R(**u**) refers to the image of a defect at **u** under a conformal map to a flat, circular disk of radius *A*. The Green’s function method cannot be used to determine the energy of defects on a surface with a boundary. Although the conformal factor *ω*(**u**) satisfies the Poisson equation, Eq. ([eq:geompdeq]), it cannot be expressed as the integral of the curvature times the Green’s function (as in Eq. ([eq:greenself-energy])), since *ω* does not satisfy simple boundary conditions. In any case, the first term in Eq. ([eq:netenergy]) has no general expression in terms of *ω* either. Interestingly, the *total* single-particle energy satisfies a *nonlinear* differential equation (the Liouville equation):\n \n$$\\nabla\\_{\\mathbf{u}}^2E(\\mathbf{u})=-\\pi K G(\\mathbf{u})-\n\\frac{4\\pi K}{a^2}e^{-\\frac{2E(\\mathbf{u})}{\\pi K}}.\n\\label{eq:liouvilledisk}$$ \nThis result can be derived by using Eq. ([eq:Greenland]) to calculate the Laplacian of the first term and using $\\nabla^2=e^{2\\omega(\\mathbf{u})}\\frac{1}{\\mathcal{R}}\n\\frac{\\partial}{\\partial\\mathcal{R}}\\mathcal{R}\n\\frac{\\partial}{\\partial\\mathcal{R}}$ to calculate the Laplacian of the second term. *E*(**u**) also satisfies an asymptotic boundary condition:\n \n$$e^{\\frac{E(\\mathbf{u})}{\\pi K}}\\rightarrow \\frac{2d}{a}$$ \nas *d*, the distance from **u** to the boundary, approaches 0. Together the differential equation and the boundary condition should determine the total geometrical and boundary energy of a single vortex, although the nonlinear Eq. ([eq:liouvilledisk]) is difficult to solve.\n[sec:bumps]Periodic surfaces\n----------------------------\nIn this section, we illustrate how a periodically curved substrate distorts the flat space interaction energies between vortices, besides generating the single-particle geometric potential. This effect is shown to be a consequence of the action of a conformal map which generally will map the target surface into a periodic reference substrate with different lattice vectors from the vectors of the target substrate. According to the general relation Eq. ([eq:greentransf]), the long-distance behavior of the Green’s function is given by the logarithm of a distorted distance.\nConsider a surface with a periodic height function *z* = *h*(*x*, *y*), i.e., say *h* satisfies\n \n*h*(*x* + *λ**i*, *y* + *μ**i*) = *h*(*x*, *y*) for\\ *i* = {1, 2}, \nwhere *i* labels the two basis vectors, which are not assumed to be orthogonal. Figure ([tulipbulbs]) shows the corresponding periods (*λ**i*, *μ**i*). A conformal mapping can be chosen to preserve the fact that the substrate is periodic but not the actual values of the periods, which are therefore given on the reference substrate by two new pairs denoted by (Λ*i*, *M**i*). In other words, we suppose that a tesselation of the target substrate by congruent unit cells is mapped to a set of congruent unit cells on the reference substrate. Then the map transforming the original coordinates (*x*, *y*) into the target coordinates (X, Y) satisfies\n \n$$\\begin{aligned}\n\\mathcal{X}(x+\\lambda\\_i, y+\\mu\\_i)=\\mathcal{X}(x, y)+\\Lambda\\_i \\nonumber\\\\\n\\mathcal{Y}(x+\\lambda\\_i, y+\\mu\\_i)=\\mathcal{Y}(x, y)+M\\_i.\n\\label{eq:periods}\\end{aligned}$$ \nThere is no simple formula for the new set of primitive lattice vectors (Λ*i*, *M**i*) for the reference space. In some cases, though, precise information can be derived from the fact that the (Λ*i*, *M**i*) share the symmetry of the topography of the original substrate. For example, if the lattice is composed of bumps which have a 90∘ rotational point symmetry, then the reference lattice will be square. On the other hand, the topology of the periodic surface with a *s**q**u**a**r**e* lattice shown in the contour plot of Fig. ([fig:squaredaway]) does not posses a 90∘ rotational symmetry and hence its conformal image will have a *r**e**c**t**a**n**g**u**l**a**r* lattice.\n![[tulipbulbs] An illustration of the period lattice of the target surface, projected into the xy-plane. The coordinates \\lambda_1,\\mu_1 and \\lambda_2,\\mu_2 describe the basis vectors, which we do not assume to be orthogonal. ](tulipbulbs3)[tulipbulbs] An illustration of the period lattice of the target surface, projected into the *x**y*-plane. The coordinates *λ*1, *μ*1 and *λ*2, *μ*2 describe the basis vectors, which we do not assume to be orthogonal. \nTo get an idea how the conformal mapping behaves macroscopically, we try to decompose it into a linear transformation L with matrix coefficients {*A*, *B*; *C*, *D*} and a periodic modulation captured by the functions *ξ*(*x*, *y*) and *η*(*x*, *y*) that distort the X and Y axes respectively:\n \n$$\\begin{aligned}\n\\mathcal{X}(x,y)=Ax+By+\\xi(x,y) \\nonumber\\\\\n\\mathcal{Y}(x,y)=Cx+Dy+\\eta(x,y).\n\\label{eq:shiftperiods}\\end{aligned}$$ \n(This decomposition is justified by the self-consistency of the following calculations.) The matrix coefficients of the linear transformation can be determined by requiring consistency with Eq. ([eq:periods]). Start by evaluating the left hand sides of the two Equations ([eq:shiftperiods]) at the positions {*x* + *λ**i*, *y* + *μ**i*} which are shifted by the two pairs of periods {*λ**i*, *μ**i*}, so that the right hand sides become X(*x*, *y*) + Λ*i* and Y(*x*, *y*) + *M**i*, according to Eq. ([eq:periods]). Then subtract the resulting equations from the corresponding unshifted Equations ([eq:shiftperiods]), for each value of *i*. We then obtain two pairs of equations\n \n$$\\begin{aligned}\nA \\lambda\\_i + B \\mu\\_i&=&\\Lambda\\_i \\nonumber\\\\\nC \\lambda\\_i + D \\mu\\_i&=&M\\_i \\quad \\ \\mathrm{for\\ }i=\\{1,2\\},\n\\label{eq:int1}\\end{aligned}$$ \nwhere we have used the fact that the periodic functions *ξ*(*x*, *y*) and *η*(*x*, *y*) are unchanged when shifted by the periods. We can now solve the four equations of Eq. ([eq:int1]) simultaneously for *A*, *B*, *C*, and\\ *D* to see that the linear transformation matrix L reads\n \n$$\\mathcal{L}=\\left(\\begin{array}{cc}A&B\\\\C&D\\end{array}\\right)=\n\\left(\\begin{array}{cc}\\Lambda\\_1&\\Lambda\\_2\\\\M\\_1&M\\_2\\end{array}\\right)\n\\left(\\begin{array}{cc}\\lambda\\_1&\\lambda\\_2\\\\\\mu\\_1&\\mu\\_2\\end{array}\\right)^{-1}.\n\\label{eq:affine}$$ \n(Now we can justify the original decomposition, Eq. ([eq:shiftperiods]), by *defining* L by Eq. ([eq:affine]) and *defining* *ξ*(*x*, *y*) and *η*(*x*, *y*) as the discrepancy between the conformal map X(*x*, *y*), Y(*x*, *y*) and the linear map L(*x*, *y*) as in Eq. ([eq:shiftperiods]). We can then check that *ξ*(*x*, *y*) and *η*(*x*, *y*) *are* periodic functions of the coordinates.)\nThe linear transformation can be used to calculate approximately the long distance behavior of the Green’s function\n \n$$\\begin{aligned}\n\\Gamma(x,y;x',y')&=&-\\frac{1}{4\\pi}\n\\ln[(\\Delta\\mathcal{X})^2+(\\Delta\\mathcal{Y})^2]\\nonumber\\\\\n&\\approx&-\\frac{1}{4\\pi}\\ln[(A\\Delta x+B\\Delta y)^2\\nonumber\\\\\n&&\\ \\ \\ \\ \\ \\ +(C\\Delta x+D\\Delta y)^2],\n\\label{eq:ellipsie}\\end{aligned}$$ \nwhere we used the fact that the periodic functions *ξ*(*x*, *y*) and *η*(*x*, *y*) are bounded and hence negligible in comparison to Δ*x* and Δ*y* at long distances. This expression illustrates the fact that the matrix L captures the long distance lattice distortions induced by the conformal mapping, apart from the additional waviness described by *ξ*(*x*, *y*) and *η*(*x*, *y*). The linear transformation determined by L is by itself typically *not conformal*, meaning that it generates an anisotropic deformation of the target lattice which does not preserve the angle between the original lattice vectors.\n![[fig:squaredaway]A periodic surface with a square lattice but not square symmetry. This surface is illustrated by its contour plot. It has the two reflectional symmetries but not the 90^{\\circ} symmetry of a square lattice; hence, generically, the conformal image will have a rectangular lattice.](squared)[fig:squaredaway]A periodic surface with a square lattice but not square symmetry. This surface is illustrated by its contour plot. It has the two reflectional symmetries but not the 90∘ symmetry of a square lattice; hence, generically, the conformal image will have a *rectangular* lattice.\nThe deformation of the lattice is controlled by the curvature of the substrate. To spell out this connection and allow an explicit evaluation of the long-distance Green’s function in Eq. ([eq:ellipsie]), we explicitly evaluate the matrix elements *L**i**j* in terms of the height function *h*(*x*, *y*) of a gently curved (or low-aspect-ratio) surface, one for which *h*(*x*, *y*) ≪ {*λ**i*, *μ**i*}, {*ξ*(*x*, *y*), *η*(*x*, *y*)} ≪ 1 and {*A* − 1, *D* − 1, *B*, *C*} ≪ 1. The new set of (isothermal) coordinates X and Y, used to implement the conformal transformation, are found by solving the Cauchy-Riemann Equations ([eq:tack])\n \n$$\\begin{aligned}\n&&\\partial\\_x\\mathcal{X}\n=\\sqrt{g}g^{yy}\\partial\\_y\\mathcal{Y}+\\sqrt{g}g^{xy}\\partial\\_x\\mathcal{Y}\n\\nonumber\\\\\n&&\\partial\\_y\\mathcal{X}\n=-\\sqrt{g}g^{xx}\\partial\\_x\\mathcal{Y}-\\sqrt{g}g^{xy}\\partial\\_y\\mathcal{Y},\\end{aligned}$$ \nwhich, upon substituting from Eq. ([eq:shiftperiods]) and making the small aspect ratio approximation discussed in Appendix [app:calmseas], reduce to\n \n$$\\begin{aligned}\n&&A+\\partial\\_x\\xi=D+\\partial\\_y\\eta+\\frac{1}{2}(h\\_x^2-h\\_y^2)\\\\\n&&B+\\partial\\_y\\xi=-C-\\partial\\_x\\eta+h\\_xh\\_y.\n\\label{eq:alphabet}\\end{aligned}$$ \nWe now proceed to show that these equations do not have solutions unless the lattice is distorted, that is to say the matrix L cannot be the identity for a generic periodic function *h*(*x*, *y*). Note that the periodicity of *ξ*(*x*, *y*) and *η*(*x*, *y*) implies that the integral of either one over any unit cell, e.g.,\n \n∬cell*d**x**d**y* *ξ*(*a* + *x*, *b* + *y*) \nis independent of the quantities (*a*, *b*) by which the unit cell is shifted. Upon differentiating the integral with respect to *a*, one obtains\n \n∬cell*d**x**d**y* *ξ**x*(*x*, *y*) = 0. \nSimilarly, the averages of *η**x*, *η**y*, and *ξ**y* are also equal to zero. Hence, upon averaging Eq. ([eq:alphabet]) over a unit cell we obtain the key relations\n \n$$\\begin{aligned}\nA-D&=&\\frac{1}{2}<h\\_x^2-h\\_y^2>\\nonumber\\\\\nB+C&=&<h\\_xh\\_y>,\n\\label{eq:approxperiods}\\end{aligned}$$ \nwhich prove our assertion that *L* cannot be a simple dilation or rotation. Some shear is naturally introduced by the non trivial metric (or height function) of the underlying surface.\nThe L matrix is undetermined up to a dilation and a rotation but this is of no consequence to the determination of the Green’s function. In fact, to find the Green’s function, note that Eq. ([eq:approxperiods]) allows us to write the matrix coefficients in terms of two undetermined constants *ε*1 and *ε*2 that will drop out of the final answer:\n \n$$\\begin{aligned}\n&&A=1+\\epsilon\\_1+\\frac{1}{4}<h\\_x^2-h\\_y^2>\\\\\n&&D=1+\\epsilon\\_1+\\frac{1}{4}<h\\_y^2-h\\_x^2>\\\\\n&&B=\\epsilon\\_2+\\frac{1}{2}<h\\_xh\\_y>\\\\\n&&C=-\\epsilon\\_2+\\frac{1}{2}<h\\_xh\\_y>,\\end{aligned}$$ \nso that consistency with Equations ([eq:approxperiods]) is guaranteed. (The variables *ε*1 and *ε*2 parameterize an overall infinitesimal scaling (by 1 + *ε*1) and a rotation (by angle *ε*2) respectively.) Substitution of these equations into Eq. ([eq:ellipsie]) gives the desired long-distance behavior of the Green’s function purely in terms of derivatives of the height function, which we assume to be known:\n \n$$\\begin{aligned}\n\\Gamma(x,y;x',y')&\\approx& -\\frac{1}{4\\pi}\\ln [\\Delta x^2+\\Delta y^2\\nonumber\\\\\n&&\\ \\ \\ \\\n+\\frac{1}{2}<h\\_x^2-h\\_y^2>((\\Delta x)^2-(\\Delta y)^2)\\nonumber\\\\\n&&\\ \\ \\ \\ \\ +2<h\\_xh\\_y>\\Delta x\\Delta y].\\end{aligned}$$ \nThis is the central result of this section; it can also be applied to interactions between disclinations in liquid crystals and dislocations in crystals. The anisotropic correction to the Green’s function, captured by the second and third term, suggests that a distorted version of the triangular lattice of vortices expected on a flat substrate may form when the helium-coated surface is rotated slowly enough that there is only one vortex to several unit cells. However, the actual ground state is likely to be difficult to observe, as the geometric potential will try to trap the vortices near saddles as discussed in Section [BS].\nBand-flows on elongated surfaces[sec:zucchini]\n----------------------------------------------\nIn this section, we show that the quantization of circulation can induce an extremely long-range force on a stretched-out sphere (such as a surface with the shape of a zucchini or a very prolate spheroid). We first demonstrate the main result in the context of a simple example before presenting a general formula for the forces experienced by vortices on azimuthally symmetric surfaces. Details are presented in Appendix [app:deimos]. Consider a cylinder of length 2*H* and radius *R* < < *H* with hemispherical caps of radius *R* at the ends, depicted in Fig. [fig:goodnplenty], and imagine a symmetric arrangement of a vortex (*n*=1) and an anti-vortex (*n*=-1) at the north and south poles respectively.\n![[fig:goodnplenty]A capped cylinder; a cylinder of length 2H is closed off by hemispheres at the north and south poles of radius R. The circulation around every lattitude is the same.](goodnplenty)[fig:goodnplenty]A capped cylinder; a cylinder of length 2*H* is closed off by hemispheres at the north and south poles of radius *R*. The circulation around every lattitude is the same.\nExtrapolating our intuition from flat space suggests that the energy of the vortex and anti-vortex is $2\\pi K\\ln\\frac{D}{a}$, where *D* is the distance between the vortices. However, more careful reasoning shows that the energy grows *linearly* rather than logarithmically with *D*. The reason is that, unlike in flat space, the velocity field does not fall off like the inverse of the distance from each vortex. Note that the azimuthal symmetry of the arrangement of the vortices implies that the flow is parallel to the lines of latitude of the surface. Since the circulation around each latitude must be $\\frac{h}{m}$, the flow speed on the cylindrical part reads\n \n$$v=\\frac{h}{2\\pi m R}.\n\\label{eq:farina}$$ \nThe kinetic energy of this part of the flow is $[4\\pi RH]\\frac{1}{2}\\rho\\_s v^2=2\\pi\\frac{KH}{R}$. Since this cylindrical part of the flow forms the main contribution to the kinetic energy when *H* > > *R* we find that the energy of a vortex-antivortex pair situated at opposite poles is linear,\n \n$$E\\_{poles}\\approx 2\\pi\\frac{KH}{R}.\n\\label{eq:lateral}$$ \n(The exact expression also includes a near-vortex energy of approximately $2\\pi K\\ln\\frac{R}{a}$.)\nIn contrast, when the vortices forming the neutral pair are across from each other on the same latitude, the aforementioned long-range persistence of the velocity field is absent because the vorticity is screened within a distance of order *R*. The resulting kinetic energy follows the familiar logarithmic growth\n \n$$E\\_{equator}\\approx 2\\pi K\\ln\\frac{2R}{a}.\n\\label{eq:opposite}$$ \nMore generally, consider an azimuthally symmetric surface described by the radial distance, *r*(*z*), as a function of height, *z*, as indicated in Fig. [fig:kitestring] A. If the north and south poles of the surface are at *z**s* and *z**n*, then *r*(*z**s*) = *r*(*z**n*) = 0 since the surface closes at the top and bottom. A point on such a surface can be identified by the coordinates (*ϕ*, *σ*) where *ϕ* is the azimuthal angle and *σ* is the distance to the point from the north pole along one of the longitudes such as the one shown in Fig. [fig:kitestring]A. In Appendix [app:deimos], we develop an approximation scheme which rests on the observation that the flow pattern becomes mostly azimuthally symmetric if $\\frac{dr}{dz}<<1$, even if the vortices break the azimuthal symmetry of the surface because they are not at the poles. If a pair of *n* = ± 1 vortices are present at different heights *z*1, 2, then the fluid in the band between them flows almost horizontally and at a nearly *ϕ*-independent speed (except for irregularities near the vortices) while the fluid beyond them is approximately stagnant (see Fig. [fig:kitestring]).\n![[fig:kitestring]The flow on an azimuthally symmetric surface, described by the coordinates (z,r) and an azimuthal angle \\phi (not shown). A) The surface is defined as the surface of revolution of the curve r(z) in the r-z plane. The other two images show the flow on an ellipsoid and compare the flow pattern predicted by the band model (B) and the exact solution determined by conformal mapping (C). The flow lines in the band between the two vortices become close to horizontal and are approximately azimuthally symmetric. Beyond the vortices, they are spaced far apart indicating a vanishingly small speed for a greatly elongated surface. ](kitestring3)[fig:kitestring]The flow on an azimuthally symmetric surface, described by the coordinates (*z*, *r*) and an azimuthal angle *ϕ* (not shown). A) The surface is defined as the surface of revolution of the curve *r*(*z*) in the *r* − *z* plane. The other two images show the flow on an ellipsoid and compare the flow pattern predicted by the band model (B) and the exact solution determined by conformal mapping (C). The flow lines in the band between the two vortices become close to horizontal and are approximately azimuthally symmetric. Beyond the vortices, they are spaced far apart indicating a vanishingly small speed for a greatly elongated surface. \nAlong any latitude inside the band the circulation is *exactly* $\\frac{h}{m}$, while it is zero above and below it. These properties approximately determine the flow away from the vortices since the asymmetric irregularities near the vortices decay exponentially, giving a speed of\n \n$$\\frac{h}{2\\pi m r(z)}\n\\label{eq:simplebandspeed}$$ \nin between *z*1 and *z*2, the locations of the vortices, and zero elsewhere. We describe this approximation as the ``band model\". This expression shows that constrictions in the surface cause the speed to increase. For the cylinder with spherical caps and arbitrarily placed vortices, Eq. ([eq:simplebandspeed]) shows that the speed is approximately constant within the band. (It increases within a distance on the order of *R* from the vortices, which are on the edges of the band.)\nThe kinetic energy can be determined approximately by noting that the energy in a thin ring on the surface between the vortices (extending from the longitudinal arclength *σ* to *σ* + *d**σ*) is\n \n$$[2\\pi r(z)d\\sigma][\\frac{1}{2}\\rho\\_s (\\frac{\\hbar}{m r(z)})^2]=\\pi K\n\\frac{d\\sigma}{r(z)}\n\\nonumber$$ \nwhere the first factor represents the area of the ring since *σ* is the geodesic distance along the surface and the second factor represents the included kinetic energy. The flow is zero past the two vortices, so the total energy is\n \n$$E=K\\pi \\int\\_{\\sigma\\_1}^{\\sigma\\_{2}}\\frac{d\\sigma}{r}.\n\\label{eq:bandenergy1}$$ \nFor the capped cylinder with a constant *r*(*z*) the integral is $\\pi\\frac{KD}{R}$, where *D* is the vertical distance between the vortices. This expression generalizes the result for vortices at the poles of the surface, Eq. ([eq:lateral]). However, when the vortices are on opposite ends of an equator, they are too close for the asymmetries to be neglected. The energy in this case is calculated in Appendix [app:deimos]. The force *F*1, *b**a**n**d*, experienced by vortex 1 can be determined by taking the gradient of Eq. ([eq:bandenergy1]). If vortex 1 is moved downward, the band of moving fluid shrinks so the energy drops. The force on the vortex therefore reads\n \n$$F\\_{1,band}=\\frac{\\pi K}{r(\\sigma\\_1)}\n\\bm{\\hat{\\sigma}}.\n\\label{eq:bandforce1}$$ \nOn the capped cylinder, this force is independent of the positions of the vortices. Even on an arbitrary elongated surface, a noteworthy feature is that the force on vortex 1 does not depend on the position of vortex 2! This force can be explained with the familiar phenomenon of lift: the vortex is on the boundary between stationary and moving fluid, so there is a pressure difference due to the Bernoulli principle.\nApproximating the flow pattern generated by multiple vortices in a similar fashion requires only minor modifications of the previous argument. In a low-resolution snapshot of the flow, the point-vortices would appear as circles of discontinuity in the velocity field that go all the way around the axis (the analogue for a layer of superfluid of a two dimensional vortex sheet). If the vortices are labeled in order of decreasing *z* a loop just below the *l*th vortex contains\n \n*N**l* = ∑*i* = 1*l**n**i* \nunits of circulation above it. Approximate azimuthal symmetry of the flow then implies that,\n \n$$\\mathbf{v}(z,\\phi)\\_{band}=N\\_l\\frac{\\hbar}{m r(z)}\\mathbf{\\hat{\\phi}}\\ \\ \\ \n(\\mathrm{for\\ }z\\_l<z<z\\_{l+1}),\n\\label{eq:bandspeed}$$ \na natural generalization of Eq. ([eq:simplebandspeed]) that is proved in Appendix [app:deimos].\nConformal mapping can be employed to justify (without detailed calculations) the decay of the nonazimuthally symmetric parts of the flow that are not determined by the quantization condition. We sketch the basic reasoning here by focusing (for simplicity) on the flow pattern *near the equator* of the surface, at a distance *σ**e**q* from the north pole. The conformal transformation that maps the elongated sphere onto a regular reference sphere with coordinates Θ, Φ reads (see Appendix [app:deimos])\n \n$$\\begin{aligned}\n&&\\sin\\Theta=\\mathrm{sech}\\int\\_{\\sigma}^{\\sigma\\_{eq}}\n{\\frac{d\\sigma'}{r(\\sigma')}}\\nonumber\\\\\n&&\\Phi=\\phi.\n\\label{eq:deimosmap}\\end{aligned}$$ \nThe upper and lower halves of the elongated sphere can be mapped to the upper and lower hemispheres by choosing appropriately between the two values of Θ that correspond to a given value of sinΘ. Near the equator the integral can be approximated by $\\frac{\\sigma-\\sigma\\_{eq}}{r\\_{eq}}$ since *r* varies slowly. Suppose the vortices are far from the equator, at a distance greater than *k**r**e**q* for a large *k*. Then the vortices above the equator are mapped exponentially close (at a distance less than *e*− *k*) to the sphere’s north pole. Likewise vortices on the southern half of the surface map exponentially close to the sphere’s south pole. We have thus reduced the task of finding the flow due to a complicated arrangement of vortices to a symmetric case. In fact, after mapping the flow on the long, thin surface to the reference sphere, nothing can be resolved beyond a pair of multiply quantized vortices at the north and south poles containing *N* and − *N* units of circulation respectively, where *N* is the total circulation number of all the vortices above the equator. Since the image vortices are very close to the poles, their flow pattern on the reference surface is approximately azimuthally symmetric near the equator. When mapped back to the elongated surface, the flow retains its approximate azimuthal symmetry in the region around the equator, completing our argument. A similar argument proves the approximate azimuthal symmetry of the flow near lines of latitude other than the equator; one simply adjusts the conformal map in Eq. ([eq:deimosmap]) so that another latitude of the target surface is mapped to the equator of the reference sphere.\nNow the geometrical force derived in Eq. ([eq:bandforce1]) can compete with physical forces such as those induced by rotating an ellipsoid about its long axis with angular velocity $\\bm{\\Omega} = \\Omega \\!\\!\\!\\! \\quad \\bm{\\hat{z}}$. Let us extend the treatment of rotational forces on curved substrates, introduced in section [sec:Rotation], to the case of an ellipsoid described by the radial function\n \n$$r(z)=R\\sqrt{1-\\frac{z^2}{H^2}}.\n\\label{eq:prolate}$$ \nLet us use the aspect ratio $\\alpha=\\frac{H}{R}$ to describe how elongated this ellipsoid is, and determine how a vortex-anti-vortex pair is torn apart by the rotation as the angular frequency is increased. As in Section [sec:Rotation], metastable vortex configurations can often be found, so we will consider transitions between different *local* minima of the vortex-energy function. Recall that the effect of the rotation on the superfluid energy is expressed in terms of the angular momentum *L**z* by an extra term − Ω*L**z* in Eq. ([eq:Erot]) which must be evaluated for (at least) a pair of opposite-signed vortices to satisfy the topological constraints imposed by the spherical topology of the surface. We find, in analogy to Eq. ([eq:Lz-c]), that the extra contribution to the vortex energetics is additive and reads\n \n$$-\\Omega L\\_z=n\\_1\\hbar \\Omega \\frac{\\rho\\_s}{m} [A(\\sigma\\_1)-A(\\sigma\\_2)],\n\\label{eq:Lzvortex}$$ \nwhere *A*(*σ*) represents the area of the ellipsoid out to a distance *σ* from the north pole while *σ*1 and *σ*2 represent the positions of the two defects (*σ*1 < *σ*2). If *n*1 = 1, the energy in Eq. ([eq:Erot]) is decreased by moving the positive vortex closer to the north pole and the negative one closer to the south pole. To understand this, note that the sense of rotation of the superfluid around a vortex is defined by an observer facing the surface. Hence, a negative vortex on the southern half of the surface rotates in the same direction as a positive one on the northern half (relative to the positive *z*-axis), and both agree with the sense of rotation of the substrate. The rotational force on vortex 1 derived from Eq. ([eq:Erot]) is\n \n$$\\mathbf{F\\_{\\Omega}}=-\\frac{n\\_1\\hbar\\rho\\_s}{m}2\\pi r(\\sigma)\\mathbf{\\hat{\\sigma}}.\n\\label{eq:Frotsequel}$$ \nAs Ω increases, *pairs* of positive and negative vortices will appear in this geometry. As each vortex pair is created, the positive vortex will move to the top side of the surface, and the negative one to the bottom. There is a critical frequency, Ω*b*, at which a single pair of vortices, once created, can exist metastably in a configuration symmetric about the *x**y*-plane. As the angular frequency is increased, the vortices are gradually pulled apart until at the frequency Ω*a*, they reach the poles. (The latter transition is analogous to the center/off-center transition for a single vortex described in Section [subsec:single].) When Ω*a* > Ω > Ω*b*, the equilibrium condition, obtained by balancing the forces in Equations ([eq:bandforce1]) and ([eq:Frotsequel]), reads\n \n$$\\frac{\\pi K}{r\\_1}=2\\pi\\frac{\\hbar\\rho\\_s}{m}\\Omega r\\_1\n\\label{eq:heartgrowsfonder}$$ \nSince *r* and *z* are connected by the equation of the ellipsoid, the vortices are located at heights\n \n$$\\pm z\\approx \\pm\\alpha\\sqrt{R^2-\\frac{\\hbar}{2m\\Omega}}.\n\\label{eq:prize}$$ \nThe vortices first become metastable when force balance is achieved with both vortices close to the equator. Upon substituting the equatorial value *r*1 = *R* into Eq. ([eq:heartgrowsfonder]) an estimate of Ω*b* is obtained\n \n$$\\Omega\\_b\\approx\\frac{\\hbar}{2mR^2}.\n\\label{eq:newadventure}$$ \nWhen the pair first appears, there will actually be a non-zero defect separation, although substituting Eq. ([eq:newadventure]) into Eq. ([eq:prize]) suggests otherwise. Imagine slowing the rotation speed through Ω*b*. The vortices will approach each other gradually; within the large vortex-separation approximation of Eq. ([eq:bandforce1]), the attraction between them will *decrease* as they become closer because *r*(*z*) increases. However, when the vortices become close enough, the attraction between them starts increasing and the vortices are suddenly pulled together. The minimum *z*-coordinate for metastable vortices is derived along these lines in Appendix [app:deimos] (which also discusses what happens at Ω = Ω*a*) and reads\n \n*z*1 = − *z*2 = *z**b* ≈ *R*ln*α*. \nThe transition through Ω*b* is illustrated pictorially in Fig. [fig:spndl] which shows how the local minimum in the energy function disappears as the frequency decreases.\n![[fig:spndl]The rotational and fluid energy (units of K) as a function of \\sigma_1=\\sigma_{tot}-\\sigma_2=\\sigma (units of R) for H=3.5 R and \\frac{m\\omega}{\\hbar}=.49,.61,.74 R^{-2}. The middle curve, roughly at \\omega=\\omega_b, shows the last position where the vortex is stable as \\omega is decreased.](spndl2)[fig:spndl]The rotational and fluid energy (units of *K*) as a function of *σ*1 = *σ**t**o**t* − *σ*2 = *σ* (units of *R*) for *H* = 3.5*R* and $\\frac{m\\omega}{\\hbar}=.49,.61,.74 R^{-2}$. The middle curve, roughly at *ω* = *ω**b*, shows the last position where the vortex is stable as *ω* is decreased.\n[sec:noodles]Interactions on a closed surface\n---------------------------------------------\nTo understand interactions between vortices on an arbitrary deformed sphere one must come to terms with the neutrality constraint on the total circulation of a flow. On any compact surface,\n \n∑*i**n**i* = 0. \nThis constraint on the sum of the circulation indices {*n**i*} always holds: if the surface is divided into two pieces by a curve, the sum of the quantum numbers on the top and bottom half must be equal and opposite (because they are both equal to the circulation around the dividing curve). As we shall see, this relation implies that there are multiple ways of splitting up the energy into single-particle energies and two-particle interaction energies, despite the fact that the total energy is well-defined. The behavior of the one-particle and interaction terms depends on how the splitting is carried out. To illustrate this ambiguity, multiply Eq. ([eq:neutrality]) by $4\\pi^2 K n\\_1 f(\\bm{u}\\_1)$ and separating out the *i* = 1 term, to obtain\n \n4*π*2*n*12*f*(**u**1) = − ∑*i* ≠ 14*π*2*n*1*n**i**f*(**u**1). \nHence, a portion 4*π*2*K**f*(**u**1) of the ``geometrical energy\" of vortex 1 can be reattributed to this vortex’s interaction with all the other vortices. This can be seen explicitly by checking that the *net* energy according to Eqs. ([eq:greenself-energy]) and ([eq:greenpair-energy]),\n \n$$\\begin{aligned}\n{1}\nE(\\{n\\_i,\\bm{u}\\_i\\})=\\sum\\_{i<j}&4\\pi^2 Kn\\_in\\_j\\Gamma(\\bm{u}\\_i,\\bm{u}\\_j)\\\\\n&-\\sum\\_i\\pi n\\_i^2 KU\\_G(\\bm{u}\\_i)\n\\label{eq:redouble}\\end{aligned}$$ \nis not changed by the following transformation:\n \n$$\\begin{aligned}\n&&\\Gamma'(\\mathbf{u}\\_1,\\mathbf{u}\\_2)=\\Gamma(\\mathbf{u}\\_1,\\mathbf{u}\\_2)\n-f(\\mathbf{u}\\_1)-f(\\mathbf{u}\\_2)\\\\\n&&U\\_G'(\\mathbf{u})=U\\_G(\\mathbf{u})+4\\pi f(\\mathbf{u})\n\\label{eq:ces}\\end{aligned}$$ \nThis flexibility is reflected in the possibility of choosing different Green’s functions Γʹ(**u**1, **u**2) for the covariant Laplacian on a deformed compact surface. A detailed discussion of Green’s functions is given in Appendix [app:CG]. Here we higlight this ambiguity by performing explicit calculations using two distinct choices of Green’s functions on a model surface formed from a unit sphere. First cut the sphere in halves along a great circle. Choose one of the hemispheres and bring opposite sides of the great circle bounding it together and glue them. The result is a pointed sphere resembling a conchigliette noodle (i.e., a shell noodle) sealed shut (see Fig. [fig:noodlemachine]A). The surface closes up smoothly since opposite sides of the seal have matching curvatures; the surface also turns out to be rotationally symmetric. (A similar shape forms when a pollen grain with a weak sector (such as the pollen of a lily) dries out.)\n![[fig:noodlemachine] A) The process of folding a hemisphere into a pointed sphere, bounded at the north and south poles by two 180^{\\circ} disclinations. The north and south poles move outward along the axis, while the latitudes stay horizontal. The Gaussian curvature is invariant because the decreasing curvature of the lines of longitude is compensated by the tighter curvature around the lines of lattitude. B) A top view of vortices during the furling-up of one hemisphere. The first stage shows both the (0<\\alpha<\\pi) hemisphere that is furled up and the other hemisphere together with two defects and their images. The hemisphere is furled up into the pointed sphere so that the left and right halves of the cut (which appears as a horizontal diameter in this top view) are brought together to form the seam on the pointed sphere; simultaneously, the defects Q_1,Q_2 move to positions \\mathbf{u}_1 and \\mathbf{u}_2 on the pointed sphere. The furling process leads to a continuous flow pattern on the pointed sphere. For example, the two points marked with an x on the cut in the original sphere are sealed together. Both points feel a strong flow, the one on the right because it is close to the vortex at Q_1 and the one on the left because it is close to this vortex’s image. ](noodleseal4)[fig:noodlemachine] A) The process of folding a hemisphere into a pointed sphere, bounded at the north and south poles by two 180∘ disclinations. The north and south poles move outward along the axis, while the latitudes stay horizontal. The Gaussian curvature is invariant because the decreasing curvature of the lines of longitude is compensated by the tighter curvature around the lines of lattitude. B) A top view of vortices during the furling-up of one hemisphere. The first stage shows both the (0 < *α* < *π*) hemisphere that is furled up and the other hemisphere together with two defects and their images. The hemisphere is furled up into the pointed sphere so that the left and right halves of the cut (which appears as a horizontal diameter in this top view) are brought together to form the seam on the pointed sphere; simultaneously, the defects *Q*1, *Q*2 move to positions **u**1 and **u**2 on the pointed sphere. The furling process leads to a continuous flow pattern on the pointed sphere. For example, the two points marked with an x on the cut in the original sphere are sealed together. Both points feel a strong flow, the one on the right because it is close to the vortex at *Q*1 and the one on the left because it is close to this vortex’s image. \nThe geometrical interaction on this surface can be given an appealing interpretation in terms of the method of images from electrostatics. When one uses the method of images to study charges in a half-space bounded by a conducting plane, one completes the space, with the other half-space. Then one introduces charges into this fictitious region to ensure that the right boundary conditions (orthogonality of the field lines to the original boundary plane) are satisfied. For the pointed sphere, we first complete the surface by opening it up again and adding back the second hemisphere. We can describe points on the pointed sphere by (*ϕ*, *σ*) (where *σ* is the distance from the north point and *ϕ* is the azimuthal angle, as for the general case described in Sec. [sec:zucchini]). Points on the completed sphere are naturally labeled by the standard spherical coordinates (*α*, *θ*) that represent the azimuthal and polar angles respectively. The mapping from the pointed sphere that returns each point to its position on the original sphere is given by\n \n$$\\begin{aligned}\n\\alpha=\\frac{\\phi}{2}\\nonumber\\\\\n\\theta=\\sigma.\n\\label{eq:half-rations}\\end{aligned}$$ \nAs *ϕ* ranges from 0 to 2*π*, *α* goes from 0 to *π*, across the hemisphere used to make the pointed sphere. Note that *θ* = *σ* since the pointed sphere is formed by isometrically bending the hemisphere (the angle *θ* is equal to the geodesic distance to the north pole of the hemisphere).\nNow for each vortex at **u** = (*ϕ*, *σ*) on the pointed sphere, we introduce two vortices on the sphere, one at *Q* with $(\\alpha,\\theta)=(\\frac{\\phi}{2},\\sigma)$ and one at $Q^\\*=(\\frac{\\phi}{2}+\\pi,\\sigma)$. The latter is the image vortex of the former, obtained by rotating *Q* by 180∘ around the *z*-axis.\nThe energy of a defect configuration on the pointed sphere is derived by halving the energy of the flow pattern produced by the doubled set of vortices on the full sphere. In analogy with the electrostatic problem, the purpose of situating the image defects in the way just described is to preserve the continuity of flows across the seam. Imagine drawing the flow pattern of all the vortices on the sphere. Focus on the hemisphere 0 < *α* < *π*. Because the vortices are placed symmetrically about the sphere’s axis, the flow near *α* = 0 will match the flow near *α* = *π* when the surface is sealed. (See Fig. [fig:noodlemachine]B.)\nThe flow pattern on the pointed sphere results from rolling up half of the flow pattern on the sphere. Once the positions of the image defects are chosen, the flow pattern is found by deriving it from the stream function $\\chi(\\bm{u})$ introduced in Sec. [subsec:effective]. The stream function at a point $\\bm{u}$ on the pointed sphere can be expressed in terms of the Green’s function of the sphere, according to Eq. ([eq:green-stream1]):\n \n$$\\chi(\\mathbf{u})\n=\\sum\\_i{\\frac{n\\_ih}{m}[\\Gamma^{sphere}(Q,Q\\_i)+\\Gamma^{sphere}(Q,Q\\_i^\\*)]}\n\\label{eq:darkandlight}$$ \nwhere *Q* is the point on the sphere corresponding to **u** on the pointed sphere and *Q**i* and *Q**i*\\* are the locations of the *i*th pair of vortices on the sphere.\nThe energy of the vortices on the sphere takes up the familiar electrostatic form of a sum of the interactions between all pairs of defects and/or their images. The energy stored in the flow pattern on the pointed sphere (which is half as large as on the complete sphere) reads\n \n$$\\begin{aligned}\n\\frac{E\\_N}{K}&=\\frac{1}{4}\\sum\\_{i\\neq j} 4\\pi^2n\\_in\\_j[\\Gamma^{sphere}(Q\\_i,Q\\_j)\n+\\Gamma^{sphere}(Q\\_i,Q\\_j^\\*)\\nonumber\\\\\n&\\ \\ \\ \\ \\ \\ \\ \\ \\ +\\Gamma^{sphere}(Q\\_i^\\*,Q\\_j)\n+\\Gamma^{sphere}(Q\\_i^\\*,Q\\_j^\\*)]\\nonumber\\\\\n&\\ \\ \\ +\\frac{1}{2}\\sum 4\\pi^2n\\_i^2\\Gamma^{sphere}(Q\\_i,Q\\_i^\\*)\\nonumber\\\\\n&=\\frac{1}{2}\\sum\\_{i\\neq j} 4\\pi^2 n\\_in\\_j[\\Gamma^{sphere}(Q\\_i,Q\\_j)\n+\\Gamma^{sphere}(Q\\_i,Q\\_j^\\*)]\\nonumber\\\\\n&\\ \\ \\ +\\sum 2\\pi^2n\\_i^2\\Gamma^{sphere}(Q\\_i,Q\\_i^\\*)\n\\label{eq:shadows}\\end{aligned}$$ \nIn the second expression, we note that the terms in the first line are equal in pairs, so that a factor of $\\frac{1}{2}$ cancels. This energy is given the same form as Eq. ([eq:redouble]) by separating out the part which depends on the positions of *two* vortices, proportional to\n \nΓ*s*(**u**1, **u**2) = Γ*s**p**h**e**r**e*(*Q*1, *Q*2) + Γ*s**p**h**e**r**e*(*Q*1, *Q*2\\*) \nand the part which depends on one vortex,\n \n*U**s*(**u**) = − 2*π*Γ*s**p**h**e**r**e*(*Q*, *Q*\\*). \nThe function in Eq. ([eq:standnoodleg]) is a Green’s function for the pointed sphere, as the placement of the images guarantees that this function is well-defined on the *pointed sphere*. It appears in the stream function, Eq. ([eq:darkandlight]), as well as in the energetics, as expected for a Green’s function.\nThe potential which describes the single-particle energy of a vortex becomes singular as the vortex approaches the apex of the cone at the north or south pole, since then the vortex *Q* approaches its image *Q*\\*. This is in accord with the result, Eq. ([eq:greenself-energy]), that the Gaussian curvature is the source of the single-particle energy since the pointed sphere has delta-function concentrations of curvature at its north and south poles:\n \n*G*(**u**) = 1 + *π**δ**N*(**u**) + *π**δ**S*(**u**). \nwhere *δ**N*(**u**) and *δ**S*(**u**) are the appropriate delta funnctions. The geometric repulsion from the positive curvature points arises from the repulsion between vortices and their images! We can check step-by-step that *U**s* is sourced by the Gaussian curvature,\n \n*U**s*(**u**) = − ∬Γ*s*(**u**, **u'**)*G*(**u'**)*d*2**u'**. \nWe substitute for *G*(**u'**) from Eq. ([eq:ouch]) and for Γ*s* from Eq. ([eq:standnoodleg]) which can be written in the form,\n \n$$\\begin{aligned}\n&\\Gamma\\_{s}(\\sigma\\_1,\\phi\\_1;\\sigma\\_2,\\phi\\_2)=\n\\Gamma^{sphere}(\\sigma\\_1,\\frac{\\phi\\_1}{2};\\sigma\\_2,\\frac{\\phi\\_2}{2})\\nonumber\\\\\n&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ +\\Gamma^{sphere}\n(\\sigma\\_1,\\frac{\\phi\\_1}{2};\\sigma\\_2,\\frac{\\phi\\_2}{2}+\\pi)\\nonumber\\\\\n&=-\\frac{1}{4\\pi}\\ln\n4[(1-\\cos\\sigma\\_1\\cos\\sigma\\_2)^2\n\\nonumber\\\\\n&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -\\sin^2\\sigma\\_1\\sin^2\\sigma\\_2\\cos^2\\frac{\\phi\\_1-\\phi\\_2}{2}].\n\\label{eq:orbitrig}\\end{aligned}$$ \nIn the last line, we have evaluated the Green’s function for the sphere by writing the chordal distance between, e.g., *Q*1 and *Q*2 in terms of the spherical coordinates (*α*1, 2, *θ*1, 2), *D*2 = 2[1 − cos*θ*1cos*θ*2 − sin*θ*1sin*θ*2cos(*α*1 − *α*2)] (see ), and then combining the two terms together. To evaluate the integral in Eq. ([eq:selfconsistent]), we have to note that the area element of this integral is $d^2\\mathbf{u}=\\frac{1}{2}\n\\sin\\sigma d\\sigma d\\phi$. The area of a region on the pointed sphere is the *same* as the area sin*θ**d**θ**d**α* of the corresponding region on the original sphere, and the factor of $\\frac{1}{2}$ results from how the angles are related, $\\alpha=\\frac{\\phi}{2}$, see Eq. ([eq:half-rations]). Now the integral on the right-hand side of Eq. ([eq:selfconsistent]) can be shown to be equal to the left-hand side using the identities\n \n$$\\begin{aligned}\n\\int\\_0^{2\\pi}\\ln|A+B\\cos t|dt\n&=&2\\pi\\ln\\frac{A+\\sqrt{A^2-B^2}}{2}\\ \\mathrm{if}\\ B<A\\nonumber\\\\\n&=&2\\pi\\ln\\frac{B}{2}\\ \\mathrm{if}\\ B>A\\nonumber.\\end{aligned}$$ \nWe have now derived one formulation of the energetics in terms of Γ*s* and *U**s*, the corresponding geometric potential. Let us contrast this isometric mapping method with the conformal mapping method in order to illustrate how different approaches can naturally lead to different delineations between vortex-vortex and vortex-curvature interactions. (The net result is of course the same from either point of view.) As a result of the isometric mapping each point is doubled, whereas the distance-distorting conformal mapping transforms each point on the pointed sphere to *one* point on the reference sphere.\nWe first use Eq. ([eq:deimosmap]) to find that the conformal map is given by\n \n$$\\tan\\frac{\\Theta}{2}=\\tan^2\\frac{\\sigma}{2}.$$ \nComparing the conformal mapping results, Eqs. ([eq:conformalself-energy]) and ([eq:conformalpair-energy]) to the Green’s function formulation, Eqs. ([eq:greenself-energy]) and ([eq:greenpair-energy]) suggests the following identification of the interaction potential (or Green’s function) and single-particle potential:\n \n$$\\begin{aligned}\n&&\\Gamma\\_c(\\mathbf{u}\\_1,\\mathbf{u}\\_2)\n=\\Gamma^{sphere}(\\Theta(\\sigma\\_1),\\phi\\_1;\\Theta(\\sigma\\_2),\\phi\\_2)\\nonumber\\\\\n&&U\\_c(\\mathbf{u})=\\omega=\\ln\\frac{2\\sin\\sigma}{1+\\cos^2\\sigma}.\n\\label{eq:confnoodle}\\end{aligned}$$ \nThese expressions differ from Equations ([eq:standnoodleg]) and ([eq:standnoodlev]). Nevertheless, as promised, the net energy is the same whether the pairs (Γ*s*, *U**s*) or (Γ*c*, *U**c*) are used in place of Γ and *U**G*. In fact,\n \n$$\\begin{aligned}\n&&\\Gamma\\_c(\\mathbf{u}\\_1,\\mathbf{u}\\_2)=\\Gamma\\_s(\\mathbf{u}\\_1,\\mathbf{u}\\_2)\n-f(\\mathbf{u}\\_1)-f(\\mathbf{u}\\_2)\\nonumber\\\\\n&&U\\_c(\\mathbf{u})=U\\_s(\\mathbf{u})+4\\pi f(\\mathbf{u})\n\\label{eq:cesnoodle}\\end{aligned}$$ \nwhere $f(\\mathbf{u})=-\\frac{1}{4\\pi}\\ln(1+\\cos^2\\sigma)$. This transforms the energy from the single-particle to the interaction terms consistently as described at the beginning of the section. Appendix [app:CG] shows that the Green’s function formulation is generally equivalent to the conformal mapping result derived in Section [sec:map], even when there is no method of images that can be used to determine the Green’s function explicitly in general.\n[sec:geomineq] Limits on the Strength and Range of Geometrical Forces\n=====================================================================\nGeometrical forces are limited in strength due to the nonlinear relation between the curvature and the geometric potential. Curvature affects both the *source* of the geometrical force and the *force law*, as illustrated in the examples of Secs. [sec:bumps] and [sec:zucchini]. As a consequence, even on a wildly distorted surface (with planar topology), there is a precise limit on the strength of the force on a single vortex. This result has the character of a geometrical optimization problem, like maximizing the capacitance of a solid when the surface area is given. Consider a vortex located at the center of a geodesic disk of radius *R*. Assume that the Gaussian curvature is zero within the disk, but may be different from zero elsewhere. Then the force **F** due to the curvature satisfies\n \n$$|\\mathbf{F}|\\leq\\frac{4\\pi Kn\\_1^2}{R}.\n\\label{eq:limit1}$$ \nwhere *n*1 is the number of circulation quanta in the vortex. This relation between *R* and **F** is proven in Appendix [app:geomineq].\nIf one warps a surface in a vain attempt to overcome the limit, but the force gets diluted because the distortion of the region around the curvature pulls the force-lines apart, as we can understand from the simple example of vortices on cones.\nA cone of *cone-angle* *θ* is obtained by taking a segment of paper with an angle *θ* and gluing the opposite edges of the angle together. This is most familiar when *θ* < 2*π*. If *θ* = 2*π**m* + *β*, such a cone can be produced by adding *m* extra sheets of paper, as illustrated in Fig. [fig:firstcontortion]. We slit the *m* sheets of paper and put them together with an angle of size *β* cut out of an additional sheet. By gluing the edges of the slits together cyclically, a cone of arbitrary angle *θ* is made.\n![[fig:firstcontortion]How to form cones of negative curvature. One complete sheet of paper is slit and an angle is cut out of an additional sheet of paper. The edges labeled 1 are taped together, and then the edges labeled 2 are taped. The circular arcs join together to form an extra-large circle. The cone angle is \\theta=2\\pi+\\beta, and cones with even larger cone angles can be formed by using additional sheets.](torment3)[fig:firstcontortion]How to form cones of negative curvature. One complete sheet of paper is slit and an angle is cut out of an additional sheet of paper. The edges labeled 1 are taped together, and then the edges labeled 2 are taped. The circular arcs join together to form an extra-large circle. The cone angle is *θ* = 2*π* + *β*, and cones with even larger cone angles can be formed by using additional sheets.\nA cone has a delta function of curvature at its apex, but no Gaussian curvature elsewhere because the surface can be formed from a flat piece of paper without stretching. The weight 2*π* − *θ* of the delta function is expressed, according to the Gauss-Bonnet theorem, as an integral of the Gaussian curvature in any region containing the apex\n \n∬*G*(**u**)*d*2**u** = 2*π* − ∮*κ**d**s* \nwhere *κ* is the geodesic curvature along the boundary of the region and *s* its arc length. Apply this formula to the circle of radius *D* centered at the apex of the cone. Imagine the circle as it would appear on the original sheets of paper, as in Fig. [fig:firstcontortion]. Its measure in radians is *β* + 2*π**m* = *θ* since it consists of *m* complete circles together with an additional arc. The length is therefore *S* = *D**θ*. The *geodesic* curvature of the circle does not change when the cone is unfolded, so it is equal to $\\frac{1}{D}$. Upon substituting in Eq. ([eq:paralleltrans]), we obtain\n \n$$\\iint G(\\mathbf{u})d^2\\mathbf{u}=2\\pi-S\\frac{1}{D}=2\\pi-\\theta.\n\\label{eq:conepoint}$$ \nWhen *θ* > 2*π* the curvature is negative.\nNow imagine a vortex (with *n*1 = ± 1, say) at a distance *D* from the cone point, on the circle of circumference *S* just considered. The arbitrarily large negative curvature which is possible by making *m* large seems to defy the general upper bound on the geometric force. According to Newton’s theorem, applied to the radius *D* circle centered at the cone’s apex and passing through the vortex, the force on the vortex is $F=\\frac{\\pi K\\iint G d^2\\mathbf{u}}{S}$. Since the circumference *S* = *D**θ* is larger than it would be in the plane, the force is diluted; substituting the integrated curvature from Eq. ([eq:conepoint]), we find that it is given by\n \n$$F=\\pi \\frac{K}{D} \\frac{2\\pi-\\theta}{\\theta}.\n\\label{eq:cone}$$ \nThis satisfies Eq. ([eq:limit1]) for all negatively curved cones (*θ* > 2*π*); even when *θ* → ∞ the magnitude of the force is less than $4\\pi\\frac{K}{D}$ because the large circumference in the denominator of the Newton’s theorem expression cancels the large integrated curvature in the numerator.\nIn the opposite limit *θ* → 0, the theorem described by Eq. ([eq:limit1]) is still correct of course. One has to be careful about applying it, however. The force on a vortex at radius *D* (given by Eq. ([eq:cone])) is *not* bounded by $\\frac{4\\pi K}{R}$ with *R* set equal to *D* when *θ* is small enough (in fact, for an extremely pointed cone, *θ* ≪ 1, the force given by Eq. ([eq:cone]) diverges), but this does not contradict the inequality because the circle of radius *D* centered at the vortex is pathological: although it does not contain any curvature, the circle wraps around the cone and intersects itself. Taking *R* to be the radius of the largest circle centered at the defect which does not intersect itself, one finds that the inequality *is* satisfied, with room to spare, for all values of the cone angle *θ* (see Appendix [app:geomineq]). One can describe a more awkwardly shaped surface such that the force on a singly-quantized vortex is arbitrarily close to the upper bound $\\frac{4\\pi K}{R}$ (see Appendix [app:geomineq]).\nOne can also provide limits to the strength of the geometric force from a localized source of curvature. Rotationally symmetric surfaces such as the Gaussian bump have force fields that do not extend beyond the bump, since the net Gaussian curvature is zero, and Newton’s theorem says that only the *i**n**t**e**g**r**a**t**e**d* Gaussian curvature can have a long range effect for a rotationally symmetric surface. To get a longer-range force, one must focus on non-symmetric surfaces, like the saddle surface of Sec. [BS]. The integration methods of Appendix [app:calmseas] can be used to show that this surface’s potential has a quadrupole form at long distance. Let us consider, more generally, a plane which is flat except for a non-rotationally symmetric deformation confined within radius *R* of the origin. (The result will not apply directly to the saddle surface since its curvature extends out to infinity.) In this case, the total integrated Gaussian curvature is zero, implying that the long-range force law cannot have any monopole component. A dipole component is not ruled out by this simple reasoning, but Appendix [app:geomineq] shows that the limiting form of the potential is at least a *quadrupole* (or a faster decaying field),\n \n$$E(\\mathbf{r})\\sim n\\_1^2\\frac{\\mu\\_2\\cos(2\\phi-\\gamma\\_2)}{r^2},\n\\label{eq:quadrupole}$$ \nwhere *r* and *ϕ* are the polar coordinates of the vortex relative to the origin, and *μ*2 and *γ*2 are constants that depend on the shape of the deformation in the vicinity of the origin. As in the previous case, there is an upper limit on the quadrupole moment *μ*2, no matter how strong the curvature of the deformation is:\n \n*μ*2 ≤ *π**K**R*2. \nFor electrostatics in the plane, the maximum quadrupole moment of *N* particles with charge 2*π* and *N* with charge − 2*π* in a region of radius *R* is at most of the order of *K**N**R*2, which has the same form as the bound in Eq. ([eq:limitq]), except for the factor of *N*. Because of the nonlinearity of the geometrical force and restrictions on how much positive and negative curvature can be separated from each other, the quadrupole moment is bounded no matter how drastically curved the surface is.\nThese results describe key physical *differences* (resulting from the fact that the curvature cannot be adjusted without changing the surface) between the geometrical forces discussed in this work and their electrostatic counterparts despite the close resemblance from a formal viewpoint.\nConclusion\n==========\nl|c|c|c &Deformed Plane&Deformed Disk&Deformed Sphere \nInteraction & *K**n**i**n**j*[4*π*2Γ(**u***i*, **u***j*)& 4*π*2*K**n**i**n**j*Γ*D*(**u***i*, **u***j*)& 4*π*2*K**n**i**n**j*Γ(**u***i*, **u***j*) \n*E*2(**u***i*, **u***j*) &$-2\\pi \\ln\\frac{R}{a}]$ && \nGreen’s & $-\\frac{1}{2\\pi}\\ln\\frac{|\\bm{\\mathcal{U}}\\_i-\\bm{\\mathcal{U}}\\_j|}{a}$ &$\\frac{1}{4 \\pi} \\ln\\left(\\frac{1+\n\\mathcal{R}\\_{i}^{2} \\mathcal{R}\\_{j}^{2} - \n2 \\mathcal{R}\\_{i} \\mathcal{R}\\_{j} \\cos \\left(\\Phi\\_{i}-\n\\Phi\\_{j}\\right) }{\\mathcal{R}\\_{i}^{2}+\\mathcal{R}\\_{j}^{2}-2\\mathcal{R}\\_{i} \n\\mathcal{R}\\_{j} \\cos \\left(\n\\Phi\\_{i}-\n\\Phi\\_{j}\\right) }\\right) $ & $-\\frac{1}{2\\pi}\\ln\\frac{\\mathcal{D}\\_{ij}}{a}+\\frac{1}{4\\pi}[\\omega(\\mathbf{u}\\_i)+\\omega(\\mathbf{u}\\_j)]$ \nFunction &&& \nGeom.-Potl. & *K**π*[∬*d*2**u**Γ(**u***i*, **u**)*G*(**u**)& $\\pi K\\ln \\frac{1-\\mathcal{R}\\_i^2}{a}\n-\\pi K\\omega(\\mathbf{u})$& 0 \n*E*1(**u***i*)& $+\\ln\\frac{R}{a}]$ && \nDiff’l Eq. & ∇2*E*1(**u***i*)& ∇2*E*1(**u***i*) \nfor *E*1(**u***i*) & = − *π**K**G*(**u***i*) & $=-\\pi K G(\\mathbf{u}\\_i)-\\frac{4\\pi K}{a^2}e^{-\\frac{2E\\_1(\\mathbf{u}\\_i)}{\\pi K}}$ & \nApplications& Gaussian bumps and saddles: &Interactions with images: & Band-Flows: \n&Secs. [subsec:anomalous]-[earnshaw]; & Sec. [subsec:multiple]; &Sec. [sec:zucchini]; \n&Periodic surfaces:&Bounded Enneper surface &Multiple Green’s functions: \n&Sec. [sec:bumps];&Sec. [sec:bubble]&App. [app:CG] \n&Nearly flat geometries:&& \n&App. [app:calmseas]&& \n| |\n| --- |\n| image |\nIn this article, we have laid out a mathematical formalism based on the method of conformal mapping that allows one to calculate the energetics of topological defects on arbitrary deformed substrates with a focus on applications to superfluid helium films. The starting point of our approach is the observation that upon a change of coordinate the metric tensor of a complicated surface can be brought in the diagonal form *g**a**b* = *e*2*ω*(**u**)*δ**a**b*. This corresponds to the metric of a *flat* plane which is locally stretched or compressed by the conformal factor *e*2*ω*(**u**). Many of the geometric interactions experienced by topological defects on curved surfaces are simply determined once the function *ω*(**u**) is known. Vortices in thin helium layers wetting a curved surface are a natural arena to explore this interplay between geometry and physics but our approach is of broader applicability.\nThe curved geometry results in a modified law for defect interaction as well as in a one body geometric potential. On a deformed plane, the latter is obtained by solving a covariant Poisson equation with the Gaussian curvature as a source. Table [table:chair] presents a summary of the general form that the defect interaction (first row) and the geometric potential (third row) take up in curved spaces with the topology of a deformed plane (first column), disk (second column) and sphere (third column). These results can be derived starting from the differential equations that the geometric potential satisfies or the appropriate Green’s functions that we list in the second and fourth row respectively for each of the three surface topologies. The fifth row of Table [table:chair] directs the reader towards the relevant sections and appendices of the paper where he will be able to find some concrete applications of the formalism and technical derivations.\nFor example, the geometric potential of an Enneper disk (a minimal surface with negative curvature described in Sec. [sec:bubble]) is given by the conformal factor $\\omega({\\bf u})$ evaluated at the point *P* = {*u*1, *u*2} where the vortex is located combined with an ``electrostatic-like\" interaction with an image defect located at the inverse of *P* with respect to the circular boundary. The geometric potential satisfies the Liouville (non-linear differential) equation that reduces to the Poisson equation derived for the plane in the limit of an infinitely large disk. In the case of deformed spheres, we showed in Appendix [app:CG] that one can make a convenient choice of Green’s function so that *all* the geometric effects are included in the defect-defect interactions without introducing a one-body geometric potential explicitly. An interesting application naturally arises on vesicles deformed into an elongated shape, like a zucchini. The range of the defect interaction becomes much longer and its functional form different from the logarithmic dependence expected in flat two dimensional spaces.\nWe hope that the discussion of the geometric effects presented in this work may pave the way for their observation in thin superfluid or liquid crystal layers on a curved substrate. A useful starting point could be the design of experiments to detect the geometric potential by balancing it with forces exerted on the defects by external fields or rotation of the sample as discussed in Section [sec:Rotation]. Such experiments should focus on single vortices, or on situations where the separation between vortices is comparable to the length scale of the geometry. Signatures of the geometric interactions described here may also survive in defect pinning experiments carried out in some bounded three dimensional geometries.\nAn infinite plane is not really infinite; it is just a surface where the boundary is very far away. A distant boundary adds only constants to the energy, and these are given in the table for the case where the boundary is a radius *R* circle.\nConformal mapping is the only way to find the energy of defects on a deformed disk. The conformal factor *ω*(**u**) is not given by the integral of the curvature times the Green’s function, since *ω* does not satisfy Dirichlet boundary conditions. Besides this, the interaction of a defect with its image has to be included, and images have to be defined in the reference plane. Interestingly, the *total* single-particle energy satisfies a *nonlinear* differential equation (the Liouville equation), which is given in the table.\nWe have used the Green’s function which eliminates self-energies on the sphere (see Appendix [app:CG] for other formulations).\nAcknowledgments\n===============\nWe thank B. Halperin, R. D. Kamien, S. Trugman and R. Zieve for helpful suggestions. AMT, VV and DRN acknowledge financial support from the National Science Foundation, through Grant DMR-0654191, and through the Harvard Materials Research Science and Engineering Center through Grant DMR-0213805. VV acknowledges financial support from NSF Grant DMR05-47230. It is a pleasure to acknowledge the Aspen Center for Physics for providing an interactive research environment where this article was completed.\n[app:calmseas]Nearly Flat Surfaces\n==================================\nThe calculations in Section [BS] are based on perturbations about near flatness (see and references therein). The perturbation theory will be in powers of an aspect ratio, *α*, which measures the ratio of surface height to width of the landscape features. (We imagine that the height of the surface is given in the form *α**m*(*x*, *y*) where *m* is a fixed function.) The leading corrections to the flat space energies are second-order in *α*. There are two of these; one is the geometric potential. When there are at least two vortices present, there is also a second order correction to the Green’s function, which ought to be retained since it is comparable to the geometric potential. The latter could be calculated by expanding the metric in Eq. ([eq:BasicGreen]) in powers of *α*. Nevertheless, because the perturbations are singular, we prefer to use conformal mapping for this step just as we use in Sec. [sec:map] to derive the geometric potential. Our calculations are limited to the case of an infinite deformed plane.\nWe use the *x* and *y* coordinates of a plane parallel to the surface for our coordinate system (the ``Monge Gauge\"). The metric is then *d**s*2 = *d**x*2 + *d**y*2 + *d**z*2 = (1 + *h**x*2)*d**x*2 + (1 + *h**y*2)*d**y*2 + 2*h**x**h**y**d**x**d**y*. Subscripts on *h* indicate derivatives, so that *h**x**x* = ∂*x*2*h* etc. Upon calculating the curvature tensor we find the Gaussian curvature in the second order approximation \n \n*G*(*x*, *y*) = *h**x**x**h**y**y* − *h**x**y*2. \nThe geometric potential is found by approximating the exact expression Eq. ([eq:curvature-defect]). The Green’s function may be replaced by the planar one since *G* is already quadratic in *α*:\n \n$$\\begin{aligned}\nU\\_G(x,y)&=&-\\frac{E}{\\pi K}\\nonumber\\\\\n&\\approx&-\\iint dx'dy'{\\Gamma\\_{flat}(x,y;x',y')(h\\_{xx}h\\_{yy}-h\\_{xy}^2)}\\nonumber\\\\\n\\label{eq:smallgeom}\\end{aligned}$$ \nTo do a conformal mapping to an equivalent flat space problem, we must solve the curved space generalization of the Cauchy-Riemann equations which define isothermal coordinates, X,Y, namely\n \n$$\\nabla^a\\mathcal(Y)=-\\gamma^a\\_b\\nabla^b\\mathcal(X).\n\\label{eq:tack}$$ \nwhere $\\gamma^{a}\\_b=g^{ac}\\sqrt{g}\\epsilon\\_{cb}$. (In words, the gradients of X and Y are at right angles to each other and have equal magnitudes at every point.) We insert the expression for the metric in terms of *h* into Eq. ([eq:tack]) and expand to second order under the assumption that X = *x* + *ξ*,Y = *y* + *η* where the deformation parameters *ξ* and *η* are second order in *α*:\n \n$$\\begin{aligned}\n\\eta\\_x+\\xi\\_y\\approx h\\_xh\\_y\\label{eq:antidiag}\\\\\n\\eta\\_y-\\xi\\_x\\approx\\frac{1}{2}(h\\_y^2-h\\_x^2)\\label{eq:diag}.\n\\label{eq:smallmess}\\end{aligned}$$ \nBy taking the derivative of Eq. ([eq:antidiag]) with respect to *x* and Eq. ([eq:diag]) with respect to *y* and adding the results, we obtain\n \n∇*f**l**a**t*2*η* ≈ *h**y*∇*f**l**a**t*2*h*, \nwhich may be solved by means of the Green’s function (keeping in mind the boundary condition that the conformal map must approach the identity at infinity; i.e., *η* → 0), giving the result\n \n*η*(*x*, *y*) ≈ − ∬*d**x*ʹ*d**y*ʹΓ*f**l**a**t*(*h**y*∇*f**l**a**t*2*h*). \nSimilarly, we may solve for *ξ* (and then check that ([eq:smallmess]) is satisfied).\nWe can use our expression for the conformal mapping in conjunction with the conformal invariance of the Green’s function Eq. ([eq:greentransf]), which implies that Γ(*x*1, *y*1; *x*2, *y*2) = Γ*f**l**a**t*(*x*1 + *ξ*1, *y*1 + *η*1; *x*2 + *ξ*2, *y*2 + *η*2). Upon expanding the flat space Green’s function in *ξ* and *η*, we have\n \n$$\\Gamma(x,y;x',y')\\approx-\\frac{1}{4\\pi}\\ln(\\Delta x^2+\\Delta y^2)-\\frac{\\Delta\\xi\\Delta x+\\Delta\\eta\\Delta y}{2\\pi(\\Delta x^2+\\Delta y^2)},\n\\label{eq:smallgreen}$$ \nwhere Δ*Q* indicates taking the difference between the values of *Q* at the two points where the Green’s function is evaluated. Eq. ([eq:eta]), and its analogue for *ξ* now show that Eq. ([eq:smallgreen]) gives the interaction eneragy to the same order as the geometric potential.\nTwo vortices of opposite signs that are very near to one another (a distance *l* such that *a* < < *l* < < *r*0 for a Gaussian bump, say) cannot tell whether they are in curved or flat space. Indeed, the flow fields cancel one another outside a range moderately greater than *l*. If *l* is much less than the curvature scale, the effects of curvature are negligible. The energy of the vortices should therefore be $2\\pi K\\ln \\frac{l}{a}$ as in flat space. This conclusion can be checked explicitly in the small aspect ratio approximation by combining the expressions for the geometric interaction and the correction to the Green’s function Eq. ([eq:smallgeom]) and Eq. ([eq:smallgreen]). (One can take the Δ*x*, Δ*y* → 0 limit of $\\frac{\\Delta\\xi\\Delta x+\\Delta\\eta\\Delta y}{(\\Delta x^2+\\Delta y^2)}$ with the help of Eqs. ([eq:antidiag]),([eq:diag]) and ([eq:eta]).) As expected, the dependence on the surface profile *h* cancels. This consistency check was behind our original suspicion about the existence of a geometric interaction. The energy of two vortices at **u**1, **u**2, without the geometric interaction included is − 4*π*2*K*Γ(**u**1, **u**2). This energy differs from the flat space energy even when the vortices are very close to one another by a position dependent contribution:\n \n$$E\\_{int}(\\mathbf{u}\\_1,\\mathbf{u}\\_2)\\approx\n2\\pi K\\ln s\\_{12}-4\\pi^2 K g(\\frac{1}{2}(\\mathbf{u}\\_2+\\mathbf{u}\\_1)).\n\\label{eq:inequality}$$ \nThis cannot be the correct expression, since as just argued, the energy should be the same as in flat space. Single particle energies give a simple resolution. If the total energy were\n \n*E* = − 4*π*2*K*Γ(**u**1, **u**2) + 2*π*2*K**g*(**u**1) + 2*π*2*K**g*(**u**2), \nthen all the *g*’s will cancel when **u**1 → **u**2. The Green’s function calculations in and the conformal mapping calculations in Section [sec:map] show that this is actually the correct resolution, and that $g(\\mathbf{u})=-\\frac{U\\_G(\\mathbf{u})}{2\\pi}$.\n[app:multipole]The Saddle Surface’s Potential\n=============================================\nFor the saddle surface with a small aspect ratio (see Eq. ([eq:saddlemesa]), we may determine the entire geometric potential analytically as a function of position. We will only outline the procedure here. We would like to evaluate\n \n$$U\\_{\\rho}(\\mathbf{r})=-\\int\\frac{1}{2\\pi} \\ln|\\mathbf{r}-\\mathbf{r'}|\\rho(\\mathbf{r'})dx'dy'\n\\label{eq:coulrho}$$ \nwhen *ρ*(*x*, *y*) = *G*(*x*, *y*) is the curvature of the surface (at the point vertically above (*x*, *y*); we are using the small-aspect ratio approximation of Appendix [app:calmseas]). Some thought shows that the curvature given by Eq. ([eq:smallcurv]) for the surface Eq. ([eq:saddlemesa]) takes the form of a polynomial times $G\\_0=e^{-\\frac{x^2+y^2}{r\\_0^2}}$. We will therefore discuss how to evaluate the potential *U**ρ* for ``charge\" distributions of the form\n \n$$\\rho(x,y)=P(x,y)e^{-\\frac{x^2+y^2}{r\\_0^2}}.\n\\label{eq:gaussy}$$ \nWe start with *ρ* = *G*0; as discussed above, the azimuthal of this distribution symmetry allows its potential to be determined by Gauss’s Law:\n \n$$-\\nabla U\\_{G\\_0}=\\frac{\\mathbf{r}}{2\\pi r^2}\\int\\_0^r 2\\pi r'G\\_0(r')dr'\n\\label{eq:vrho0}$$ \nThis integral is elementary and *U**G*0 can be evaluated by one further integration, although this cannot be done in closed form.\nConveniently, the potential due to a distribution of the form ([eq:gaussy]) can be determined from the special case of *ρ* = *G*0 by differentiation. (Intuitively, derivative charge distributions such as ∂*x**G*0 are superpositions of infinitesimally shifted copies of *G*0. We can therefore apply superposition to find potentials for such distributions. This is analogous to finding the electric fields of multipoles by differenting the monopole field.) To this end, we rewrite Eq. ([eq:coulrho]) for the special case *ρ* = *G*0 as\n \n$$U\\_{G\\_0}(\\mathbf{r})=-\\int\\frac{1}{2\\pi} \\ln |\\mathbf{\\Delta}|\nG\\_0(\\mathbf{r}-\\mathbf{\\Delta}) d\\Delta\\_x d\\Delta\\_y.\n\\label{eq:coulrho2}$$ \nwhere (Δ*x*, Δ*y*) are the components of **\\Delta** = **r** − **r'**. It follows that\n \n$$\\partial\\_{x}^n\\partial\\_{y}^mU\\_{G\\_0}(\\mathbf{r})=\n-\\int\\frac{1}{2\\pi} \\ln |\\mathbf{\\Delta}|\n\\partial\\_{x}^n\\partial\\_{y}^mG\\_0(\\mathbf{r}\n-\\mathbf{\\Delta}) d\\Delta\\_x d\\Delta\\_y.\n\\label{eq:mess}$$ \nThe right hand side represents the potential corresponding to the source in Eq. ([eq:gaussy]) with a special degree *k* polynomial in place of *P*. This polynomial, obtained by multiple differentiations of a Gaussian, is very complicated, but we will show that polynomials of this specific form can be superimposed to give any desired polynomial (including the degree 8 polynomial appropriate to our Gaussian saddle-surface). We will then, in principle, be able to express *U**G* as a superposition of *U**G*0 and its derivatives.\nThe expansion of the charge distribution of Eq. ([eq:gaussy]) in terms of the derivatives of *G*0 can be carried out with the help of Fourier integrals. Our goal is to find an expression of the form\n \n*ρ*(*x*, *y*) = *P*(*x*, *y*)*e*− *x*2 − *y*2 = *Q*(∂*x*, ∂*y*)*e*− *x*2 − *y*2 \nwhere *P*(*x*, *y*) is the polynomial appearing in Eq. ([eq:gaussy]). We have to determine a polynomial operator *Q*(∂*x*, ∂*y*) = ∑*n*, *m**q**n**m*∂*x**n*∂*y**m* so that Eq. ([eq:opeq]) is true. We have set *r*0 = 1 for convenience. Applying the Fourier transform to both sides of Eq. ([eq:opeq]) gives\n \n$$P(i\\partial\\_{p\\_x},i\\partial\\_{p\\_y})e^{-(\\frac{p\\_x^2+p\\_y^2}{4})}=Q(i p\\_x,i p\\_y)e^{-(\\frac{p\\_x^2+p\\_y^2}{4})},$$ \nor (by substituting *u* = *i**p**x*,*v* = *i**p**y*),\n \n$$Q(u,v)=e^{-(\\frac{u^2+v^2}{4})}P(-\\partial\\_u,-\\partial\\_v) e^{(\\frac{u^2+v^2}{4})}\n\\label{eq:herm}$$ \nThe operator *Q*(∂*x*, ∂*y*) which satisfies Eq. ([eq:opeq]) can be produced by working out the derivatives in this expression and replacing *u* and *v* by ∂*x* and ∂*y*. Now the potential can be worked out using\n \n*U**G*(**r**) = *Q*(∂*x*, ∂*y*)*U**G*0(*r*). \nIn fact, multiplying Eq. ([eq:mess]) by *q**n**m* and summing over *n* and *m* shows (with the help of Eq. ([eq:opeq]) that $Q(\\partial\\_x,\\partial\\_y)U\\_{G\\_0}(\\mathbf{r})=\n-\\frac{1}{2\\pi}\\iint \\ln|\\bm{\\delta}|\\rho(\\bm{r}-\\bm{\\Delta}|)\nd\\Delta\\_xd\\Delta\\_y$.\nSince all *derivatives* of *U**G*0 can be calculated analytically starting from Eq. ([eq:vrho0]), Eq. ([eq:multipole2]) will yield an analytic expression for *U**G*, provided we can show that *Q* has no constant term. To show this, we integrate both sides of ([eq:opeq]) to see that\n \n*π**Q*(0, 0) = ∫*P*(*x*, *y*)*e*− (*x*2 + *y*2)*d**x* *d**y* = ∫*G*(*x*, *y*)*d**x**d**y*. \nThat is, *Q*’s constant term is proportional to the net Gaussian curvature; since the net curvature is zero for any surface which flattens out at infinity, *Q* has no constant term.\nThe potential of the saddle surface can thus be determined in closed form by the following procedure: expand the curvature to determine the polynomial *P*. Calculate *Q* from ([eq:herm]). Since ([eq:sumrule]) guarantees that *Q* has no constant term, we may calculate the geometrical potential by differentiating ([eq:vrho0]) repeatedly. This method is not much more practical for human calculations than is numerically integrating ([eq:coulrho]) by hand. A computer program, like Mathematica (which produced 272 terms), can use Eqs. ([eq:multipole2]) and ([eq:herm]) to calculate the values rapidly and make the graphs shown in Figs. [fig:dangerous] and [fig:peacefulvalley]. There is one comprehensible consequence of these calculations: the long distance potential is dominated by a quadrupole, attracting vortices from some directions and repelling them towards others. Hence there are four additional local minima *outside* of the central trap!\n[app:vdW]Van der Waals Attraction on a Curved Surface\n=====================================================\nBecause the van der Waals force is very short-ranged, falling off like $\\frac{1}{r^6}$, one can approximate the integral expression for the disjoining pressure $\\Pi({\\bf x})$ in Eq. ([eq:wet-4]) by corrections depending only on the local curvature of the substrate. The integral is the total van der Waals interaction energy between a point *P* at **x** which is above the helium film and all the atoms in the substrate:\n \nΠ(**x**) = ∫*w*(∣**x** − **x'**∣)*d*3**x'** \nWe now choose a simpler coordinate system (see Fig. [fig:curvwet]) by rotating space so that the tangent plane to the substrate at the point of the substrate closest to *P* becomes horizontal. Let us take the point of tangency to be the origin of our new coordinates, (*ξ*1, *ξ*2, *ξ*3) In this coordinate system, *P* is the point (*ξ*1, *ξ*2, *ξ*3) = (0, 0, *D*) (where *D* is the thickness of the film at *P*). The rotated substrate can be described by its height above the new ``horizontal plane\" (an arbitrary plane parallel to the tangent plane) using the equation *ξ*3 = *h**r**o**t*(*ξ*1, *ξ*2). The disjoining pressure is\n \n$$\\Pi(P)=\\iint {d\\xi\\_1d\\xi\\_2\\int\\_{-\\infty}^{h\\_{rot}(\\xi\\_1,\\xi\\_2)}{d\\xi\\_3\nw(\\sqrt{\\xi\\_1^2+\\xi\\_2^2+(D-\\xi\\_3)^2})}}\n=-\\frac{\\pi \\alpha}{6 D^3}-\\iint d\\xi\\_1d\\xi\\_2\\int\\_{h\\_{rot}(\\xi\\_1,\\xi\\_2)}^{h\\_{rot}(0,0)}\n{d\\xi\\_3 w(\\sqrt{\\xi\\_1^2+\\xi\\_2^2+(D-\\xi\\_3)^2})}$$ \nwhere we have first integrated over the entire region below the plane *ξ*3 = 0, thereby getting the van der Waals interaction between a point and a flat substrate as the first term. We then subtract the surplus energy that has been included by integrating over the shaded region (see Fig. [fig:curvwet]).\n![[fig:curvwet]The disjoining pressure is the interaction between P and the substrate, which is below the surface S in the figure. This can be calculated by using the result for the interaction with an imaginary flat substrate (below the tangent plane S_{tan}) and subtracting the very narrow shaded region.](pink4)[fig:curvwet]The disjoining pressure is the interaction between *P* and the substrate, which is below the surface *S* in the figure. This can be calculated by using the result for the interaction with an imaginary flat substrate (below the tangent plane *S**t**a**n*) and subtracting the very narrow shaded region.\nSince the force is short-ranged, we use the quadratic approximation to *h**r**o**t*, $h\\_{rot}(\\xi\\_1,\\xi\\_2)=h\\_{rot}(0,0)\n-\\frac{\\xi\\_1^2}{2}\\kappa\\_1-\\frac{\\xi\\_2^2}{2}\\kappa\\_2$ where we have assumed the axes to be aligned with the principle curvatures. Finally since *κ*1, 2*D* ≪ 1 this wedge-shaped region is extremely thin close to the origin and the remainder term can therefore be approximated by ignoring the dependence of *w* on *ξ*3:\n \n$$\\Delta\\Pi=-\\iint {d\\xi\\_1d\\xi\\_2 (\\frac{\\xi\\_1^2}{2}\\kappa\\_1\n+\\frac{\\xi\\_2^2}{2}\\kappa\\_2)w(\\sqrt{\\xi\\_1^2+\\xi\\_2^2+D^2})}.$$ \nThis integral can be evaluated in polar coordinates:\n \n$$\\begin{aligned}\n\\Delta \\Pi&=&\\alpha\\iint {\\frac{rdrd\\phi}{(r^2+D^2)^3}\n(\\frac{\\kappa\\_1}{2}r^2\\cos^2\\phi+\\frac{\\kappa\\_2}{2}r^2\\sin^2\\phi)}\\nonumber\\\\\n&=&\\alpha\\frac{\\kappa\\_1+\\kappa\\_2}{2}\\pi\\int\\_0^\\infty{\\frac{r^3 dr}{(r^2+D^2)^3}}\\nonumber\n\\\\\n&=&\\alpha\\frac{H\\pi}{4D^2},\\end{aligned}$$ \nsince the mean curvature *H* is given by $\\frac{1}{2}(\\kappa\\_1+\\kappa\\_2)$. Upon combining this expression with the flat substrate result, we obtain Eq. ([eq:wet-7]).\n[app:CG]Consumer’s Guide to Green’s Functions on Compact Surfaces\n=================================================================\nTHIS PART OF THE SECTION CAN BE ADDED IF WE REMOVE THE NOODLE SECTION The circulations of the vortices on a closed surface must satisfy the neutrality constraint\n \n∑*i**n**i* = 0. \nThe circulations must always be related this way. To see this, imagine dividing the surface into two pieces with a curve. The sum of the quantum numbers on the top half is proportional to the circulation measured counterclockwise around the dividing curve and the sum for the bottom half is proportional to the circulation measured the opposite way around. (On the spherical Earth, it is not worth arguing whether the northern or southern hemisphere is inside the equator!) This relation implies that there are multiple ways of splitting up the energy into single-particle energies and two-particle interaction energies, all of which give the same total energy. In fact, an arbitrary portion 4*π*2*K**f*(**u**1) of the ``geometrical energy\" of vortex 1 can be reinterpreted as coming from this vortex’s interaction with all the other vortices. This can be seen explicitly by checking that the following transformation does not change the *net* energy:\n \n$$\\begin{aligned}\n&&\\Gamma'(\\mathbf{u}\\_1,\\mathbf{u}\\_2)=\\Gamma(\\mathbf{u}\\_1,\\mathbf{u}\\_2)\n-f(\\mathbf{u}\\_1)-f(\\mathbf{u}\\_2)\\\\\n&&U\\_G'(\\mathbf{u})=U\\_G(\\mathbf{u})+4\\pi f(\\mathbf{u})\n\\label{eq:ces}\\end{aligned}$$ \nHence there is no particularly natural choice of a Green’s function on the sphere. With so many choices out there, you’ll be greatful for this friendly guide to help you focus on the important features and possible pitfalls of the different Green’s functions.\nThe ambiguity in the one-vortex energy (Eq. ([eq:ces])) on the sphere also implies that there is no particularly natural choice of a Green’s function on the sphere. With so many choices out there, you’ll be greatful for this friendly guide to help you focus on the important features and possible pitfalls of these different functions.\nThe first point you need to know is that all of them work pretty much just as well, provided they are used consistently; one should not use the single-vortex energy Eq. ([eq:greenself-energy]) designed to work with a different Green’s function from the one used to calculate the pair interaction Eq. ([eq:greenpair-energy]). The general definition of of a Green’s Function, broadened from Eq. ([eq:BasicGreen]), is that it is a symmetric function of two points on the deformed sphere satisfying the equation\n \n∇**x**2Γ(**x**, **y**) = − *δ*(**x**, **y**) + *F*(**x**). \nThe only restriction on the function *F* is that its integral over the deformed sphere must equal 1. (Integrating the Laplacian on the left shows that there is no solution unless the right-hand side integrates to zero.) The Green’s function on a sphere, $-\\frac{1}{2\\pi}\\ln\\frac{\\mathcal{D}(\\mathcal{X},\\mathcal{Y})}{a}$ has $\\frac{1}{4\\pi}-\\delta(\\mathcal{X},\\mathcal{Y})$ as its Laplacian. Eq. ([eq:generalgreen]) is a more versatile vision of what a Green’s function should be, using a function *F* in place of the constant.\nThat Eqs. ([eq:greenself-energy]) and ([eq:greenpair-energy]) give the correct net energy follows from the result proven in Sec. [sec:map] by conformal mapping to the unit sphere:\n \n$$E(\\{n\\_i,\\mathbf{u}\\_i\\})=\\sum\\_{i<j}4\\pi^2Kn\\_in\\_j\n\\Gamma\\_{sphere}(\\mathbf{\\mathcal{U}}\\_i,\\mathbf{\\mathcal{U}}\\_j)\n-\\sum\\_i\\pi Kn\\_i^2\\omega(\\mathbf{u}\\_i)\n\\label{eq:confenergy}$$ \nThe interaction potential in this equation\n \n$$\\Gamma\\_c(\\mathbf{x},\\mathbf{y})=-\\frac{1}{2\\pi}\\ln|\\mathbf{\\mathcal{X}}-\n\\mathbf{\\mathcal{Y}}|\n\\label{eq:hildegaard}$$ \nsatisfies\n \n$$\\begin{aligned}\n\\nabla\\_{\\mathbf{u}}^2\\Gamma\\_c(\\mathbf{u},\\mathbf{u'})&=&e^{2\\omega(\\mathbf{u})})\\nabla\\_{\\mathbf{\\mathcal{U}}}^2\n\\Gamma\\_{sphere}(\\mathbf{\\mathcal{U}},\\mathbf{\\mathcal{U'}})\\nonumber\\\\\n&=&e^{2\\omega}(-\\delta\\_{R}(\\mathbf{\\mathcal{U}},\\mathbf{\\mathcal{U'}})+\n\\frac{1}{4\\pi})\\nonumber\\\\\n&=&-\\delta\\_T(\\mathbf{u},\\mathbf{u'})+\\frac{e^{2\\omega(\\mathbf{u})}}{4\\pi}.\n\\label{eq:e2om}\\end{aligned}$$ \nIn the first step, the scale factor is introduced to compensate for the change from the reference to the target surface. In the second step, the Laplacian of the sphere’s Green’s function $-\\frac{1}{2\\pi}\\ln\\frac{\\mathcal{D}\\_{ij}}{a}$ is substituted. In the third step, the *δ*-function is transformed back to the target surface. The last line shows that Γ*c* is a Green’s function as set out by Eq. ([eq:generalgreen]), which we call the ``conformal Green’s function.\" The *F*-function that goes with this Green’s function gets its spatial dependence from the conformal factor.\nThe single particle potential *ω* in Eq. ([eq:confenergy]) satisfies\n \n∇**u**2*ω* = *G**T*(**u**) − *e*2*ω*(**u**) \nwhich follows from Eq. ([eq:Greenland]) with the curvature of the unit sphere, *G**R* = 1, substituted.\nNow any Green’s function Γ can be used to solve Poisson’s equation for any *net-neutral* function *ρ* on the target surface,\n \n∇**u**2∬*d*2**u'**Γ(**u**, **u'**)*ρ*(**u'**) = − *ρ*(**u**). \nThis follows from Eq. ([eq:generalgreen]). It can be used to derive Eqs. ([eq:greenpair-energy]) and ([eq:greenself-energy]) from their special case, the energy derived by conformal mapping. We first use the Poisson"},"data_id":{"kind":"string","value":"arxiv_0002053"}}},{"rowIdx":497,"cells":{"content":{"kind":"string","value":"Γ-Convergence involving nonlocal gradients with varying horizon: Recovery of local and fractional models\n========================================================================================================\nThis work revolves around the rigorous asymptotic analysis of models in nonlocal hyperelasticity. The corresponding variational problems involve integral functionals depending on nonlocal gradients with a finite interaction range *δ*, called the horizon. After an isotropic scaling of the associated kernel functions, we prove convergence results in the two critical limit regimes of vanishing and diverging horizon. While the nonlocal gradients localize to the classical gradient as *δ* → 0, we recover the Riesz fractional gradient as *δ* → ∞, irrespective of the nonlocal gradient we started with. Besides rigorous convergence statements for the nonlocal gradients, our analysis in both cases requires compact embeddings uniformly in *δ* as a crucial ingredient. These tools enable us to derive the Γ-convergence of quasiconvex integral functionals with varying horizon to their local and fractional counterparts, respectively.\nMSC (2020): 49J45, 35R11, 74A70\nKeywords: nonlocal variational problems, nonlocal and fractional gradients, localization, Γ-convergence\nDate:.\nIntroduction\n============\nNonlocal-to-local limits constitute a pivotal aspect in nonlocal modeling. They can provide a useful consistency check in confirming the compatibility of a (new) nonlocal model with a local counterpart that is covered by well-established theories. More generally speaking, the asymptotic analysis of critical parameter regimes can yield new insights about limit models from their approximations and vice versa. In this paper, we address these topics in the context of models with nonlocal gradients, which have seen a great rise in interest in the last few years, see e.g., .\nFor a general radial kernel *ρ*, the nonlocal gradient of a function *u* : R*n* → R*m* associated to *ρ* is defined as\n \n$$\\begin{aligned}\n\\label{Drho\\_intro}\n D\\_{\\rho} u(x)=\\int\\_{{{\\mathbb{R}}^{n}}} \\frac{u(x)-u(y)}{|x-y|}\\otimes \\frac{x-y}{|x-y|} \\rho(x-y) \\, dy \\quad \\text{for $x \\in {{\\mathbb{R}}^{n}}$},\\end{aligned}$$ \nwhenever this integral exists. A widely studied special case is the Riesz fractional gradient given by *D**s* : = *D**ρ**s* with *ρ**s* = ∣ ⋅ ∣− (*n* + *s* − 1) for the fractional parameter *s* ∈ (0, 1). It satisfies natural physical invariance requirements and was brought to the spotlight by Shieh & Spector, who established useful counterparts of results from classical Sobolev space theory and showed that the associated function spaces coincide with the Bessel potential spaces *H**s*, *p*(R*n*; R*m*). This paved the way for the forthcoming works on fractional variational problems, which were proposed as alternatives to the standard models in continuum mechanics. Due to the reduced regularity requirements imposed by the fractional derivatives in comparison to the classical ones, these models admit a broader class of admissible deformations, which allows to account also for discontinuity effects, such as fracture and cavitation.\nDespite the desirable properties of the Riesz fractional gradient, an intrinsic drawback from the perspective of continuum mechanical modeling is that it involves integration over the whole space R*n*, which is not suitable for models on bounded domains. This shortcoming can be resolved by considering kernel functions *ρ* with compact support, meaning that the range of interactions between individual points, called the horizon *δ* > 0, is finite. In , Bellido, Cueto & Mora-Corral used finite-horizon fractional gradients as a basis to propose models of nonlocal hyperelasticity.\nNote that the concept of a horizon stems originally from peridynamics, a nonlocal formulation of continuum mechanics that, in contrast to classical modeling, avoids the use of derivatives, and instead considers the interaction between individual particles that are not necessarily at an infinitesimal distance. Since its introduction in the 2000s, it has led to a vast literature, ranging from applied to theoretical contributions, see e.g. . While energetic approaches in the context of bond-based peridynamics typically involve double-integrals, the energy functionals of nonlocal hyperelasticity, which are integrals depending on nonlocal gradients, can be interpreted in the context of state-based peridynamics ; one of the advantages of this framework is that it allows to model a broad range of material properties, e.g., general Poisson ratios in isotropic elastic materials, in contrast to the bond-based formulation .\nA first step in putting the nonlocal hyperelastic models on a solid mathematical foundation is to guarantee the existence of solutions, which has been addressed in for the case of finite-horizon gradients. The arguments rely essentially on two key techniques: a nonlocal version of the fundamental theorem of calculus , which is needed to prove Poincaré inequalities and compact embeddings, and a translation method established in that expresses the nonlocal gradient as the classical gradient composed with a convolution. These tools were recently extended in to general nonlocal gradients with compactly supported radial kernels, and we utilize this in Section [sec:existence] to develop an existence theory in this broadened setting.\nOur focus in this work lies on the study of the critical parameter regimes for the horizon *δ*, which can be seen as an important next step towards understanding and validating nonlocal hyperelasticity. While the widely used bond-based models are only able to recover a considerably restrictive class of models through a nonlocal-to-local limit passage, we establish in this paper that the models involving nonlocal gradients are compatible with their local counterpart via a vanishing horizon limit. For a complete picture of the horizon-dependence, we also analyze the other extreme regime of diverging horizon, providing a rigorous connection with purely fractional models.\nIn the following, we adopt the framework for general nonlocal gradients from the recent paper by Bellido, Mora-Corral & Schönberger. Starting with a fixed radial kernel *ρ* satisfying the hypotheses of (see [itm:h0]-[itm:upper] in Section [sec:nonlocalgradients]) with horizon equal to 1, i.e., $\\operatorname{supp}\\rho = \\overline{B\\_1(0)}$, we apply an isotropic rescaling to obtain the kernels\n \n$$\\label{eq:scalingintro}\n\\rho\\_{\\delta}= c\\_{\\delta}\\rho\\left( \\frac{\\, \\cdot\\,}{{\\delta}}\\right)$$ \nwith horizon *δ* > 0 and scaling constants *c**δ* > 0 to be chosen suitably depending on the targeted parameter regime for *δ*, cf. (i) and (ii) below; examples of admissible realizations of kernel functions *ρ* can be found in Example [ex:kernels].\nOur aim in this work is to study the asymptotic behavior of the nonlocal gradients associated to the kernels *ρ**δ* in for both limits of vanishing and diverging horizon, that is,\n(i) *δ* → 0 (ii) *δ* → ∞,\nand prove the convergence of minimizers via Γ-convergence (cf. ) for the corresponding families of *δ*-dependent functionals F*δ*. They consist of vectorial integrals of the form\n \nF*δ*(*u*) : = ∫Ω*δ**f*(*x*, *D**ρ**δ**u*) *d**x*, \nwhere the involved quantities are given as follows: The set Ω ⊂ R*n* is a bounded domain, Ω*δ* = Ω + *B**δ*(0) is its expansion by the horizon parameter, and the integrand *f* : R*n* × R*n* × *m* → R is assumed to have *p*-growth for *p* ∈ (1, ∞) and to be quasiconvex in its second argument. The space of admissible functions for is *H*0*ρ*, *p*, *δ*(Ω; R*m*), the natural nonlocal Sobolev space associated to the gradient *D**ρ**δ* with a zero complementary-value conditions, meaning that the functions vanish in Ω*c*, see Section [se: preliminaries] for more details. Let us now give a brief overview of our findings on the two limit passages *δ* → 0 and *δ* → ∞.\n**(i) Localization via shrinking horizon *δ* → 0.** With the scaling factors *c**δ* = *δ*− *n* in , which preserves the mass of the kernel *ρ**δ*, we confirm for each *u* ∈ *W*1, *p*(R*n*; R*m*) the convergence of the nonlocal gradients to the classical one, precisely,\n \n$$\\begin{aligned}\nD\\_{\\rho\\_{\\delta}}u \\to \\nabla u \\quad \\text{ in $L^p({\\mathbb{R}}^n;{\\mathbb{R}}^{m\\times n})$ as $\\delta\\to 0$,}\\end{aligned}$$ \nsee Lemma [le:recovery]; for sufficiently smooth functions, the convergence is uniform and the optimal rate of convergence is explicitly determined by *δ*2.\nAs our main result within (i), Theorem [th: Gamma convergence to 0] states the Γ-convergence of (F*δ*)*δ* from with respect to the strong *L**p*-topology as *δ* → 0 to the limit functional F0 given by \n$${\\mathcal{F}}\\_0 (u) = \\int\\_{{\\Omega}} f(x, \\nabla u)\\,dx \\quad \\text{ for $u \\in W^{1,p}\\_0({\\Omega};{\\mathbb{R}}^m)$,}$$ \n and provides along with this, the corresponding equi-coercivity of (F*δ*)*δ*; the latter constitutes the major novelty of Theorem [th: Gamma convergence to 0], as explained below. Consequently, the minimizers of F*δ*, which exist by Theorem [th:existence], converge (up to subsequences) in *L**p* to a minimizer of F0.\nThe key technical ingredient for our proof of equi-coercivity for (F*δ*)*δ* is the estimate , which can be seen as an enhanced Poincaré-type inequality. The proof of uses the isotropic scaling from to identify the dependence of the Fourier symbol of *D**ρ**δ* on *δ*. Together with the results on the Fourier symbol of *D**ρ* for a fixed kernel, we then deduce from the Mihlin-Hörmander theorem that the spaces *H**ρ*, *p*, *δ*(Ω; R*m*) do not change with *δ* > 0 and that there is a *δ*-independent constant *C* > 0 such that\n \n$$\\label{eq:coercivityintro}\n\\norm{u}\\_{H^{\\sigma,p}({{\\mathbb{R}}^{n}};{\\mathbb{R}}^m)} \\leq C \\norm{D\\_{\\rho\\_{\\delta}} u}\\_{L^p({{\\mathbb{R}}^{n}};{\\mathbb{R}^{m \\times n}})} \\quad \\text{for all $u \\in H^{\\rho,p,{\\delta}}({\\Omega};{\\mathbb{R}}^m)$ and ${\\delta}\\in (0,1]$,}$$ \nwhere *σ* > 0 is related to the kernel *ρ*, see Theorem [th:comparison] and Corollary [cor:compactness].\nTo set our contribution into context with the existing literature, we mention that closely related localization results for nonlocal gradients in various relevant topologies can be found in or deduced from the results in the recent paper by Arroyo-Rabasa, which addresses more general nonlocal first-order linear operators. Mengesha & Spector in also present a first Γ-convergence result for scalar and convex variational problems in their setting. Beyond the fact that we consider more general quasiconvex integrands in the vectorial case, the only minor difference with our set-up (see and ) lies in the definition of the nonlocal gradient, where they consider interactions only between points within the domain Ω. Accordingly, some of our arguments regarding the liminf-inequality and the construction of recovery sequences share similarities with . However, in contrast to our work, does not contain any compactness results (uniformly in *δ*) or equi-coercivity results, and, therefore, cannot guarantee the existence or the convergence of minimizers for the involved integral functionals. On the other hand, in a different setting of non-symmetric half-space nonlocal gradients for the case *p* = 2, such compactness results have recently been obtained uniformly in *δ*.\n**(ii) Fractional models via diverging horizon *δ* → ∞.** For the regime of large horizons, we identify the scaling factors $c\\_{\\delta}= \\overline{\\rho}(1/{\\delta})^{-1}$ in , with $\\overline{\\rho}$ the radial representation of *ρ*, and assume additionally that the kernels *ρ**δ* converge pointwise to a function *ρ*∞; the scaling factors ensure that *ρ*∞ is equal to 1 on the unit sphere. It turns out that the limit kernel *ρ*∞ will always be a fractional kernel, that is,\n \n$$\\begin{aligned}\n\\label{rhoinfty\\_intro}\n\\rho\\_\\infty=|\\cdot|^{-(n+s\\_\\infty-1)}\\end{aligned}$$ \nwith some *s*∞ ∈ (0, 1) characteristic for *ρ*, see Lemma [le: fractional kernel as limit]. Even though this observation may be surprising, given that *ρ* will generally not have a fractional singularity at the origin, it follows because of the fact that the radial representation of *ρ*∞ gains multiplicativity through the limit process, and is thus, a power function.\nIn view of , we then deduce in Proposition [prop:convergenceinfinite] the convergence of the nonlocal gradients to a Riesz fractional gradient. Precisely, it holds for *u* ∈ *W*1, *p*(R*n*; R*n*) that\n \n$$\\begin{aligned}\n\\label{convergence\\_Ddelta\\_intro}\nD\\_{\\rho\\_{\\delta}} u\\to D^{s\\_\\infty}u \\quad \\text{ in $L^p({\\mathbb{R}}^n;{\\mathbb{R}}^{m\\times n})$ as ${\\delta}\\to \\infty$.}\n \\end{aligned}$$ \nUsing a similar strategy as in (i), we can specify the dependence of the Fourier symbol of *D**ρ**δ* on *δ* and show the analogue of for large *δ*, see Proposition [prop:poincare2]. This facilitates along with the proof of our main theorem on Γ-convergence and equi-coercivity of the family (F*δ*)*δ* (cf. ) as *δ* → ∞; explicitly, Theorem [th:infty] yields the Γ-limit Γ(*L**p*)-lim*δ* → ∞F*δ* = F∞ with\n \n$$\\begin{aligned}\n\\label{Fcal\\_frac}\n{\\mathcal{F}}\\_\\infty(u) = \\int\\_{{\\mathbb{R}}^n} f(x,D^{s\\_\\infty}u)\\,dx \\qquad \\text{for $u \\in H^{s\\_\\infty,p}\\_0({\\Omega};{\\mathbb{R}}^m)$.}\\end{aligned}$$ \nIn particular, we have that the minimizers of F*δ* converge (up to subsequence) in *L**p* to a minimizer of a variational integral depending on Riesz fractional gradients.\nNote that the resulting limit objects, that is, fractional functionals of the type , have been well-studied in the last years under different assumptions on the integrand, see e.g. . The aspects addressed include weak lower semicontinuity, relaxation, existence of minimizers, and Euler-Lagrange equations.\nThe prototypical example that illustrates our results is a truncated version of the Riesz fractional kernel, that is, \n$$\\rho = \\frac{w}{\\abs{\\,\\cdot\\,}^{n+s-1}}\\ \\quad \\text{for $s \\in (0,1)$},$$ \n where *w* : R*n* → [0, ∞) is a suitable smooth, radial cut-off function compactly supported in the closed unit ball of R*n*, cf. Example [ex:kernels]a). Applying the two scaling choices of (i) and (ii) gives the scaled kernels \n$$\\rho\\_{\\delta}= {\\delta}^{s-1} \\frac{w(\\,\\cdot\\,/{\\delta})}{\\abs{\\, \\cdot\\, }^{n+s-1}} \\ \\quad \\text{and} \\quad \\rho\\_{\\delta}= \\overline{w}(1/{\\delta})^{-1} \\frac{w(\\,\\cdot\\,/{\\delta})}{\\abs{\\, \\cdot\\,}^{n+s-1}},$$ \n respectively, where $\\overline{w}$ denotes the radial representation of *w*. Both these kernels are supported in the ball $\\overline{B\\_\\delta(0)}$ of radius *δ* around the origin and give rise to the finite-horizon fractional gradients *D**δ**s* studied in ; we remark that in those references the dependence of the kernel on *δ* is not made explicit, since the horizon was always considered fixed. As a consequence of the results in this paper, variational problems involving the gradients *D**δ**s* are confirmed to approximate their local analogue with classical gradients in the limit of vanishing horizon. As *δ* → ∞, we justify the intuitive connection with the fractional case, which reflects an infinite range of interaction.\nLet us close the introduction by briefly pointing out some interesting further connections for broader context. We observe that for fractional gradients, localization also occurs by letting the fractional index *s* tend to 1. This was demonstrated for the Riesz fractional gradient *D**s* in and for its finite-horizon version *D**δ**s* in. Among the first works on nonlocal-to-local limit passages were those in the context of bond-based peridynamics , which successfully recover various classical models. However, as shown in, there are classical energies that cannot be obtained from double-integral functionals as *δ* goes to 0, as opposed to the setting of this paper. Finally, the limit of diverging horizon is less common in the literature. Nevertheless, the convergence of finite-horizon versions of the fractional *p*-Laplacian has been established in for both *δ* → 0 and *δ* → ∞, recovering the classical and fractional *p*-Laplacian, respectively.\nThe manuscript is organized as follows. After introducing in Section [se: preliminaries] the nonlocal gradients and associated function spaces that we are going to work with, we specify the required conditions on the kernel *ρ* and collect some technical tools and preliminary results. The core of the paper are Sections [se: localization] and [se: convergence in infinity], where we address the limit analysis of *δ* → 0 and *δ* → ∞ to recover the classical and fractional models, respectively. These two sections, which are each presented in a self-contained way, share a parallel structure: First, showing the convergence of the varying horizon nonlocal gradients, then, compactness statements uniformly in *δ*, and, finally, the Γ-convergence of the energy functionals in.\nPreliminaries\n=============\nNotation\n--------\nWe write $\\abs{x}=\\left(\\sum\\_{i=1}^n x\\_i^2\\right)^{1/2}$ for the Euclidean norm of a vector *x* = (*x*1, ⋯, *x**n*) ∈ R*n* and $\\abs{A}$ for the Frobenius norm of a matrix *A* ∈ R*m* × *n*. The ball centered at *x* ∈ R*n* and with radius *r* > 0 is denoted by $B\\_r(x)=\\{ y \\in {\\mathbb{R}}^n : \\abs{x-y}<r\\}$ and the distance between *x* ∈ R*n* and a set *E* ⊂ R*n* is written as dist(*x*, *E*). For an open set Ω ⊂ R*n* and *δ* > 0, we define\n \n$$\\begin{aligned}\n\\label{Omegadelta}\n\\Omega\\_\\delta:= \\Omega+B\\_{\\delta}(0)=\\{x \\in {\\mathbb{R}}^n\\,:\\, \\operatorname{dist}(x,\\Omega) <\\delta\\}\\end{aligned}$$ \nand set Ω− *δ* : = {*x* ∈ Ω : dist(*x*, Ω*c*) > *δ*}. The complement of a set *E* ⊂ R*n* is indicated by *E**c* : = R*n* \\ *E*, its closure by $\\overline{E}$, and its boundary by ∂*E*. We take \n$$\\mathbbm{1}\\_E(x) = \\begin{cases} 1 &\\text{for} \\ x \\in E,\\\\\n0 &\\text{otherwise},\n\\end{cases}\\qquad x\\in {\\mathbb{R}}^n,$$ \n to be the indicator function of a set *E* ⊂ R*n*. The extension of a function *u* : *E* → R to R*n* as zero is sometimes explicitly denoted as $\\mathbbm{1}\\_E\\, u$. For a function *u* : R*n* → R, we denote its support by supp*u*, and, if *u* is Lipschitz continuous, its Lipschitz constant by Lip(*u*).\nFor *U* ⊂ R*n* open, we adopt the standard notation for the space of smooth functions with compact support *C**c*∞(*U*), the Lebesgue space *L**p*(*U*) and the Sobolev space *W*1, *p*(*U*) with *p* ∈ [1, ∞]. The spaces can be extended componentwise to vector-valued functions; the target space is explicitly mentioned in the notation, like, for example, *L**p*(*U*; R*m*). We use the usual multi-index notation for partial derivatives ∂*α* with *α* ∈ N0*n*. Our convention for the Fourier transform of *f* ∈ *L*1(R*n*) is\n \n*f̂*(*ξ*) = ∫R*n**f*(*x*) *e*− 2*π**i**x* ⋅ *ξ* *d**x*, *ξ* ∈ R*n*, \nsee e.g., for more details.\nFor real functions, we use the monotonicity properties of being increasing and decreasing in the non-strict sense. A function *f* : R → R is called almost decreasing if there is a *C* > 0 such that *f*(*t*) ≥ *C**f*(*s*) for *t* ≤ *s*, and an analogous definition holds for almost increasing. For a radial function *p* : R*n* → R, we denote its radial representation by $\\overline{p}:[0,\\infty) \\to {\\mathbb{R}}$, i.e., $p(x) = \\overline{p}(\\abs{x})$ for *x* ∈ R*n*, and call *p* radially decreasing or increasing, if its radial representation is decreasing or increasing, respectively.\nFinally, throughout the manuscript, we use generic constants, which may change from line to line without renaming.\nNonlocal gradients\n------------------\nWe now introduce the key elements of our setting, that is, the nonlocal gradients for general kernels *ρ* as recently studied in ; for related work see also , as well as on the special case of finite-horizon fractional gradients.\nAssume throughout that *ρ* : R*n* \\ {0} → [0, ∞) is a radial kernel such that\n1. [itm:h0] $\\inf\\_{\\overline{B\\_{{\\varepsilon}}(0)}} \\rho >0 \\ \\text{for some $>0$} \\quad \\text{and} \\quad \\rho\\, \\min\\{1,|\\cdot|^{-1}\\} \\in L^1({\\mathbb{R}}^n).$\nUnder this condition, the kernel gives rise to an associated nonlocal gradient.\n[Nonlocal gradient] The nonlocal gradient *D**ρ**φ* : R*n* → R*n* for *φ* ∈ *C**c*∞(R*n*) is given by\n \n$$D\\_\\rho {\\varphi}(x) = \\int\\_{{\\mathbb{R}}^n} \\frac{{\\varphi}(x)-{\\varphi}(y)}{\\abs{x-y}}\\frac{x-y}{\\abs{x-y}}\\rho(x-y)\\,dy, \\quad x \\in {\\mathbb{R}}^n.$$ \nWe collect here some key properties of *D**ρ* that will be used later on. First of all, with $\\overline{\\rho}:(0,\\infty) \\to [0,\\infty)$ the radial representation of *ρ*, i.e., $\\rho=\\overline{\\rho}(\\abs{\\,\\cdot\\,})$, the nonlocal gradient can be written as the convolution of the classical gradient with the locally integrable function \n$$Q\\_\\rho(x) := \\int\\_{\\abs{x}}^{\\infty} \\frac{\\overline{\\rho}(r)}{r}\\,dr, \\quad x \\in {\\mathbb{R}}^n \\setminus \\{0\\},$$ \n that is,\n \n$$\\begin{aligned}\n\\label{eq:translationsmooth}\nD\\_\\rho {\\varphi}&= Q\\_\\rho \\* \\nabla {\\varphi}= \\nabla (Q\\_\\rho \\* {\\varphi}) \\quad \\text{for all ${\\varphi}\\in C\\_c^{\\infty}({\\mathbb{R}}^n)$}, \\end{aligned}$$ \nsee . When *ρ* ∈ *L*1(R*n*), then also *Q**ρ* ∈ *L*1(R*n*) and one obtains after taking the Fourier transform that\n \n$$\\label{eq:drhofourier}\n\\widehat{D\\_\\rho {\\varphi}}(\\xi) = 2\\pi i \\xi \\widehat{Q}\\_\\rho(\\xi) \\widehat{{\\varphi}}(\\xi) \\quad \\text{for $\\xi \\in {\\mathbb{R}}^n$},$$ \nsee .\nAn important property of the nonlocal gradient is the presence of a duality relation with the nonlocal divergence, as conveyed by the following integration by parts formula, cf. .\n[Integration by parts, ] Let *φ* ∈ *C**c*∞(R*n*) and *ψ* ∈ *C**c*∞(R*n*; R*n*). Then\n \n$$\\begin{aligned}\n\\label{eq:integration by parts}\n \\int\\_{{{\\mathbb{R}}^{n}}} D\\_{\\rho}{\\varphi}\\cdot \\psi \\, dx =- \\int\\_{{{\\mathbb{R}}^{n}}} {\\varphi}\\operatorname{div}\\_{\\rho}\\psi \\, dx,\n \\end{aligned}$$ \nwhere\n \n$$\\operatorname{div}\\_{\\rho} \\psi(x):=\\int\\_{{\\mathbb{R}}^n} \\frac{\\psi(x)-\\psi(y)}{\\abs{x-y}}\\cdot\\frac{x-y}{\\abs{x-y}}\\rho(x-y)\\,dy.$$ \nNote that the integration over the whole space in is sufficient for our purpose, since we will be working mainly with compactly supported functions; an alternative version of integration by parts over a bounded domain giving rise to boundary terms was recently proven in .\nThe previous lemma motivates a distributional definition of nonlocal gradients.\n[Weak nonlocal gradients] Let *u* ∈ *L*1(R*n*) + *L*∞(R*n*). We say that *V* ∈ *L*loc1(R*n*; R*n*) is the weak nonlocal gradient of *u*, and write *D**ρ**u* = *V*, if\n \n∫R*n**V* ⋅ *ψ* *d**x* = − ∫R*n**u*div*ρ**ψ* *d**x* for all *ψ* ∈ *C**c*∞(R*n*; R*n*). \nIn analogy to classical Sobolev spaces, one introduces for *p* ∈ (1, ∞) the *ρ*-nonlocal Sobolev spaces as \n*H**ρ*, *p*(R*n*) : = {*u* ∈ *L**p*(R*n*) : *D**ρ**u* ∈ *L**p*(R*n*; R*n*)}, \n endowed with the norm\n \n$$\\label{eq: nonlocal norm}\n \\norm{u}\\_{H^{\\rho,p}({\\mathbb{R}}^n)} :=\\norm{u}\\_{L^p({\\mathbb{R}}^n)} + \\norm{D\\_\\rho u}\\_{L^p({\\mathbb{R}}^n;{\\mathbb{R}}^n)},$$ \nsee . Note that these spaces can be equivalently characterized as the closure of *C**c*∞(R*n*) under the norm in in light of. Additionally, we define for an open Ω ⊂ R*n* the subspaces \n$$H^{\\rho,p}\\_0(\\Omega):=\\overline{C\\_c^{\\infty}(\\Omega)}^{\\norm{\\cdot}\\_{H^{\\rho,p}({\\mathbb{R}}^n)}},$$ \n where the elements of *C**c*∞(Ω) are interpreted as extended to R*n* by zero. If Ω is a bounded Lipschitz domain, *H*0*ρ*, *p*(Ω) agrees with the complementary-value space of the functions in *H**ρ*, *p*(R*n*) that are zero in Ω*c*, see . Prescribed complementary values can be viewed as the nonlocal analogue of Dirichlet boundary conditions in the local setting.\n[Riesz fractional gradient and Bessel potential spaces][ex:Rieszfracgrad] The special choice of kernel function\n \n$$\\begin{aligned}\n\\label{eq:rhoskernel}\n\\rho^s:= \\frac{1}{|\\cdot|^{n+s-1}}\\quad \\text{ with $s \\in (0,1)$}\\end{aligned}$$ \ngives rise to the Riesz *s*-fractional gradient *D**s* : = *D**ρ**s*, given for *φ* ∈ *C**c*∞(R*n*) by \n$$D^s \\varphi(x)= \\int\\_{{{\\mathbb{R}}^{n}}} \\frac{\\varphi(x)-\\varphi(y)}{|x-y|^{n+s}}\\frac{x-y}{|x-y|}\\, dy, \\quad x\\in {\\mathbb{R}}^n,$$ \n cf. . Commonly, *D**s* features a normalization constant *c**n*, *s*, which we omit here for the sake of a cleaner presentation in Section [se: convergence in infinity]. The associated nonlocal Sobolev space $H^{\\rho^s\\hspace{-0.07cm},p}({{\\mathbb{R}}^{n}})$ coincides with the Bessel potential space *H**s*, *p*(R*n*) as shown in . A property we will often exploit is that\n \n$$\\begin{aligned}\n\\label{embeddingBessel}\n\\text{$H^{s,p}({{\\mathbb{R}}^{n}})$ is compactly embedded into $L^p\\_{{\\mathrm{loc}}}({{\\mathbb{R}}^{n}})$, }\\end{aligned}$$ \nsee e.g., or. Moreover, we set $H^{s,p}\\_0(\\Omega)=\\{u\\in H^{s,p}({\\mathbb{R}}^n): u=0 \\ \\text{a.e.~in $c$}\\}$.\nBesides the previously introduced hypothesis [itm:h0] on the kernel *ρ*, used for the definition of the nonlocal gradient, we require a few more properties in order to have a wider variety of technical tools, such as compact embeddings and Poincaré inequalities, at our disposal. In accordance with (see also ), we make the following assumptions:\nLet *ɛ* be as in (H0), *ν* > 0 and 0 < *σ* ≤ *γ* < 1.\n1. [itm:h1] The function $f\\_\\rho:(0,\\infty)\\to {\\mathbb{R}}, \\ r \\mapsto r^{n-2}\\overline{\\rho}(r)$ is decreasing on (0, ∞) and *r* ↦ *r**ν**f**ρ*(*r*) is decreasing on (0, *ɛ*);\n2. [itm:derivatives] *f**ρ* is smooth outside the origin and for every *k* ∈ N there exists a *C**k* > 0 with \n$${\\left|\\frac{d^k}{dr^k}f\\_\\rho(r)\\right|} \\leq C\\_k \\frac{f\\_\\rho(r)}{r^k} \\quad \\text{for $r\\in (0, {\\varepsilon})$};$$\n3. [itm:lower] the function $r \\mapsto r^{n+\\sigma-1}\\overline{\\rho}(r)$ is almost decreasing on (0, *ɛ*);\n4. [itm:upper] the function $r \\mapsto r^{n+\\gamma-1}\\overline{\\rho}(r)$ is almost increasing on (0, *ɛ*).\nMost of the time, we will not work directly with these hypotheses, but instead make use of the results and tools proven in ; we refer to that paper for a more detailed discussion of the assumptions [itm:h0]-[itm:upper]. The Riesz potential kernel from satisfies all these properties, as one can easily check. Beyond that, we list here a few examples with compactly supported kernels from that fit into the setting. These will be revisited also in the later sections to illustrate our findings.\n[Selected kernel functions *ρ*][ex:kernels] Let *w* ∈ *C**c*∞(R*n*) be a non-negative radial function with *w*(0) > 0.\na) Let *s* ∈ (0, 1) and suppose that $w/\\abs{\\,\\cdot\\,}^{1+s}$ is radially decreasing. Then,\n \n$$\\begin{aligned}\n\\label{ex:finitehorizonkernel}\n\\rho(x) = \\frac{w(x)}{\\abs{x}^{n+s-1}}, \\quad x\\in {\\mathbb{R}}^n\\setminus \\{0\\}, \\end{aligned}$$ \nsatisfies [itm:h0]-[itm:upper] with *σ* = *γ* = *s*. The associated nonlocal gradient *D**ρ* is referred to as a finite-horizon fractional gradient. In fact, it holds that *H**ρ*, *p*(R*n*) = *H**s*, *p*(R*n*) with equivalent norms by.\nb) Let *s* ∈ (0, 1) and *κ* ∈ { − 1, 1}. If $\\operatorname{supp}(w) \\subset \\overline{B\\_1(0)}$ and $w\\log^\\kappa (1/\\abs{\\,\\cdot\\,})/\\abs{\\, \\cdot\\,}^{1+s}$ is radially decreasing, then the kernel function given by \n$$\\rho(x) = \\frac{w(x)\\log^\\kappa(1/\\abs{x})}{\\abs{x}^{n+s-1}}, \\quad\\text{ $x\\in {\\mathbb{R}}^n\\setminus \\{0\\}$,}$$ \n satisfies [itm:h0]-[itm:upper] with *σ* = *s* and any *γ* ∈ (*s*, 1) if *κ* = 1 and with any *σ* ∈ (0, *s*) and *γ* = *s* if *κ* = − 1.\nc) Consider a smooth function *s* : [0, ∞) → (0, 1) and let $w/\\abs{\\,\\cdot\\,}^{1+s(\\abs{\\, \\cdot\\,})}$ be radially decreasing. Then, \n$$\\rho(x)=\\frac{w(x)}{\\abs{x}^{n+s(\\abs{x})-1}}, \\quad x\\in {\\mathbb{R}}^n\\setminus\\{0\\},$$ \n is a kernel with spatially varying fractional parameter satisfying [itm:h0]-[itm:upper] with *σ* = min[0, *ɛ*]*s* and *γ* = max[0, *ɛ*]*s* for any *ɛ* > 0.\nThe following auxiliary result from will be exploited in Sections [se: localization] and [se: convergence in infinity] to prove compactness results uniformly in the horizon parameter. It provides bounds on the Fourier transform of *Q**ρ* and its derivatives in terms of the radial representation of *ρ*.\n[Estimates on *Q̂**ρ* and its derivatives, ][lem:bms] Let *ρ* : R*n* \\ {0} → [0, ∞) be a radial kernel with compact support satisfying [itm:h0]-[itm:upper]. Then *Q̂**ρ* is smooth, positive, and there exists a constant *C* > 0 such that\n \n$$\\begin{aligned}\n\\label{eq:bms1}\n\\frac{1}{C} \\frac{\\bar{\\rho}(1/\\abs{\\xi})}{\\abs{\\xi}^n} \\leq \\widehat{Q}\\_\\rho(\\xi) \\leq C \\frac{\\bar{\\rho}(1/\\abs{\\xi})}{\\abs{\\xi}^n} \\quad \\text{for all $\\abs{\\xi} \\geq 1/{\\varepsilon}$}.\\end{aligned}$$ \nMoreover, for every *α* ∈ N0*n*, one has\n \n$$\\begin{aligned}\n\\label{eq:bms2}\n{\\Bigl|\\partial^{\\alpha} \\widehat{Q}\\_\\rho(\\xi)\\Bigr|} \\leq C\\_{\\alpha} \\abs{\\xi}^{-\\abs{\\alpha}} {\\left|\\widehat{Q}\\_\\rho(\\xi)\\right|} \\quad \\text{for all $\\xi \\not = 0$}\\end{aligned}$$ \nwith constants *C**α* > 0.\nFinally, Poincaré-type inequalities will be indispensible tools for our analysis. We present here a particular consequence of , that suffices for our setting.\n[Poincaré inequalities and compact embedding, ][lem:compactness] Let Ω ⊂ R*n* be open and bounded and suppose that the radial kernel function *ρ* satisfies [itm:h0]-[itm:upper] and has compact support. Then there is a *C* > 0 such that\n \n$$\\begin{aligned}\n\\label{Poincare}\n\\norm{u}\\_{L^p({{\\mathbb{R}}^{n}})} \\leq C \\norm{D\\_\\rho u}\\_{L^p({{\\mathbb{R}}^{n}};{{\\mathbb{R}}^{n}})} \\quad \\text{for all $u \\in H^{\\rho,p}\\_0({\\Omega})$},\\end{aligned}$$ \nand *H*0*ρ*, *p*(Ω) is compactly embedded into *L**p*(R*n*).\nIn fact, comparing *ρ* as in the previous lemma with the kernel from Example [ex:kernels]a) with *s* = *σ* leads to a stronger estimate that will be utilized several times in what follows. Precisely, by using [itm:lower] and, we find that there is a *C* > 0 such that\n \n$$\\begin{aligned}\n\\label{estimateHsigmap}\n\\norm{u}\\_{H^{\\sigma,p}({{\\mathbb{R}}^{n}})} \\leq C \\norm{D\\_\\rho u}\\_{L^p({{\\mathbb{R}}^{n}};{{\\mathbb{R}}^{n}})} \\quad \\text{for all $u \\in H^{\\rho,p}\\_0({\\Omega})$}.\\end{aligned}$$ \n[Relaxed assumptions on *ρ*][rem:relaxed] Note that according to , there is an equivalence between the function spaces and Poincaré inequalities associated to kernels that agree around the origin. Hence, and hold even when the smoothness in [itm:h1] only holds locally, or when the assumption of *ρ* having compact support is dropped. For example, one could replace the function *w* in Example [ex:kernels]a) by an indicator function $\\mathbbm{1}\\_{B\\_\\delta(0)}$ with *δ* > 0 or by an exponentially decaying function *e*− *α*∣ ⋅ ∣ with *α* > 0, which leads to the truncated and tempered fractional kernel of , respectively.\nExistence theory for nonlocal variational problems\n--------------------------------------------------\nLet us address next the solvability of vectorial variational problems involving the nonlocal gradients as introduced in the previous section. Besides being of general interest, the existence statement of Theorem [th:existence] is needed below to conclude the convergence of minimizers for the variational problems with varying horizon in Sections [se: localization] and [se: convergence in infinity]. We remark that the results presented here are new in this generality, but can be derived by following closely the techniques of, where the direct method in the calculus of variations is applied to the special case of functionals depending on finite-horizon fractional gradients. The adaptation of the proofs is straightforward and left to the reader.\nThroughout this section, we assume that *p* ∈ (1, ∞), Ω ⊂ R*n* is a bounded Lipschitz domain, and the kernel *ρ* satisfies [itm:h0]-[itm:upper] and has compact support. The following result, which allows us to translate the nonlocal gradients into classical gradients, can be proven by extending via density as in.\n[From nonlocal to local gradients][le:translation] The linear map Q*ρ* : *H**ρ*, *p*(R*n*) → *W*1, *p*(R*n*), *u* ↦ *Q**ρ* \\* *u* is bounded and it holds for all *u* ∈ *H**ρ*, *p*(R*n*) that \n*D**ρ**u* = ∇(Q*ρ**u*). \nAnother ingredient is the strong convergence of nonlocal gradients in the complement of Ω, which follows as in by utilizing the compact embedding of *H*0*ρ*, *p*(Ω) into *L**p*(R*n*) (see Lemma [lem:compactness]) and the Leibniz rule in.\n[Strong convergence in the complement] Let (*u**j*)*j* ⊂ *H*0*ρ*, *p*(Ω) be a sequence that converges weakly to *u* in *H**ρ*, *p*(R*n*). Then, for any *η* > 0 it holds that \n$$D\\_\\rho u\\_j \\to D\\_\\rho u \\quad \\text{in $L^p(({\\Omega}\\_\\eta)^c;{{\\mathbb{R}}^{n}})$ as $j\\to \\infty$},$$ \n recalling the definition Ω*η* = Ω + *B**η*(0), see .\nWith these two technical tools and the Poincaré inequality from Lemma [lem:compactness] at hand, one can argue as in the sufficiency part of and to obtain the existence of minimizers for vectorial variational problems with quasiconvex integrands.\n[Existence of minimizers][th:existence] Let *δ* > 0 be such that $\\operatorname{supp}\\rho = \\overline{B\\_{\\delta}(0)}$ and let *f* : Ω*δ* × R*m* × *n* → R be a Carathéodory integrand such that \n$$c\\abs{A}^p-C\\leq f(x,A) \\leq C(1+\\abs{A}^p) \\quad \\text{for a.e.~$x \\in \\Omega\\_{\\delta}$ and all $A \\in {\\mathbb{R}^{m \\times n}}$}.$$ \n If *A* ↦ *f*(*x*, *A*) is quasiconvex for a.e. *x* ∈ Ω, then the functional\n \n$$\\begin{aligned}\n\\label{Fcal}\n{\\mathcal{F}}:H^{\\rho,p}\\_0({\\Omega};{\\mathbb{R}}^m) \\to {\\mathbb{R}}, \\quad {\\mathcal{F}}(u):= \\int\\_{{\\Omega}\\_{\\delta}} f(x,D\\_\\rho u)\\,dx\\end{aligned}$$ \nadmits a minimizer.\nNote that taking *δ* > 0 such that $\\operatorname{supp}\\rho = \\overline{B\\_{\\delta}(0)}$ ensures that *D**ρ**u* is zero in Ω*δ**c* for all *u* ∈ *H*0*ρ*, *p*(Ω). Hence, the functional F in defined as an integral over the bounded set Ω*δ* captures all the non-trivial parts of *D**ρ*.\nQuasiconvexity, which is well-known to characterize the weak lower semicontinuity of integral functionals in the classical case, is indeed the natural convexity notion also in the context of variational integrals depending on nonlocal gradients. This observation can be seen as a generalization of and and relies on Lemma [le:translation] and the following inverse translation operator.\n[From local to nonlocal gradients] There is a bounded linear operator P*ρ* : *W*1, *p*(R*n*) → *H**ρ*, *p*(R*n*) such that P*ρ* = (Q*ρ*)− 1. In particular, for all *v* ∈ *W*1, *p*(R*n*) we have \n∇*v* = *D**ρ*(P*ρ**v*). \nWe define the operator \nP*ρ* : S(R*n*) → S(R*n*), P*ρ**v* : = (*v*/*Q̂**ρ*)∨, \n which is well-defined given that 1/*Q̂**ρ* is smooth and polynomially bounded by Lemma [lem:bms] and [itm:upper]. It is also clear that P*ρ* = (Q*ρ*)− 1 and *D**ρ* ∘ P*ρ* = ∇ on the space S(R*n*), so it is sufficient to prove that P*ρ* extends to a bounded operator from *W*1, *p*(R*n*) to *L**p*(R*n*). By Lemma [lem:bms], [itm:lower] and the Mihlin-Hörmander theorem (cf. e.g. ), it can be verified that ⟨ ⋅ ⟩*σ* − 1/*Q̂**ρ* is an *L**p*-multiplier, where $\\langle \\xi \\rangle:=\\sqrt{1+|\\xi|^2}$ for *ξ* ∈ R*n*. Hence, arguing as in, we find that P*ρ* extends to a bounded operator from *H*1 − *σ*, *p*(R*n*) to *L**p*(R*n*) and, in particular, it is also bounded from *W*1, *p*(R*n*) to *L**p*(R*n*).\nThe following can now be proven as in, given that we have both translation operators Q*ρ*, P*ρ*, the Leibniz rule from, and the compact embedding of *H*0*ρ*, *p*(Ω) into *L*∞(R*n*) for *p* > *n*/*σ* (cf. ).\n[Nonlocal representation of quasiconvexity] Let *δ* > 0 be such that $\\operatorname{supp}\\rho = \\overline{B\\_{\\delta}(0)}$. A continuous function *h* : R*m* × *n* → R is quasiconvex if and only if \n$$f(A) \\leq \\frac{1}{\\abs{{\\Omega}\\_{\\delta}}}\\int\\_{{\\Omega}\\_{\\delta}} f(A+D\\_\\rho u)\\,dx \\quad \\text{for all $A \\in {\\mathbb{R}^{m \\times n}}$ and $u \\in H^{\\rho,\\infty}\\_0({\\Omega};{\\mathbb{R}}^m)$},$$ \n with $H^{\\rho,\\infty}\\_0({\\Omega};{\\mathbb{R}}^m):=\\{u \\in L^{\\infty}({\\mathbb{R}}^n;{\\mathbb{R}}^m)\\,:\\, D\\_\\rho u \\in L^{\\infty}({\\mathbb{R}}^n;{\\mathbb{R}^{m \\times n}}), \\ u=0 \\ \\text{a.e.~in $Øc$}\\}$.\nScaled kernels\n--------------\nWe introduce here the setting and notations for varying horizon nonlocal gradients obtained via scaling of a fixed nonlocal gradient, as they will be used in the limits of vanishing and diverging horizon in Sections [se: localization] and [se: convergence in infinity].\nOur starting point is a radial kernel *ρ* that satisfies [itm:h0]-[itm:upper] and is normalized in the sense that\n \n$$\\label{eq:normalized}\n\\operatorname{supp}\\rho = \\overline{B\\_1(0)} \\quad \\text{and} \\quad \\int\\_{{{\\mathbb{R}}^{n}}} \\rho \\,dx =n.$$ \nOne can then compute that\n \n$$\\begin{aligned}\n\\label{normalization\\_Qrho}\n\\int\\_{{\\mathbb{R}}^n} Q\\_\\rho\\,dx=1, \\quad \\text{ or equivalently, } \\quad \\widehat{Q}\\_\\rho(0)=1.\\end{aligned}$$ \nNotice that the kernels from Example [ex:kernels] can all be rescaled and normalized to satisfy.\nThe rescaled family of kernels (*ρ**δ*)*δ* for horizons *δ* > 0 is then defined by\n \n$$\\begin{aligned}\n\\label{eq:scaledprelim}\n\\rho\\_{\\delta}(x) = c\\_{\\delta}\\rho\\left(\\frac{x}{{\\delta}}\\right), \\quad x \\in {{\\mathbb{R}}^{n}},\\end{aligned}$$ \nwith (*c**δ*)*δ* ⊂ (0, ∞) a suitable sequence of scaling factors. Precisely, they are chosen as *c**δ* = *δ*− *n* for the limit *δ* → 0 and as $c\\_{\\delta}= \\overline{\\rho}(1/{\\delta})^{-1}$ for the limit *δ* → ∞.\nWe collect here a few general observations about the rescaled kernels and associated gradients. First, it follows that $\\operatorname{supp}\\rho\\_{\\delta}= \\overline{B\\_{\\delta}(0)}$, which makes *D**ρ**δ* a nonlocal gradient with horizon *δ*; in particular, the initial gradient *D**ρ* corresponds with the gradient *D**ρ*1 with horizon distance equal to 1. Moreover, the rescaling preserves the key properties of the kernel function, that is, for any *δ* > 0, the kernel *ρ**δ* also satisfies [itm:h0]-[itm:upper]. This makes all the results in the previous sections applicable to these kernels as well, in particular, the existence result of Theorem [th:existence]. Finally, we observe that the kernel associated to *ρ**δ* satisfies\n \n$$\\label{eq:qrhoscaling}\nQ\\_{\\rho\\_{\\delta}} = c\\_{\\delta}Q\\_\\rho \\left( \\frac{\\cdot}{{\\delta}}\\right) \\quad \\text{and} \\quad \\widehat{Q}\\_{\\rho\\_{\\delta}} = c\\_{\\delta}{\\delta}^{n}\\widehat{Q}\\_\\rho ({\\delta}\\cdot),$$ \nwhere we have used for the interaction between scaling and Fourier transforms.\nTo highlight the dependence on the horizon parameter, we denote the spaces associated to *D**ρ**δ* by \n*H**ρ*, *p*, *δ*(R*n*) = {*u* ∈ *L**p*(R*n*) : *D**ρ**δ**u* ∈ *L**p*(R*n*; R*n*)}, \n and similarly for *H*0*ρ*, *p*, *δ*(Ω); it holds specifically that *H**ρ*, *p*(R*n*) = *H**ρ*, *p*, 1(R*n*).\nLocalization when *δ* → 0\n=========================\nThis section is devoted the localization process, that is, to the asymptotic analysis in the limit of vanishing horizon. We start by showing that the suitably scaled nonlocal gradients converge to the classical one as *δ* → 0, and subsequently prove compactness results uniformly in the horizon parameter *δ*. Finally, we utilize these tools to establish the Γ-convergence of integral functionals depending on scaled nonlocal gradients to their local counterparts as the horizon tends to zero.\nFor this analysis, we fix a radial kernel *ρ* that satisfies [itm:h0]-[itm:upper] and, and consider for *δ* ∈ (0, 1], the scaled kernels \n$$\\rho\\_\\delta = \\frac{1}{\\delta^n}\\rho\\left(\\frac{\\,\\cdot\\,}{\\delta}\\right) \\quad \\text{and} \\quad Q\\_{\\rho\\_\\delta}= \\frac{1}{\\delta^n}Q\\_{\\rho}\\left(\\frac{\\,\\cdot\\,}{\\delta}\\right),$$ \n which corresponds to with the scaling factors *c**δ* = *δ*− *n*. Observe that this choice of scaling preserves the normalizations ∫R*n**ρ**δ* *d**x* = *n* and ∫R*n**Q**ρ**δ* *d**x* = 1 for each *δ* ∈ (0, 1], and specifies to\n \n*Q̂**ρ**δ* = *Q̂**ρ*(*δ* ⋅ ). \nThroughout this section, we take *p* ∈ (1, ∞) and assume Ω to be a bounded Lipschitz domain.\nLocalization of the nonlocal gradient\n-------------------------------------\nHere, we present the convergence of the scaled nonlocal gradients to the classical gradient. Starting with the case of smooth functions, which features an explicit convergence rate, we subsequently extend the analysis to Sobolev functions on bounded domains and the whole space R*n*. In the case of bounded domains, the nonlocal gradient is defined on a smaller set than the classical one, but this difference vanishes as *δ* → 0. Closely related localization results can be found in.\n[Localization of nonlocal gradients][le:recovery] The following statements hold:\n* For each *φ* ∈ *C**c*∞(R*n*) and for all *δ* ∈ (0, 1], one has that\n \n$$\\begin{aligned}\n\\label{eq:quadratic convergence}\n \\norm{D\\_{\\rho\\_{\\delta}} {\\varphi}- \\nabla {\\varphi}}\\_{L^{\\infty}({\\mathbb{R}}^n;{\\mathbb{R}}^n)} \\leq {\\delta}^2 {\\mathrm{Lip}}(\\nabla^2 {\\varphi}).\n \\end{aligned}$$ \nIn particular, *D**ρ**δ**φ* → ∇*φ* uniformly on R*n* as *δ* → 0.\n* For each *u* ∈ *W*1, *p*(Ω), it holds that \n$$\\mathbbm{1}\\_{{\\Omega}\\_{-{\\delta}}} D\\_{\\rho\\_{\\delta}}u \\to \\nabla u \\quad \\text{in } L^p({\\Omega};{\\mathbb{R}}^n) \\text{ as } {\\delta}\\to 0;$$ \n recall that Ω− *δ* : = {*x* ∈ Ω : dist(*x*, Ω*c*) > *δ*}.\n* For each *u* ∈ *W*1, *p*(R*n*), one has that *u* ∈ *H**ρ**δ*, *p*(R*n*) for all *δ* ∈ (0, 1], and \n*D**ρ**δ**u* → ∇*u* in *L**p*(R*n*; R*n*)as *δ* → 0.\n*Part (*i*):* Let *ψ* ∈ *C**c*∞(R*n*; R*n*). Then the multivariate version of Taylor’s theorem with integral remainder shows for *x* ∈ R*n* that\n \n$$\\begin{aligned}\n \\abs{Q\\_{\\rho\\_{\\delta}}\\*\\psi(x)- \\psi(x)} &= {\\left|\\int\\_{B\\_{\\delta}(x)} Q\\_{\\rho\\_{\\delta}}(x-y)\\bigl(\\psi(y) - \\psi(x)\\bigr)\\,dy\\right|}\\\\\n &\\leq {\\left|\\int\\_{B\\_{\\delta}(x)} Q\\_{\\rho\\_{\\delta}}(x-y)\\nabla \\psi(x) (y-x)\\,dy\\right|}\\\\\n &\\qquad +\\int\\_{B\\_{\\delta}(x)} Q\\_{\\rho\\_{\\delta}}(x-y){\\left|\\int\\_0^1\\bigl(\\nabla \\psi(x+t(y-x))-\\nabla\\psi(x)\\bigr) (y-x)\\,dt\\right|}\\,dy\\\\\n &\\leq {\\delta}^2 {\\mathrm{Lip}}(\\nabla \\psi)\\int\\_{B\\_{\\delta}(x)} Q\\_{\\rho\\_{\\delta}}(x-y)\\,dy= {\\delta}^2 {\\mathrm{Lip}}(\\nabla \\psi),\\end{aligned}$$ \nwhere we have used that $\\norm{Q\\_{\\rho\\_{\\delta}}}\\_{L^1({\\mathbb{R}}^n)}=1$, and also the radiality of *Q**ρ**δ* to cancel the term in the second line. Applying this estimate with *ψ* = ∇*φ* for any *φ* ∈ *C**c*∞(R*n*) proves the claim in light of.\n*Part (*i**i*)*: Since *u* ∈ *W*1, *p*(Ω), the nonlocal gradient *D**ρ**δ**u* is well-defined in Ω− *δ* and coincides with *Q**ρ**δ* \\* ∇*u* on this set (cf. ). Let *j* ∈ N and choose *φ**j* ∈ *C**c*∞(R*n*) such that\n \n$$\\begin{aligned}\n\\norm{{\\varphi}\\_j - u}\\_{W^{1,p}({\\Omega})} \\leq \\frac{1}{j}, \\end{aligned}$$ \nwhich is possible as Ω is a bounded Lipschitz domain by assumption. Additionally, we can choose *δ* = *δ*(*j*) small enough in light of Part (*i*) such that \n$$\\norm{D\\_{\\rho\\_{\\delta}}{\\varphi}\\_j - \\nabla {\\varphi}\\_j}\\_{L^p({\\Omega};{\\mathbb{R}}^n)} \\leq \\frac{1}{j} \\quad \\text{and} \\quad \\norm{\\nabla u}\\_{L^p({\\Omega}\\setminus {\\Omega}\\_{-{\\delta}};{{\\mathbb{R}}^{n}})} \\leq \\frac{1}{j}.$$ \n The previous estimates along with then imply\n \n$$\\begin{aligned}\n\\norm{\\mathbbm{1}\\_{{\\Omega}\\_{-{\\delta}}} D\\_{\\rho\\_{\\delta}}u-\\nabla u}\\_{L^p({\\Omega};{{\\mathbb{R}}^{n}})} &\\leq \\norm{D\\_{\\rho\\_{\\delta}}u-D\\_{\\rho\\_{\\delta}}{\\varphi}\\_j}\\_{L^p({\\Omega}\\_{-{\\delta}};{{\\mathbb{R}}^{n}})}+\\norm{D\\_{\\rho\\_{\\delta}}{\\varphi}\\_j-\\nabla{\\varphi}\\_j}\\_{L^p({\\Omega}\\_{-{\\delta}};{{\\mathbb{R}}^{n}})} \\\\\n& \\qquad +\\norm{\\nabla {\\varphi}\\_j - \\nabla u}\\_{L^p({\\Omega}\\_{-{\\delta}};{{\\mathbb{R}}^{n}})}+\\norm{\\nabla u}\\_{L^p({\\Omega}\\setminus {\\Omega}\\_{-{\\delta}};{{\\mathbb{R}}^{n}})}\\\\\\\n& \\leq \\norm{Q\\_{\\rho\\_\\delta} \\ast \\nabla u - Q\\_{\\rho\\_\\delta}\\ast \\nabla {\\varphi}\\_j}\\_{L^p({\\Omega}\\_{-{\\delta}};{{\\mathbb{R}}^{n}})} + \\frac{3}{j} \\\\ & \\leq \\norm{Q\\_{\\rho\\_{\\delta}}}\\_{L^1({{\\mathbb{R}}^{n}})} \\norm{\\nabla u - \\nabla {\\varphi}\\_j}\\_{L^p({\\Omega};{{\\mathbb{R}}^{n}})} + \\frac{3}{j} \\leq \\frac{4}{j},\\end{aligned}$$ \nwhere the last line is due to Young’s convolution inequality.\n*Part (*i**i**i*):* This follows with similar arguments as in Part(*i**i*) or, alternatively, from with A = ∇.\n[rem: localization of NL gradients] a) In view of estimate , *D**ρ**δ* converges to ∇ quadratically in *δ*, given that ∇2*φ* is Lipschitz continuous. More generally, if *φ* is twice differentiable such that ∇2*φ* is *α*-Hölder continuous with *α* ∈ (0, 1], a similar argument induces the convergence rate *δ*1 + *α*, while for a differentiable *φ* with *α*-Hölder continuous gradient, convergence takes place at a rate of *δ**α*.\nb) Our Γ-convergence result in Section [subsec:Gamma delta0] is formulated for admissible functions with prescribed Dirichlet conditions in the complement of Ω. Therefore, Lemma [le:recovery](*i**i**i*) is sufficient for these purposes. However, the sharper result in Part(*i**i*) for bounded domains can be useful in the future for studying vanishing-horizon limits in more general settings, such as the Neumann-type problems considered in.\nCompactness uniformly in *δ* ∈ (0, 1]\n-------------------------------------\nIn this section, we establish a compactness result for the nonlocal gradients that hold uniformly in the horizon parameter. The following theorem, which is also interesting in its own right (cf. below), serves as a technical basis by providing a comparison between the norms of the nonlocal gradients with different horizons; this includes also the classical gradient, denoted for consistency by *D**ρ*0 : = ∇. Our proof relies on Fourier multiplier theory and takes inspiration from the one of for comparing Sobolev spaces associated to different nonlocal gradients.\n[Comparison between scaled nonlocal gradients][th:comparison] Let *δ̄* > 0 and (*δ*1, *δ*2) ∈ [*δ̄*, 1] × [0, 1]. Then, there exists a constant *C* = *C*(*ρ*, *n*, *p*, *δ̄*) > 0 such that \n$$\\norm{D\\_{\\rho\\_{\\delta\\_1}}\\varphi}\\_{L^p({\\mathbb{R}}^n;{\\mathbb{R}}^n)} \\leq C \\norm{D\\_{\\rho\\_{\\delta\\_2}}\\varphi}\\_{L^p({\\mathbb{R}}^n;{\\mathbb{R}}^n)} \\quad \\text{for all $\\varphi \\in C\\_c^{\\infty}({\\mathbb{R}}^n)$}.$$ \nDefine the function *m**δ*1, *δ*2 : R*n* \\ {0} → (0, ∞) by \n$$m\\_{\\delta\\_1, \\delta\\_2}(\\xi) := \\frac{\\widehat{Q}\\_\\rho(\\delta\\_1 \\xi)}{\\widehat{Q}\\_\\rho(\\delta\\_2 \\xi)},$$ \n recalling that *Q̂**ρ* is non-negative. Then, we find in view of and that \n$$\\widehat{D\\_{\\rho\\_{\\delta\\_1}}\\varphi} = m\\_{\\delta\\_1, \\delta\\_2}\\widehat{D\\_{\\rho\\_{\\delta\\_2}}\\varphi}.$$ \nfor every *φ* ∈ *C**c*∞(R*n*); in particular, the case *δ*2 = 0 is valid given that *Q̂**ρ*(0) = 1. It now suffices to show with the help the Mihlin-Hörmander theorem (cf. e.g. ) that *m**δ*1, *δ*2 are *L**p*-multipliers with estimates independent of *δ*1 and *δ*2. To this aim, we need to prove that for every *α* ∈ N0*n* with $\\abs{\\alpha} \\leq \\tfrac{n}{2}+1$,\n \n$$\\begin{aligned}\n\\label{eq:mihlin}\n {\\bigl|\\partial^{\\alpha} m\\_{\\delta\\_1, \\delta\\_2}(\\xi)\\bigr|} \\leq C \\abs{\\xi}^{-\\abs{\\alpha}} \\quad \\text{for all $\\xi \\not =0$}\n \\end{aligned}$$ \nwith a constant *C* > 0 depending only on *n*, *ρ* and *δ̄*.\nTo this aim, note that the second part of Lemma [lem:bms] together with the Leibniz and quotient rules for differentiation imply \n$$\\abs{\\partial^{\\alpha} m\\_{\\delta\\_1, \\delta\\_2}(\\xi)} \\leq C \\abs{\\xi}^{-\\abs{\\alpha}} \\abs{m\\_{\\delta\\_1, \\delta\\_2}(\\xi)} \\quad \\text{for all $\\xi \\not =0$}$$ \n with *C* = *C*(*n*) > 0. Therefore, it only remains to verify for *α* = 0, that is, we need to show *m**δ*1, *δ*2 is uniformly bounded independent of *δ*1 and *δ*2. We prove this by distinguishing two cases.\n*Case 1: *δ*1 ≥ *δ*2.* For $0<\\abs{\\xi} \\leq \\tfrac{1}{\\delta\\_2{\\varepsilon}}$ with *ɛ* > 0 the parameter in the hypotheses (H0)-(H4), one can estimate \n$${\\bigl|m\\_{\\delta\\_1, \\delta\\_2}(\\xi)\\bigr|}={\\left|\\frac{\\widehat{Q}\\_\\rho(\\delta\\_1 \\xi)}{\\widehat{Q}\\_\\rho(\\delta\\_2 \\xi)}\\right|} \\leq\\textstyle \\bigl(\\min\\_{\\overline{B\\_{1/{\\varepsilon}}(0)}}\\widehat{Q}\\_\\rho\\bigr)^{-1},$$ \n considering that $\\norm{\\widehat{Q}\\_\\rho}\\_{L^\\infty({\\mathbb{R}}^n)}\\leq \\norm{Q\\_\\rho}\\_{L^1({\\mathbb{R}}^n)}=1$ by . On the other hand, we infer for $\\abs{\\xi} \\geq \\tfrac{1}{\\delta\\_2{\\varepsilon}}$ from Lemma [lem:bms] that\n \n$$\\begin{aligned}\n|m\\_{\\delta\\_1, \\delta\\_2}(\\xi)| \n&={\\left|\\frac{\\widehat{Q}\\_\\rho(\\delta\\_1 \\xi)}{\\widehat{Q}\\_\\rho(\\delta\\_2 \\xi)}\\right|} \\leq C\\left(\\frac{\\delta\\_2}{\\delta\\_1}\\right)^n \\frac{\\overline{\\rho}\\Bigl(\\frac{1}{\\delta\\_1\\abs{\\xi}}\\Bigr)}{\\overline{\\rho}\\Bigl(\\frac{1}{\\delta\\_2\\abs{\\xi}}\\Bigr)} = C\\left(\\frac{\\delta\\_2}{\\delta\\_1}\\right)^{1-\\gamma} \\frac{\\overline{\\rho}\\Bigl(\\frac{1}{\\delta\\_1\\abs{\\xi}}\\Bigr) \\Bigl(\\frac{1}{\\delta\\_1\\abs{\\xi}}\\Bigr)^{n+\\gamma-1}}{\\overline{\\rho}\\Bigl(\\frac{1}{\\delta\\_2\\abs{\\xi}}\\Bigr)\\Bigl(\\frac{1}{\\delta\\_2\\abs{\\xi}}\\Bigr)^{n+\\gamma-1}} \\\\ &\\leq C \\left(\\frac{\\delta\\_2}{\\delta\\_1}\\right)^{1-\\gamma} \\leq C\\bar{\\delta}^{\\gamma-1},\n \\end{aligned}$$ \nwhere the second inequality uses the almost monotonicity in [itm:upper].\n*Case 2: *δ*1 ≤ *δ*2.* Similarly as in Case 1, we obtain for $0<\\abs{\\xi} \\leq \\tfrac{1}{\\delta\\_1{\\varepsilon}}$ that \n$${\\bigl|m\\_{\\delta\\_1, \\delta\\_2}(\\xi)\\bigr|}={\\left|\\frac{\\widehat{Q}\\_\\rho(\\delta\\_1 \\xi)}{\\widehat{Q}\\_\\rho(\\delta\\_2 \\xi)}\\right|} \\leq\\textstyle \\bigl(\\min\\_{\\overline{B\\_{\\delta\\_2/(\\delta\\_1{\\varepsilon})}(0)}}\\widehat{Q}\\_\\rho\\bigr)^{-1}\\leq \\bigl(\\min\\_{\\overline{B\\_{1/(\\bar{\\delta}{\\varepsilon})}(0)}}\\widehat{Q}\\_\\rho\\bigr)^{-1},$$ \n and for $\\abs{\\xi} \\geq 1/(\\delta\\_1{\\varepsilon})$ by Lemma [lem:bms] that \n$$\\abs{m\\_{\\delta\\_1, \\delta\\_2}(\\xi)} = {\\left|\\frac{\\widehat{Q}\\_\\rho(\\delta\\_1 \\xi)}{\\widehat{Q}\\_\\rho(\\delta\\_2 \\xi)}\\right|} \\leq C\\left(\\frac{\\delta\\_2}{\\delta\\_1}\\right)^n \\frac{\\overline{\\rho}\\Bigl(\\frac{1}{\\delta\\_1\\abs{\\xi}}\\Bigr)}{\\overline{\\rho}\\Bigl(\\frac{1}{\\delta\\_2\\abs{\\xi}}\\Bigr)} \\leq C \\left(\\frac{\\delta\\_2}{\\delta\\_1}\\right)^{1-\\sigma} \\leq C\\bar{\\delta}^{\\sigma-1},$$ \n with the second inequality due to [itm:lower].\nFinally, combining the two cases shows that ${\\bigl|m\\_{\\delta\\_1, \\delta\\_2}(\\xi)\\bigr|}\\leq C=C(\\rho, \\bar{\\delta})$ for all *ξ* ≠ 0, which concludes the proof.\nTogether with a density argument, the previous theorem shows that *H**ρ*, *p*, *δ*1(R*n*) = *H**ρ*, *p*, *δ*2(R*n*) and *H*0*ρ*, *p*, *δ*1(Ω) = *H*0*ρ*, *p*, *δ*2(Ω) for all *δ*1, *δ*2 ∈ (0, 1] or equivalently, that\n \n$$\\begin{aligned}\n\\label{eq:spaces delta independent}\nH^{\\rho,p,{\\delta}}({\\mathbb{R}}^n)=H^{\\rho,p}({\\mathbb{R}}^n) \\quad \\text{and} \\quad H^{\\rho,p,{\\delta}}\\_0(\\Omega) = H^{\\rho,p}\\_0(\\Omega) \\qquad \\text{ for all ${\\delta}\\in (0,1]$.}\\end{aligned}$$ \nIn fact, by inspecting the proof of Theorem [th:comparison], it is not hard to see that holds for all *δ* > 0, which shows that our nonlocal function spaces do not depend on the horizon parameter *δ*. Based on this observation, we obtain the following corollary as a consequence of .\n[cor:compactness] There exists a constant *C* = *C*(*ρ*, *n*, Ω, *p*) > 0 such that \n$$\\norm{u}\\_{H^{\\sigma,p}({\\mathbb{R}}^n)} \\leq C\\norm{D\\_{\\rho\\_\\delta}u}\\_{L^p(\\Omega\\_{\\delta};{\\mathbb{R}}^n)} \\quad \\text{for all $u \\in H^{\\rho,p,{\\delta}}\\_0(\\Omega)$ and $\\delta \\in (0,1]$.}$$ \nWe observe that there is another way of proving Corollary [cor:compactness] that does not pass through the stronger statement of Theorem [th:comparison]. For this alternative argument, it suffices to require that the kernel *ρ* satisfies [itm:h0]-[itm:derivatives] and \n$$\\liminf\\_{\\abs{x} \\to 0} \\rho(x)\\abs{x}^{n+\\sigma-1} > 0.$$ \n Indeed, one can compare *ρ**δ* with the kernel from Example [ex:kernels]a) for *s* = *σ* by arguing as in and checking that the constants are independent of *δ*. When *p* = 2, even [itm:derivatives] is not necessary (cf. ), so that also the truncated fractional gradients in Remark [rem:relaxed] are covered.\nIn order to ensure the existence of convergent subsequences required for the forthcoming Γ-convergence result (see Theorem [th: Gamma convergence to 0]), we proceed with the following compactness statement.\n[Convergent subsequences for vanishing horizon][le:compactness] Let (*δ**j*)*j* ⊂ (0, 1] be a sequence with *δ**j* → 0 and suppose that *u**j* ∈ *H*0*ρ*, *p*, *δ**j*(Ω) for each *j* ∈ N with \n$$\\sup\\_{j \\in {\\mathbb{N}}} \\norm{D\\_{\\rho\\_{{\\delta}\\_j}} u\\_j}\\_{L^p(\\Omega\\_{{\\delta}\\_j};{\\mathbb{R}}^n)} < \\infty.$$ \n Then, there is a *u* ∈ *W*01, *p*(Ω) (extended to R*n* as zero) such that, up to a non-relabeled subsequence, \n$$u\\_j \\to u \\ \\ \\text{in $L^p({\\mathbb{R}}^n)$} \\quad \\text{and} \\quad D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j {\\rightharpoonup}\\nabla u \\ \\ \\text{in $L^p({\\mathbb{R}}^n;{\\mathbb{R}}^n)$ as $j \\to \\infty$.}$$ \nBy Corollary [cor:compactness], the sequence (*u**j*)*j* is bounded in *H**σ*, *p*(R*n*). Since each *u**j* is supported in $\\overline{\\Omega}$, we conclude from the compact embedding $H^{\\sigma,p}({{\\mathbb{R}}^{n}})\\hookrightarrow\\hookrightarrow L^p\\_{{\\mathrm{loc}}}({{\\mathbb{R}}^{n}})$ (cf. [embeddingBessel]) that there is a *u* ∈ *L**p*(R*n*) with *u* = 0 a.e. in Ω*c* such that, up to a non-relabeled subsequence,\n \n$$\\begin{aligned}\nu\\_j \\to u\\qquad \\text{ in $L^p({\\mathbb{R}}^n)$ as $j\\to \\infty$.}\\end{aligned}$$ \nMoreover, up to extracting a potential further subsequence, we find that $D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j {\\rightharpoonup}V$ in *L**p*(R*n*; R*n*) for some *V* ∈ *L**p*(R*n*; R*n*). To deduce that *V* = ∇*u*, we compute for *φ* ∈ *C**c*∞(R*n*) that\n \n$$\\begin{aligned}\n \\int\\_{{\\mathbb{R}}^n} V\\,{\\varphi}\\,dx &= \\lim\\_{j \\to\\infty}\\int\\_{{\\mathbb{R}}^n} D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j\\,{\\varphi}\\,dx\\\\\n &= - \\lim\\_{j \\to \\infty}\\int\\_{{\\mathbb{R}}^n} u\\_j\\,D\\_{\\rho\\_{{\\delta}\\_j}}{\\varphi}\\,dx\\\\\n &=-\\int\\_{{\\mathbb{R}}^n} u\\,\\nabla{\\varphi}\\,dx,\n \\end{aligned}$$ \nwhere the last equality follows from the localization result for the nonlocal gradients in Lemma [le:recovery](*i*). This shows that *u* ∈ *W*01, *p*(Ω) (extended to R*n* as zero) with ∇*u* = *V*, which finishes the proof.\nΓ-convergence *δ* → 0\n---------------------\nWe are now in the position to make the conjectured localization of our variational problems in the limit *δ* → 0 rigorous, choosing Γ-convergence as a natural framework.\nBefore stating the theorem, let us collect the relevant objects. The family of vectorial energy functionals (F*δ*)*δ* ∈ (0, 1] with F*δ* : *L**p*(R*n*; R*m*) → R∞ : = R ∪ {∞} for *δ* ∈ (0, 1] is given by\n \n$$\\label{eq:Fcald}\n{\\mathcal{F}}\\_{\\delta}(u):=\\begin{cases}\n\\displaystyle\\int\\_{\\Omega\\_{\\delta}} f(x,D\\_{\\rho\\_{\\delta}} u)\\,dx &\\text{for $u \\in H^{\\rho,p,{\\delta}}\\_0(\\Omega;{\\mathbb{R}}^m)$},\\\\\n\\infty &\\text{else,}\n\\end{cases}$$ \nwhere *f* : Ω1 × R*m* × *n* → R is a suitable Carathéodory integrand; considering that the functions in the domain of F*δ* are defined on R*n* with zero Dirichlet conditions in Ω*c*, we may take, without loss of generality, the integrals over the bounded set Ω*δ*.\nAs prospective limit functional, we introduce F0 : *L**p*(R*n*; R*m*) → R∞ given by\n \n$$\\label{eq:Fcal0}\n{\\mathcal{F}}\\_0(u):=\\begin{cases}\n\\displaystyle\\int\\_{\\Omega} f(x,\\nabla u)\\,dx &\\text{for $u \\in W^{1,p}\\_0(\\Omega;{\\mathbb{R}}^m)$},\\\\\n\\infty &\\text{else;}\n\\end{cases}$$ \nhere, functions in *W*01, *p*(Ω; R*m*) are identified with their extension to R*n* as zero.\n[Localization for vanishing horizon via Γ-convergence] [th: Gamma convergence to 0] Let *f* : Ω1 × R*m* × *n* → R be a Carathéodory integrand such that \n$$c\\abs{A}^p-C\\leq f(x,A) \\leq C(1+\\abs{A}^p) \\quad \\text{for a.e.~$x \\in \\Omega\\_1$ and all $A \\in {\\mathbb{R}^{m \\times n}}$}$$ \n with *c*, *C* > 0. If *f*(*x*, ⋅ ) is quasiconvex for a.e. *x* ∈ Ω, then the family (F*δ*)*δ* ∈ (0, 1] in Γ-converges with respect to *L**p*(R*n*; R*m*)-convergence to the functional F0 in as *δ* → 0, that is, \nΓ(*L**p*)-lim*δ* → 0F*δ* = F0. \n Additionally, (F*δ*)*δ* is equi-coercive with respect to convergence in *L**p*(R*n*; R*m*).\nLet (*δ**j*)*j* ⊂ (0, 1] be a sequence converging to 0 as *j* → ∞.\n*Equi-coercivity:* By the growth bound on *f* from below and Corollary [cor:compactness], we deduce that there are constants *c*ʹ, *C*ʹ > 0 such that \n$${\\mathcal{F}}\\_{{\\delta}\\_j}(u)\\geq c' \\norm{u}\\_{H^{\\sigma,p}({\\mathbb{R}}^n)} -C'$$ \n for all *j* ∈ N and *u* ∈ *H*0*ρ*, *p*, *δ**j*(Ω; R*m*). This yields the equi-coercivity, given the compact embedding of *H**σ*, *p*(R*n*) into $L\\_{\\rm loc}^p({\\mathbb{R}}^n)$, cf. .\n*Liminf-inequality:* Let (*u**j*)*j* ⊂ *L**p*(R*n*; R*m*) with *u**j* → *u* in *L**p*(R*n*; R*m*). Assuming without loss of generality that \nliminf*j* → ∞F*δ**j*(*u**j*) = lim*j* → ∞F*δ**j*(*u**j*) < ∞, \n we have that *u**j* ∈ *H*0*ρ*, *p*, *δ**j*(Ω; R*m*) for each *j* ∈ N and\n \n$$\\begin{aligned}\n\\sup\\_{j \\in {\\mathbb{N}}}\\norm{D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j}\\_{L^p(\\Omega\\_{{\\delta}\\_j};{\\mathbb{R}}^m)}< \\infty,\\end{aligned}$$ \ndue to the lower bound on *f*. Lemma [le:compactness] therefore yields a *u* ∈ *W*01, *p*(Ω; R*m*) (extended to R*n* as zero) such that $D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j {\\rightharpoonup}\\nabla u $ in *L**p*(R*n*; R*m* × *n*). If we use translation operators as in Lemma [le:translation] to define\n \n$$\\begin{aligned}\nv\\_j:={\\mathcal{Q}}\\_{\\rho\\_{{\\delta}\\_j}}u\\_j = Q\\_{\\rho\\_{{\\delta}\\_j}}\\ast u\\_j \\quad \\text{ for $j \\in {\\mathbb{N}}$,}\\end{aligned}$$ \nthen (*v**j*)*j* is a bounded sequence in *W*1, *p*(R*n*; R*m*) with $\\nabla v\\_j = D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j {\\rightharpoonup}\\nabla u$ in *L**p*(R*n*; R*m* × *n*) as *j* → ∞. Consequently, it even holds that $v\\_j {\\rightharpoonup}u$ in *W*1, *p*(R*n*; R*m*), considering that the functions *v**j* are zero outside of Ω1 for each *j* ∈ N. A standard lower semicontinuity result for functionals with quasiconvex integrands (cf. ) then implies\n \n$$\\begin{aligned}\n\\liminf\\_{j \\to\\infty} {\\mathcal{F}}\\_{{\\delta}\\_j}(u\\_j)&= \\liminf\\_{j \\to \\infty} \\int\\_{\\Omega\\_{{\\delta}\\_j}} f(x,D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j)\\,dx \\\\\n&\\geq \\liminf\\_{j \\to \\infty} \\int\\_{\\Omega} f(x,D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j)\\,dx -C\\abs{\\Omega\\_{{\\delta}\\_j}\\setminus \\Omega}\\\\\n&=\\liminf\\_{j \\to \\infty} \\int\\_{\\Omega} f(x,\\nabla v\\_j)\\,dx\\geq \\int\\_{\\Omega}f(x,\\nabla u)\\,dx = {\\mathcal{F}}\\_0(u),\\end{aligned}$$ \nwhich is the desired liminf-inequality.\n*Recovery sequence:* Without loss of generality, consider *u* ∈ *W*01, *p*(Ω; R*m*). Then, we infer from Lemma [le:recovery](*i**i**i*) that *u* ∈ *H*0*ρ*, *p*, *δ**j*(Ω; R*m*) for all *j* ∈ N with *D**ρ**δ**j**u* → ∇*u* in *L**p*(R*n*; R*m* × *n*) as *j* → ∞. The upper bound on *f* enables the application of Lebesgue’s dominated convergence theorem to find \nlim*j* → ∞F*δ**j*(*u*) = lim*j* → ∞∫Ω*δ**j**f*(*x*, *D**ρ**δ**j**u*) *d**x* = ∫Ω*f*(*x*, ∇*u*) *d**x* = F0(*u*). \n This shows that the constant sequence is a suitable recovery sequence.\na) Under the assumptions of Theorem [th: Gamma convergence to 0], for every *δ* ∈ (0, 1], the functional F*δ* admits a minimizer according to Theorem [th:existence]. Therefore, standard properties of Γ-convergence imply that these minimizers converge, up to subsequence, to a minimizer of F0 as *δ* → 0.\nReferring to the literature, a closely related Γ-convergence result involving similar nonlocal gradients in the case *m* = 1 can be found in. However, the latter does not feature the equi-coercivity required for the convergence of minimizers.\nb) Since the definition of *D**ρ**δ**u* on Ω*δ* only depends on the values of *u* in Ω2*δ*, the Dirichlet condition in Ω*c* can be equivalently replaced by prescribing zero values in Ω2*δ* \\ Ω. We remark that the papers and on finite-horizon fractional gradients use a slightly different convention by considering the gradients on Ω and requiring Dirichlet conditions in Ω*δ* \\ Ω− *δ*. Clearly, both settings are equivalent by a suitable renaming of the domain. The reason for our choice is that only the setting used here is meaningful for both limit passages *δ* → 0 and *δ* → ∞.\nc) Theorem [th: Gamma convergence to 0] can readily be extended to non-zero complementary values, that is, to admissible functions in the spaces *g* + *H*0*ρ*, *p*, *δ*(Ω; R*m*) for any given *g* ∈ *W*1, *p*(R*n*; R*m*). Indeed, since *g* ∈ *H**ρ*, *p*, *δ*(Ω; R*m*) for all *δ* > 0 with \n$$\\norm{D\\_{\\rho\\_{\\delta}}g}\\_{L^p({\\mathbb{R}}^n;{\\mathbb{R}^{m \\times n}})} \\leq \\norm{Q\\_{\\rho\\_{\\delta}}}\\_{L^1({{\\mathbb{R}}^{n}})}\\norm{\\nabla g}\\_{L^p({\\mathbb{R}}^n;{\\mathbb{R}^{m \\times n}})}=\\norm{\\nabla g}\\_{L^p({\\mathbb{R}}^n;{\\mathbb{R}^{m \\times n}})},$$ \n the argument follows through in the same manner with the domain of the Γ-limit being *g* + *W*01, *p*(Ω; R*m*).\nBy applying Theorem [th: Gamma convergence to 0] to the kernels of Example [ex:kernels], we obtain the localization of functionals as in with nonlocal gradients associated to the following scaled kernels:\na) For *ρ* as in Example [ex:kernels]a), one finds that \n$$\\rho\\_{\\delta}(x) = {\\delta}^{-n} \\frac{w(x/{\\delta})}{\\abs{x/{\\delta}}^{n+s-1}} = {\\delta}^{s-1} \\frac{w(x/{\\delta})}{\\abs{x}^{n+s-1}}, \\quad x\\in {\\mathbb{R}}^n\\setminus \\{0\\}.$$ \n These scaled kernels coincide (up to a constant) with finite-horizon fractional gradients *D**δ**s* from. Theorem [th: Gamma convergence to 0] then complements the localization result for *s* ↑ 1 in.\nb) The scaled versions of the kernels *ρ* in Example [ex:kernels]b) read as \n$$\\rho\\_{\\delta}(x) = {\\delta}^{-n} \\frac{w(x/{\\delta})\\log^\\kappa({\\delta}/\\abs{x})}{\\abs{x/{\\delta}}^{n+s-1}} = {\\delta}^{s-1} \\frac{w(x/{\\delta})(\\log({\\delta})-\\log(\\abs{x}))^\\kappa}{\\abs{x}^{n+s-1}}, \\quad x\\in {\\mathbb{R}}^n\\setminus \\{0\\}.$$ \nc) With *ρ* as in Example [ex:kernels]c), the scaled kernels are given by \n$$\\rho\\_{\\delta}(x) = {\\delta}^{-n} \\frac{w(x/{\\delta})}{\\abs{x/{\\delta}}^{n+s(\\abs{x/{\\delta}})-1}} = {\\delta}^{s(\\abs{x/{\\delta}})-1} \\frac{w(x/{\\delta})}{\\abs{x}^{n+s(\\abs{x/{\\delta}})-1}}, \\quad x\\in {\\mathbb{R}}^n\\setminus \\{0\\}.$$ \nΓ-convergence *δ* → ∞\n=====================\nWe focus now on the asymptotics of the nonlocal gradient as the horizon *δ* diverges to infinity. As proven below, the associated limiting object is the Riesz fractional gradient. While this is to be expected for finite-horizon fractional gradients, surprisingly, the same holds when starting from any general nonlocal gradient within our setting. This section is structured in parallel to Section [se: localization], showing first the convergence of the nonlocal gradients to the Riesz fractional gradient, then providing a uniform compactness result, and finally, proving the Γ-convergence of the associated energy functionals.\nLet *ρ* again be a non-negative radial kernel that satisfies [itm:h0]-[itm:upper] and, and assume throughout that *p* ∈ (1, ∞) and Ω is a bounded Lipschitz domain. The lack of integrability of the fractional kernel on R*n* calls for a different scaling procedure compared with Section [se: localization]. Precisely, for *δ* ∈ (1/*ɛ*, ∞) (so that $\\overline{\\rho}(1/{\\delta})\\not=0$), we now define\n \n$$\\begin{aligned}\n\\label{eq:kernelrhodelta}\n\\rho\\_{\\delta}(x):=\\overline{\\rho}\\left(\\frac{1}{{\\delta}}\\right)^{-1}\\rho\\left(\\frac{x}{{\\delta}}\\right),\\end{aligned}$$ \nwhich corresponds to with $c\\_{\\delta}:=\\overline{\\rho}(1/{\\delta})^{-1}$. In this way, the values of *ρ**δ* on the unit sphere ∂*B*1(0) are normalized to 1 for any *δ*. In addition, we require that these kernels converge pointwise on R*n* \\ {0} as *δ* → ∞, and set\n \n$$\\begin{aligned}\n\\label{eq:convergenceassumption}\n\\rho\\_{\\infty}(x) := \\lim\\_{{\\delta}\\to \\infty}\\rho\\_{\\delta}(x)=\\lim\\_{{\\delta}\\to \\infty}\\overline{\\rho}\\left(\\frac{1}{{\\delta}}\\right)^{-1}\\rho\\left(\\frac{x}{{\\delta}}\\right), \\quad x\\in {\\mathbb{R}}^n\\setminus\\{0\\}.\\end{aligned}$$ \nWith the scaling , the kernel function *Q**ρ**δ* and its Fourier transform satisfy\n \n$$\\label{eq:qrhohatinfty}\nQ\\_{\\rho\\_{\\delta}} = \\overline{\\rho}\\left(\\frac{1}{{\\delta}}\\right)^{-1}Q\\_\\rho\\left(\\frac{\\,\\cdot\\,}{{\\delta}}\\right) \\quad \\text{and} \\quad \\widehat{Q}\\_{\\rho\\_{\\delta}}= {\\delta}^n\\overline{\\rho}\\left(\\frac{1}{{\\delta}}\\right)^{-1} \\widehat{Q}\\_{\\rho}({\\delta}\\, \\cdot\\,).$$ \nLet us point out that our chosen scaling is, up to a constant, the only relevant one. Indeed, if there is a sequence (*c̄**δ*)*δ* of positive reals such that (*c̄**δ**ρ*( ⋅ /*δ*))*δ* converges pointwise as *δ* → ∞, then for *x* ∈ ∂*B*1(0), \n$$\\lim\\_{{\\delta}\\to \\infty}\\bar{c}\\_{\\delta}\\rho\\left(\\frac{x}{{\\delta}}\\right) = \\lim\\_{{\\delta}\\to \\infty} \\bar{c}\\_{\\delta}\\overline{\\rho}\\left(\\frac{1}{{\\delta}}\\right) =: \\bar{c}.$$ \n Therefore, we obtain for all *x* ∈ R*n* \\ {0} that \n$$\\lim\\_{{\\delta}\\to \\infty}\\bar{c}\\_{\\delta}\\rho\\left(\\frac{x}{{\\delta}}\\right)=\\lim\\_{{\\delta}\\to \\infty} \\bar{c}\\_{\\delta}\\overline{\\rho}\\left(\\frac{1}{{\\delta}}\\right)\\overline{\\rho}\\left(\\frac{1}{{\\delta}}\\right)^{-1}\\rho\\left(\\frac{x}{{\\delta}}\\right)=\\bar{c}\\rho\\_\\infty(x).$$ \nConvergence of nonlocal gradients as *δ* → ∞\n--------------------------------------------\nThis section is about establishing that the scaled nonlocal gradients converge to the Riesz fractional gradient as *δ* → ∞. We commence with some bounds on *ρ**δ* from and the limit kernel *ρ*∞ that will be used repeatedly later. Recall that *ɛ* > 0 is as in [itm:h0]-[itm:upper], and that *σ*, *γ* with 0 < *σ* ≤ *γ* < 1 are the parameters appearing in the hypotheses [itm:lower] and [itm:upper], respectively.\n[le:wedge] There exist constants *C*, *c* > 0 such that for every *δ* > 1/*ɛ* and all *x* ∈ *B**δ**ɛ*(0) \\ {0},\n \n$$\\label{eq:rhodwedge}\nc\\min\\left\\{\\frac{1}{\\abs{x}^{n+\\sigma-1}},\\frac{1}{\\abs{x}^{n+\\gamma-1}}\\right\\} \\leq \\rho\\_{\\delta}(x) \\leq C\\max\\left\\{\\frac{1}{\\abs{x}^{n+\\sigma-1}},\\frac{1}{\\abs{x}^{n+\\gamma-1}}\\right\\}.$$ \nIn particular, it holds for all *x* ∈ R*n* \\ {0} that\n \n$$\\begin{aligned}\n c\\min\\left\\{\\frac{1}{\\abs{x}^{n+\\sigma-1}},\\frac{1}{\\abs{x}^{n+\\gamma-1}}\\right\\} \\leq \\rho\\_\\infty(x) \\leq C\\max\\left\\{\\frac{1}{\\abs{x}^{n+\\sigma-1}},\\frac{1}{\\abs{x}^{n+\\gamma-1}}\\right\\}.\\end{aligned}$$ \nObserve first that by [itm:lower] and [itm:upper], there are constants *c*ʹ, *C*ʹ > 0 such that\n \n$$\\label{eq:comparingrho}\nc'\\left(\\frac{t}{r}\\right)^{n+\\gamma-1}\\overline{\\rho}(t) \\leq \\overline{\\rho}(r)\\leq C' \\left(\\frac{t}{r}\\right)^{n+\\sigma-1}\\overline{\\rho}(t)$$ \nfor all *t*, *r* ∈ (0, *ɛ*) with *r* ≥ *t*.\nLet *x* ∈ *B**δ**ɛ*(0). If $\\abs{x} \\geq 1$, we can apply with the choice $r=\\abs{x}/{\\delta}$ and *t* = 1/*δ* to find \n$$\\frac{c'}{\\abs{x}^{n+\\gamma-1}}\\leq \\overline{\\rho}\\left(\\frac{1}{{\\delta}}\\right)^{-1}\\overline{\\rho}\\left(\\frac{\\abs{x}}{{\\delta}}\\right)\\leq \\frac{C'}{\\abs{x}^{n+\\sigma-1}}.$$ \n As for the case $0<\\abs{x} \\leq 1$, we resort to as well, but take *r* = 1/*δ* and $t=\\abs{x}/{\\delta}$ instead, which gives \n$$\\frac{1}{C'\\abs{x}^{n+\\sigma-1}}\\leq \\overline{\\rho}\\left(\\frac{1}{{\\delta}}\\right)^{-1}\\overline{\\rho}\\left(\\frac{\\abs{x}}{{\\delta}}\\right)\\leq \\frac{1}{c'\\abs{x}^{n+\\gamma-1}}.$$ \n We conclude that holds for suitably chosen constants *c*, *C*.\nAs the next lemma shows, *ρ*∞ must be a fractional kernel, no matter the specific choice of *ρ*. This finding is a key ingredient for proving that only Riesz fractional gradients can be obtained as the limit of increasing horizon nonlocal gradients with the scaled sequence of kernels *ρ**δ*.\n[Limit kernel *ρ*∞ is fractional] [le: fractional kernel as limit] There is *s*∞ ∈ [*σ*, *γ*] such that *ρ*∞ in satisfies \n$$\\rho\\_\\infty(x)=\\rho^{s\\_\\infty}(x):= \\frac{1}{\\abs{x}^{n+s\\_\\infty-1}} \\quad \\text{for all $x \\in {\\mathbb{R}}^n\\setminus\\{0\\}$.}$$ \nOur argument relies on proving that $\\overline{\\rho}\\_{\\infty}$ is a multiplicative function. To this aim, we consider *r*, *t* > 0 and compute that\n \n$$\\begin{aligned}\n \\overline{\\rho}\\_\\infty(r\\cdot t) &= \\lim\\_{{\\delta}\\to\\infty}\\overline{\\rho}\\left(\\frac{1}{{\\delta}}\\right)^{-1}\\overline{\\rho}\\left(\\frac{r \\cdot t}{{\\delta}}\\right) = \\lim\\_{{\\delta}\\to\\infty}\\overline{\\rho}\\left(\\frac{1}{{\\delta}}\\right)^{-1}\\overline{\\rho}\\left(\\frac{r}{{\\delta}}\\right)\\overline{\\rho}\\left(\\frac{r}{{\\delta}}\\right)^{-1}\\overline{\\rho}\\left(\\frac{r \\cdot t}{{\\delta}}\\right)\\\\\n &=\\overline{\\rho}\\_\\infty(r) \\lim\\_{{\\delta}\\to\\infty}\\overline{\\rho}\\left(\\frac{1}{{\\delta}/r}\\right)^{-1}\\overline{\\rho}\\left(\\frac{ t}{{\\delta}/r}\\right) = \\overline{\\rho}\\_\\infty(r)\\overline{\\rho}\\_\\infty(t).\n \\end{aligned}$$ \nSince $\\overline{\\rho}\\_\\infty$ is also locally bounded away from 0 by Lemma [le:wedge], we deduce that $\\overline{\\rho}\\_\\infty$ must be a power function (cf. ). Together with Lemma [le:wedge], it follows therefore that $\\rho\\_\\infty(x)=1/\\abs{x}^{n+s\\_\\infty-1}$ for some *s*∞ ∈ [*σ*, *γ*].\n[rem:rhobars] The parameter *s*∞ associated to a limit kernel *ρ*∞ can be determined directly from *ρ* via the limit \n$$s\\_\\infty=\\log(\\overline{\\rho}\\_\\infty(1/e))-n+1=\\lim\\_{{\\delta}\\to \\infty} \\log\\left(\\overline{\\rho}(1/{\\delta})^{-1}\\overline{\\rho}\\left(1/(e{\\delta})\\right)\\right)-n+1.$$ \n[ex:sbar] One observes that the kernels *ρ* from Example [ex:kernels] satisfy, i.e., their rescaled versions converge pointwise. We identify the limiting fractional exponent *s*∞ for illustration:\na) Let *ρ* be as in Example [ex:kernels]a). Then, for *x* ∈ R*n* \\ {0}, \n$$\\rho\\_\\infty(x)=\\lim\\_{{\\delta}\\to \\infty}\\frac{\\overline{w}(1/{\\delta})^{-1}}{{\\delta}^{n+s-1}}\\frac{w(x/{\\delta})}{\\abs{x/{\\delta}}^{n+s-1}}=\\frac{1}{\\abs{x}^{n+s-1}} =\\rho^s(x),$$ \n which yields *s*∞ = *s*.\nb) For the kernel *ρ* of Example [ex:kernels]b), one obtains the same limit as in a), that is, *ρ*∞ = *ρ**s*, and hence, *s*∞ = *s*. The detailed calculation reads\n \n$$\\begin{aligned}\n \\rho\\_\\infty(x)&=\\lim\\_{{\\delta}\\to \\infty}\\overline{w}(1/{\\delta})^{-1}\\frac{1}{\\log^\\kappa({\\delta}){\\delta}^{n+s-1}}\\frac{w(x/{\\delta})\\log^\\kappa({\\delta}/\\abs{x})}{\\abs{x/{\\delta}}^{n+s-1}}=\\frac{1}{\\abs{x}^{n+s-1}}\\lim\\_{{\\delta}\\to \\infty}\\frac{\\log^\\kappa({\\delta}/\\abs{x})}{\\log^\\kappa({\\delta})}\\\\\n &=\\frac{1}{\\abs{x}^{n+s-1}}\\lim\\_{{\\delta}\\to \\infty} \\left(\\frac{\\log({\\delta})-\\log(\\abs{x})}{\\log({\\delta})}\\right)^\\kappa=\\frac{1}{\\abs{x}^{n+s-1}}\n \\end{aligned}$$ \nfor *x* ∈ R*n* \\ {0}.\nc) In the case of *ρ* from Example [ex:kernels]c), the limit fractional exponent becomes *s*∞ = *s*(0), as\n \n$$\\begin{aligned}\n \\rho\\_\\infty(x)&=\\lim\\_{{\\delta}\\to \\infty}\\overline{w}(1/{\\delta})^{-1}\\frac{1}{{\\delta}^{n+s(1/{\\delta})-1}}\\frac{w(x/{\\delta})}{\\abs{x/{\\delta}}^{n+s(\\abs{x}/{\\delta})-1}}\\\\\n &=\\lim\\_{{\\delta}\\to \\infty}\\frac{1}{{\\delta}^{s(1/{\\delta})-s(\\abs{x}/{\\delta})}\\abs{x}^{n+s(\\abs{x}/{\\delta})-1}} =\\frac{1}{\\abs{x}^{n+s(0)-1}}\n \\end{aligned}$$ \nfor *x* ∈ R*n* \\ {0} shows.\nIn the next step, we show that the nonlocal gradients converge to the fractional gradient induced by *ρ*∞ as *δ* → ∞, see Proposition [prop:convergenceinfinite]. The proof involves the following auxiliary result, which allows to control the integrability of the kernels during the limit passage.\n[le:l1convergence] It holds that \n$$\\rho\\_{\\delta}\\,\\min\\{1,|\\cdot|^{-1}\\} \\to \\rho\\_{\\infty}\\, \\min\\{1,|\\cdot|^{-1}\\} \\quad \\text{in $L^1({\\mathbb{R}}^n)$ as ${\\delta}\\to \\infty$.}$$ \nWe already know the pointwise a.e. convergence by, and it follows from that\n \n$$\\begin{aligned}\n\\label{est45}\n\\mathbbm{1}\\_{B\\_{{\\delta}{\\varepsilon}}(0)} \\rho\\_{\\delta}\\min\\{1,|\\cdot|^{-1}\\} \\leq C\\min\\{1,|\\cdot|^{-1}\\}\\max\\left\\{\\frac{1}{\\abs{\\,\\cdot\\,}^{n+\\sigma-1}},\\frac{1}{\\abs{\\,\\cdot\\,}^{n+\\gamma-1}}\\right\\}.\\end{aligned}$$ \nSince the right-hand side is integrable, Lebesgue’s dominated convergence theorem implies \n$$\\mathbbm{1}\\_{B\\_{{\\delta}{\\varepsilon}}(0)}\\rho\\_{\\delta}\\, \\min\\{1,|\\cdot|^{-1}\\} \\to \\rho\\_{\\infty}\\, \\min\\{1,|\\cdot|^{-1}\\} \\quad \\text{in $L^1({\\mathbb{R}}^n)$ as ${\\delta}\\to \\infty$.}$$ \nIt remains to show that\n \n$$\\label{eq:remainder}\n\\mathbbm{1}\\_{B\\_{{\\delta}{\\varepsilon}}(0)^c}\\rho\\_{\\delta}\\min\\{1,|\\cdot|^{-1}\\} \\to 0 \\quad \\text{in $L^1({\\mathbb{R}}^n)$ as ${\\delta}\\to \\infty$.}$$ \nConsidering that, in light of [itm:h1], *ρ*( ⋅ /*δ*) ≤ *C* in *B**δ**ɛ*(0)*c* for some *C* > 0, and *ρ*( ⋅ /*δ*) = 0 on *B**δ*(0)*c* by , we find \n$$\\mathbbm{1}\\_{B\\_{{\\delta}{\\varepsilon}}(0)^c}\\rho\\_{\\delta}\\leq C\\mathbbm{1}\\_{B\\_{\\delta}(0)\\setminus B\\_{{\\delta}{\\varepsilon}}(0)}\\overline{\\rho}(1/{\\delta})^{-1} \\leq C\\mathbbm{1}\\_{B\\_{\\delta}(0)\\setminus B\\_{1}(0)}\\overline{\\rho}(1/{\\delta})^{-1},$$ \n given that *δ**ɛ* > 1. This yields\n \n$$\\begin{aligned}\n\\int\\_{B\\_{{\\delta}{\\varepsilon}/2}(0)^c}\\rho\\_{\\delta}(x) \\min\\{1,\\abs{x}^{-1}\\}\\,dx &\\leq C\\overline{\\rho}\\left(\\frac{1}{{\\delta}}\\right)^{-1} \\int\\_{B\\_{\\delta}(0)\\setminus B\\_1(0)} \\abs{x}^{-1}\\,dx \\nonumber \\\\ &\\leq \\begin{cases}\nC\\overline{\\rho}(1/{\\delta})^{-1}({\\delta}^{n-1}-1) &\\text{if $n>1$}, \\\\\nC\\overline{\\rho}(1/{\\delta})^{-1}\\log({\\delta}) &\\text{if $n=1$.}\\label{rs}\n\\end{cases}\\end{aligned}$$ \nIn either case, the expression in converges to 0 as *δ* → ∞ in view of [itm:lower], which gives rise to and finishes the proof.\nAs a consequence, we now obtain the convergence of the nonlocal gradients to the Riesz fractional gradient as *δ* → ∞ in the case of Sobolev functions.\n[Convergence to fractional gradient as *δ* → ∞][prop:convergenceinfinite] For any *u* ∈ *W*1, *p*(R*n*) it holds that \n$$D\\_{\\rho\\_{\\delta}}u \\to D\\_{\\rho\\_{\\infty}}u=D^{s\\_\\infty}u \\quad \\text{in $L^p({\\mathbb{R}}^n;{\\mathbb{R}}^n)$ as ${\\delta}\\to \\infty$.}$$ \nIn light of (cf. also the proof of ), we deduce the estimate \n$$\\norm{D\\_{\\rho\\_{\\delta}}u-D\\_{\\rho\\_{\\infty}}u}\\_{L^p({{\\mathbb{R}}^{n}};{{\\mathbb{R}}^{n}})}\\leq C \\norm{u}\\_{W^{1,p}({{\\mathbb{R}}^{n}})}{\\bigl\\|(\\rho\\_{\\delta}-\\rho\\_\\infty)\\min\\{1,|\\cdot|^{-1}\\}\\bigr\\|}\\_{L^1({{\\mathbb{R}}^{n}})},$$ \n for all *u* ∈ *W*1, *p*(R*n*), and the statement follows via Lemma [le:l1convergence].\nCompactness uniformly in *δ* ∈ (1/*ɛ*, ∞)\n-----------------------------------------\nNext, we address the issue of compactness with the goal of deriving a counterpart of Lemma [le:compactness] in the setting of diverging horizon. This relies on the following analogue of the Poincaré-type inequality in Corollary [cor:compactness]. The proof is based on the comparison of the scaled nonlocal gradients *D**ρ**δ* with a suitable finite-horizon fractional gradient.\n[prop:poincare2] There exists a constant *C* = *C*(*ρ*, *n*, Ω, *p*) > 0 such that \n$$\\norm{u}\\_{H^{\\sigma,p}({\\mathbb{R}}^n)} \\leq C\\norm{D\\_{\\rho\\_\\delta}u}\\_{L^p(\\Omega\\_{\\delta};{\\mathbb{R}}^n)} \\quad \\text{for all $u \\in H^{\\rho,p,{\\delta}}\\_0(\\Omega)$ and $\\delta \\in (1/{\\varepsilon},\\infty)$.}$$ \nConsider *D*1*σ* : = *D**ρ*1*σ* induced by the kernel function\n \n$$\\begin{aligned}\n\\label{eq:rho1sigma}\n \\rho\\_1^\\sigma= \\frac{w}{| \\cdot |^{n+\\sigma-1}},\n \\end{aligned}$$ \nwhere *w* ∈ *C**c*∞(R*n*) is a non-negative radially decreasing function with *w*(0) > 0 and supp*w* = *B*1(0); note that *ρ*1*σ* falls into the setting of Example [ex:kernels]a) with *s* = *σ* and recall that *σ* is the parameter appearing in the hypothesis (H3) on *ρ*.\nThen, by there is a constant *C* > 0 such that \n$$\\norm{u}\\_{H^{\\sigma,p}({\\mathbb{R}}^n)} \\leq C\\norm{D^\\sigma\\_1 u}\\_{L^p(\\Omega\\_1;{\\mathbb{R}}^n)} \\quad \\text{for all $u \\in H^{\\rho^\\sigma\\_1,p}\\_0(\\Omega)$}.$$ \n The remaining proof shows that there is a constant *C* > 0 independent of *δ* such that\n \n$$\\begin{aligned}\n\\label{eq:estimates\\_graddelta}\n \\norm{D^\\sigma\\_1 \\varphi}\\_{L^p({\\mathbb{R}}^n;{\\mathbb{R}}^n)} \\leq C\\norm{D\\_{\\rho\\_{\\delta}}\\varphi}\\_{L^p({\\mathbb{R}}^n;{\\mathbb{R}}^n)} \\quad \\text{for all $\\varphi \\in C\\_c^{\\infty}({\\mathbb{R}}^n)$ and ${\\delta}\\in (1/{\\varepsilon},\\infty)$,}\n \\end{aligned}$$ \nfrom which the claim follows after a density argument.\nLet us define *m**δ* : R*n* \\ {0} → R as \n$$m\\_\\delta(\\xi) := \\frac{\\widehat{Q}\\_{\\rho^\\sigma\\_1}(\\xi)}{\\widehat{Q}\\_{\\rho\\_{\\delta}}(\\xi)},$$ \n and observe that, in light of , \n$$\\widehat{D^\\sigma\\_1 \\varphi}= m\\_\\delta \\widehat{D\\_{\\rho\\_{\\delta}}\\varphi} \\qquad \\text{for $\\varphi\\in C\\_c^\\infty({\\mathbb{R}}^n)$}.$$ \n The estimate follows then directly from Fourier multiplier theory, once *m**δ* is confirmed to satisfy the Mihlin-Hörmander condition with uniform constants, that is,\n \n$$\\label{eq:mihlin2}\n \\abs{\\partial^{\\alpha} m\\_\\delta(\\xi)} \\leq C \\abs{\\xi}^{-\\abs{\\alpha}}$$ \nfor all *α* ∈ N0*n* with $\\abs{\\alpha} \\leq n/2+1$ and *C* > 0 a constant independent of *δ*.\nTo this aim, observe that implies for $\\xi \\not =0$ that\n \n$$\\begin{aligned}\n {\\left|\\partial^{\\alpha} \\widehat{Q}\\_{\\rho\\_{\\delta}}(\\xi)\\right|} &={\\left|\\overline{\\rho}(1/{\\delta})^{-1}{\\delta}^{n+\\abs{\\alpha}}\\partial^{\\alpha}\\widehat{Q}\\_\\rho({\\delta}\\xi)\\right|}\\\\\n &\\leq C \\overline{\\rho}(1/{\\delta})^{-1}{\\delta}^{n+\\abs{\\alpha}}\\abs{{\\delta}\\xi}^{-\\abs{\\alpha}}{\\left|\\widehat{Q}\\_\\rho({\\delta}\\xi)\\right|}\n =C\\abs{\\xi}^{-\\abs{\\alpha}}{\\left|\\widehat{Q}\\_{\\rho\\_{\\delta}}(\\xi)\\right|},\n \\end{aligned}$$ \nwith *C* > 0 independent of *δ*. Since the same holds for *Q̂**ρ*1*σ*, we deduce via the Leibniz and quotient rule that \n$$\\abs{\\partial^{\\alpha} m\\_{\\delta}(\\xi)} \\leq C \\abs{\\xi}^{-\\abs{\\alpha}} \\abs{m\\_{\\delta}(\\xi)} \\quad \\text{for all $\\xi \\not =0$}.$$ \nTherefore, it remains to verify for *α* = 0, which corresponds to showing that *m**δ* is bounded independent of *δ*. For simpler notation, we write $\\langle \\xi \\rangle:=\\sqrt{1+|\\xi|^2}$ for *ξ* ∈ R*n*. Since the estimate in Lemma [lem:bms] along with allows us to deduce \n$$\\widehat{Q}\\_{\\rho^{\\sigma}\\_1}(\\xi) \\leq C{\\left\\langle \\xi \\right\\rangle}^{\\sigma-1} \\quad \\text{for all $\\xi \\in {\\mathbb{R}}^n$,}$$ \n the proof of for *α* = 0 can be reduced to verifying that\n \n$$\\begin{aligned}\n\\label{eq:estFourierQrho}\n \\widehat{Q}\\_{\\rho\\_{\\delta}}(\\xi) \\geq C{\\left\\langle \\xi \\right\\rangle}^{\\sigma-1} \\quad \\text{for all $\\xi \\in {\\mathbb{R}}^n$.}\n \\end{aligned}$$ \nLet us first consider $\\abs{\\xi} \\leq 1/({\\delta}{\\varepsilon})$. Then, in view of ,\n \n$$\\begin{aligned}\n\\label{eq:estQrhodelta11}\n \\widehat{Q}\\_{\\rho\\_{\\delta}}(\\xi) \\geq \\overline{\\rho}(1/{\\delta})^{-1}{\\delta}^n \\textstyle \\min\\_{\\overline{B\\_{1/{\\varepsilon}}(0)}} \\widehat{Q}\\_{\\rho} \\geq C\\geq C{\\left\\langle \\xi \\right\\rangle}^{\\sigma-1},\n \\end{aligned}$$ \nwhere *C* is independent of *δ* because $\\overline{\\rho}(1/{\\delta})^{-1}{\\delta}^n \\to \\infty$ as *δ* → ∞ by [itm:upper]. For the case $\\abs{\\xi} > 1/({\\delta}{\\varepsilon})$, we use Lemma [lem:bms] and Lemma [le:wedge] to infer\n \n$$\\begin{aligned}\n\\label{eq:estQrhodelta12}\n \\begin{split}\n \\widehat{Q}\\_{\\rho\\_{\\delta}}(\\xi) &\\geq C\\overline{\\rho}(1/{\\delta})^{-1}{\\delta}^n\\abs{{\\delta}\\xi}^{-n}\\overline{\\rho}(1/\\abs{{\\delta}\\xi})\\\\\n &\\geq C \\abs{\\xi}^{-n}\\min\\bigl\\{\\abs{\\xi}^{n+\\sigma-1},\\abs{\\xi}^{n+\\gamma-1}\\bigr\\}\\\\\n &=C \\min\\bigl\\{\\abs{\\xi}^{\\sigma-1},\\abs{\\xi}^{\\gamma-1}\\bigr\\}\\geq C{\\left\\langle \\xi \\right\\rangle}^{\\sigma-1}.\n \\end{split}\n \\end{aligned}$$ \nFinally, together with gives, and thus, . This finishes the proof in light of the Mihlin-Hörmander theorem (see e.g. ).\nWhile the previous proof is built on, we mention that a statement parallel to Theorem [th:comparison] cannot be expected to hold for an unbounded parameter range of *δ*. Indeed, this is due to the fact that the singular behavior of *ρ*∞ and *ρ**δ* at the origin may be different, as one can see, for instance, from the two kernels in Example [ex:sbar]b); they feature a stronger and weaker singularity than *ρ*∞, respectively.\nBy combining Proposition [prop:poincare2] and Proposition [prop:convergenceinfinite], we can now deduce the following compactness statement.\n[Convergent subsequences for diverging horizon][le:compactness2] Let (*δ**j*)*j* ⊂ (1/*ɛ*, ∞) be a sequence with *δ**j* → ∞ and suppose that *u**j* ∈ *H*0*ρ*, *p*, *δ**j*(Ω) for each *j* ∈ N with \n$$\\sup\\_{j \\in {\\mathbb{N}}} \\norm{D\\_{\\rho\\_{{\\delta}\\_j}} u\\_j}\\_{L^p(\\Omega\\_{{\\delta}\\_j};{\\mathbb{R}}^n)} < \\infty.$$ \n Then, there is a *u* ∈ *H*0*s*∞, *p*(Ω), such that, up to a non-relabeled subsequence, \n$$u\\_j \\to u \\quad \\text{in $L^p({\\mathbb{R}}^n)$} \\quad \\text{and} \\quad D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j {\\rightharpoonup}D^{s\\_\\infty} u \\quad \\text{in $L^p({\\mathbb{R}}^n;{\\mathbb{R}}^n)$ as $j \\to \\infty$.}$$ \n Moreover, for every *η* > 0 it holds that \n$$D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j \\to D^{s\\_\\infty} u \\quad \\text{in $L^p((\\Omega\\_\\eta)^c;{\\mathbb{R}}^n)$ as $j \\to \\infty$.}$$ \nWe infer from Proposition [prop:poincare2] that (*u**j*)*j* is a bounded sequence in *H**σ*, *p*(R*n*). Then, the compact embedding $H^{\\sigma,p}({\\mathbb{R}}^n)\\hookrightarrow\\hookrightarrow L\\_{{\\mathrm{loc}}}^p({\\mathbb{R}}^n)$ (see ) together with the fact that each *u**j* is supported in $\\overline{\\Omega}$ yields the existence of a *u* ∈ *L**p*(R*n*) with *u* = 0 a.e. in Ω*c* such that, up to a non-relabeled subsequence,\n \n$$\\begin{aligned}\n\\label{eq:convergenceuj}\nu\\_j \\to u \\qquad\\text{ in $L^p({\\mathbb{R}}^n)$ as $j\\to \\infty$. }\\end{aligned}$$ \nAfter selecting a potential further subsequence, we find that $D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j {\\rightharpoonup}V$ in *L**p*(R*n*; R*n*) for some *V* ∈ *L**p*(R*n*; R*n*). One can compute that\n \n$$\\begin{aligned}\n\\int\\_{{\\mathbb{R}}^n} V\\,{\\varphi}\\,dx &= \\lim\\_{j \\to\\infty}\\int\\_{{\\mathbb{R}}^n} D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j\\,{\\varphi}\\,dx\\\\\n&= - \\lim\\_{j \\to \\infty}\\int\\_{{\\mathbb{R}}^n} u\\_j\\,\\operatorname{div}\\_{\\rho\\_{{\\delta}\\_j}}{\\varphi}\\,dx=-\\int\\_{{\\mathbb{R}}^n} u\\,\\operatorname{div}^{s\\_\\infty}{\\varphi}\\,dx,\\end{aligned}$$ \nfor any *φ* ∈ *C**c*∞(R*n*; R*n*), where the last equality employs Proposition [prop:convergenceinfinite] adapted to the nonlocal divergence. This allows us conclude that *V* = *D**s*∞*u* and *u* ∈ *H*0*s*∞, *p*(Ω).\nTo prove the second part of the statement, we exploit that the nonlocal gradients on (Ω*η*)*c* can be expressed as a convolution. Precisely, let us define \n$$d\\_{\\delta}(z):=-\\frac{z\\rho\\_{\\delta}(z)}{\\abs{z}^2} \\quad \\text{and} \\quad d\\_\\infty(z):=-\\frac{z\\rho\\_\\infty(z)}{\\abs{z}^2} \\quad \\text{for $z \\in {\\mathbb{R}}^n\\setminus \\{0\\}$.}$$ \n For *φ* ∈ *C**c*∞(Ω), we can compute in view of the radiality of *ρ* that for any *x* ∈ (Ω*η*)*c*,\n \n$$\\begin{aligned}\n\\label{eq:Drhod}\nD\\_{\\rho\\_{\\delta}}{\\varphi}(x) = \\int\\_{{\\mathbb{R}}^n} -\\frac{{\\varphi}(y)}{\\abs{x-y}}\\frac{x-y}{\\abs{x-y}}\\rho\\_{\\delta}(x-y)\\,dy = (\\mathbbm{1}\\_{B\\_{\\eta}(0)^c}d\\_{\\delta}) \\* {\\varphi}(x);\\end{aligned}$$ \nsince $\\mathbbm{1}\\_{B\\_{\\eta}(0)^c}d\\_{\\delta}\\in L^1({\\mathbb{R}}^n)$, the identity can be extended via density to all *u* ∈ *H*0*ρ*, *p*, *δ*(Ω). In the same way, there is an analogous representation when considering the kernels *ρ*∞, that is,\n \n$$\\begin{aligned}\nD^{s\\_\\infty} u = D\\_{\\rho\\_\\infty} u = (\\mathbbm{1}\\_{B\\_{\\eta}(0)^c}d\\_\\infty) \\ast u \\qquad \\text{on $(\\Omega\\_\\eta)^c$} \\end{aligned}$$ \nfor *u* ∈ *H*0*s*∞, *p*(Ω) = *H*0*ρ*∞, *p*(Ω).\nFurthermore, we observe that Lemma [le:l1convergence] induces the convergence\n \n$$\\begin{aligned}\n\\mathbbm{1}\\_{B\\_{\\eta}(0)^c}d\\_{\\delta}\\to \\mathbbm{1}\\_{B\\_{\\eta}(0)^c}d\\_\\infty \\qquad \\text{ in $L^1({\\mathbb{R}}^n)$ as ${\\delta}\\to \\infty$.}\\end{aligned}$$ \nThis allows us to conclude by Young’s convolution inequality and that \n$$\\norm{D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j - D^{s\\_\\infty} u}\\_{L^p((\\Omega\\_\\eta)^c;{\\mathbb{R}}^n)}=\\norm{(\\mathbbm{1}\\_{B\\_{\\eta}(0)^c}d\\_{\\delta})\\*u\\_j - (\\mathbbm{1}\\_{B\\_{\\eta}(0)^c}d\\_\\infty)\\*u}\\_{L^p((\\Omega\\_\\eta)^c;{\\mathbb{R}}^n)} \\to 0 \\ \\text{as ${\\delta}\\to 0$}.$$ \nΓ-convergence *δ* → ∞\n---------------------\nBased on the technical foundations provided in the previous sections, we are now in the position to prove the Γ-convergence for diverging horizon. We consider for *δ* ∈ (1/*ɛ*, ∞) the functionals F*δ* : *L**p*(R*n*; R*m*) → R∞ : = R ∪ {∞} given by\n \n$$\\label{eq:Fcald2}\n{\\mathcal{F}}\\_{\\delta}(u):=\\begin{cases}\n\\displaystyle\\int\\_{\\Omega\\_{\\delta}} f(x,D\\_{\\rho\\_{\\delta}} u)\\,dx &\\text{for $u \\in H^{\\rho,p,{\\delta}}\\_0(\\Omega;{\\mathbb{R}}^m)$},\\\\\n\\infty &\\text{else,}\n\\end{cases}$$ \nwhere *f* : R*n* × R*m* × *n* → R is a suitable Carathéodory integrand and *ρ**δ* is the scaled version of the kernel *ρ*, cf. As made precise in Theorem [th:infty] below, the limiting object for *δ* → ∞ is the functional F∞ : *L**p*(R*n*; R*m*) → R∞,\n \n$$\\label{eq:Fcalinf}\n{\\mathcal{F}}\\_\\infty(u):={\\mathcal{F}}^{s\\_\\infty}(u):=\\begin{cases}\n\\displaystyle\\int\\_{{\\mathbb{R}}^n} f(x,D^{s\\_\\infty} u)\\,dx &\\text{for $u \\in H^{s\\_\\infty,p}\\_0(\\Omega;{\\mathbb{R}}^m)$},\\\\\n\\infty &\\text{else;}\n\\end{cases}$$ \nThe fractional parameter *s*∞ is here related to the kernel *ρ* via lim*δ* → ∞*ρ**δ* = *ρ**s*∞, see , Lemma [le: fractional kernel as limit], and also Remark [rem:rhobars].\n[Γ-convergence for diverging horizon][th:infty] Let *f* : R*n* × R*m* × *n* → R be a Carathéodory integrand such that \n$$c\\abs{A}^p - a(x)\\leq f(x,A) \\leq a(x)+C\\abs{A}^p \\quad \\text{for a.e.~$x \\in {\\mathbb{R}}^n$ and all $A \\in {\\mathbb{R}^{m \\times n}}$}$$ \n with *c*, *C* > 0 and *a* ∈ *L*1(R*n*). If *f*(*x*, ⋅ ) is quasiconvex for a.e. *x* ∈ Ω, then the family (F*δ*)*δ* ∈ (1/*ɛ*, ∞) in Γ-converges with respect to *L**p*(R*n*; R*m*)-convergence to the functional F∞ in as *δ* → ∞, that is, \nΓ(*L**p*)-lim*δ* → ∞F*δ* = F∞. \n Additionally, the sequence (F*δ*)*δ* is equi-coercive with respect to convergence in *L**p*(R*n*; R*m*).\nLet (*δ**j*)*j* ⊂ (1/*ɛ*, ∞) be a sequence converging to ∞ as *j* → ∞.\n*Equi-coercivity:* From Proposition [prop:poincare2] and the lower bound on *f*, we deduce that \n$${\\mathcal{F}}\\_{{\\delta}\\_j}(u)\\geq C \\norm{u}\\_{H^{\\sigma,p}({\\mathbb{R}}^n)} - \\norm{a}\\_{L^1({\\mathbb{R}}^n)}$$ \n for all *j* ∈ N and *u* ∈ *H*0*ρ*, *p*, *δ**j*(Ω; R*m*). The embedding now immediately gives the stated equi-coercivity.\n*Liminf-inequality:* Consider a sequence (*u**j*)*j* ⊂ *L**p*(R*n*; R*m*) with *u**j* → *u* in *L**p*(R*n*; R*m*) satisfying, without loss of generality, \nliminf*j* → ∞F*δ**j*(*u**j*) = lim*j* → ∞F*δ**j*(*u**j*) < ∞. \n Then, *u**j* ∈ *H*0*ρ*, *p*, *δ**j*(Ω; R*m*) for each *j* ∈ N and $\\sup\\_{j \\in {\\mathbb{N}}}\\norm{D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j}\\_{L^p(\\Omega\\_{{\\delta}\\_j};{\\mathbb{R}}^m)}< \\infty$, so that Lemma [le:compactness2] yields that *u* lies in *H*0*s*∞, *p*(Ω; R*m*) with\n \n$$\\begin{aligned}\n\\label{eq:convDrhoj}\n D\\_{\\rho\\_{{\\delta}\\_j}}u\\_j {\\rightharpoonup}D^{s\\_\\infty} u\\qquad \\text{ in $L^p({\\mathbb{R}}^n;{\\mathbb{R}^{m \\times n}})$ as $j\\to \\infty$.}\n \\end{aligned}$$ \nSimilarly to the proof of Theorem"},"data_id":{"kind":"string","value":"arxiv_0002054"}}},{"rowIdx":498,"cells":{"content":{"kind":"string","value":"The Schrödinger Cat: Physics, Myth, and Philosophy\n==================================================\nThe most discussed of “live” metaphors in physical sciences is that of the Schrödinger cat. Introduced in the early, but mature days of the new quantum theory, in 1935, the parable of the cat has provoked and enlivened debates on the meaning of the quantum theory, on the ontology of the quantum wavefunction, on the puzzle of the collapse of the wavefunction, on the meaning of quantum measurement, and on the boundary between the quantum world and the classical world. In this article, I will discuss these issues of the quantum theory focussing on various aspects of the metaphor of the Schrödinger cat.\n> \n> This is a slightly revised version of my article written in 2005, and published in the compiled volume, ‘History of Science and Philosophy of Science’, p. 313-338 (Ed. P. K. Sengupta, Centre for Studies in Civilization, PHISPC, Pearson-Longman, Delhi, 2010). It was also published in *Sandhaan*, Vol. V, No.2, 1-52 (2005), the journal of the Centre for the Studies in Civilizations, New Delhi).\n> \n> \n> \nPreface\n=======\nIn 1935, Erwin Schrödinger was inspired to think deeply about the quantum mechanical description of two systems that interacted and then separated, by the discussion in the Einstein-Podolsky-Rosen (EPR) paper that questioned the completeness of standard quantum mechanics. In several papers he discussed the central and universal trait of ‘entanglement’ in quantum mechanics. He realized that interactions that form correlations ‘entangle’ the individual descriptions, resulting in only a joint and less detailed description of the physical state, and that the individual wavefunctions ceased to exist. He wrote,\n> \n> When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (or *ψ*-functions) have become entangled.\n> \n> \n> \nAn observation on one system then disentangles the description (*ψ*-function) and restores individual realities, in the quantum mechanical sense, in terms of individual wavefunctions. *One of these systems could be macroscopic*, in terms of size, mass etc., and then new conceptual issues arise. He discussed a theory of quantum measurement based on these insights. He demonstrated that the quantum uncertainty was not a blurring of physical variables, confined to the microscopic domain of atomic systems; the possibility of an interaction expanded the domain of uncertainty and fuzziness to the macro-world. In these studies arose the ever-living fable of the life of a cat that got entangled with the randomly timed radioactive decay of an atom. The graphic scenario exemplified since then the central tenets of the problem of ‘quantum measurement’, as well as the troubling notion of a quantum-classical divide. The ‘Schrödinger cat’ (also referred to as “Schrödinger’s cat”) challenged the universal applicability of quantum mechanics and its ability to represent physical states faithfully, further supporting Einstein’s views that were expressed in the EPR paper. Indirectly, it raised new questions about the role of the conscious observer in quantum mechanics. With life mixed with physics and quantum mysteries, it generated large popular interest. Philosophers discussed the nature and meaning of physical reality in quantum mechanics and its relation to the observer, in the domain well beyond the microscopic world of quantum mechanics.\nThere were also intellectual compromises: the superposition and interference of the quantum states of a single ‘large enough’ system being confused as the realization of a quantum state of the Schrödinger cat is common. The perennial confusion between ‘a cat in a superposition of its non-commensurate physical states’, and the very different idea of ‘an entangled state of a Schrödinger cat’ is serious and widespread. Some of these myths are here to stay, being ingrained in the collective psyche by repeated statements.\nThe foundational problems raised in the early years of quantum mechanics, especially the problem of quantum measurement, remain open. The aim of this article is to explore the genuine issues in physics and philosophy that are prompted by Schrödinger’s entangled cat.\nThe Quantum Theory\n==================\nThe years between 1924 and 1927 saw accelerated revolutionary changes in the understanding of the microscopic atomic physics. The old quantum theory that slowly developed in the work of Planck, Einstein, Bohr and others saw a sudden change in its physical and mathematical understanding and formalism. One line of thought starting with Louis de Broglie’s proposal that particles have wave nature matured in Erwin Schrödinger succeeding in writing a mathematically and physically consistent “wave equation” for the entity possessing the wave-particle duality. With this equation, soon named the Schrödinger equation, accurate calculations of quantized energies and spectral lines in atoms could be made. Also, the equation enabled calculations of many other phenomena pertaining to microscopic physics. In a parallel development, Werner Heisenberg wrote down quantum rules for calculating these quantities based on a mathematical formalism in which physical quantities were represented by matrices – arrays of complex numbers. An important mathematical property of these matrices were that they were non-commuting; the order of their multiplication was important and reversing the order gave a different value for their product in general. This property accounted for the wave-particle duality in the Schrödinger formalism. Soon, Schrödinger showed that the two formalisms were equivalent, and the mathematical foundations of the new quantum theory were established. In 1927, the important work by Paul Dirac extended the quantum formalism and linked it with the mathematics of classical mechanics, and he also wrote down the quantum wave equation for the electron, consistent with the theory of relativity. Of crucial importance were the contributions by Max Born and John von Neumann, for the physical interpretation of the mathematical elements of the theory.\nOne important aspect of the Heisenberg rules is that some combinations of physical quantities (‘observables’) cannot be determined accurately simultaneously. Multiple observations yield values with a statistical dispersion, and the product of the dispersion in the values of repeated measurements of two such observables on similar systems have a minimum universal value, given by the Planck’s constant ℏ; this is called the uncertainty relation. For example, the position and the momentum of a particle are such a pair. It is important to note that the uncertainty relation in quantum mechanics is defined only for repeated measurements in a statistical ensemble, and never for a single particle or a physical system. On the other hand, the uncertainty relation can be applied to a single system for *estimating the expected uncertainty in the physical quantities*, when the physical state is known.\nIn the present usage both the Schrödinger and Heisenberg approaches are practised, though the fundamental equation of quantum motion is usually written down in the Schrödinger form, as a differential equation. The Schrödinger equation describes how the wavefunction (or the *ψ*-function) evolves in time under the action of external interactions. The wavefunction itself is an unobservable, but all observables of the physical system are related to it by certain statistical relations. Usually, solving the equation for the specific problems gives rise to quantized discrete energy states, as well as some un-quantized states. The uncertainty relation follows in this approach from the properties of the wavefunction, and its relation to the various observables.\nA crucial piece of interpretational discovery was made by M. Born regarding the Schrödinger wavefunction, and this discovery provided the most essential and at the same time most problematic aspect of the quantum theory; Born realized that the connection between the wavefunction and results of a measurement or observation was essentially statistical. Thus, the square of the wavefunction provides a probability density for various observables associated with the system. It became clear after Born’s work that quantum mechanics had no predictive power for single events. It is a theory of statistical predictions of the properties of a microscopic system, like an atom. This turn in the meaning of the theory, which had a deterministic differential equation as its basic equation, came as an unpleasant shock to the pioneers like Einstein and Schrödinger.\nWhile it is possible to think of the wavefunction for a single particle as somehow a wave associated with the particle, as envisaged by de Broglie originally, this picture breaks down when one considers several particles. The wave that one is referring to is not a real wave in space, propagating as time progresses. It is a wave in the sense of an abstract mathematical function that admitted the properties of ‘superposition’ and ‘interference’, but one that merely represents the potentialities of measurement results that can be obtained from the physical system. In other words, the wavefunction itself is not an observable. Then ‘Born’s rule’ associates the square of the wavefunction to the probability of the realization of a particular result being obtained. The quantum theory has only statistical predictions and the wavefunction provides the mathematical link between the physical system and these probabilities. Right at this point it should have been clear that one should not physically associate the wavefunction and the quantum system, and *one should not see the physical system or the particle as an entity possessing the dual property of a wave and the particle*.\nThe Uncomfortable Quantum Rules\n===============================\nThe basic idea that is operative in the new mechanics is that the representations of two distinct physical states of a system can be added linearly in various ways to get a new valid physical state of the system. This never happens in the familiar world of particles around us. For example, a familiar system that exhibits just two discrete possible physical states is a coin that is tossed. It can either turn a `Head’ or a `Tail’. This happens randomly. But there is no concept of a physical state that is a sum or difference of these two states in a mathematically literal sense. However, when one considers a physical wave (on water for example) it makes perfect sense to think of the state of a new wave that is a sum or difference of two other waves – wherever the crests add together, the new wave just gets amplified and when a crest and trough coincide the result is the absence of a wave. So waves can cancel or augment each other, and their sum or difference is also a physical wave. But if a coin, the states of which are not waves, were to be described quantum mechanically, then a state that is a sum or difference of `Heads’ and `Tails’ is also valid and one needs to imagine what that might mean. At the same time, when an observation is made to determine the state of the coin, one and only one of these two possibilities turn up, and that too randomly. *It is this conceptual wave-particle duality that quantum mechanics embodies, and this duality is unfamiliar to our experience, even though each aspect separately is very familiar*. It is this uncomfortable consequence of the supposed applicability of quantum mechanics to the everyday physical world that Schrödinger tried to express dramatically in his discussion of the quantum states of a cat. His analysis revealed an even more puzzling aspect of the superposition of the wavefunctions, which he called the ‘entanglement’.\nPhysicists who took the wave-particle duality of matter seriously in the sense of associating both a particle nature and also a wave nature to a single entity, thought of microscopic particles as some fuzzy (blurred, or distributed) entity. Schrödinger had spent a great deal of effort in trying to understand the quantum fuzziness and its implication to the larger world. Does the uncertain fuzziness suggested by the uncertainty principle and the mathematical shape of the wavefunction of an electron or an atom become sharply defined matter, without any fuzziness, in the macroscopic world? He discussed the following “burlesque” (translated as ‘ridiculous’) example to bring out the strange implications associated with the mathematical machinery of the new quantum theory \n> \n> One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following diabolical devise (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small that *perhaps* in the course of 1 hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat is still alive if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The wavefunction of the entire system would express this by having in it the living and the dead cat (pardon my expression) mixed or smeared out in equal parts.\n> \n> \n> It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be *resolved* by direct observation. That prevents us from so naively accepting as valid a “blurred model” for representing reality.\n> \n> \n> \nThis example, now called ‘the Schrödinger cat’, is the central theme of our discussion. Before we discuss the various aspects of the Schrödinger cat, it might be useful to have a brief look at the mathematical machinery that forms the basis of all the discussions related to this example.\nThe Mathematics and Physics of Linear Superposition\n===================================================\nThe basic mathematical machinery of quantum mechanics is simpler than that of the older classical physics. The feature that is most important is that of the linear superposition, familiar from the mathematics of vectors and physics of waves. In quantum mechanics, the physical state of a particle has a meaning only after a measurement. But its mathematical state can be represented by a wavefunction, or more generally as a vector in an abstract space. By ‘vector’ we mean an abstract entity with various components along different directions, representing physical quantities like velocity, force etc. The difference in quantum mechanics is that these components are complex numbers in general and the components do not refer to spatial directions; the ‘space’ in which the vectors represent the physical system is an abstract mathematical space, which could even have infinite dimensions. The fundamental premise of the quantum theory is that if A and B are two legitimate vectorial representations of possible physical states of a particle, then their linear addition, *a**A* + *b**B*, is also a legitimate state of the particle. The vectors are added after arbitrarily stretching or contracting them by complex number scaling factors ‘a’ and ‘b’, with the restriction ∣*a*∣2 + ∣*b*∣2 = 1. Then the Born rule says that a measurement will reveal the property A with the probability ∣*a*∣2 = *a**a*\\* and B with the probability ∣*b*∣2.\n![The double-slit experiment. An interference pattern (regions where there are particles separated by regions where there are no particles detected) is observed with light, atoms and electrons. The natural explanation is in terms of waves that split and superpose. But the same physical behaviour is observed even when the particles are sent one by one forcing us to attribute a wave with even single particles.](Fig1.jpg \"fig:\") [fig:fig1]\nFirst we mention a physical example that illustrates how superpositions become necessary. In the familiar example of a double-slit interference experiment, light falling on a screen with two slits forms an interference pattern with bright and dark fringes on a detector screen far away (Figure 1). If light is treated as waves, there is an immediate explanation in terms of the familiar interference patterns involving waves, like waves on water. It is possible to do the experiment such that the source of light is a single atom or a small number of atoms, and then light is received at the screens as discrete photons, much like particles arriving, well separated in time (as ‘seen’ by an electronic detector), and not as a continuous wave. Yet, the final pattern of interference obtainable on a long exposure camera is identical to the one obtained by the instantaneous wave description. *The only logical possibility is to think of each photon being associated with a wave that interferes independent of when and where the next photon arrives*. In such a picture, the state of the photon after the slits has to be very different from our usual description of the state of a particle with a single unique trajectory. Somehow one has to account for the waves that will split and propagate through both the slits and then interfere even though we have only one indivisible particle to deal with (figure [fig:fig1]). That is where the concept of superposition becomes necessary. The state of the photon is described as containing two parts, one corresponding to the propagation through the upper slit (U), and another through the lower slit (D), in equal proportions. These two parts can then interfere and represent the observed pattern. This does not imply that the particle itself splits at the slits and propagates in the two paths simultaneously – the associated waves do that job, if one needs to form a graphical representation. In fact, the particle cannot be in both slits in any manner without grossly violating all conservation principles in physics. So, the state of the particle after the slits is *represented* as a wavefunction,\n \n∣*S**t**a**t**e*⟩ ∼ ∣*W**a**v**e*-*U*⟩ + ∣*W**a**v**e*-*D*⟩ \nSince the crest of the upper wave can have an arbitrary shift relative to the crest of the lower in general (due to an extra piece of glass near the upper slit, for example), a ‘phase factor’ could also be added after the ‘+’ in this expression. (This can even make the ` + ` into ‘ − ’, corresponding to the trough instead of the crest at a particular position). All such general expressions are called linear superpositions. The wavefunctions contain complex number quantities to accommodate the needs of the representation. What is observable is related to the square of the wavefunction, ∣∣*S**t**a**t**e*⟩∣2, which will involve the ‘interference term’ consisting of the bilinear product 2∣*W**a**v**e*-*U*⟩ ⋅ ∣*W**a**v**e*-*D*⟩\\*, agreeing with the empirical observations accurately. It turns out that one can completely forget about the picture of a physical wave, and replace it with that of the concept of a ‘vector’ that can be added and subtracted with appropriate complex factors. In fact, modern quantum mechanics is done in the language of these ‘state vectors’, which are equivalent to the representation with the wavefunctions.\nThe basic idea is that the *representation* of the state of a physical system is by means of a vector in an abstract space. For our purpose this vector is very similar to other familiar vectors, a directed arrow pointing in some direction in space, whose length is agreed to be one unit. (See figure [fig:fig2]).\n![The concept of a vector and projections on different basis. The thick arrow is the state vector, and the light arrows along the two directions are the projections. On the right is the same vector projected on a different basis.](Fig2.jpg \"fig:\") [fig:fig2]\nWe can choose some coordinate system in space, and work with just two axes for clarity; x-axis and y-axis (called the ‘basis’), which can be any two perpendicular directions for convenience (though I refer to ‘space’ here, the vectors of quantum mechanics are in an abstract space of any dimensions). Some immediate obvious facts about this construction can be noted: 1) the length of the vector does not depend on either rotating it, or on rotating the coordinate system, 2) the projections of the vector on the two axes depend on the rotation of the vector or the coordinate system, and the sum of the squares of the projections is, by the Pythagoras theorem, just the square of the length of the vector, 3) the vector itself can be represented as the vectorial sum of its projections, *V⃗* = *V**x**x̂* + *V**y**ŷ*; this is a linear superposition. A general linear superposition corresponds to scaling up or down each vector and then adding them with the correct directions.\n![The meaning of a linear superposition – the vector along the diagonal is a superposition, a sum, of the other two orthogonal (perpendicular) vectors, one stretched more than the other by a scaling factor. On the right is the superposition with a negative sign for the second vector.](Fig3.jpg \"fig:\") [fig:fig3]\nIn classical mechanics several quantities have been described by vectors, like velocity, force etc. Given a vector quantity for a classical system we can talk about values of its projections on various directions, and we think of the physical system as possessing all these values simultaneously as its physical properties, irrespective of what measurements we make on the system as long as the measurement itself has not changed the properties. In quantum mechanics, the state vector does represent the properties of the system, but in a way very different from what one is used to in classical mechanics. The essential concept is the following. The state vector represents the physical state and the basis that we choose defines the property one wants to observe. It is sufficient to illustrate this with a simple example.\nIf ∣*s*⟩ represents the state vector relevant for the observation of the angular momentum of a particle, then one has to choose a basis corresponding to the components of the angular momentum in various directions in space. This aspect is similar to the case in classical mechanics where if *S⃗* is the angular momentum vector for a physical system, then the angular momentum in a direction at some angle is given by the projection of the vector *S⃗* along that direction. *In quantum mechanics the projection of the state vector along a direction does not correspond to the value of the angular momentum in that direction*. *Indeed the value is usually quantized and random in each measurement*; irrespective of what the projection is, the value measured is one of the possible discrete values, at random, say + 1 or − 1 in relevant units. But *the square of the length of the projection can be associated with the probability of realizing a particular value of angular momentum when measured in that direction*. This appearance of the notion of probabilities makes the quantum projections very different from the classical vector projection. In classical physics, the projection of a particular vector quantity along some direction is the component of that quantity itself, along that direction. Given the original vector, one can say with certainty that a particular value will be reproducibly obtained if projected along some direction. For observations on a quantum system, the projection is not directly related to results of individual observations. If the original state vector has unit length corresponding to a measured quantized value *s*, and if the projection has a length of 1/2, one can say that there is a probability of (1/2)2 = 1/4, that a measurement in that direction yields the value *s* for the projected angular momentum. So there is a shift from ‘value’ to ‘probability’.\nFor example, suppose the physical system (say, an electron) is prepared by some means with the value of its angular momentum along a particular direction equal to +1, in units of ℏ/2. It turns out that for this system, the values of the projections of the angular momentum in *any direction* are always either + 1 or − 1, and nothing else. (There are other particles where the values could be + 1, 0 and − 1). This of course is very strange, but we will not try to find a ‘reason’ for this strange behaviour since there is no possibility of succeeding in explaining such behaviour with the familiar reasoning. Mathematically we will represent the original state vector as ∣*s*⟩. Now suppose that the direction in which the system was prepared with the definite value + 1 is along a direction that is symmetrical relative to the directions x and y, somewhat like the diagonal in figure 2 (the mathematically correct representation of this example is slightly different from the graphical example in the figure, but for our purpose the present discussion will suffice). Subsequently we want to measure the angular momentum in the directions x or y, shown as the horizontal and the vertical axes in the figure. Geometrically, the projections of the original state vector on either of these axes are equal to $1/\\sqrt{2}$. Therefore, if a measurement of the angular momentum is done along the x-axis, say, then we will either get + 1 with a *probability 1/2*, which is the square of the projection, or − 1 with a *probability 1/2*, like when tossing a coin.\nTwo important clarifications are necessary. The fact that one gets a particular value in a measurement, for example + 1 along the x direction, does not mean that it had that value just before that measurement. Obviously, we had prepared the system with its angular momentum +1 along the diagonal. Yet, in a measurement along the horizontal direction, we got a value + 1 (or − 1). After the measurement, the angular momentum projection in the horizontal direction remains + 1, and any further measurement in the same direction gives + 1 with certainty. Therefore, by the Born rule, the state vector’s projection along the horizontal should now be equal to + 1, so that its square corresponds to the probability 1, or ‘certainty’. The revised description is now with a new state vector along x, with the y-component ceasing to exist – this is called the ‘collapse of the state’, or the ‘collapse of the wavefunction’. The state vector has changed suddenly on a measurement, and the vertical component has gone to zero after the measurement resulted in a definite value in the orthogonal direction. Mathematically, the situation is similar for the wavefunction corresponding to the particle passing through a screen with two slits (labelled + 1 and − 1). A wavefunction with the symmetrical superposition of equal components ‘collapses’ to a single component ( + 1 and − 1) randomly, when an observation is attempted. If we had associated the two-component wavefunction physically with the particle (as one often does with the waves of the wave-particle duality), then the vanishing of one component has the implications of the vanishing of some physical entity from space, spontaneously and instantaneously during the measurement. This is what makes the collapse of the wavefunction mysterious and troublesome.\nThe Collapse of the Wavefunction and Other Disasters\n====================================================\nThe ‘collapse of the wavefunction’ is undoubtedly the most discussed and the most dreaded consequence of the mathematical machinery of quantum mechanics, as interpreted today. Valid physical states of a system are routinely described by a general sum of several distinguishable, and mutually incompatible states (if one occurs physically, the other is excluded, like the ‘Heads’ and ‘Tails’ states). A multi-component wavefunction with the observable possibilities *S*1, *S*2 and *S*3, for example, can be represented as the linear superposition\n \n$$\\label{key}\n\\left\\vert \\psi\\right\\rangle =a\\left\\vert S1\\right\\rangle +b\\left\\vert\nS2\\right\\rangle+c\\left\\vert S3\\right\\rangle$$ \nA measurement on a single physical system turns up just one of the possibilities contained in the superposition, statistically, and with a probability related to the length of each vector contributing to the superposition. For instance, a measurement might result in the outcome *S*2 (with the probability ∣*c*∣2). But as soon as a single measurement is completed, the superposition of several states changes into just one of the states, corresponding to the value realized in the measurement. In this case, the wavefunction becomes\n \n$$\\label{key}\n \\left\\vert \\psi'\\right\\rangle =\\left\\vert\n S2\\right\\rangle$$ \nAll other components vanish. If one had attributed a physical (material) reality in space and time to the components present in the superposition, then these elements had to physically vanish when a measurement is made. It is as if a physical state that was distributed and spread out into different physical states suddenly condenses into one state, when observed. In the example of the double-slit experiment, one can make an observation near one of the slits to answer the query, ‘through which slit is the particle passing?’ If the detector registers the particle, then the particle could not be passing though the other slit. If it did not register anything, then the particle is certainly passing through the other slit. Either way, the path is determined. Having detected the path with certainty, there is no room for probabilities any more, and then the wave function or state vector component in the other path should vanish to zero immediately. Otherwise there will be inconsistency with the Born’s rule which says that the probability of detection is related to the amplitude of the wave. In the case of the classical coins, just before the coin lands in the final state, its state is alternating between the two possibilities and it is no surprise or paradox for us if it finally lands randomly in one of the states. But the quantum superposition is very different – as it evolves, it is not alternating between the two states. It remains in a superposition which is neither of the two definite possibilities at any instant. Then an observation reduces it to one of the states. In fact, there are examples where one of the states involved in the superposition is physically far away from the other state. Then a measurement annihilates one of the states even when this state is physically far away from the place where the observation is made! This is a (mild) example of ‘nonlocality’ in quantum physics.\nFrom Superposition to Entanglement\n==================================\nSuperposition is a simple concept when one thinks in terms of vectors or waves adding up on each other. Some superpositions are special in quantum mechanics, and they lead to consequences that have shaken up the very foundations of our familiar notions of causality and locality. Consider two systems, each with its own quantum states (formally called ‘kets’) $\\left|s1\\right>$ and $\\left|s2\\right>$. Each of these could be a simple superposition of the two states ∣*u*⟩ and ∣*d*⟩, which are the two observable states of the system. Suppose these superpositions for each individual particle are $\\left|s1\\right>=\\frac{1}{\\sqrt{2}}(\\left|u1\\right>+\\left|d1\\right>)$ and $\\left|s2\\right>=\\frac{1}{\\sqrt{2}}(\\left|u2\\right>+\\left|d2\\right>)$. For example, the labels might correspond to a particle spinning clockwise (u) or anticlockwise (d) when looked at from a particular direction. Since there are two particles, one can also study the ‘joint state’ of the total system, which is written as a juxtaposition of the states, $\\left|s1\\right>\\left|s1\\right>$. The joint state would be written as\n \n$$\\begin{aligned}\n \\left|s1\\right>\\left|s2\\right>=\\left(\\frac{1}{\\sqrt{2}}(\\left|u1\\right>+\\left|d1\\right>)\\right)\\left(\\frac{1}{\\sqrt{2}}(\\left|u2\\right>+\\left|d2\\right>)\\right)\\\\\n =\\frac{1}{2}\\left(\\left|u1\\right>\\left|u2\\right>+\\left|u1\\right>\\left|d2\\right>+\\left|d1\\right>\\left|u2\\right>+\\left|d1\\right>\\left|d2\\right>\\right)\\end{aligned}$$ \nIn this expression the first ket ∣*s*1⟩ stands for the first particle and the second one ∣*s*2⟩ for the second. Each is in a state of superposition and the joint state is a simple juxtaposition of all the possibilities of the superposition. There are totally four possibilities and each one will happen with a probability (1/2)2 = 25%. This is a general superposition, and it is a ‘product state’ since it can be written as a product of two factors, $\\left|s1\\right>$ and $\\left|s1\\right>$, each of which is a sum (superposition) of two factors. The important point to note is that each particle is said to have its own definite state, $\\left|s1\\right>$ and $\\left|s2\\right>$. The joint state is ‘separable’ into $\\left|s1\\right>$ and $\\left|s1\\right>$. Now suppose we put some restriction on the possible states – say, when one particle is up the other should always be down (this might come from the requirement of some conservation principle, like that total spin is zero). This could happen due to some brief interaction between the particles. Then two of the four possibilities are prohibited, and the joint state would be\n \n$$\\label{key}\n \\left|S=0\\right>=\\frac{1}{\\sqrt{2}}\\left(\\left|u1\\right>\\left|d2\\right>\\right)+\\frac{1}{\\sqrt{2}}\\left(\\left|d1\\right>\\left|u2\\right>\\right)$$ \nNow there are only two possibilities for the joint state, each with a probability of 50%. *This cannot be written as a product of the individual states $\\left|s1\\right>$ and $\\left|s2\\right>$. The joint state $\\left|S=0\\right>$, which is also just a superposition of two product states, is not ‘separable’ anymore*. This is called an ‘entangled state’ with very peculiar properties. The outcome of a measurement on each particle is entangled with that of the other; ∣*u*1⟩ with ∣*d*2⟩ and ∣*d*1⟩ with ∣*u*2⟩. *Since it is not a simple juxtaposition of the two individual states, it turns out that each of the particles does not have any physical state by itself, in the quantum theory!* The independent quantum status of each particle is lost and there is only a joint state. Neither particle is in state ‘up’ or ‘down’, nor are they in any superposition of ‘up’ and ‘down’ states. However, the most general individual state possible in the quantum theory can only be some general superposition of the ‘u’ and ‘d’ states. Here, neither particle is in such a state. All one can say is that if one is found in ‘u’ state, the other will be found in ‘d’ state and vice versa. A measurement will get one of these two joint possibilities, each with a 50% probability. If a measurement is made, the entangled state collapses into one of the two joint states, $\\left|u1\\right>\\left|d2\\right>$ or $\\left|u1\\right>\\left|d2\\right>$. Once this happens, each particle has a definite state and the joint state is a simple juxtaposition of the individual states. The individual states and separability are regained only after a measurement on at least one particle. So, a particle that had no description in terms of a definite state before the measurement acquires a definite state as a result of measurement. Moreover, only a measurement on one of the particles is sufficient – the other particle acquires a definite state automatically without any measurement or observation on it! For example, if the measurement shows that the first particle is in the state $\\left|d\\right>$, then definitely the joint state is $\\left|d1\\right>\\left|u2\\right>$, and not $\\left|u1\\right>\\left|d2\\right>$. This means that *the second particle, on which no observation was made, is now certainly in the state $\\left|u\\right>$ after the measurement on the first particle, though neither had any specific state whatsoever before the measurement on the first particle*.\nTherefore, the quantum entanglement in the state makes the properties of the particles inseparable, and one gets definite properties that are emergent only after an observation. *The bizarre consequence is when the two particles are not in the same spatial region*. The particles can be separated after the interaction that made the entangled state. Then a measurement on one particle instantaneously gives it a definite state *and also a definite state to the other particle that is far away from the first particle*! This is the much discussed violation of locality in quantum mechanics. On the other hand, if we assume that one measurement should not influence the physical state of the other particle that is spatially separated, then we see that quantum mechanics has no description of such independent states. Quantum mechanics corresponds to an incomplete description because its states can be affected by an arbitrarily distant measurement, whereas the requirement of locality is that a physical state in what ever form should not be influenced instantaneously by processes that are spatially distinct and separated. It was Einstein who pointed this severe difficulty in the well known Einstein-Podolsky-Rosen (EPR) paper in 1935. Schrödinger’s papers on the properties of such joint states, entanglement, and its crucial place in the problem of measurement were partly inspired by the EPR paper.\nThe Schrödinger Cat\n===================\nWe now come back to the description of the Schrödinger cat. In his paper that summarized the “Present state of quantum mechanics” he wrote,\n> \n> One can even set up quite ridiculous cases... The wavefunction of the entire system would express this by having in it the living and the dead cat (pardon my expression) mixed or smeared out in equal parts.\n> \n> \n> \nThe colloquial expression “the living and the dead cat mixed or smeared out in equal parts” is the source of a serious confusion to those who have not seen Schrödinger’s precise discussion in a follow up paper.\nThere are descriptions of the general physical situation implied here without any mention of a live body, and yet completely equivalent to it from the point of view of the mathematical theory of quantum mechanics. I mention one such by Einstein, written in 1948 :\n> \n> As far as I know, it was E. Schrödinger who first called attention... Rather than considering a system which comprises only a radioactive atom (and its process of transformation), one considers a system which includes also the means for ascertaining the radioactive transformation – for example, a Geiger-counter with automatic registration-mechanism. Let this latter include a registration-strip, moved by a clockwork, upon which a mark is made by tripping the counter. True, from the point of view of quantum mechanics this total system is very complex and its configuration space is of very high dimension. But there is in principle no objection to treating this entire system from the standpoint of quantum mechanics... The reason for the introduction of the system supplemented by the registration-mechanism lies in the following. The location of the mark on the registration-strip is a fact which belongs entirely within the sphere of macroscopic concepts, in contradistinction to the instant of disintegration of a single atom.\n> \n> \n> \nThe way Einstein describes the situation stresses the mathematical equivalence of the example of the Schrödinger cat, and the quantum measurement problem that involves a macroscopic ‘apparatus’. It is possible to represent systems like the Schrödinger cat mathematically in a straightforward way; the initial state of the system while the cat is being placed in the box along with the undecayed atoms is $\\left|S\\_i\\right>=\\left|A+\\right>\\left|CL\\right>$ where we denoted ‘*A* + ’ for the undecayed atom, and ‘CL’ for the live cat. This is a product state. We have written the total state as simply the state of the atom AND the state of the cat together. If there is definite knowledge that the atom has decayed, then of course the cat is dead soon. So, given the knowledge that the atom has decayed, the joint state is $\\left|S\\_f\\right>=\\left|A-\\right>\\left|CD\\right>$, where ‘*A* − ’ stands for the decayed atom. The peculiarity of quantum physics is that if there is no definite knowledge of the state, when the box is closed with the cat, atoms and the poison inside, the state of the system can be some (time dependent) linear superposition of these two possible joint states,\n \n$$\\label{key}\n \\left|S\\_t\\right>=a(t)\\left|A+\\right>\\left|CL\\right>+b(t)\\left|A-\\right>\\left|CD\\right>$$ \nNow, what does this mean? Unlike the other two states ($\\left|S\\_i\\right>$ and $\\left|S\\_f\\right>$), one cannot say whether the atom has decayed or not (or equivalently, whether there is a mark on a paper strip after the Geiger counter). Correspondingly, one cannot say whether the cat is dead or not. It will be incorrect to say (in the quantum theory) that the atom is either decayed or not and the cat is either alive or dead. That is not what the state says. Most importantly, the atom is not in a superposition of its two possible states, and the cat is not in any superposition of the states ‘alive’ and ‘dead’! But, there are no other individual states in the quantum mechanical description – we have exhausted all possibilities. The joint state clearly says that the state is a superposition of the two joint possibilities, and that neither the atom nor the cat is in any definite state. *In particular, it does not say that the atom is both decayed and not, or that the cat is both dead and live*. (It is this incorrect description, *in which the quantum cat is in a superposed state of both alive and dead*, that is often debated as the ‘paradox of the Schrödinger cat’!)\nWe are talking about the state or the wavefunction representing the state of the system in some abstract space. The only definite thing quantum mechanics says at this point is that IF a measurement is performed (opening the box), one will either see the state $\\left|A+\\right>\\left|CL\\right>$ (the atom undecayed AND the cat alive), OR the state $\\left|A-\\right>\\left|CD\\right>$ (the atom decayed AND the cat dead), corresponding to the definite states $\\left|S\\_i\\right>$ and $\\left|S\\_f\\right>$. It is very important to note that the neither the atom nor the cat are in states of superposition separately. *The cat is definitely not in the state $\\alpha \\left|CL\\right>+\\beta \\left|CD\\right>$. The cat has no state at all*! So hasn’t the atom. But together they have some quantum state. And that state has no implication on what the state of the cat or the atom is until a measurement is made. Therefore, typical statements, commonly made by almost everybody, like `the cat is both dead and alive’, or the `cat is in a superposition of being dead and alive’ etc. are wrong. They completely miss the meaning of the simple entangled state in quantum mechanics. It is unfortunate that such serious misrepresentation of a fundamental point is made in spite of Schrödinger’s clear description (italics added),\n> \n> When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, *then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own*. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (or *ψ*-functions) have become entangled.\n> \n> \n> \nCommon Misconceptions\n=====================\nI would like to summarize the common misconceptions about the Schrödinger’s cat parable before proceeding further. Even professional physicists often tend to loosely say, either because they assume that the meaning is understood or because some of them genuinely lack clarity, that the superposed state means the cat is both dead and alive. I quote from some recent technical papers:\n“...an unfortunate cat is placed in a quantum superposition of being dead and alive… This situation defies our sense of reality because we only observe live or dead cats, and we expect that cats are either alive or dead independent of our observation”.\n“Applying quantum theory to our macroscopic world, Schrödinger proposed a thought experiment with an unfortunate cat lingering in the twilight zone between life and death…”\n“The paradox arises because the atom, being a microscopic object, must be described by quantum mechanics. After one hour, and before it is observed, the atom is in an equal superposition of being decayed and undecayed. However, if quantum mechanics is a universal and complete theory, it must describe the whole system. And, since the state of the cat is correlated with the state of the atom, the cat must also be in a superposition of being dead and alive. This clearly contradicts our everyday experience of cats! This sort of apparent contradiction arises whenever the state of macroscopic objects is correlated with that of microscopic objects. It comes about because in our experience of our macroscopic surroundings, objects are either in one state or another, but never in a superposition of several states at the same time.”\n“…the quantum state must ultimately involve such a complex number superposition of a dead cat and a live cat: so the cat is both dead and alive at the same time!” …“Of course such a situation is an absurdity for the behaviour of a cat sized object in the actual physical world as we experience it”.\nAny statement which asserts that the cat is both dead and alive at the same time in the entangled state (or even in a simple superposition) is the misconception that is shared by a large number of people. The confusion is at two levels: 1) the central point about entanglement that neither system has an individual quantum mechanical state whatsoever is missed, 2) even the notion of superposition in standard quantum mechanics or in quantum mechanical experiments known so far never implied that the physical entity itself carried both the properties represented by the state at the same time. This needs to be clarified once and for all.\nIn standard quantum mechanics, if an electron has its spin pointing in the up direction, it can be described also as an equal superposition of states corresponding to ‘right’ and ‘left’ in any horizontal direction. If a measurement is made on the electron with an apparatus sensitive to the spin direction in the horizontal direction, then one gets ‘left’ or ‘right’ with equal chance, though originally it was prepared in the state ‘up’. In fact, after getting ‘right’ or ‘left’, if an attempt is made to measure it in the vertical direction, one ends up getting ‘up’ or ‘down’ with equal chance! In any case, one does not say that the particle with spin ‘up’ is both in the ‘left’ and ‘right’’ direction at the same time. It is in the ‘up’ state. And *nobody has any empirical experience of seeing an electron with its spin both pointing ‘left’ and right’, in spite of its microscopic status!* What is seen empirically is that an ensemble of electrons contains both ‘left’ and ’right’ pointing electrons. This example is mathematically and physically similar to that of any superposition in quantum mechanics.\nThe relevant question then is why isn’t there any physical state in the macroscopic world that is a *superposition of states* corresponding to being alive and being dead. That would be some new state of ‘being’ that has equal projection to the state representing ‘being dead’ and ‘being live’. But that still does not mean that some organism in that state of being is both dead and alive. Indeed, nobody has seen an atom both decayed and undecayed. Nobody has seen an electron at two different places at the same time. The superposition one is always referring to is the superposition of the wavefunctions, or the state vectors, representing the two distinguishable states of the quantum entity. The connection between the quantum entity (whether it is an atom or a cat) and the wavefunctions representing the various states of being is not a physical identity or equivalence. When an observation is made ``to see’, only one of the observable possibilities are seen, with the probabilities corresponding to the magnitude of the wavefunction for each state making up the total wavefunction. Theoretically it may be correct to say that the cat cannot be described or represented as either dead or alive, but it is certainly incorrect to say that the cat is neither dead nor alive, or to say that the cat is both dead and alive. The total wavefunction in superposition is often interpreted as meaning both the above statements, and clearly, from a logical point of view, this lack of uniqueness goes against such interpretations.\nPressing a bit further, one might want to ask the question during some point of time in the experiment, “is the cat alive or dead inside the box?” Quantum mechanics has no way of answering this question. *In fact QM never gives a definite answer to a question that is asked about the exact physical state of a single physical entity*. It just gives the probabilities of the cat being found dead or alive when the box is opened. *In fact, quantum mechanics has nothing to say about any individual cat*. The predictions of QM are statistical, and if the experiment is performed repeatedly, the fraction of cats that are dead and the fraction that are alive will match well with the squares of the corresponding wavefunctions.\nSo, Schrödinger was not discussing a paradox of quantum physics. He was just discussing a ridiculous example that needed the correct interpretation. If one associated the wavefunction, as a physical equivalence, to the quantum entity, then one has to say that the quantum entity itself is in a superposition and therefore could also be in a superposition of being dead and alive at the same time. Therefore, this ridiculous example warns us from wrongly interpreting the wavefunction. As Schrödinger himself said, “It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be *resolved* by direct observation. That prevents us from so naively accepting as valid a “blurred model” for representing reality.” One need not bring up the example of the cat for being sufficiently warned. When an atom passes through a screen with two closely spaced slits, the quantum wavefunction after the slits is very similar to the superposed wavefunction we can write for any macroscopic object. But this does not mean that the atom itself got split at the slits, or that there are atoms in both paths after the slits – that would violate a whole lot of known physics, including the fundamental conservation laws. So, while we do not know what is split into two at the slits (it is called the wavefunction!) what we know for sure is that the atom itself is not split. Same logic applies to the Schrödinger’s cat.\nWhy is the Cat Special?\n=======================\nSchrödinger chose to describe the issues that involve a linear superposition of states representing life and death. This was perhaps the burlesque aspect he refereed to, because it is otherwise common and natural in quantum mechanics to have exactly this kind of entangled superpositions whenever two systems interact.\nThis makes the original Schrödinger’s cat very different from the ‘cat states’ described in most of the current discussions, though by and large physicists have ignored this difference. The cat had two special aspects- it is really big on the scale of atoms, and it is a living being. We also think it has a reasonably developed intelligent brain, thought and awareness. Each of these aspects make the original Schrödinger cat state different from the states experimentalists deal with while discussing the Schrödinger cat problem. It is a routine context in experimental and theoretical quantum mechanics to come across entangled states. Consider an atom entering a small cavity in which there is no light. The atom enters in an excited state (*A* + ). While passing through the cavity the atom has some probability to decay (*A* − ) and deposit a photon in the cavity (*C* + ). It also has a similar probability not to decay during the passage (*C* − ). Now this situation is very similar to the case of the Schrödinger’s cat, except that we are talking about a microscopic change in the state of the macroscopic cavity (one photon deposited or not). The corresponding mathematical states are $\\left|A+\\right>\\left|C-\\right>$ and $\\left|A-\\right>\\left|C+\\right>$. The entangled state is simply the superposition of these two joint states. Again, the atom or the cavity does not have a specific state, and neither of them is in a superposition of any specific states. But the two together are in a superposition of possibilities of togetherness. One may use this inanimate, soul-less, consciousness-less state as a comparison to the original Schrödinger’s cat state. First of all, though the cavity itself might be macroscopic, there is only a microscopic change between its two possible states, differing by one photon. *The live and dead states of an animal differ in so many aspects that it cannot be considered as a microscopic change*. Then, there is the aspect of the irreversible relation between life and death. While it is natural and relatively easy to evolve from a state of being live to a state of being dead, the reverse evolution is never seen. So, in the preparation of the Schrödinger cat state, one cannot start with a dead cat and then hope to get a superposition of states corresponding to being alive and dead. But in the case of atom and the cavity, one can start with a cavity with one photon and an atom in the ground state (the second state above) and can have a superposition with a state in which the cavity has zero photon and the atom is in the excited state. This essential irreversibility is unique to the Schrödinger’s cat state. On the other hand, I should state that technically such irreversibility can be mimicked by using a bad cavity where the photon cannot be held for sufficiently long time.\nThe third difference is perhaps the most significant from the point of view of the interpretation of quantum mechanics. Whether or not the observer outside the closed box has decided to make the observation, the conscious animal inside will ‘know’ when it is dying, and when the glass vial with poison has broken. As far as the animal is concerned, though totally unaware of Hilbert spaces and quantum superpositions, or even of Schrödinger, its own state or the state of the glass vial containing the poison is known with certainty. If one insists on an abstract representation, the state is either one or the other, and never a superposition, just as for an observer sitting inside the box and watching carefully, there is no quantum superposition. One may propose that for the observer outside who is forced to use the quantum formalism, nothing has changed if a secret observer or the cat himself is making the observations inside the box – neither the mathematical representation nor the physical results in measurements could change, whereas the description of the system is completely different for the observer inside and for the observer outside. This then would show that the wavefunction is not a property related to the state of the physical system – it merely represents the unobserved possibilities and *subjective information* in a way consistent with various statistical probabilities inherent in the system. But this view turns out to be incorrect. If there is any means, hidden or known, of breaking the superposition – be it a secret observer inside the box, be it the consciousness of the cat, or be it a purely inanimate surveillance system that registers the state of the cat – then the superposition is broken and the system is reduced to a definite state. As soon as the ‘information’ is accessed physically from the quantum system once, the wavefunction changes and no subjective element is left in the description. We will now discuss this important aspect in some more detail.\nThe Schrödinger Cat and the Quantum Measurement Problem\n=======================================================\nThe observation of a physical quantity in a system involves a subtle conceptual issue, which becomes a core physical issue of quantum mechanics itself, when fully formulated. Every observation or ‘measurement’ needs an ‘apparatus’, which we usually designate as the experimental set up. This is invariably a macroscopic physical system, but could also include the live ‘observer’. In any case, let us focus first on the microscopic quantum system and the measuring apparatus. *The process of the measurement involves a physical interaction between the system and the apparatus*. We take for granted that the apparatus is a ‘classical object’, playing its role of extracting the information on the observable quantity, without entering in the theoretical machinery of quantum mechanics. But such an assumption amounts to admitting that quantum mechanics is not a universal theory, and that it is applicable only to microscopic system. Then, one is asserting that a separate and distinct theory describes macroscopic objects like the apparatus. On the other hand, if we claim that quantum mechanics is the universal theory applicable to the dynamics of all objects, then there are serious consequences for which no solution has been found. This is the quantum measurement problem.\nIt was Schrödinger who first discussed the core issue. In the same paper that discussed the concept of entanglement, he wrote,\n> \n> By the interaction the two representatives (or *ψ*-functions) have become entangled. To disentangle them we must gather further information by experiment, although we knew as much as anybody could possibly know about all that happened. Of either system, taken separately, all previous knowledge may be entirely lost, leaving us but one privilege: to restrict the experiments to one only of the two systems. After reestablishing one representative by observation, the other one can be inferred simultaneously. In what follows the whole of this procedure will be called the *disentanglement*. Its sinister importance is due to its being involved in every measuring process and therefore forming the basis of the quantum theory of measurement, threatening us thereby with at least a *regressus in infinitum*, since it will be noticed that the procedure itself involves measurement.\n> \n> \n> \nThus, every measurement involves the interaction of the system and an apparatus, each of which will be described by its own *ψ*-function, before the interaction. This will be some general superposition of the possible physical states, in each case. However, the interaction results in the entangled state, exactly of the same nature as the joint state of the atom and the cat in the example of the Schrödinger cat. This will be an entangled state of the quantum system ‘s’ and an apparatus ‘A’,\n \n$$\\label{key}\n \\left|\\psi\\right>=a\\left|s+\\right>\\left|A+\\right>+b\\left|s-\\right>\\left|A-\\right>$$ \nThen, neither the system nor the apparatus has any definite physical states of its own! There is only a joint state, with less information, as Schrödinger noticed. The problem is even more severe. To determine the state of the apparatus, which is the measurement, the observer ‘O’ has to access the ‘reading’, which involves an interaction of the observer with the apparatus. However, if the theory of quantum mechanics applies to everything including the observer, all that we will manage is the larger entanglement involving the system, the apparatus, and the observer!\n \n$$\\label{key}\n \\left|\\psi'\\right>=a\\left|s+\\right>\\left|A+\\right>\\left|O+\\right>+b\\left|s-\\right>\\left|A-\\right>\\left|O-\\right>$$ \nAdding more observers simply enlarges the entanglement. This is the *regressus in infinitum* Schrödinger referred to. There is no way to realize a final observation of a definite disentangled physical state, if quantum mechanics is the correct theory and if it is a universal theory. This is a devastating conclusion, and that is why the quantum measurement problem is treated as the foremost unsolved problem in the formulation of quantum mechanics.\nWigner’s Friend\n===============\nThe act of observation on a quantum particle that is described by a superposition of states reduces the superposition to just one of the terms in the superposition (in general the new state could be a different superposition, but here we will consider the case when there are only two possibilities and the measurement is designed to reduce the superposition to one of the states). When that happens, the system is described by a definite well defined state. This reduction from the general superposition to one of the component states after a measurement is called reduction of the wavefunction or more graphically, the collapse of the wavefunction. In the context of the Schrödinger’s cat, opening the box and looking into is the act of measurement and this collapses the wavefunction to one of the two possibilities. Let us say in a particular experiment the reduction took place to the state $\\left|CD\\right>$. Then the atom has definitely decayed and the cat is dead. But QM does not allow one to conclude that this was the state of the system even the smallest time interval before the box was opened. Before looking into the box, the system is described by the superposition of joint states and after opening it, the system will be described by one of the joint states. The interesting thing about the interpretation of QM is that if there is no superposition, one could claim without inconsistency that the system is in that particular definite physical state, and in this special case one can attribute a physical equivalence between the wavefunction (state) and the system without inconsistency.\nIn 1956, Eugene Wigner modified the Schrödinger parable to include more characters. Instead of an observer directly looking in, suppose a ‘friend’ is asked to keep a watch and then answer queries about the outcome in the box. Again one can think of the friend as a part of the system and formally write a wavefunction to include the friend. The outcome observed by the friend is correlated with some state of neurons in his brain, and one can imagine that the total wavefunction now is the tri-partite entangled state,\n \n$$\\label{key}\n \\left|S\\_t\\right>=a(t)\\left|A+\\right>\\left|F+\\right>\\left|CL\\right>+b(t)\\left|A-\\right>\\left|F-\\right>\\left|CD\\right>$$ \nFor the query by the observer, the friend answers Yes or No ( + or − ) and then the system collapses to one of the definite states.\nThe complication arises because the friend is supposed to have a conscious mind, and his observing the cat has the same role physically as the final observer looking at the cat. Therefore, if the wavefunction collapse is a physical process then it would have happened when the friend looked at the system, and the wavefunction written above is not valid. The friend is never to be described in a superposed state. His consciousness has completed the act of observation and his own ideas about the description of the system and himself are to be described even within quantum mechanics by definite states that are not in superposition. But as far as the outside observer is concerned, consistency demands ignoring the consciousness and writing the formal wavefunction; this gives results consistent with a limited set of experiments that asks only the probabilities regarding the two states ‘dead’ and ‘alive’.\nThe cat itself may have a conscious mind, being a living and highly evolved animal. Then writing a superposed wavefunction to describe a cat will be absurd. QM is supposed to be applicable to the entire world, irrespective of the size and weight of the objects, but examples like that of Wigner’s friend prompt us to think that there is perhaps a very important divide between macroscopic physics and microscopic physics. Perhaps there is a grey area as well, a transition region, where quantum mechanical description slowly merges into the classical physics description of particles and cats where superposition has no meaning.\nThere is a discussion of these issues by B. d’Espagnat, who explored the relation between physical reality and ‘mentality’. Essentially, he finds the possibility of relating the hidden variables in the de Broglie-Bohm version of the quantum theory to an ‘internal state of consciousness’, thereby bringing the consciousness of Wigner’s friend and the unobserved paths of the particles on a similar level of physical discussion. He makes these internal states of consciousness elements of physical reality, but distinguishes them from elements of empirical reality by stating that the internal state of consciousness is not communicable. These two notions of consciousness are then linked through mechanisms of physical interactions. A closer look at such approaches reveals that no new significant insight is achieved, and that in effect a different set of words is used in a description that remains conceptually and empirically similar to the older discussions.\nFrom the earlier discussion of entanglement, and the *regressus in infinitum* that Schrödinger mentioned, it is clear that Wigner’s introduction of a conscious being in the process of observation was rather naive. It amounts to the statement (by a physicist) that there are elements in physical nature that cannot be described within physical theories like quantum mechanics.\nIs Quantum Superposition Subjective and Information Based?\n==========================================================\nThe conscious cat is never in a superposition. But for the unaware observer the superposition is broken only after the observation. Consciousness enters the formalism, though there is no argument that can prove that it enters the physical process. The question arises whether the quantum formalism can survive up to conscious observers capable of first person self-observations of their own state. This has been already addressed in the example of Wigner’s friend. An examination of this issue in detail in not intended in this article. However, it is relevant to describe the essential ideas concisely.\nThe wavefunction of a physical system is intimately related to the physical configuration of the experimental apparatus that observes the physical system. For example, for a light beam that approaches a screen with slits, the subsequent superposition of states corresponding to the passage through each of the slits crucially depends on the properties of the slits. Therefore, to describe the quantum state of the particle, the conscious observer should know the experimental system well; otherwise the quantum state that he writes to describe the physical processes and observations could be incorrect and then the observations will not agree with the theoretical description. Now let us consider the situation where a second observer puts a detector near one of the slits, but does not detect a particle. Therefore, the particle has gone through the other slit, and for this observer the state of the particle at a distant screen is not a superposition of the states corresponding to the passage through each of the slits. As far he is concerned, the wavefunction corresponds to the passage through only one of the slits, and then there is no superposition. He concludes also that there will not be any interference at the screen. It is possible to do this kind of a measurement without destroying (absorbing) the particle itself, and therefore the observer who is unaware of this prior observation will not be able to say whether somebody is observing the particles or not by merely looking at the number of particles that reach the screen. Also, if the second observer makes observations to see through which path each particle arrives, he will get the 50-50 probability expected from the experimental situation, and as predicted by quantum mechanics itself. Transferred to the case of the cat, an observer sitting inside the box can see at each moment whether the cat is dead or alive, whereas the person outside cannot. Irrespective of this situation, the quantum mechanical prediction regarding the probability of observing it dead or alive comes out correctly, as calculated from the superposed state. At this point one might think that it is reasonable to associate the superposition with ‘information’ or ‘knowledge’. The observer inside the box has the information that the cat is either dead or alive, and therefore for him its description is never in a superposition. But for the observer outside, the information on the state of the cat is not available till an observation is made, and therefore he could write a superposition and predict from that the probability for seeing the cat alive or dead. If this were the case, then we could conclude that the wavefunction represents information as obtainable by a conscious being, and this might resolve a number of issues in the interpretation of quantum mechanics. Indeed, there are several new generation quantum physicists who think along these lines, especially those who work in areas related to quantum information and quantum computation. But this is an incorrect view. In the case of the double-slit experiment, the first observation that determined the path of each of the particle destroys the entire possibility of interference at the screen, irrespective of whether the observer who set up the screen to do the experiment has the knowledge of the state of the system. *If he writes a superposition for the state because of the lack of his knowledge of the first observation, he will find a contradiction with his quantum mechanical prediction.* No matter what he writes for the state, there will not be any physical interference pattern. That possibility is permanently destroyed after the first attempt at the determination of the path of the particles. In fact, the interference possibility is destroyed even if the first observer never looked at the information collected near the slits. The very positioning of the working detectors capable of detecting the path destroys all subsequent possibility of interference and hence the possibility of a coherent superposition is destroyed right after the first detectors. This example clearly shows that a coherent superposition has nothing to do with the information in the brain of a conscious observer.\nFor the case of the cat, therefore, if there were a possibility to write a superposition state corresponding to its physical state, any sort of experimental apparatus that can interact with the cat and determine with certainty its state of being will destroy the coherent superposition. However this does not alter the probability of observing it alive, just as the probability of detecting the particle near one of the slits remains 50% even for the second observer. Hence we conclude that observing it alive or dead is not the signature of a coherent quantum superposition. It is the possibility of an interference. This is related to the possibility of an observation of a state of being that is different from being alive and being dead. In a sense, every living being has the presence of death, among other things, inherent in life, and this is indeed the natural state. But we are discussing about formalising a quantum physical state that is described as superposition of the states representing life and death. Since it is impossible to elaborate on this further I will stop this part of the discussion here.\nMacroscopic Superpositions\n==========================\nAt this point a second kind of the Schrödinger cat should be mentioned, which is different from the original Schrödinger cat, but quantum theoretically related to it. In fact in many experimental situations physicists attempt to create this second kind of ‘cat’. It is our general notion that microscopic objects like atoms and electrons can be such that their physical states could be in a linear superposition. For example, the electron has a spin angular momentum, and the two possible states are the spin pointing up and the spin pointing down. But a superposition of these two states is also a valid state. The question naturally arises whether it is possible to have a linear superposition of the states of a macroscopic object: a cricket ball can be spinning in such a way that it is either in a state of ‘off-spin’ or in a state of ‘leg-spin’ – is it possible to have a superposition of these two states? Such a possibility will imply that the ball will not be spinning in either of these states till an attempt is made to decide which direction it is spinning – and the attempt will reveal with equal probability that it does the leg-spin or the off-spin. While spin bowlers might love to learn this quantum spin superposition, this does not seem possible. All our experience shows that macroscopic objects always are in definite states whether or not we are observing them. This is of course a belief, but there are reasons for holding such a belief. However, quantum mechanics, as a formalism, allows with complete generality a linear superposition of states, irrespective of the size of the object. In principle, the state of a cricket ball could be in a superposition of the leg-spin and the off-spin. The two important questions to be answered are a) is it possible to prepare (and observe) such states, b) if not, why is it impossible?\nQuantum physicists often call a superposition of states corresponding to a macroscopic object, like a ball, or even more modestly, a pearl, a Schrödinger cat state. There is no entanglement, the system is in some definite state, but it is also in a superposition of states when considered in some specific basis. An example could be a current flowing in a loop of wire, and the possible states are a clockwise current and an anti-clockwise current. A Schrödinger cat state in this situation will be a state corresponding to a superposition of states corresponding to a clockwise and an anti-clockwise currents (again, this does not mean that the currents are actually flowing clockwise and anticlockwise). Going one step further, one can ask the question whether the cat or any living being could be in a state of superposition of states corresponding to being alive and being dead. In other words, is the state $\\left|\\psi\\right>=a\\left|CL\\right>+b\\left|CD\\right> $ possible?\n![A two-state system with the possibility of quantum tunnelling between the states 1 and 2. If the system is initially prepared in the state 1, it has some probability to leak into the state 2, and the probability can oscillate back and forth. The quantum description of the state of the system is in terms of a superposition. Classically, if the system is prepared in the well of state 1, it remains there for ever.](Fig4 \"fig:\") [fig:fig4]\nIf a macroscopic familiar physical object has two possible distinguishable states labelled as 1 and 2 (Heads and Tails for a coin, for example) we will first assume that there is a quantum mechanical description of these states employing state vectors, and we call them $\\left|1\\right>$ and $\\left|2\\right>$. So far the correspondence is formal. But once the applicability of quantum mechanics is assumed for objects like coins, the state $a\\left|1\\right>+b\\left|2\\right>$ becomes legitimate. Classically this has no operational meaning. In quantum mechanics, this implies that when observations are made in repeated experiments the distinguishable states 1 and 2 will show up randomly with the probabilities *a*2 and *b*2. Only when *a* = *b*, the values will show up with equal probabilities of 1/2 each, as in the case of a normal coin. The quantum state implies an even more bizarre consequence; it implies that the coin or the two-state physical object can be prepared in special situations such that if we start with the state $\\left|1\\right>$, there is a nonzero probability that after some time it is in state $\\left|2\\right>$. This is called tunnelling. Indeed, the radioactive atom naturally is in a situation where if we start with the undecayed atom, its quantum state becomes a superposition of the states corresponding to decayed and undecayed atoms. The same physical picture should apply to (special) coins as well if they obey quantum mechanics. In the case of the atom, once the decay happens, the chance that the decay products will combine back to form the atom again is essentially zero. This is because the decay products are allowed to escape. One can imagine situations where the radioactive atom as well as the decay products are confined in space such that there is some chance that the atom is restored to its original state. When that is possible the behaviour of the system becomes oscillatory, in the sense that the decay is followed by a restoration of the atom and then again a decay, and so on.\nTherefore, *one experimental signature of being able to prepare the superposition of macroscopic states is the oscillations between two possible states* as a result of what is called the ‘macroscopic quantum coherence’. This is similar to certain phenomena involving superconductivity, like the Josephson effect. But there, what oscillates is the density of a large number of paired electrons. For a single particle, the quantity that oscillates is the probability. Can the quantum mechanical probability oscillate between two distinct states for a macroscopic system? In the case of the coins this will mean that the certainty of seeing the coin ‘Heads’ will slowly change to the certainty of seeing it ‘Tails’ and then back to Heads and so on, provided the state of the coin can tunnel between these two states somehow. Most of the experiments seeking to study the Schrödinger’s cat problem look at this kind of physical effects in macroscopic quantities, like a current.\nThere are two views that one can take at this point. One is to say that quantum mechanics applies to all objects, and that coins can certainly tunnel between two states. But since the coin is big on the atomic scale, its quantum mechanical parameters like the de Broglie wavelength are extremely small compared to its own size as well as to the size of all scales in the problem. Therefore, the probability of tunnelling is extremely small – so small that there is practically no chance that one will see this tunnelling over any reasonable time scales. Indeed a calculation using standard quantum rules shows that this explanation is certainly plausible. Then the problem is closed. Another view is that coins are fundamentally different from atoms, and they obey different rules of dynamics. There is a transition of rules from the microscopic world to the macroscopic world. The research on this possibility is considered very important, but the progress is slow.\nExperiments on the Schrödinger Cat\n==================================\nNow I will briefly review the experimental activity related to the discussions on the Schrödinger’s cat. A much discussed early experiment that dealt with a superposition at a scale larger than the mass scale of electrons or atoms was a double-slit experiment that was performed using the molecule Carbon-60 or the ‘Bucky-ball’. Each molecule contains 60 Carbon atoms and hence one can claim that its scale is much larger than what one is used to in quantum mechanical or optical double-slit experiments. An interference pattern was obtained in this experiment, as expected if quantum mechanics applied to molecules of this size. Of course, nobody was surprised at this result, and it is treated as an important step towards testing for superpositions on an even larger scale as in the case of double-slit interference experiments with molecules of DNA, or even ‘live’ systems like a virus. These states are called ‘mesoscopic’ Schrödinger’s cat states. Physicists retain the names despite scaling down enormously the physical system as well as the content of the concern expressed by Schrödinger. It should be obvious that a genuine Schrödinger cat state is an entangled state with another system, and a double-slit experiment with a simple superposition, *a*∣*ψ*1⟩ + *b*∣*ψ*2⟩, is not an experiment on a Schrödinger cat state.\nA notional analogue of the original entangled state that Schrödinger himself discussed has been created at the atomic scale, as a superposition of two states corresponding to two widely separated spatial positions of a single Beryllium ion in an electromagnetic trap. The ion oscillates in the trap by a distance of the order of 100 nanometres or so, large compared to the typical atomic size or the size of the quantum uncertainty in each of the position states. If a classical particle oscillates in such a trap, it will have some definite position at each instant, and the extreme positions in the trap are accessed periodically back and forth. When the particle is at one extreme it is not at the other. If we denote these two extreme positions as *x*1 and *x*2, and two other internal states of the ion as (up) and (down), the experimenters managed to put the ion into a state of superposition of states corresponding to the position *x*1 entangled with the (up) internal state of the ion and position *x*2 with the (down) state of the ion. Of course when a measurement is done, the ion would be found at only one of these positions, with the corresponding internal state. However, this is the entanglement of quantum states in a single quantum system (single ion), unlike the entanglement of states of two physically distinct systems (the atom and the cat) in Schrödinger’s example. Mathematically, the state is identical to the Schrödinger’s cat state, because it is an entangled state, though there is nothing ‘ridiculous’, or even macroscopic, about the one produced in the laboratory.\nThere is another class of experiments in which coherent oscillations between two possible and classically distinguishable states of millions of electrons in a superconducting device are monitored. The superposition corresponds to that of a clockwise and an anti-clockwise current of large number of electrons. Classically it is impossible to think of a current flowing in both directions at the same time in the same wire. Quantum mechanically, the state of the electrons can be a superposition of states corresponding to currents in opposite directions. There is no entanglement involving a large number of particles here. The experimental signature in these situations is the existence of a step in the energy levels of the system and this has been measured. Again, these experiments test for the applicability of quantum mechanical superposition principle when the physical system has a size (number of particles) much larger than what one is used to in a typical quantum mechanical experiment. Except for this size factor, the system is similar to other quantum systems like atoms or electrons and does not contain the subtleties of the original discussion by Schrödinger. Further, the *state of the original Schrödinger cat consists of one atom and one cat, and not a coherent ensemble of identically prepared atoms and cats*. There is a vast conceptual difference between the two situations.\nIrreversibility of Death and the Schrödinger Cat\n================================================\nThe examples of superpositions discussed above contained the possibility that due to quantum tunnelling, if the state is prepared in one of the states, it has a probability to evolve into the other state; eventually the system oscillates back and forth between the two possible states. As we discussed, *what oscillates back and forth regularly is the probability, and not the physical system itself*. A physical system with its state represented by the state vector in a superposition can only make quantum jumps between the two possible states, randomly. But, if a large number of similar physical systems are prepared in one of the states, after some time more and more physical systems start appearing in the other state and the number in the other state increases smoothly. Then the process reverses and keeps on oscillating. But this is possible only if the states are reversible – there should not be any fundamental limitation in reaching from one state to the other. While it is physically possible and indeed easy to go from a state of being alive to a state of being dead, the other possibility does not exist. Therefore, the superposed state of the Schrödinger’s cat differs subtly, in a fundamental sense, from the kind of superposed states people create in the laboratory.\nThe Quantum Zeno Effect: The Eternal Cat\n========================================\nThe quantum Zeno effect was highlighted by E. C. G. Sudarshan and B. Misra. The original Zeno (of Elea) paradox referred to the apparent impossibility of motion, and there is not a good logical way of presenting this paradox because, with our present evolved intuition on motion, the Zeno paradox does not appeal to the modern common sense. But the essential idea is that if one subdivides the motion into smaller and smaller parts indefinitely, the distance travelled at each instant approaches zero and then, there is no motion. This freezing of instantaneous position in motion is suggested as the freezing of the whole motion. But the argument is not valid for uniform motion since the time taken to move is proportionately smaller, and therefore the smallest divisions of space just takes the correspondingly smallest durations of time, with the same velocity, never freezing.\nHowever, consider some type of ‘motion’ which starts off sluggishly and then speeds up. So, in the initial durations there is hardly any movement and then the speed picks up and it starts moving uniformly, as in a traffic stop. If the traffic light stays green only for a short time, comparable to the time required to start moving even a short distance, the speed will never really pick up, and the time to move a given distance keeps increasing faster and faster, with a decreasing duration of the green signal. Finally when the light stays on for a duration smaller than the time to press the accelerator, the motion really freezes. This pseudo-Zeno effect happens because the motion is nonlinear in time and not uniform – starts off slowly and then picks up. This simple analogy, though not perfect, helps to understand the nature of the quantum Zeno effect. In quantum mechanics, a change of a state, or the quantum motion, is equivalent to a rotation of the state vector in an abstract space. A measurement is a projection of the vector along a chosen direction in its space. If the vector rotates by a very small amount and if a projection to its original direction (initial state) is taken, the length of the projection is almost the same as the length of the vector itself; the projection is different from the original vector only in the second order (square) in small quantities, and hence the difference is negligible. So, if the system is not allowed to evolve for sufficient time before a measurement, that is equivalent to a small rotation and a projection, and the projection is almost the same as the original state. If there are frequent small rotations and projections (measurements) then the projections remain close to the original vector, and the state really does not evolve. It freezes in the original state, and this is the quantum Zeno effect. This can happen in any frequent quantum measurement. Sometimes this is paraphrased by saying that the watched kettle never boils.\nWhat is the relevance of the quantum Zeno effect for the Schrödinger’s cat? If there are frequent observations made on the system, does the cat live to its natural life expectation? (of course, the Zeno effect is not expected to prolong the natural life expectancy of a living being, though this is a theoretical possibility if quantum mechanics applies also to the live macroscopic state!) A more important question is to ask whether the quantum Zeno freezing occurs if the observation is done on just any part of the quantum system that would indicate the change of state, or whether it has to be on the initial member of the causal chain leading to the cat’s death. Can one delay the cat’s death by watching the cat continuously, or does one have to make observations on the quantum mechanical atom for the quantum Zeno freezing to occur?\nThis problem has not been analyzed to my knowledge. But we can make some immediate common sense statements. The causal chain involved in the evolution of the state of the Schrödinger cat is well defined and the starting point of the avalanche is the decay of the atom. If the cat dies of some other causes while in the box, that has no causal implication to the state of the atom. So, by looking at the state of the cat it is not logically and causally necessary to conclude that the atom has decayed, whereas if the atom is found decayed then the cat is most likely dead, if the apparatus is working as indented. So, we can safely come to the technical conclusion without going into the machinery of quantum mechanics that the quantum Zeno freezing of the possibility of the cat’s death can happen only by the frequent observations on the state of the atom, and not by the frequent observations of the state of the cat.\nBut this conclusion seems to depend on the existence of a definite causal chain with time delays between various events leading to the cat’s demise. A strictly quantum mechanical analysis using quantum states and their superposition on the other hand shows that the system never evolves into an entangled superposition from the initial state $\\left|\\psi\\right>=\\left|A+\\right>\\left|CL\\right>$, in which the atom has not decayed and the cat is alive, if repeated and frequent observations are made on just the cat. The cat can only ‘jump’ from the state $\\left|CL\\right>$ to the state $\\left|CD\\right>$ without ever forming an entangled state with the atom! This contradiction with our earlier common sense conclusion happens because the quantum mechanical changes in the wavefunctions implied in the entangled state does not incorporate the causal time delays, and this is a deficiency of the way these states are written formally. This can be modified within the formalism of quantum mechanics and mutual consistency can be achieved in principle.\nThe Quantum Muddle\n==================\nKarl Popper had drawn attention to a serious problem in the interpretation of quantum mechanics. He accused quantum physicists of being confused by the ‘quantum muddle’. Briefly stated, the quantum muddle refers to physically associating the wavefunction with the physical system, as a physical equivalence. The wave-particle duality is one such association often used – physicists refer to a particle having a wave property and the wavefunction is associated with this physical wave property of the particle. Popper argued that this is the source of most of confusions and paradoxes of the quantum theory. He reasoned that when one talks about a probability distribution function applicable to a population, one means that every statistical property of the population can be derived using the probability function. The formulae for the calculation resemble the ones used in quantum mechanics. But nobody, rightly, physically equates the probability distribution function with the individuals making up the population. Nobody would say that distribution function is a physical property of the individual. Seeing the individual with specific properties – a particular value for height, weight, hair colour etc., does not change the probability distribution function itself, and one would not talk about a ‘collapse of the function’ when an observation of an individual results in a set of definite values of the properties of the individual.\nThe importance of Popper’s observation is the following: if a tight physical equivalence is not attributed to the correspondence between the wavefunction and the physical system, then the wavefunction is more of a tool for description. Even if a genuine ontology is associated with the wavefunction, if it has only a statistical role then we can stop attributing the simultaneous existence of contradictory properties in the same physical object (when the wavefunction splits at a double-slit, the particle does not). While the wavefunction of the Schrödinger’s cat is in a superposition of two states corresponding to its being alive and being dead, the cat itself is not in a physical state of being dead and being alive. This is fully consistent with the mathematics of quantum mechanics and its predictions, and it takes away the confusion and perhaps the popular surprise from the often quoted and debated paradoxes.\nEinstein had emphasized a similar point about quantum mechanics, especially in his later writings. In the essay ‘Physics and Reality’, Einstein wrote,\n> \n> The *ψ*-function does not in any way describe a condition which could be that of a single system; it relates rather to many systems, to ‘an ensemble of systems’ in the sense of statistical mechanics. If, except for certain special cases, the *ψ*-function furnishes only statistical data concerning measurable magnitudes, the reason lies not only in the fact that the operation of measuring introduces unknown elements, which can be grasped only statistically, but also because of the very fact that the *ψ*-function does not, in any sense, describe the condition of one single system. The Schrödinger equation determines the time variations which are experienced by the ensemble of systems which may exist with or without external action on the single system.\n> \n> \n> \nI have argued elsewhere that the extreme view taken by Popper – that one should dissociate the single physical system (like a particular atom of an ensemble of atoms) from the wavefunction that describes the quantum behaviour of the ensemble – is not consistent with quantum mechanical experiments. One can devise experimental situations from which one has to conclude that even at the single particle level ‘something’ splits into two at the double-slit arrangement in the experiment. It is not the particle, and it does not carry energy. But it holds and encodes the properties of the physical system including information on its energy, momentum etc. What is its physical reality is a question that needs to be formulated and addressed in future. Answering this question will also clarify almost all contentious issues of interpretations of quantum mechanics.[1](#fn1)\nThe Boundary Between the Classical and the Quantum World\n========================================================\nOne of the important issues that have been discussed extensively in the context of the Schrödinger’s cat is that of a boundary between the microscopic quantum world and the macroscopic classical world. Nobody knows whether there is such a boundary where the physicals laws are different. At present the understanding is that the physical laws are uniform across the two worlds, and they are the quantum mechanical rules, but the observed behaviour changes as one moves up to the microscopic world. Thus one does not see interferences of relatively large objects passing through small holes, and one does not see unexplainable randomness or correlations when one deals with macroscopic objects, whereas observations of microscopic atomic entities give results that are unfamiliar in the macroscopic world of common experience.\nThe question arises, how large or massive an object can be described by a linear superposition of its physical states? Quantum mechanics would suggest that there is no limit and any object in this Universe, and perhaps the Universe itself, should be described by a wavefunction that could be in some complex superposition. But we also need to explain adequately why we do not see routinely the manifestations of a description in terms of superposition, like coherent oscillations between various possible states, drastic uncertainties and unpredictability in measurement results etc. One of the important issues that is of relevance in the experiments on macroscopic systems is the need to isolate the system from its immediate environment or interactions of various kinds. Only then the quantum mechanical aspects become visible in the experiment. The loss of quantum mechanical superposition due to a large number of tiny interactions with the external environment is called ‘decoherence’. This is thought to be the fundamental reason why large objects familiar in the everyday world are not in any quantum mechanical superposition of states. Mathematically it is possible to show that the decoherence mechanisms do spoil *coherent superpositions*. *But the same mathematical analysis also implies that the system is still described by a superposition, albeit ‘incoherent’*. What this means is that the phase factor that appears in the addition of the two states fluctuates and does not have any fixed value, but it is still a superposition. For the case of the Schrödinger’s cat (‘superposition version’) an incoherent superposition results if the sign in between the states $\\left|CL\\right>$ and $\\left|CD\\right>$ fluctuates randomly between ‘ + ’ and ‘ − ’. This still gives what would be seen classically – either a dead cat or a live cat, but the state of the individual cat is not a definite $\\left|CL\\right>$ or a definite $\\left|CD\\right>$. Of course, one can claim without inconsistency that in reality the state of affairs is like this and that all living and nonliving things in this world are in some incoherent quantum mechanical superposition till an observation is made to determine the state.\nThe Schrödinger Cat and the Interpretations of Quantum Mechanics\n================================================================\nBefore we end the discussion we will have a brief look at how the Schrödinger’s cat is viewed by different interpretations of quantum mechanics. The official view is that of the old Copenhagen interpretation in which the state of the cat is not entitled to a physical reality until an observation is made. (As an aside, the interpretation of the current theory of quantum mechanics was developed in Göttingen, by the school led by Max Born in which W. Heisenberg and P. Jordan were the early contributors, and not in Bohr’s school in Copenhagen). In this view, the superposed or entangled wavefunctions allow a description of the state of the cat in an abstract sense, but do not imply anything further in terms of ontology. An observation is defined as a suitable interaction with a classical macroscopic device that reduced the state to a real and definite physical state. The view is entirely consistent, though unsatisfactory to many at the level of physical ontology and the physics and philosophy of (physical) reality.\nAnother consistent and yet physically empty (in the sense of being beyond the grasp of empirical methods) view is that of the ‘Many-Worlds’ interpretation of quantum physics. In this view, each component of the wavefunction is a physical reality in a Universe of its own, and therefore superposition signifies a totality of these definite states in different Universes. The evolution of the state is a coherent branching of the state, and of an observer himself, into different Universes, everything being subject to the laws of quantum mechanics. ‘Naturally’, some definite result is obtained in the measurement in each of the non-overlapping Universes. Thus the same cat is dead in some Universe, and alive in another, and there is no issue of the collapse of the wavefunction, nonlocality etc. Only, Schrödinger would have found the entire set up more ridiculous than his example of the superposed cat itself.\nAn important interpretational view is that of L. de Broglie and D. Bohm, in which there is a hidden physical reality to the wavefunction and there is an observable physical reality to the particle. The associated quantum wave goes into a superposition of states, and the particle is guided by the superposed wave in a complicated trajectory. This view is also consistent for single particle quantum mechanics, whereas the many questions that arise in multi-particle situations have not been not been answered satisfactorily. I feel that the de Broglie-Bohm view runs into some problems handling the Schrödinger’s cat. In the example of the double-slit, the particle passes through one of the slits and propagates to the screen guided by the wavefunction. But in each trial of the experiment employing particles one by one, the particle passes either through the upper slit or through the lower slit, and generally remains in the upper or lower part of the screen. If one translates this to the case of the cat, the state of being dead or alive materializes during the evolution itself, and any quantum mechanically specific property like interference should come from the interference of the guiding waves. In the machinery of quantum mechanics it is possible to retrace the evolution of the wavefunction without making any interaction or observation during the evolution and such a reversible evolution should be accountable in any valid formalism. The Copenhagen view has no problems in this regard, but the de Broglie-Bohm view seems to face some problems because once the cat is dead in a physically real sense, there is no possibility of reviving it even with the magic of quantum mysteries. Therefore, any interpretation of quantum mechanics that ascribes physical reality to the states before an observation is made will run into severe problems and inconsistency when it has to handle the Schrödinger’s cat (the Many-Worlds interpretation passes this test because the reality is branched into different Universes). On the other hand, even the Schrödinger’s cat can be brought consistently within the Bohmian quantum mechanics if we treat the state of death as a mere physical rearrangement of a set of particles and molecules obeying the laws of quantum mechanics. Then the evolution from being live to being dead is a quantum mechanical evolution that is in principle reversible, and can be treated consistently within all these interpretations. This aspect needs further study and thought.\nCan the Cat Teach Us Further?\n=============================\nThe Schrödinger’s cat has played a significant role in making us think about several issues related to the interpretation of quantum mechanics and its applicability at various scales of physical phenomena. It has also motivated some challenging experimental activity in which people try to observe phenomena that are specific to quantum mechanics, in progressively larger and larger physical systems. At present there seems to be some clarity and much confusion on many of the issues, and all these can be traced to our lack of understanding of the meaning and ontology of the wavefunction. It is only when we understand the nature of the wavefunction, that we will be able to talk with clarity about superposition, entanglement, collapse, measurement etc.[2](#fn2) Without doubt, the Schrödinger cat will remain alive, provoking and motivating interesting ideas, experiments and debates in the foundational aspects of the most challenging theory of our times.\nEpilogue\n========\nThis article was written 15 years ago, for a volume edited by P. K. Sengupta in the remarkable series ‘History of Science and Philosophy of Science’, of the Centre for Studies in Civilization, India. No changes in the descriptive part of the article seem to be required even now. However, a definite advance happened quite recently, which redraws the entire canvas of the understanding of quantum mechanics, clarifying and resolving all the foundational issues, including the concerns expressed in examples like the Schrödinger cat. This consists of two breakthroughs about the correct understanding of both the mathematical structure and the physical content of quantum mechanics :\n1. I have been able to prove in detail the hunch expressed by Einstein and Popper, that the Schrödinger equation and its wavefunction describe an ensemble of system, and not a single quantum system. In fact, the Schrödinger equation is readily shown to be the continuity equation for the probability density, ∂*ρ*/∂*t* = ∇ ⋅ (*ρ**v*), written in terms of its complex single valued square root, $\\psi(t)=\\sqrt{\\rho}exp(i\\phi)$. Then the Born relation *ρ* = *ψ**ψ*\\* is an exact mathematical relation, and not an additional postulate. The quantity *ϕ*(*t*) enables the representation of both *ρ* and the dynamical velocity *v* in the same function *ψ*. The *ψ*-function pertains to the ensemble, and it is a hybrid quantity consisting of the (real) square root of the ensemble probability density and the ensemble averaged complex valued periodic function of action. Then, it turns out that the superposition and the entanglement do not involve the physical states of the particle (or the cat); what superposes is merely the action function with a periodic property.\n2. The second result addresses the question, what then is the evolution equation for quantum dynamics if the Schrödinger equation is factually the continuity equation? The single particle dynamical equation is a modified Hamilton’s equation, simply written as ∂*ζ*(*S*)/∂*t* = *i*ℏ*H**ζ*, where *ζ*(*S*) = *e**x**p*(*i**S*/ℏ) and *H* is the Hamiltonian. This is universally applicable at all scales of masses, sizes and velocities. There is no fundamental distinction between the dynamics of microscopic and macroscopic bodies, and there is no quantum-classical divide. The entire probability space from zero to one is mapped in the values of the action spanning the small value ℏ, because of the periodicity of the function *ζ*(*S*). This new equation implies that the fundamental uncertainty relation is in fact Δ*S* ≥ ℏ.\nTogether, these two results completely eliminate all the foundational and interpretational issues of quantum mechanics, while reproducing all the results involving quantum interference and correlation.\nThus, the alleged wave-particle duality of a particle is not the physical fact. A particle has no wave nature and there is no ‘matter-wave’; instead, the action of its dynamics manifests as a periodic complex valued entity, which defines the multiple possibilities of observables and their superposition. Since this ‘action-wave’ carries action, and not energy or momentum, there is no collapse of the quantum state when observations are made. Just as a probability density of an ensemble is not altered or perturbed when an individual statistical event is realized, the wavefunction of the Schrödinger equation does not collapse when an observation is realized. The same remarks apply for an entangled wavefunction as well. The collapse of the wavefunction is eliminated from the quantum mechanics, and with it, the quantum measurement problem is solved entirely.\nFor the example of the Schrödinger cat, any single radioactive atom and the individual cat are always in their definite states, *either* $\\left|A+\\right>\\left|CL\\right>$ *or* $\\left|A-\\right>\\left|CD\\right>$, and never in a superposition or entanglement. What is superposed are the relevant complex exponential functions of the action corresponding to the two systems. A single atom-cat pair is always in a definite joint state that will eventually jump from the initial state $\\left|A+\\right>\\left|CL\\right>$ to the fateful state $\\left|A-\\right>\\left|CD\\right>$, whether or not one is observing either system. What the Schrödinger equation and the evolving wavefunction describe is the situation in the statistical ensemble, consisting of large number of atom-cat pairs. This definitely and completely answers all the questions raised in the context of the Schrödinger cat. All the ridiculous aspects disappear. The conscious observer and his friend play no role in the dynamical evolution.\nThe details of this universal picture of clarity about quantum mechanics, the Schrödinger equation, and its wavefunction are described in the articles and.\nAcknowledgements\n================\nSome of the thoughts presented in this article evolved in the presence of ‘Jaldi’, the playful cat in Paris (still very much alive, which is very reassuring). I sincerely thank Prof. Pradip Sengupta for skillfully provoking me to write the original article, about 16 years ago, and for thus helping me to acquire some important insights. Martine Armand helped in structuring the updated sections.\n99 [1]#1 [1]`#1` urlstyle[1]doi: #1\nA. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? *Physical Review*, 470 (10):0 777–780, may 1935. doi: 10.1103/physrev.47.777.\nE. Schrödinger. Discussion of probability relations between separated systems. *Mathematical Proceedings of the Cambridge Philosophical Society*, 310 (4):0 555–563, oct 1935. doi: 10.1017/s0305004100013554.\nE. Schrödinger. Die gegenwärtige situation in der quantenmechanik. *Die Naturwissenschaften*, 230 (48):0 807–812, nov 1935. doi: 10.1007/bf01491891. English translation by J. D. Trimmer reprinted in Quantum Theory and Measurement, Eds., J. A. Wheeler and W. H. Zurek, (Princeton University Press, Princeton, 1983).\nJ. D. Trimmer. The present situation in quantum mechanics: A translation of Schrödinger’s ‘cat paradox’ paper. *Proceedings of the American Philosophical Society*, 124:0 323–338, 1980. url: <www.jstor.org/stable/986572>.\nE. Schrödinger. An undulatory theory of the mechanics of atoms and molecules. *Physical Review*, 280 (6):0 1049–1070, dec 1926. doi: 10.1103/physrev.28.1049.\nP. A. M. Dirac. The physical interpretation of the quantum dynamics. *Proceedings of the Royal Society of London A*, 1130 (765):0 621–641, jan 1927. doi: 10.1098/rspa.1927.0012.\nA. Einstein. Remarks to the essays appearing in this collective volume. In Paul Arthur Schilpp, editor, *Albert Einstein: Philosopher-Scientist*. MJF Books, NY, 1949.\nE. P. Wigner. Remarks on the mind-body question. In I. J. Good, editor, *The Scientist Speculates*. Heinemann, London, 1961.\nB. d’Espagnat. A tentative new approach to the Schrödinger cat problem. arXiv:quant-ph/0301160, 2003.\nM. Arndt, O. Nairz, J. vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger. Waveparticle duality of C60 molecules. *Nature*, 4010 (6754):0 680–682, oct 1999. doi: 10.1038/44348.\nC. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland. A “Schrödinger cat” superposition state of an atom. *Science*, 2720 (5265):0 1131–1136, may 1996. doi: 10.1126/science.272.5265.1131.\nJonathan R. Friedman, V. Patel, W. Chen, S. K. Tolpygo, and J. E. Lukens. Quantum superposition of distinct macroscopic states. *Nature*, 4060 (6791):0 43–46, jul 2000. doi: 10.1038/35017505.\nC. H. van der Wal. Quantum superposition of macroscopic persistent-current states. *Science*, 2900 (5492):0 773–777, oct 2000. doi: 10.1126/science.290.5492.773.\nB. Misra and E. C. G. Sudarshan. The Zeno’s paradox in quantum theory. *Journal of Mathematical Physics*, 180 (4):0 756–763, apr 1977. doi: 10.1063/1.523304.\nK. Popper. *Quantum Theory and the Schism in Physics*. Routledge, 1992.\nA. Einstein. Physics and reality. *Journal of the Franklin Institute*, 221:0 349–382, 1936.\nC. S. Unnikrishnan. Reconstructing quantum mechanics without foundational problems. arXiv:1812.06088, 2018.\nW. R. Hamilton. On a general method of expressing the path of light, and of the planets, by the coefficients of a characteristic function. *Dublin University Review and Quarterly Magazine*, 1:0 795–826, 1833.\nD. Bohm and B. J. Hiley. *The Undivided Universe: An Ontological Interpretation of Quantum Theory*. Routledge, London, 1993.\nC. S. Unnikrishnan. The ensemble nature of the Schrödinger equation and its wavefunction, and a new universal action mechanics. in *Prof. G Ramachandran Memorial Volume*. S. Sirsi A. R. Usha Devi G. Umesh K. S. Mallesh, M. V. N. Murthy (eds), 2021.\n---\n1. I found the definite answer only recently, more than a decade after writing this article, that the physical entity that superposes in quantum mechanics is the ‘action’, the physical quantity introduced by W. R. Hamilton in 1833 . This also lead to completion of Hamilton’s mechanics. A full account of the formalism that completely eliminates all foundational problems of quantum mechanics is now at hand, and it can be found in the reference.[↩](#fnref1)\n2. A complete understanding of the wavefunction and the Schrödinger equation was achieved recently. See the section, ’Epilogue’.[↩](#fnref2)\n\nThe Schrödinger Cat: Physics, Myth, and Philosophy\n==================================================\nThe most discussed of “live” metaphors in physical sciences is that of the Schrödinger cat. Introduced in the early, but mature days of the new quantum theory, in 1935, the parable of the cat has provoked and enlivened debates on the meaning of the quantum theory, on the ontology of the quantum wavefunction, on the puzzle of the collapse of the wavefunction, on the meaning of quantum measurement, and on the boundary between the quantum world and the classical world. In this article, I will discuss these issues of the quantum theory focussing on various aspects of the metaphor of the Schrödinger cat.\n> \n> This is a slightly revised version of my article written in 2005, and published in the compiled volume, ‘History of Science and Philosophy of Science’, p. 313-338 (Ed. P. K. Sengupta, Centre for Studies in Civilization, PHISPC, Pearson-Longman, Delhi, 2010). It was also published in *Sandhaan*, Vol. V, No.2, 1-52 (2005), the journal of the Centre for the Studies in Civilizations, New Delhi).\n> \n> \n> \nPreface\n=======\nIn 1935, Erwin Schrödinger was inspired to think deeply about the quantum mechanical description of two systems that interacted and then separated, by the discussion in the Einstein-Podolsky-Rosen (EPR) paper that questioned the completeness of standard quantum mechanics. In several papers he discussed the central and universal trait of ‘entanglement’ in quantum mechanics. He realized that interactions that form correlations ‘entangle’ the individual descriptions, resulting in only a joint and less detailed description of the physical state, and that the individual wavefunctions ceased to exist. He wrote,\n> \n> When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (or *ψ*-functions) have become entangled.\n> \n> \n> \nAn observation on one system then disentangles the description (*ψ*-function) and restores individual realities, in the quantum mechanical sense, in terms of individual wavefunctions. *One of these systems could be macroscopic*, in terms of size, mass etc., and then new conceptual issues arise. He discussed a theory of quantum measurement based on these insights. He demonstrated that the quantum uncertainty was not a blurring of physical variables, confined to the microscopic domain of atomic systems; the possibility of an interaction expanded the domain of uncertainty and fuzziness to the macro-world. In these studies arose the ever-living fable of the life of a cat that got entangled with the randomly timed radioactive decay of an atom. The graphic scenario exemplified since then the central tenets of the problem of ‘quantum measurement’, as well as the troubling notion of a quantum-classical divide. The ‘Schrödinger cat’ (also referred to as “Schrödinger’s cat”) challenged the universal applicability of quantum mechanics and its ability to represent physical states faithfully, further supporting Einstein’s views that were expressed in the EPR paper. Indirectly, it raised new questions about the role of the conscious observer in quantum mechanics. With life mixed with physics and quantum mysteries, it generated large popular interest. Philosophers discussed the nature and meaning of physical reality in quantum mechanics and its relation to the observer, in the domain well beyond the microscopic world of quantum mechanics.\nThere were also intellectual compromises: the superposition and interference of the quantum states of a single ‘large enough’ system being confused as the realization of a quantum state of the Schrödinger cat is common. The perennial confusion between ‘a cat in a superposition of its non-commensurate physical states’, and the very different idea of ‘an entangled state of a Schrödinger cat’ is serious and widespread. Some of these myths are here to stay, being ingrained in the collective psyche by repeated statements.\nThe foundational problems raised in the early years of quantum mechanics, especially the problem of quantum measurement, remain open. The aim of this article is to explore the genuine issues in physics and philosophy that are prompted by Schrödinger’s entangled cat.\nThe Quantum Theory\n==================\nThe years between 1924 and 1927 saw accelerated revolutionary changes in the understanding of the microscopic atomic physics. The old quantum theory that slowly developed in the work of Planck, Einstein, Bohr and others saw a sudden change in its physical and mathematical understanding and formalism. One line of thought starting with Louis de Broglie’s proposal that particles have wave nature matured in Erwin Schrödinger succeeding in writing a mathematically and physically consistent “wave equation” for the entity possessing the wave-particle duality. With this equation, soon named the Schrödinger equation, accurate calculations of quantized energies and spectral lines in atoms could be made. Also, the equation enabled calculations of many other phenomena pertaining to microscopic physics. In a parallel development, Werner Heisenberg wrote down quantum rules for calculating these quantities based on a mathematical formalism in which physical quantities were represented by matrices – arrays of complex numbers. An important mathematical property of these matrices were that they were non-commuting; the order of their multiplication was important and reversing the order gave a different value for their product in general. This property accounted for the wave-particle duality in the Schrödinger formalism. Soon, Schrödinger showed that the two formalisms were equivalent, and the mathematical foundations of the new quantum theory were established. In 1927, the important work by Paul Dirac extended the quantum formalism and linked it with the mathematics of classical mechanics, and he also wrote down the quantum wave equation for the electron, consistent with the theory of relativity. Of crucial importance were the contributions by Max Born and John von Neumann, for the physical interpretation of the mathematical elements of the theory.\nOne important aspect of the Heisenberg rules is that some combinations of physical quantities (‘observables’) cannot be determined accurately simultaneously. Multiple observations yield values with a statistical dispersion, and the product of the dispersion in the values of repeated measurements of two such observables on similar systems have a minimum universal value, given by the Planck’s constant ℏ; this is called the uncertainty relation. For example, the position and the momentum of a particle are such a pair. It is important to note that the uncertainty relation in quantum mechanics is defined only for repeated measurements in a statistical ensemble, and never for a single particle or a physical system. On the other hand, the uncertainty relation can be applied to a single system for *estimating the expected uncertainty in the physical quantities*, when the physical state is known.\nIn the present usage both the Schrödinger and Heisenberg approaches are practised, though the fundamental equation of quantum motion is usually written down in the Schrödinger form, as a differential equation. The Schrödinger equation describes how the wavefunction (or the *ψ*-function) evolves in time under the action of external interactions. The wavefunction itself is an unobservable, but all observables of the physical system are related to it by certain statistical relations. Usually, solving the equation for the specific problems gives rise to quantized discrete energy states, as well as some un-quantized states. The uncertainty relation follows in this approach from the properties of the wavefunction, and its relation to the various observables.\nA crucial piece of interpretational discovery was made by M. Born regarding the Schrödinger wavefunction, and this discovery provided the most essential and at the same time most problematic aspect of the quantum theory; Born realized that the connection between the wavefunction and results of a measurement or observation was essentially statistical. Thus, the square of the wavefunction provides a probability density for various observables associated with the system. It became clear after Born’s work that quantum mechanics had no predictive power for single events. It is a theory of statistical predictions of the properties of a microscopic system, like an atom. This turn in the meaning of the theory, which had a deterministic differential equation as its basic equation, came as an unpleasant shock to the pioneers like Einstein and Schrödinger.\nWhile it is possible to think of the wavefunction for a single particle as somehow a wave associated with the particle, as envisaged by de Broglie originally, this picture breaks down when one considers several particles. The wave that one is referring to is not a real wave in space, propagating as time progresses. It is a wave in the sense of an abstract mathematical function that admitted the properties of ‘superposition’ and ‘interference’, but one that merely represents the potentialities of measurement results that can be obtained from the physical system. In other words, the wavefunction itself is not an observable. Then ‘Born’s rule’ associates the square of the wavefunction to the probability of the realization of a particular result being obtained. The quantum theory has only statistical predictions and the wavefunction provides the mathematical link between the physical system and these probabilities. Right at this point it should have been clear that one should not physically associate the wavefunction and the quantum system, and *one should not see the physical system or the particle as an entity possessing the dual property of a wave and the particle*.\nThe Uncomfortable Quantum Rules\n===============================\nThe basic idea that is operative in the new mechanics is that the representations of two distinct physical states of a system can be added linearly in various ways to get a new valid physical state of the system. This never happens in the familiar world of particles around us. For example, a familiar system that exhibits just two discrete possible physical states is a coin that is tossed. It can either turn a `Head’ or a `Tail’. This happens randomly. But there is no concept of a physical state that is a sum or difference of these two states in a mathematically literal sense. However, when one considers a physical wave (on water for example) it makes perfect sense to think of the state of a new wave that is a sum or difference of two other waves – wherever the crests add together, the new wave just gets amplified and when a crest and trough coincide the result is the absence of a wave. So waves can cancel or augment each other, and their sum or difference is also a physical wave. But if a coin, the states of which are not waves, were to be described quantum mechanically, then a state that is a sum or difference of `Heads’ and `Tails’ is also valid and one needs to imagine what that might mean. At the same time, when an observation is made to determine the state of the coin, one and only one of these two possibilities turn up, and that too randomly. *It is this conceptual wave-particle duality that quantum mechanics embodies, and this duality is unfamiliar to our experience, even though each aspect separately is very familiar*. It is this uncomfortable consequence of the supposed applicability of quantum mechanics to the everyday physical world that Schrödinger tried to express dramatically in his discussion of the quantum states of a cat. His analysis revealed an even more puzzling aspect of the superposition of the wavefunctions, which he called the ‘entanglement’.\nPhysicists who took the wave-particle duality of matter seriously in the sense of associating both a particle nature and also a wave nature to a single entity, thought of microscopic particles as some fuzzy (blurred, or distributed) entity. Schrödinger had spent a great deal of effort in trying to understand the quantum fuzziness and its implication to the larger world. Does the uncertain fuzziness suggested by the uncertainty principle and the mathematical shape of the wavefunction of an electron or an atom become sharply defined matter, without any fuzziness, in the macroscopic world? He discussed the following “burlesque” (translated as ‘ridiculous’) example to bring out the strange implications associated with the mathematical machinery of the new quantum theory \n> \n> One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following diabolical devise (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small that *perhaps* in the course of 1 hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat is still alive if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The wavefunction of the entire system would express this by having in it the living and the dead cat (pardon my expression) mixed or smeared out in equal parts.\n> \n> \n> It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be *resolved* by direct observation. That prevents us from so naively accepting as valid a “blurred model” for representing reality.\n> \n> \n> \nThis example, now called ‘the Schrödinger cat’, is the central theme of our discussion. Before we discuss the various aspects of the Schrödinger cat, it might be useful to have a brief look at the mathematical machinery that forms the basis of all the discussions related to this example.\nThe Mathematics and Physics of Linear Superposition\n===================================================\nThe basic mathematical machinery of quantum mechanics is simpler than that of the older classical physics. The feature that is most important is that of the linear superposition, familiar from the mathematics of vectors and physics of waves. In quantum mechanics, the physical state of a particle has a meaning only after a measurement. But its mathematical state can be represented by a wavefunction, or more generally as a vector in an abstract space. By ‘vector’ we mean an abstract entity with various components along different directions, representing physical quantities like velocity, force etc. The difference in quantum mechanics is that these components are complex numbers in general and the components do not refer to spatial directions; the ‘space’ in which the vectors represent the physical system is an abstract mathematical space, which could even have infinite dimensions. The fundamental premise of the quantum theory is that if A and B are two legitimate vectorial representations of possible physical states of a particle, then their linear addition, *a**A* + *b**B*, is also a legitimate state of the particle. The vectors are added after arbitrarily stretching or contracting them by complex number scaling factors ‘a’ and ‘b’, with the restriction ∣*a*∣2 + ∣*b*∣2 = 1. Then the Born rule says that a measurement will reveal the property A with the probability ∣*a*∣2 = *a**a*\\* and B with the probability ∣*b*∣2.\n![The double-slit experiment. An interference pattern (regions where there are particles separated by regions where there are no particles detected) is observed with light, atoms and electrons. The natural explanation is in terms of waves that split and superpose. But the same physical behaviour is observed even when the particles are sent one by one forcing us to attribute a wave with even single particles.](Fig1.jpg \"fig:\") [fig:fig1]\nFirst we mention a physical example that illustrates how superpositions become necessary. In the familiar example of a double-slit interference experiment, light falling on a screen with two slits forms an interference pattern with bright and dark fringes on a detector screen far away (Figure 1). If light is treated as waves, there is an immediate explanation in terms of the familiar interference patterns involving waves, like waves on water. It is possible to do the experiment such that the source of light is a single atom or a small number of atoms, and then light is received at the screens as discrete photons, much like particles arriving, well separated in time (as ‘seen’ by an electronic detector), and not as a continuous wave. Yet, the final pattern of interference obtainable on a long exposure camera is identical to the one obtained by the instantaneous wave description. *The only logical possibility is to think of each photon being associated with a wave that interferes independent of when and where the next photon arrives*. In such a picture, the state of the photon after the slits has to be very different from our usual description of the state of a particle with a single unique trajectory. Somehow one has to account for the waves that will split and propagate through both the slits and then interfere even though we have only one indivisible particle to deal with (figure [fig:fig1]). That is where the concept of superposition becomes necessary. The state of the photon is described as containing two parts, one corresponding to the propagation through the upper slit (U), and another through the lower slit (D), in equal proportions. These two parts can then interfere and represent the observed pattern. This does not imply that the particle itself splits at the slits and propagates in the two paths simultaneously – the associated waves do that job, if one needs to form a graphical representation. In fact, the particle cannot be in both slits in any manner without grossly violating all conservation principles in physics. So, the state of the particle after the slits is *represented* as a wavefunction,\n \n∣*S**t**a**t**e*⟩ ∼ ∣*W**a**v**e*-*U*⟩ + ∣*W**a**v**e*-*D*⟩ \nSince the crest of the upper wave can have an arbitrary shift relative to the crest of the lower in general (due to an extra piece of glass near the upper slit, for example), a ‘phase factor’ could also be added after the ‘+’ in this expression. (This can even make the ` + ` into ‘ − ’, corresponding to the trough instead of the crest at a particular position). All such general expressions are called linear superpositions. The wavefunctions contain complex number quantities to accommodate the needs of the representation. What is observable is related to the square of the wavefunction, ∣∣*S**t**a**t**e*⟩∣2, which will involve the ‘interference term’ consisting of the bilinear product 2∣*W**a**v**e*-*U*⟩ ⋅ ∣*W**a**v**e*-*D*⟩\\*, agreeing with the empirical observations accurately. It turns out that one can completely forget about the picture of a physical wave, and replace it with that of the concept of a ‘vector’ that can be added and subtracted with appropriate complex factors. In fact, modern quantum mechanics is done in the language of these ‘state vectors’, which are equivalent to the representation with the wavefunctions.\nThe basic idea is that the *representation* of the state of a physical system is by means of a vector in an abstract space. For our purpose this vector is very similar to other familiar vectors, a directed arrow pointing in some direction in space, whose length is agreed to be one unit. (See figure [fig:fig2]).\n![The concept of a vector and projections on different basis. The thick arrow is the state vector, and the light arrows along the two directions are the projections. On the right is the same vector projected on a different basis.](Fig2.jpg \"fig:\") [fig:fig2]\nWe can choose some coordinate system in space, and work with just two axes for clarity; x-axis and y-axis (called the ‘basis’), which can be any two perpendicular directions for convenience (though I refer to ‘space’ here, the vectors of quantum mechanics are in an abstract space of any dimensions). Some immediate obvious facts about this construction can be noted: 1) the length of the vector does not depend on either rotating it, or on rotating the coordinate system, 2) the projections of the vector on the two axes depend on the rotation of the vector or the coordinate system, and the sum of the squares of the projections is, by the Pythagoras theorem, just the square of the length of the vector, 3) the vector itself can be represented as the vectorial sum of its projections, *V⃗* = *V**x**x̂* + *V**y**ŷ*; this is a linear superposition. A general linear superposition corresponds to scaling up or down each vector and then adding them with the correct directions.\n![The meaning of a linear superposition – the vector along the diagonal is a superposition, a sum, of the other two orthogonal (perpendicular) vectors, one stretched more than the other by a scaling factor. On the right is the superposition with a negative sign for the second vector.](Fig3.jpg \"fig:\") [fig:fig3]\nIn classical mechanics several quantities have been described by vectors, like velocity, force etc. Given a vector quantity for a classical system we can talk about values of its projections on various directions, and we think of the physical system as possessing all these values simultaneously as its physical properties, irrespective of what measurements we make on the system as long as the measurement itself has not changed the properties. In quantum mechanics, the state vector does represent the properties of the system, but in a way very different from what one is used to in classical mechanics. The essential concept is the following. The state vector represents the physical state and the basis that we choose defines the property one wants to observe. It is sufficient to illustrate this with a simple example.\nIf ∣*s*⟩ represents the state vector relevant for the observation of the angular momentum of a particle, then one has to choose a basis corresponding to the components of the angular momentum in various directions in space. This aspect is similar to the case in classical mechanics where if *S⃗* is the angular momentum vector for a physical system, then the angular momentum in a direction at some angle is given by the projection of the vector *S⃗* along that direction. *In quantum mechanics the projection of the state vector along a direction does not correspond to the value of the angular momentum in that direction*. *Indeed the value is usually quantized and random in each measurement*; irrespective of what the projection is, the value measured is one of the possible discrete values, at random, say + 1 or − 1 in relevant units. But *the square of the length of the projection can be associated with the probability of realizing a particular value of angular momentum when measured in that direction*. This appearance of the notion of probabilities makes the quantum projections very different from the classical vector projection. In classical physics, the projection of a particular vector quantity along some direction is the component of that quantity itself, along that direction. Given the original vector, one can say with certainty that a particular value will be reproducibly obtained if projected along some direction. For observations on a quantum system, the projection is not directly related to results of individual observations. If the original state vector has unit length corresponding to a measured quantized value *s*, and if the projection has a length of 1/2, one can say that there is a probability of (1/2)2 = 1/4, that a measurement in that direction yields the value *s* for the projected angular momentum. So there is a shift from ‘value’ to ‘probability’.\nFor example, suppose the physical system (say, an electron) is prepared by some means with the value of its angular momentum along a particular direction equal to +1, in units of ℏ/2. It turns out that for this system, the values of the projections of the angular momentum in *any direction* are always either + 1 or − 1, and nothing else. (There are other particles where the values could be + 1, 0 and − 1). This of course is very strange, but we will not try to find a ‘reason’ for this strange behaviour since there is no possibility of succeeding in explaining such behaviour with the familiar reasoning. Mathematically we will represent the original state vector as ∣*s*⟩. Now suppose that the direction in which the system was prepared with the definite value + 1 is along a direction that is symmetrical relative to the directions x and y, somewhat like the diagonal in figure 2 (the mathematically correct representation of this example is slightly different from the graphical example in the figure, but for our purpose the present discussion will suffice). Subsequently we want to measure the angular momentum in the directions x or y, shown as the horizontal and the vertical axes in the figure. Geometrically, the projections of the original state vector on either of these axes are equal to $1/\\sqrt{2}$. Therefore, if a measurement of the angular momentum is done along the x-axis, say, then we will either get + 1 with a *probability 1/2*, which is the square of the projection, or − 1 with a *probability 1/2*, like when tossing a coin.\nTwo important clarifications are necessary. The fact that one gets a particular value in a measurement, for example + 1 along the x direction, does not mean that it had that value just before that measurement. Obviously, we had prepared the system with its angular momentum +1 along the diagonal. Yet, in a measurement along the horizontal direction, we got a value + 1 (or − 1). After the measurement, the angular momentum projection in the horizontal direction remains + 1, and any further measurement in the same direction gives + 1 with certainty. Therefore, by the Born rule, the state vector’s projection along the horizontal should now be equal to + 1, so that its square corresponds to the probability 1, or ‘certainty’. The revised description is now with a new state vector along x, with the y-component ceasing to exist – this is called the ‘collapse of the state’, or the ‘collapse of the wavefunction’. The state vector has changed suddenly on a measurement, and the vertical component has gone to zero after the measurement resulted in a definite value in the orthogonal direction. Mathematically, the situation is similar for the wavefunction corresponding to the particle passing through a screen with two slits (labelled + 1 and − 1). A wavefunction with the symmetrical superposition of equal components ‘collapses’ to a single component ( + 1 and − 1) randomly, when an observation is attempted. If we had associated the two-component wavefunction physically with the particle (as one often does with the waves of the wave-particle duality), then the vanishing of one component has the implications of the vanishing of some physical entity from space, spontaneously and instantaneously during the measurement. This is what makes the collapse of the wavefunction mysterious and troublesome.\nThe Collapse of the Wavefunction and Other Disasters\n====================================================\nThe ‘collapse of the wavefunction’ is undoubtedly the most discussed and the most dreaded consequence of the mathematical machinery of quantum mechanics, as interpreted today. Valid physical states of a system are routinely described by a general sum of several distinguishable, and mutually incompatible states (if one occurs physically, the other is excluded, like the ‘Heads’ and ‘Tails’ states). A multi-component wavefunction with the observable possibilities *S*1, *S*2 and *S*3, for example, can be represented as the linear superposition\n \n$$\\label{key}\n\\left\\vert \\psi\\right\\rangle =a\\left\\vert S1\\right\\rangle +b\\left\\vert\nS2\\right\\rangle+c\\left\\vert S3\\right\\rangle$$ \nA measurement on a single physical system turns up just one of the possibilities contained in the superposition, statistically, and with a probability related to the length of each vector contributing to the superposition. For instance, a measurement might result in the outcome *S*2 (with the probability ∣*c*∣2). But as soon as a single measurement is completed, the superposition of several states changes into just one of the states, corresponding to the value realized in the measurement. In this case, the wavefunction becomes\n \n$$\\label{key}\n \\left\\vert \\psi'\\right\\rangle =\\left\\vert\n S2\\right\\rangle$$ \nAll other components vanish. If one had attributed a physical (material) reality in space and time to the components present in the superposition, then these elements had to physically vanish when a measurement is made. It is as if a physical state that was distributed and spread out into different physical states suddenly condenses into one state, when observed. In the example of the double-slit experiment, one can make an observation near one of the slits to answer the query, ‘through which slit is the particle passing?’ If the detector registers the particle, then the particle could not be passing though the other slit. If it did not register anything, then the particle is certainly passing through the other slit. Either way, the path is determined. Having detected the path with certainty, there is no room for probabilities any more, and then the wave function or state vector component in the other path should vanish to zero immediately. Otherwise there will be inconsistency with the Born’s rule which says that the probability of detection is related to the amplitude of the wave. In the case of the classical coins, just before the coin lands in the final state, its state is alternating between the two possibilities and it is no surprise or paradox for us if it finally lands randomly in one of the states. But the quantum superposition is very different – as it evolves, it is not alternating between the two states. It remains in a superposition which is neither of the two definite possibilities at any instant. Then an observation reduces it to one of the states. In fact, there are examples where one of the states involved in the superposition is physically far away from the other state. Then a measurement annihilates one of the states even when this state is physically far away from the place where the observation is made! This is a (mild) example of ‘nonlocality’ in quantum physics.\nFrom Superposition to Entanglement\n==================================\nSuperposition is a simple concept when one thinks in terms of vectors or waves adding up on each other. Some superpositions are special in quantum mechanics, and they lead to consequences that have shaken up the very foundations of our familiar notions of causality and locality. Consider two systems, each with its own quantum states (formally called ‘kets’) $\\left|s1\\right>$ and $\\left|s2\\right>$. Each of these could be a simple superposition of the two states ∣*u*⟩ and ∣*d*⟩, which are the two observable states of the system. Suppose these superpositions for each individual particle are $\\left|s1\\right>=\\frac{1}{\\sqrt{2}}(\\left|u1\\right>+\\left|d1\\right>)$ and $\\left|s2\\right>=\\frac{1}{\\sqrt{2}}(\\left|u2\\right>+\\left|d2\\right>)$. For example, the labels might correspond to a particle spinning clockwise (u) or anticlockwise (d) when looked at from a particular direction. Since there are two particles, one can also study the ‘joint state’ of the total system, which is written as a juxtaposition of the states, $\\left|s1\\right>\\left|s1\\right>$. The joint state would be written as\n \n$$\\begin{aligned}\n \\left|s1\\right>\\left|s2\\right>=\\left(\\frac{1}{\\sqrt{2}}(\\left|u1\\right>+\\left|d1\\right>)\\right)\\left(\\frac{1}{\\sqrt{2}}(\\left|u2\\right>+\\left|d2\\right>)\\right)\\\\\n =\\frac{1}{2}\\left(\\left|u1\\right>\\left|u2\\right>+\\left|u1\\right>\\left|d2\\right>+\\left|d1\\right>\\left|u2\\right>+\\left|d1\\right>\\left|d2\\right>\\right)\\end{aligned}$$ \nIn this expression the first ket ∣*s*1⟩ stands for the first particle and the second one ∣*s*2⟩ for the second. Each is in a state of superposition and the joint state is a simple juxtaposition of all the possibilities of the superposition. There are totally four possibilities and each one will happen with a probability (1/2)2 = 25%. This is a general superposition, and it is a ‘product state’ since it can be written as a product of two factors, $\\left|s1\\right>$ and $\\left|s1\\right>$, each of which is a sum (superposition) of two factors. The important point to note is that each particle is said to have its own definite state, $\\left|s1\\right>$ and $\\left|s2\\right>$. The joint state is ‘separable’ into $\\left|s1\\right>$ and $\\left|s1\\right>$. Now suppose we put some restriction on the possible states – say, when one particle is up the other should always be down (this might come from the requirement of some conservation principle, like that total spin is zero). This could happen due to some brief interaction between the particles. Then two of the four possibilities are prohibited, and the joint state would be\n \n$$\\label{key}\n \\left|S=0\\right>=\\frac{1}{\\sqrt{2}}\\left(\\left|u1\\right>\\left|d2\\right>\\right)+\\frac{1}{\\sqrt{2}}\\left(\\left|d1\\right>\\left|u2\\right>\\right)$$ \nNow there are only two possibilities for the joint state, each with a probability of 50%. *This cannot be written as a product of the individual states $\\left|s1\\right>$ and $\\left|s2\\right>$. The joint state $\\left|S=0\\right>$, which is also just a superposition of two product states, is not ‘separable’ anymore*. This is called an ‘entangled state’ with very peculiar properties. The outcome of a measurement on each particle is entangled with that of the other; ∣*u*1⟩ with ∣*d*2⟩ and ∣*d*1⟩ with ∣*u*2⟩. *Since it is not a simple juxtaposition of the two individual states, it turns out that each of the particles does not have any physical state by itself, in the quantum theory!* The independent quantum status of each particle is lost and there is only a joint state. Neither particle is in state ‘up’ or ‘down’, nor are they in any superposition of ‘up’ and ‘down’ states. However, the most general individual state possible in the quantum theory can only be some general superposition of the ‘u’ and ‘d’ states. Here, neither particle is in such a state. All one can say is that if one is found in ‘u’ state, the other will be found in ‘d’ state and vice versa. A measurement will get one of these two joint possibilities, each with a 50% probability. If a measurement is made, the entangled state collapses into one of the two joint states, $\\left|u1\\right>\\left|d2\\right>$ or $\\left|u1\\right>\\left|d2\\right>$. Once this happens, each particle has a definite state and the joint state is a simple juxtaposition of the individual states. The individual states and separability are regained only after a measurement on at least one particle. So, a particle that had no description in terms of a definite state before the measurement acquires a definite state as a result of measurement. Moreover, only a measurement on one of the particles is sufficient – the other particle acquires a definite state automatically without any measurement or observation on it! For example, if the measurement shows that the first particle is in the state $\\left|d\\right>$, then definitely the joint state is $\\left|d1\\right>\\left|u2\\right>$, and not $\\left|u1\\right>\\left|d2\\right>$. This means that *the second particle, on which no observation was made, is now certainly in the state $\\left|u\\right>$ after the measurement on the first particle, though neither had any specific state whatsoever before the measurement on the first particle*.\nTherefore, the quantum entanglement in the state makes the properties of the particles inseparable, and one gets definite properties that are emergent only after an observation. *The bizarre consequence is when the two particles are not in the same spatial region*. The particles can be separated after the interaction that made the entangled state. Then a measurement on one particle instantaneously gives it a definite state *and also a definite state to the other particle that is far away from the first particle*! This is the much discussed violation of locality in quantum mechanics. On the other hand, if we assume that one measurement should not influence the physical state of the other particle that is spatially separated, then we see that quantum mechanics has no description of such independent states. Quantum mechanics corresponds to an incomplete description because its states can be affected by an arbitrarily distant measurement, whereas the requirement of locality is that a physical state in what ever form should not be influenced instantaneously by processes that are spatially distinct and separated. It was Einstein who pointed this severe difficulty in the well known Einstein-Podolsky-Rosen (EPR) paper in 1935. Schrödinger’s papers on the properties of such joint states, entanglement, and its crucial place in the problem of measurement were partly inspired by the EPR paper.\nThe Schrödinger Cat\n===================\nWe now come back to the description of the Schrödinger cat. In his paper that summarized the “Present state of quantum mechanics” he wrote,\n> \n> One can even set up quite ridiculous cases... The wavefunction of the entire system would express this by having in it the living and the dead cat (pardon my expression) mixed or smeared out in equal parts.\n> \n> \n> \nThe colloquial expression “the living and the dead cat mixed or smeared out in equal parts” is the source of a serious confusion to those who have not seen Schrödinger’s precise discussion in a follow up paper.\nThere are descriptions of the general physical situation implied here without any mention of a live body, and yet completely equivalent to it from the point of view of the mathematical theory of quantum mechanics. I mention one such by Einstein, written in 1948 :\n> \n> As far as I know, it was E. Schrödinger who first called attention... Rather than considering a system which comprises only a radioactive atom (and its process of transformation), one considers a system which includes also the means for ascertaining the radioactive transformation – for example, a Geiger-counter with automatic registration-mechanism. Let this latter include a registration-strip, moved by a clockwork, upon which a mark is made by tripping the counter. True, from the point of view of quantum mechanics this total system is very complex and its configuration space is of very high dimension. But there is in principle no objection to treating this entire system from the standpoint of quantum mechanics... The reason for the introduction of the system supplemented by the registration-mechanism lies in the following. The location of the mark on the registration-strip is a fact which belongs entirely within the sphere of macroscopic concepts, in contradistinction to the instant of disintegration of a single atom.\n> \n> \n> \nThe way Einstein describes the situation stresses the mathematical equivalence of the example of the Schrödinger cat, and the quantum measurement problem that involves a macroscopic ‘apparatus’. It is possible to represent systems like the Schrödinger cat mathematically in a straightforward way; the initial state of the system while the cat is being placed in the box along with the undecayed atoms is $\\left|S\\_i\\right>=\\left|A+\\right>\\left|CL\\right>$ where we denoted ‘*A* + ’ for the undecayed atom, and ‘CL’ for the live cat. This is a product state. We have written the total state as simply the state of the atom AND the state of the cat together. If there is definite knowledge that the atom has decayed, then of course the cat is dead soon. So, given"},"data_id":{"kind":"string","value":"arxiv_0002055"}}},{"rowIdx":499,"cells":{"content":{"kind":"string","value":"Machine Learning for Spatiotemporal Sequence Forecasting: A Survey\n==================================================================\nreal-world phenomena are spatiotemporal, such as the traffic flow, the diffusion of air pollutants and the regional rainfall. Correctly predicting the future of these spatiotemporal systems based on the past observations is essential for a wide range of scientific studies and real-life applications like traffic management, precipitation nowcasting, and typhoon alert systems.\nIf there exists an accurate numerical model of the dynamical system, which usually happen in the case that we have fully understood its rules, forecasting can be achieved by first finding the initial condition of the model and then simulate it to get the future predictions . However, for some complex spatiotemporal dynamical systems like atmosphere, crowd, and natural videos, we are still not entirely clear about their inner mechanisms. In these situations, machine learning based methods, by which we can try to learn the systems’ inner “laws” based on the historical data, have proven helpful for making accurate predictions . Moreover, recent studies have shown that the “laws” inferred by machine learning algorithms can further be used to guide other tasks like classification and control. For example, controlling a robotic arm will be much easier once you can anticipate how it will move after performing specific actions .\n[tb!] [tbl:type-spatiotemporal-forecasting]\n| Problem Name | Coordinates | Measurements |\n| --- | --- | --- |\n| Trajectory Forecasting of Moving Point Cloud | Changing | Fixed/Changing |\n| Spatiotemporal Forecasting on Regular Grid | Fixed regular grid | Changing |\n| Spatiotemporal Forecasting on Irregular Grid | Fixed irregular grid | Changing |\nIn this paper, we review these machine learning based methods for spatiotemporal sequence forecasting. We define a length-*T* spatiotemporal sequence as a series of matrices **X**1 : *T* = [**X**1, **X**2, ...**X***T*]. Each **X***t* ∈ R*K* × (*D* + *E*) contains a set of coordinates and their corresponding measurements. Here, *K* is the number of coordinates, *D* is the number of measurements, and *E* is the dimension of the coordinate. We further denote the measurement part and coordinate part of **X***t* as **M***t* ∈ R*K* × *D* and **C***t* ∈ R*K* × *E*. For many forecasting problems, we can use auxiliary information to enhance the prediction accuracy. For example, regarding wind speed forecasting, the latitudes and past temperatures are also helpful for predicting the future wind speeds. We denote the auxiliary information available at timestamp *t* as A*t*. The *Spatiotemporal Sequence Forecasting* (STSF) problem is to predict the length-*L* (*L* > 1) sequence in the future given the previous observations plus the auxiliary information that is allowed to be empty. The mathematical form is given in .\n \n$$\\label{eq:stsf}\n\\begin{aligned}\n \\mathbf{\\hat{X}}\\_{t+1:t+L} = \\operatorname\\*{arg\\,max}\\_{\\mathbf{X}\\_{t+1:t+L}} p(\\mathbf{X}\\_{t+1:t+L} \\mid \\mathbf{X}\\_{1:t}, \\mathcal{A}\\_{t}).\n\\end{aligned}$$ \nHere, we limit STSF to only contain problems where the input and output are both sequences with multiple elements. According to this definition, problems like video generation using a static image and dense optical flow prediction , in which the input or the output contain a single element, will not be counted.\nBased on the characteristics of the coordinate sequence **C**1 : *T* and the measurement sequence **M**1 : *T*, we classify STSF into three categories as shown in Table [tbl:type-spatiotemporal-forecasting]. If the coordinates are changing, the STSF problem is called *Trajectory Forecasting of Moving Point Cloud* (TF-MPC). Problems like human trajectory prediction and human dynamics prediction fall into this category. We need to emphasize here that the entities in moving point cloud can not only be human but also be general moving objects. If the coordinates are fixed, we only need to predict the future measurements. Based on whether the coordinates lie in a regular grid, we can get the other two categories, called *STSF on Regular Grid* (STSF-RG) and *STSF on Irregular Grid* (STSF-IG). Problems like precipitation nowcasting , crowd density prediction and video prediction , in which we have dense observations, are STSF-RG problems. Problems like air quality forecasting , influenza prediction and traffic speed forecasting , in which the monitoring stations (or sensors) spread sparsely across the city, are STSF-IG problems. We need to emphasize that although the measurements in TF-MPC can also be time-variant due to factors like appearance change, most models for TF-MPC deal with the fixed-measurement case and only predict the coordinates. We thus focus on the fixed-measurement case in this survey.\nHaving both the characteristics of spatial data like images and temporal data like sentences and audios, spatiotemporal sequences contain information about “what”, “when”, and “where” and provide a comprehensive view of the underlying dynamical system. However, due to the complex spatial and temporal relationships within the data and the potential long forecasting horizon, spatiotemporal sequence forecasting imposes new challenges to the machine learning community. The first challenge is how to learn a model for multi-step forecasting. Early researches in time-series forecasting have investigated two approaches: *Direct Multi-step* (DMS) estimation and *Iterated Multi-step* (IMS) estimation. The DMS approach directly optimizes the multi-step forecasting objective while the IMS approach learns a one-step-ahead forecaster and iteratively applies it to generate multi-step predictions. However, choosing between DMS and IMS involves a trade-off among forecasting bias, estimation variance, the length of the prediction horizon and the model’s nonlinearity . Recent studies have tried to find the mid-ground between these two approaches . The second challenge is how to model the spatial and temporal structures within the data adequately. In fact, the number of free parameters of a length-*T* spatiotemporal sequence can be up to *O*(*K**T**D**T*) where *K* is usually above thousands for STSF-RG problems like video prediction. Forecasting in such a high dimensional space is impossible if the model cannot well capture the data’s underlying structure. Therefore, machine learning based methods for STSF, including both classical methods and deep learning based methods, have special designs in their model architectures for capturing these spatiotemporal correlations.\nIn this paper, we organize the literature about STSF following these two challenges. Section [ch:multistep] covers the general learning strategies for multi-step forecasting including IMS, DMS, boosting strategy and scheduled sampling. Section [ch:method-classical] reviews the classical methods for STSF including the feature-based methods, state-space models, and Gaussian process models. Section [ch:dl] reviews the deep learning based methods for STSF including deep temporal generative models and feed-forward and recurrent neural network-based methods. Section [ch:discussion] summarizes the survey and discusses some future research directions.\nLearning Strategies for Multi-step Forecasting\n==============================================\nCompared with single-step forecasting, learning a model for multi-step forecasting is more challenging because the forecasting output $\\mathbf{\\hat{X}}\\_{t+1:t+L}$ is a sequence with non-i.i.d elements. In this section, we first introduce and compare two basic strategies called IMS and DMS and then review two extensions that bridge the gap between these two strategies. To simplify the notation, we keep the following notation rules in this and the following sections. We denote the information available at timestamp *t* as F*t* = {**X**1 : *t*, A*t*}. Learning an STSF model is thus to train a model, which can be either probabilistic or deterministic, to make the *L*-step-ahead prediction $\\mathbf{\\hat{X}}\\_{t+1:t+L}$ based on F*t*. We also denote the ground-truth at timestamp *t* as $\\mathbf{\\tilde{X}}\\_t$ and recursively applying the basic function *f*(*x*) for *h* times as *f*(*h*)(*x*). Setting the *k*th subscript to be “:” means to select all elements at the *k*th dimension. Setting the *k*th subscript to a letter *i* means to select the *i*th element at the *k*th dimension. For example, for matrix **X**, **X***i*, : means the *i*th row, **X**: , *j* means the *j*th column, and **X***i*, *j* means the (*i*, *j*)th element. The flattened version of matrices **A**, **B** and **C** is denoted as vec(**A**, **B**, **C**).\nIterative Multi-step Estimation\n-------------------------------\nThe IMS strategy trains a one-step-ahead forecasting model and iteratively feeds the generated samples to the one-step-ahead forecaster to get the multi-step-ahead prediction. There are two types of IMS models, the deterministic one-step-ahead forecasting model **X***t* = *m*(F*t*; *ϕ*) and the probabilistic one-step-ahead forecasting model *p*(**X***t* ∣ F*t*; *ϕ*). For the deterministic forecasting model, the forecasting process and the training objective of the deterministic forecaster are shown in and respectively where *d*(*x̃*, *x*) measures the distance between *x̃* and *x*.\n \n$$\\label{eq:ims-testing}\n \\begin{aligned}\n \\mathbf{\\hat{X}}\\_{t+h} &= m^{(h)}(\\mathcal{F}\\_t;\\phi),~~\\forall 1 \\leq h \\leq L.\\\\\n \\end{aligned}$$ \n \n$$\\label{eq:ims-training-deterministic}\n \\phi^\\star = \\operatorname\\*{arg\\,min}\\_\\phi \\mathbb{E}\\left[d(\\mathbf{\\tilde{X}}\\_{t+1}, m(\\mathcal{F}\\_t;\\phi))\\right].$$ \nFor the probabilistic forecasting model, we use the chain-rule to get the probability of the length-*L* prediction, which is given in , and estimate the parameters by *Maximum Likelihood Estimation* (MLE).\n \n$$\\label{eq:ims-testing-probabilistic}\n \\begin{aligned}\n &p(\\mathbf{X}\\_{t+1:t+L}\\mid \\mathcal{F}\\_t;\\phi) = p(\\mathbf{X}\\_{t+1} \\mid \\mathcal{F}\\_t;\\phi) \\\\\n &\\qquad\\prod\\_{i=2}^L p(\\mathbf{X}\\_{t+i} \\mid \\mathcal{F}\\_t, \\mathbf{X}\\_{t+1:t+i-1};\\phi).\\\\\n \\end{aligned}$$ \nWhen the one-step-ahead forecaster is probabilistic and nonlinear, the MLE of is generally intractable. In this case, we can draw samples from *p*(**X***t* + 1 : *t* + *L* ∣ F*t*; *ϕ*) by techniques like *Sequential Monte-Carlo* (SMC) and predict based on these samples.\nThere are two advantages of the IMS approach: 1) The one-step-ahead forecaster is easy to train because it only needs to consider the one-step-ahead forecasting error and 2) we can generate predictions of an arbitrary length by recursively applying the basic forecaster. However, there is a discrepancy between training and testing in IMS. In the training process, we use the ground-truths of the previous *L* − 1 steps to generate the *L*th-step prediction. While in the testing process, we feed back the model predictions instead of the ground-truths to the forecaster. This makes the model prone to accumulative errors in the forecasting process . For the deterministic forecaster example above, the optimal muti-step-ahead forecaster, which can be obtained by minimizing $\\mathbb{E}\\left[\\sum\\_{h=1}^L d(\\mathbf{\\tilde{X}}\\_{t+h}, m^{(h)}(\\mathcal{F}\\_t;\\phi))\\right]$, is not the same as recursively applying the optimal one-step-ahead forecaster defined in when *m* is nonlinear. This is because the forecasting error at earlier timestamps will propagate to later timestamps .\nDirect Multi-step Estimation\n----------------------------\nThe main motivation behind DMS is to avoid the error drifting problem in IMS by directly minimizing the long-term prediction error. Instead of training a single model, DMS trains a different model *m**h* for each forecasting horizon *h*. There can thus be *L* models in the DMS approach. When the forecasters are deterministic, the forecasting process and training objective are shown in and respectively where *d*( ⋅ , ⋅ ) is the distance measure.\n \n$$\\label{eq:dms-testing}\n \\begin{aligned}\n \\mathbf{\\hat{X}}\\_{t+h} &= m\\_h(\\mathcal{F}\\_t;\\phi\\_L), \\forall 1 \\leq h \\leq L.\\\\\n \\end{aligned}$$ \n \n$$\\label{eq:dms-training}\n \\begin{aligned}\n \\phi\\_1^\\star,..., \\phi\\_L^\\star &= \\operatorname\\*{arg\\,min}\\_{\\phi\\_1,...,\\phi\\_L} \\mathbb{E}\\left[d(\\mathbf{\\tilde{X}}\\_{t+1:t+L}, \\mathbf{\\hat{X}}\\_{t+1:t+L})\\right].\n \\end{aligned}$$ \n \n$$\\label{eq:dms-training-recursive}\n \\begin{aligned}\n \\phi^\\star = \\operatorname\\*{arg\\,min}\\_{\\phi} \\mathbb{E}\\left[\\sum\\_{h=1}^L d(\\mathbf{\\tilde{X}}\\_{t+h}, m^{(h)}(\\mathcal{F}\\_t;\\phi))\\right].\n \\end{aligned}$$ \nTo disentangle the model size from the number of forecasting steps *L*, we can also construct *m**h* by recursively applying the one-step-ahead forecaster *m*1. In this case, the model parameters {*ϕ*1, ..., *ϕ**L*} are shared and the optimization problem turns into . We need to emphasize here that , which optimizes the multi-step-ahead forecasting error, is different from , which only minimizes the one-step-ahead forecasting error. This construction method is widely adopted in deep learning based methods .\nComparison of IMS and DMS\n-------------------------\nChevillon compared the IMS strategy and DMS strategy. According to the paper, DMS can lead to more accurate forecasts when 1) the model is misspecified, 2) the sequences are non-stationary, or 3) the training set is too small. However, DMS is more computationally expensive than IMS. For DMS, if the *ϕ**h*s are not shared, we need to store and train *L* models. If the *ϕ**h*s are shared, we need to recursively apply *m*1 for *O*(*L*) steps . Both cases require larger memory storage or longer running time than solving the IMS objective. On the other hand, the training process of IMS is easy to parallelize since each forecasting horizon can be trained in isolation from the others . Moreover, directly optimizing a *L*-step-ahead forecasting loss may fail when the forecasting model is highly nonlinear, or the parameters are not well-initialized .\nDue to these trade-offs, models for STSF problems choose the strategies that best match the problems’ characteristics. Generally speaking, DMS is preferred for short-term prediction while IMS is preferred for long-term forecast. For example, the IMS strategy is widely adopted in solving TF-MPC problems where the forecasting horizons are long, which are generally above 12 and can be as long as 100 . The DMS strategy is used in STSF-RG problems where the dimensionality of data is usually very large and only short-term predictions are required . For STSF-IG problems, some works adopt DMS when the overall dimensionality of the forecasting sequence is acceptable for the direct approach .\nOverall, IMS is easier to train but less accurate for multi-step forecasting while DMS is more difficult to train but more accurate. Since IMS and DMS are somewhat complementary, later studies have tried to bridge the gap between these two approaches. We will introduce two representative strategies in the following two sections.\nBoosting Strategy\n-----------------\nThe boosting strategy proposed in combines the IMS and DMS for univariate time series forecasting. The strategy boosts the linear forecaster trained by IMS with several small and nonlinear adjustments trained by DMS. The model assumption is shown in .\n \n*x̃**t* + *h* = *m*(*h*)(F*t*; *ϕ*) + ∑*i* = 1*I**h**ν**l**i*(*x̃**t* − *j*, *x̃**t* − *k*; *ψ**i*) + *e**t*, *h*. \nHere, *m*(F*t*; *ϕ*) is the basic one-step-ahead forecasting model, *l**i*(*x̃**t* − *j*, *x̃**t* − *k*; *ψ**i*) is the chosen weak learner at step *i*, *I**h* is the boosting iteration number, *ν* is the shrinkage factor and *e**t*, *h* is the error term. The author set the base linear learner to be the auto-regressive model and set the weak nonlinear learners to be the penalized regression splines. Also, the weak learners were designed to only account for the interaction between two elements in the observation history. The training method is similar to the gradient boosting algorithm . *m*(F*t*; *ϕ*) is first trained by minimizing the IMS objective . After that, the weak-learners are fit iteratively to the gradient residuals.\nThe author also performed a theoretical analysis of the bias and variance of different models trained with DMS, IMS and the boosting strategy for 2-step-ahead prediction. The result shows that the estimation bias of DMS can be arbitrarily small while that of the IMS will be amplified for larger forecasting horizon. The model estimated by the boosting strategy also has a bias, but it is much smaller than the IMS approach due to the compensation of the weak learners trained by DMS. Also, the variance of the DMS approach depends on the variability of the input history F*t* and the model size which will be very large for complex and nonlinear models. The variance of the boosting strategy is smaller because the basic model is linear and the weak-learners only consider the interaction between two elements.\nAlthough the author only proposed the boosting strategy for the univariate case, the method, which lies in the mid-ground between IMS and DMS, should apply to the STSF problems with multiple measurements, which leads to potential future works as discussed in Section [ch:discussion].\nScheduled Sampling\n------------------\nThe idea of *Scheduled Sampling* (SS) is first to train the model with IMS and then gradually replaces the ground-truths in the objective function with samples generated by the model. When all ground-truth samples are replaced with model-generated samples, the training objective becomes the DMS objective. The generation process of SS is described in :\n \n$$\\label{eq:ss-training}\n \\begin{aligned}\n &\\forall 1 \\leq h \\leq L,\\\\\n &\\mathbf{\\hat{X}}\\_{t+h} \\sim p(\\mathbf{X} \\mid \\mathcal{F}\\_t, \\mathbf{\\bar{X}}\\_{t+1:t+h-1};\\phi),\\\\\n &\\mathbf{\\bar{X}}\\_{t+h} = (1 - \\tau\\_{t+h}) \\mathbf{\\hat{X}}\\_{t+h} + \\tau\\_{t+h} \\mathbf{\\tilde{X}}\\_{t+h},\\\\\n &\\tau\\_{t+h} \\sim B(1, \\epsilon\\_k).\n \\end{aligned}$$ \nHere, $\\mathbf{\\hat{X}}\\_{t+h}$ and $\\mathbf{\\tilde{X}}\\_{t+h}$ are the generated sample and the ground-truth at forecasting horizon *h*, correspondingly. $p(\\mathbf{X} \\mid \\mathcal{F}\\_t, \\mathbf{\\bar{X}}\\_{t+1:t+h-1};\\phi)$ is the basic one-step-ahead forecasting model. *τ**t* + *h* is a random variable following the binomial distribution, which controls whether to use the ground-truth or the generated sample. *ε**k* is the probability of choosing the ground-truth at the *k*th iteration. In the training phase, SS minimizes the distance between $\\mathbf{\\hat{X}}\\_{t+1:t+L}$ and $\\mathbf{\\tilde{X}}\\_{t+1:t+L}$, which is shown in . In the testing phase, *τ* is fixed to 0, meaning that the model-generated samples are always used.\n \n$$\\label{eq:ss-objective}\n\\begin{aligned}\n&\\mathbb{E}\\_{p(\\mathbf{\\hat{X}}\\_{t+1:t+L}\\mid \\mathcal{F}\\_t;\\phi)}\\big[d(\\mathbf{\\hat{X}}\\_{t+1:t+L}, \\mathbf{\\tilde{X}}\\_{t+1:t+L}) \\big] \\\\\n&= \\mathbb{E}\\_{p(\\tau\\_{t+1:t+L-1})} \\big[d(\\mathbf{\\hat{X}}\\_{t+1:t+L}, \\mathbf{\\tilde{X}}\\_{t+1:t+L})\\\\\n&\\qquad\\qquad\\prod\\_{h=1}^L p(\\mathbf{\\hat{X}}\\_{t+1} \\mid \\mathcal{F}\\_t, \\mathbf{\\bar{X}}\\_{t+1:t+h-1};\\phi)\\big].\n\\end{aligned}$$ \nIf *ε**k* equals to 1, the ground-truth will always be chosen and the objective function will be the same as in the IMS strategy. If *ε**k* is 0, the generated sample will always be chosen and the optimization objective will be the same as in the DMS strategy. Based on this observation, the author proposed to gradually decay *ε**k* during training to make the optimization objective shift smoothly from IMS to DMS, which is a type of *curriculum learning* . The paper also provided some ways to change the sampling ratio. One issue of SS is that the expectation over *τ**t* + *h*s in makes the loss function non-differentiable. The original paper obviates the problem by treating *X̂**t* + *h*s as constants, which does not optimize the true objective .\nThe SS strategy is widely adopted in DL models for STSF. In , the author applied SS to video prediction and showed that the model trained via SS outperformed the model trained via IMS. In , the author proposed a training strategy similar to SS except that the training curriculum is adapted from IMS to DMS by controlling the number of the prediction steps instead of controlling the frequency of sampling the ground-truth. In , the author used SS to train models for traffic speed forecasting and showed it outperformed IMS.\nClassical Methods for STSF\n==========================\nIn this section, we will review the classical methods for STSF, including feature-based methods, state space models, and Gaussian process models. These methods are generally based on the shallow models or spatiotemporal kernels. Since a spatiotemporal sequence can be viewed as a multivariate sequence if we flatten the **X***t*s into vectors or treat the observations at every location independently, methods designed for the general multivariate time-series forecasting are naturally applicable to STSF. However, directly doing so ignores the spatiotemporal structure within the data. Therefore, most algorithms introduced in this section have extended this basic approach by utilizing the domain knowledge of the specific task or considering the interactions among different measurements and locations.\nFeature-based Methods\n---------------------\nFeature-based methods solve the STSF by training a regression model based on some human-engineered spatiotemporal features. To reduce the problem size, most feature-based methods treat the measurements at the *K* locations to be independent given the extracted features. The high-level IMS and DMS models of the feature-based methods are given in and respectively.\n \n$$\\begin{aligned}\n \\mathbf{\\hat{X}}\\_{t+1, k} &= f\\_{\\text{IMS}}(\\Psi\\_k(\\mathcal{F}\\_t);\\phi) \\label{eq:feature-based-ims},\\\\\n \\mathbf{\\hat{X}}\\_{t+1:t+L, k} &= f\\_{\\text{DMS}}(\\Psi\\_k(\\mathcal{F}\\_t;\\phi) \\label{eq:feature-based-dms}.\\end{aligned}$$ \nHere, Ψ*k*(F*t*) represents the features at location *k*. Once the feature extraction method Ψ*k* is defined, any regression model *f* can be applied. The feature-based methods are usually adopted in solving STSF-IG problems. Because the observations are sparsely and irregularly distributed in STSF-IG, it is sometimes more straightforward to extract human-engineered features than to develop a purely machine learning based model. Despite their straightforwardness, the feature-based methods have a major drawback. The model’s performance relies heavily on the quality of the human-engineered features. We will briefly introduce two representative feature-based methods for two STSF-IG problems to illustrate the approach.\nOhashi and Torgo proposed a feature-based method for wind-speed forecasting. The author designed a set of features called spatiotemporal indicators and applied several regression algorithms including regression tree, support vector regression, and random forest. There are two kinds of spatiotemporal indicators defined in the paper: the average or weighted average of the observed historical values within a space-time neighborhood and the ratio of two average values calculated with different distance thresholds. The regression model takes the spatiotemporal indicators as the input to generate the one-step-ahead prediction. Experiment results show that the spatiotemporal indicators are helpful for enhancing the forecasting accuracy.\nZheng et al. proposed a feature-based method for air quality forecasting. The author adopted the DMS approach and combined multiple predictors, including the temporal predictor, the spatial predictor, and the inflection predictor, to generate the features at location *k* and timestamp *t*. The temporal predictor focuses on the temporal side of the data and uses the measurements within a temporal sliding window to generate the features. The spatial predictor focuses on the spatial side of the data and uses the previous measurements within a nearby region to generate the features. Also, to deal with the sparsity and irregularity of the coordinates, the author partitioned the neighborhood region into several groups and aggregated the inner measurements within a group. The inflection predictor extracts the sudden changes within the observations. For the regression model, the temporal predictor uses the linear regression model, the spatial predictor uses a 2-hidden-layer neural network and the inflection predictor used the regression tree model. A regression tree model further combines these predictors.\nState Space Models\n------------------\nThe *State Space Model* (SSM) adopts a generative viewpoint of the sequence. For the observed sequence **X**1 : *T*, SSM assumes that each element **X***t* is generated by a hidden state **H***t* and these hidden states, which are allowed to be discrete in our definition, form a Markov process. The general form of the state space model is given in . Here, *g* is the transition model, *f* is the observation model, **H***t* is the hidden state, $\\bm{\\epsilon}\\_t$ is the noise of the transition model, and $\\bm{\\sigma}\\_t$ is the noise of the observation model. The SSM has a long history in forecasting and many classical time-series models like *Markov Model* (MM), *Hidden Markov Model* (HMM), *Vector Autoregression* (VAR), and *Autoregressive Integrated Moving Average Model* (ARIMA) can be written as SSMs.\n \n$$\\begin{aligned}\n \\label{eq:ssm}\n \\mathbf{H}\\_t &= g(\\mathbf{H}\\_{t-1}, \\bm{\\epsilon}\\_t),\\\\\\nonumber\n \\mathbf{X}\\_t &= f(\\mathbf{H}\\_t, \\bm{\\sigma}\\_t).\\end{aligned}$$ \nGiven the underlying hidden state **H***t*, the observation **X***t* is independent concerning the historical information F*t*. Thus, the posterior distribution of the sequence for the next *L* steps can be written as follows:\n \n$$\\label{eq:ssm:forecasting-posterior}\n \\begin{aligned}\n &p(\\mathbf{X}\\_{t+1:t+L}|\\mathcal{F}\\_{t}) = \\int p(\\mathbf{H}\\_t \\mid \\mathcal{F}\\_t)\\\\\n &\\quad\\prod\\_{i=t+1}^{t+L} p(\\mathbf{X}\\_{i}\\mid \\mathbf{H}\\_{i}) p(\\mathbf{H}\\_{i} \\mid \\mathbf{H}\\_{i-1}) d\\mathbf{H}\\_{t:t+L}\n \\end{aligned}$$ \nThe advantage of SSM for solving STSF is that it naturally models the uncertainty of the dynamical system due to its Bayesian formulation. The posterior distribution encodes the probability of different future outcomes. We can not only get the most likely future but also know the confidence of our prediction by Bayesian inference. Common types of SSMs used in STSF literatures have either discrete hidden states or linear transition models and have limited representational power . In this section, we will focus on these common types of SSMs and leave the review of more advanced SSMs, which are usually coupled with deep architectures, in Section [ch:deep-temporal-generative-model]. We will first briefly review the basics of three common classes of SSMs and then introduce their representative spatiotemporal extensions, including the group-based method, *Space-time Autoregressive Moving Average* (STARIMA), and the low-rank tensor auto-regression.\n### Common Types of SSMs\nIn STSF literature, the following three classes of SSMs are most common.\n**First-order Markov Model:**When we set **H***t* to be **X***t* − 1 in and constrain it to be discrete, the SSM reduces to the first-order MM. Assuming **X***t*s have *C* choices, the first-order MM is determined by the transition matrix **A** ∈ R*C* × *C* where **A***i*, *j* = *p*(**X***t* + 1 = *i* ∣ **X***t* = *j*) and the prior distribution *π* ∈ R*C* where *π**i* = *p*(**X** = *i*). The maximum posterior estimator can be obtained via dynamic programming and the optimal model parameters can be solved in closed form .\n**Hidden Markov Model:**When the states are discrete, the SSM turns into the HMM. Assuming the **H***t*s have *C* choices, the HMM is determined by the transition matrix **A** ∈ R*C* × *C*, the state prior *π* and the class conditional density *p*(**X***t* ∣ **H***t* = *i*). For HMM, the maximum posterior estimator and the optimal model parameters can be calculated via the forward algorithm and the *Expectation Maximization* (EM) algorithm, respectively . It is worth emphasizing that the posterior of HMM is tractable even for non-Gaussian observation models. This is because the states of HMM are discrete and it is possible to enumerate all the possibilities.\n**Linear-Gaussian SSM:**When the states are continuous and both *g* and *f* in are linear-Gaussian, the SSM becomes the *Linear-Gaussian State Space Model* (LG-SSM) :\n \n$$\\label{eq:lg-ssm}\n \\begin{aligned}\n \\mathbf{H}\\_t &= \\mathbf{A}\\_t \\mathbf{H}\\_{t-1} + \\bm{\\epsilon}\\_t,\\\\\n \\mathbf{X}\\_t &= \\mathbf{C}\\_t \\mathbf{H}\\_t + \\bm{\\sigma}\\_t,\\\\\n \\bm{\\epsilon}\\_t &\\sim \\mathcal{N}(0, \\mathbf{Q}\\_t),\\\\\n \\bm{\\sigma}\\_t &\\sim \\mathcal{N}(0, \\mathbf{R}\\_t),\\\\\n \\end{aligned}$$ \nwhere **Q***t* and **R***t* are covariance matrices of the Gaussian noises. A large number of classical time-series models like VAR and ARIMA can be written in the form of LG-SSM . As both the transition model and the observation model are linear-Gaussian, the posterior is also a linear-Gaussian and we can use the *Kalman Filtering* (KF) algorithm to calculate the mean and variance. Similar to HMM, the model parameters can also be learned by EM.\n### Group-based Methods\nGroup-based methods divide the *N* locations into non-intersected groups and train an independent SSM for each group. To be more specific, for the location set *S* = {1, 2, ..., *N*}, the group-based methods assume that it can be decomposed as *S* = ∪ *i* = 1*K**G**i* where $G\\_i \\cap G\\_j = \\varnothing,\\forall i \\neq j$. The measurement sequences that belong to the same group, i.e., *D**i* = {{**X**1, *j*, ..., **X***t*, *j*}, *j* ∈ *G**i*}, are used to fit a single SSM. If *K* is equal to 1, it reduces to the simplest case where the measurement sequences at all locations share the same model.\nThis technique has mainly been applied to TF-MPC problems. Asahara et al. proposed a group-based extension of MM called *Mixed-Markov-chain Model* (MMM) for pedestrian-movement prediction. MMM assumes that the people belonging to a specific group satisfy the same first-order MM. To use the first-order MM, the author manually discretized the coordinates of each person. The group-assignments along with the model parameters are learned by EM. Experiments show that MMM outperforms MM and HMM in this task. Mathew et al. extended the work by using HMM as the base model. Also, in , the groups are determined by off-line clustering algorithms instead of EM.\nGroup-based methods capture the intra-group relationships by model sharing. However, it cannot capture the inter-group correlations. Also, only a single group assignment is considered while there may be multiple grouping relationships in the spatiotemporal data. Moreover, the latent state transitions of SSMs have some internal grouping effects. It is often not necessary to explicitly divide the locations into different groups. Due to these limitations, group-based methods are less mentioned in later research works.\n### STARIMA\nSTARIMA is a classical spatiotemporal extension of the univariate ARIMA model . STARIMA emphasizes the impact of a location’s spatial neighborhoods on its future values. The fundamental assumption is that the future measurements at a specific location *i* is not only related to the past measurements observed at this location but also related to the past measurements observed at its local neighborhoods. STARIMA is designed for univariate STSF-RG and STSF-IG problems where there is a single measurement at each location. If we denote the measurements at timestamp *t* as **x***t* ∈ R*N*, the model of STARIMA is shown in the following:\n \n$$\\label{eq:starima}\n \\begin{aligned}\n \\Delta^d \\mathbf{x}\\_t = &\\sum\\_{k=1}^p \\sum\\_{l=0}^{\\lambda\\_k} \\phi\\_{kl} \\mathbf{W}^{(l)} \\Delta^d \\mathbf{x}\\_{t-k} + \\bm{\\epsilon}\\_t\\\\\n &- \\sum\\_{k=1}^p \\sum\\_{l=0}^{m\\_k} \\theta\\_{kl} \\mathbf{W}^{(l)} \\bm{\\epsilon}\\_{t-k}.\\\\\n \\end{aligned}$$ \nHere, Δ is the temporal difference operator where Δ**x***t* = **x***t* − **x***t* − 1 and Δ*k***x***t* = Δ{Δ*k* − 1**x***t*}. *ϕ**k**l* and *θ**k**l* are respectively the autoregressive parameter and the moving average parameter at temporal lag *k* and spatial lag *l*. *λ**k*, *m**k* are maximum spatial orders. **W**(*l*) is a predefined *N* × *N* matrix of spatial order *l*. $\\bm{\\epsilon}\\_t$ is the normally distributed error vector at time *t* that satisfies the following relationships:\n \n$$\\begin{aligned}\n \\mathbb{E}[\\bm{\\epsilon}\\_t] & = \\bm{0},\\\\\n \\mathbb{E}\\left[\\bm{\\epsilon}\\_t \\bm{\\epsilon}\\_{t+s}^T\\right] & = \\begin{cases}\n \\sigma^2 \\mathbf{I}, &s = 0\\\\\n \\bm{0}, &\\text{otherwise}\n \\end{cases},\\\\\n \\mathbb{E}\\left[\\mathbf{x}\\_t \\bm{\\epsilon}\\_{t+s}^T\\right] &= 0, \\forall s>0.\n \\end{aligned}$$ \n2em\nThe spatial order matrix **W**(*l*) is the key in STARIMA for modeling the spatiotemporal correlations. **W***i*, *j*(*l*) is only non-zero if the site *j* is the *l*th order neighborhood of site *i*. **W**(0) is defined to be the identity matrix. The spatial order relationship in a two-dimensional coordinate system is illustrated in Figure [fig:starima-spatial-order]. However, the **W**(*l*)s need to be predefined by the model builder and cannot vary over-time. Also, the STARIMA still belongs to the class of LG-SSM. It is thus not suitable for modeling nonlinear spatiotemporal systems. Another problem of STARIMA is that it is designed for problems where there is only one measurement at each location and has not considered the case of multiple correlated measurements.\n [fig:starima-spatial-order]\n### Low-rank Tensor Auto-regression\nBahadori et al. proposed to train an auto-regression model with low-rank constraint to better capture the latent structure of the spatiotemporal data. This method is designed for STSF-IG and STSF-RG problems where the coordinates are fixed. Assuming the observed measurements **X***t* ∈ R*N* × *D* satisfy the VAR(*J*) model, the optimization problem of the low-rank tensor auto-regression is given in :\n \n$$\\begin{aligned}\n \\label{eq:low-rank-tensor}\n \\min\\_{\\mathcal{W}}&\\sum\\_{t=J+1}^T \\norm{\\hat{\\mathbf{X}}\\_t - \\mathbf{X}\\_t}\\_F^2 + \\mu \\sum\\_{t=1}^T \\text{tr}(\\hat{\\mathbf{X}}\\_t^T \\mathbf{L} \\hat{\\mathbf{X}}\\_t),\\\\\n s.t.&\\quad r(\\mathcal{W}) \\leq \\rho,\\\\\n \\quad&\\quad\\hat{\\mathbf{X}}\\_{t + 1, k} = \\mathcal{W}\\_{k}\\mathbf{X}\\_{t-J+1:t},~\\forall t, k.\n \\end{aligned}$$ \nHere, W ∈ R*N* × *D* × *J**D* is the three-dimensional weight tensor. **L** ∈ R*D* × *D* is the spatial Laplacian matrix calculated from the fixed coordinate matrix **C** ∈ R*N* × *E*. *r*(W) means the rank of the weight tensor and *ρ* is the maximum rank. The regularization term in the objective function constrains the predicted results to be spatially smooth. *μ* is the strength of this spatial similarity constraint.\nConstraining the rank of the weight tensor imposes an implicit spatiotemporal structure in the data. When the weight tensor is low-rank, the observed sequences can be described with a few latent factors.\nThe optimization problem is non-convex and NP-hard in general due to the low-rank constraint. One standard approach is to relax the rank constraint to a nuclear norm constraint and optimize the relaxed problem. However, this approach is not computationally feasible for solving because of the high dimensionality of the spatiotemporal data. Thus, researchers proposed optimization algorithms to accelerate the learning process.\nThe primary limitation of the low-rank tensor auto-regression is that the regression model is still linear even if we have perfectly learned the weights. We cannot directly apply it to solve forecasting problems that have strong nonlinearities.\nGaussian Process\n----------------\n*Gaussian Process* (GP) is a non-parametric Bayesian model. GP defines a prior *p*(*f*) in the function space and tries to infer the posterior *p*(*f* ∣ *D*) given the observed data . GP assumes the joint distribution of the function values of an arbitrary finite set {*x*1, *x*2, ..., *x**n*}, i.e., *p*(*f*(*x*1), ..., *f*(*x**n*)), is a Gaussian distribution. A typical form of the GP prior is shown in :\n \n*f* ∼ GP(*m*(*x*), *κ*(*x*, *x*′)), \nwhere *m*(*x*) is the mean function and *κ*(*x*, *x*′) is the kernel function.\nIn geostatistics, GP regression is also known as kriging , which is used to interpolate the missing values of the data. When applied to STSF, GP is usually used in STSF-IG and STSF-RG problems where the coordinates are fixed. The general formula of the GP based models is shown in :\n \n$$\\label{eq:gp:type1}\n \\begin{aligned}\n f &\\sim \\mathcal{GP}(m(X), \\kappa\\_\\theta),\\\\\n Y &\\sim p(Y \\mid f(X)).\n \\end{aligned}$$ \nHere, *X* contains the space-time positions of the samples, which is the concatenation of the spatial coordinates and the temporal index. For example, *X**k* = (*t**k*, *i**k*, *j**k*) where *t**k* is the timestamp and (*i**k*, *j**k*) is the spatial coordinate. *κ**θ* is the kernel function parameterized by *θ*. *p*(*Y* ∣ *f*(*X*)) is the observation model. Like SSM, GP is also a generative model. The difference is that GP directly generates the function describing the relationship between the observation and the future while SSM generates the future measurements one-by-one through the Markov assumption. We need to emphasize here that GP can also be applied to solve TF-MPC problems. A well-known example is the *Interacting Gaussian Processes* (IGP) , in which a GP models the movement of each person, and a potential function defined over the positions of different people models their interactions. Nevertheless, the method for calculating the posterior of IGP is similar to that of and we will stick to this model formulation.\nTo predict the measurements given a GP model, we are required to calculate the predictive posterior *p*(*f*⋆ ∣ *X*⋆, *X*D, *Y*D) where {*X*D, *Y*D} are the observed data points and *X*⋆ is the candidate space-time coordinates to infer the measurements. If the observation model is a Gaussian distribution, i.e., *p*(*Y* ∣ *f*(*X*)) = N(*f*(*X*), *σ*2), the posterior distribution is also a Gaussian distribution as shown in .\n \n$$\\label{eq:gp\\_normal}\n \\begin{aligned}\n &p(f^{\\star} \\mid X^{\\star}, X^{\\mathcal{D}}, Y^{\\mathcal{D}}) = \\mathcal{N}(\\bm{\\mu}^\\star, \\bm{\\Sigma}^\\star),\\\\\n &\\mathbf{K} = \\kappa(X^{\\mathcal{D}}, X^{\\mathcal{D}}) + \\sigma^2 \\mathbf{I}\\\\\n &\\bm{\\mu}^\\star =\\kappa(X^{\\star}, X^{\\mathcal{D}})\\mathbf{K}^{-1} Y^{\\mathcal{D}},\\\\\n &\\bm{\\Sigma}^\\star = \\kappa(X^\\star, X^\\star) - \\kappa(X^\\star, X^{\\mathcal{D}}) \\mathbf{K}^{-1} \\kappa(X^\\star, X^{\\mathcal{D}}),\n \\end{aligned}$$ \nIf the observation model is non-Gaussian, which is common for STSF problems , Laplacian approximation can be utilized to approximate the posterior . The inference process of GP is computationally expensive due to the involvement of **K**− 1 and the naive approach requires $O(\\abs{D}^3)$ computations and $O(\\abs{D}^2)$ storage. There are many acceleration techniques to speed up the inference . We will not investigate them in detail in this survey and readers can refer to .\nThe key ingredient of GP is the kernel function. For STSF problems, the kernel function should take both the spatial and temporal correlations into account. The common strategy for defining complex kernels is to combine multiple basic kernels by operations like summation and multiplication. Thus, GP models for STSF generally use different kernels to model different aspects of the spatiotemporal sequence and ensemble them together to form the final kernel function. In , the author proposed to decompose the spatiotemporal kernel as *κ**s**t*((*s*, *t*), (*s*′, *t*′)) = *κ**s*(*s*, *s*′)*κ**t*(*t*, *t*′) where *κ**s* is the spatial kernel and *κ**t* is the temporal kernel . If we denote the kernel matrix corresponding to *κ**s**t*, *κ**s*, *κ**t* as **K***s**t*, **K***s*, **K***t*, this decomposition results in **K***s**t* = **K***s* ⊗ **K***t* where ⊗ is the Kronecker product. Based on this observation, the author utilized the computational benefits of Kronecker algebra to scale up the inference of the GP models . In , the author combined two spatial kernels and one temporal kernel to construct a GP model for seasonal influenza prediction. The author defined the basic spatial kernels as the average of multiple RBF kernels. The temporal kernel was defined to be the sum of the periodic kernel, the Paciorek’s kernel, and the exponential kernel. The final kernel function used in the paper was *κ*final = *κ*space + *κ*time + *κ*time × *κ*space2.\nRemarks\n-------\nIn this section, we reviewed three types of classical methods for solving the STSF problems, feature-based methods, SSMs, and GP models. The feature-based methods are simple to implement and can provide useful predictions in some practical situations. However, these methods rely heavily on the human-engineered features and may not be applied without the help of domain experts. The SSMs assume that the observations are generated by Markovian hidden states. We introduced three common types of SSMs: first-order MM, HMM and LG-SSM. Group-based methods extend SSMs by creating groups among the states or observations. STARIMA extends ARIMA by explicitly modeling the impact of nearby locations on the future. Low-rank tensor autoregression extends the basic autoregression model by adding low-rank constraints on the model parameters. The advantage of these common types of SSMs for STSF is that we can calculate the exact posterior distribution and conduct Bayesian inference. However, the overall nonlinearity of these models are limited and may not be suitable for some complex STSF problems like video prediction. GP models the inner characteristics of the spatiotemporal sequences by spatiotemporal kernels. Although being very powerful and flexible, GP has high computational and storage complexity. Without any optimization, the inference of GP requires $O(\\abs{D}^3)$ computations and $O(\\abs{D}^2)$ storage, which makes it not suitable for cases when a huge number of training instances are available.\nDeep Learning Methods for STSF\n==============================\n*Deep Learning* (DL) is an emerging new area in machine learning that studies the problem of how to learn a hierarchically-structured model to directly map the raw inputs to the desired outputs . Usually, DL model stacks some basic learnable blocks, or layers, to form a deep architecture. The overall network is then trained end-to-end. In recent years, a growing number of researchers have proposed DL models for STSF. To better understand these models, we will start with an introduction of the relevant background of DL and then review the models for STSF.\nPreliminaries\n-------------\n### Basic Building Blocks\nIn this section, we will review the basic blocks that have been used to build DL models for STSF. Since there is a vast literature on DL, the introduction of this part serves only as a brief introduction. Readers can refer to for more details.\n**Restricted Boltzmann Machine:***Restricted Boltzmann Machine* (RBM) is the basic building block of *Deep Generative Models* (DGMs). RBMs are undirected graphical models that contain a layer of observable variables **v** and another layer of hidden variables **h**. These two layers are interconnected and form a bipartite graph. In its simplest form, **h** and **v** are all binary and contain *n**h* and *n**v* nodes. The joint distribution and the conditional distributions are defined in :\n \n$$\\label{eq:rbm\\_basic}\n \\begin{aligned}\n p(\\mathbf{v}, \\mathbf{h}) &= \\frac{1}{Z} \\exp(-E(\\mathbf{v}, \\mathbf{h})),\\\\\n E(\\mathbf{v}, \\mathbf{h}) &= -\\mathbf{b}^T \\mathbf{v} - \\mathbf{c}^T \\mathbf{h} - \\mathbf{v}^T \\mathbf{W} \\mathbf{h},\\\\\n p(\\mathbf{h} \\mid \\mathbf{v}) &= \\prod\\_{j=1}^{n\\_h} \\sigma\\left((2\\mathbf{h}-1)\\circ(\\mathbf{c} + \\mathbf{W}^T\\mathbf{v})\\right)\\_j,\\\\\n p(\\mathbf{v} \\mid \\mathbf{h}) &= \\prod\\_{j=1}^{n\\_v} \\sigma\\left((2\\mathbf{v}-1)\\circ(\\mathbf{b} + \\mathbf{W}\\mathbf{h})\\right)\\_j.\n \\end{aligned}$$ \nHere, **b** is the bias of the visible state, **c** is the bias of the hidden state, **W** is the weight between visible states and hidden states, *σ* is the sigmoid function defined in Table [tbl:activations], and *Z* is the normalization constant for energy-based models, which is also known as the partition function . RBM can be learned by Contrastive Divergence (CD) , which approximates the gradient of log*Z* by drawing samples from *k* cycles of MCMC.\nThe basic form of RBM has been extended in various ways. *Gaussian-Bernoulli RBM* (GRBM) assumes the visible variables are continuous and replaces the Bernoulli distribution with a Gaussian distribution. *Spike and Slab RBM* (ssRBM) maintains two types of hidden variables to model the covariance.\n**Activations:**Activations refer to the element-wise nonlinear mappings **h** = *f*(**x**). Activation functions that are used in the reviewed models are listed in Table [tbl:activations].\n[tb!] [tbl:activations]\n| **Name** | **Formula** |\n| --- | --- |\n| *σ* (sigmoid) | $\\sigma(x) = \\frac{1}{1 + e^{-x}}$ |\n| tanh | $\\text{tanh}(x) = \\frac{e^x - e^{-x}}{e^x + e^{-x}}$ |\n| ReLU | ReLU(*x*) = max(0, *x*) |\n| Leaky ReLU | LReLU(*x*) = max( − *α**x*, *x*) |\n**Sigmoid Belief Network:***Sigmoid Belief Network* (SBN) is another building block of DGM. Unlike RBM which is undirected, SBN is a directed graphical model. SBN contains a hidden binary state **h** and a visible binary state **v**. The basic generation process of SBN is given in the following:\n \n$$\\begin{aligned}\n p(\\mathbf{v}\\_m \\mid \\mathbf{h}) &= \\sigma(\\mathbf{w}\\_m^T \\mathbf{h} + \\mathbf{c}\\_m),\\\\\n p(\\mathbf{h}\\_j) &= \\sigma(\\mathbf{b}\\_j).\n \\end{aligned}$$ \nHere, **b***j*, **c***m* are biases and **w***m* is the weight. Because the posterior distribution is intractable, SBN is learned by methods like *Neural Variational Inference and Learning* (NVIL) and *Wake-Sleep* (WS) algorithm, which train an inference network along with the generation network.\n**Fully-connected Layer:**The *Fully-Connected* (FC) layer is the very basic building block of DL models. It refers to the linear-mapping from the input **x** to the hidden state **h**, i.e, **h** = **W****x** + **b**, where **W** is the weight and **b** is the bias.\n**Convolution and Pooling:**The convolution layer and pooling layer are first proposed to take advantage of the translation invariance property of image data. The convolution layer computes the output by scanning over the input and applying the same set of linear filters. Although the input can have an arbitrary dimensionality , we will just introduce 2D convolution and 2D pooling in this section. For input X ∈ R*C**i* × *H**i* × *W**i*, the output of the convolution layer H ∈ R*C**o* × *H**o* × *W**o*, which is also known as feature map, will be calculated as the following:\n \nH = W \\* X + **b**. \nHere, W ∈ R*C**i* × *C**o* × *K**h* × *K**w* is the weight, *K**h* and *K**w* are the kernel sizes, “ \\* ” denotes convolution, and **b** is the bias. If we denote the feature vector at the (*i*, *j*) position as H: , *i*, *j*, the flattened weight as **W** = vec(W), and the flattened version of the local region inputs as **x**N(*i*, *j*) = vec([X: , *s*, *t* ∣ (*s*, *t*) ∈ N(*i*, *j*)]), can be rewritten as :\n \nH: , *i*, *j* = **W****x**N(*i*, *j*) + **b**. \nThis shows that convolution layer can also be viewed as applying multiple FC layers with shared weight and bias on local regions of the input. Here, the ordered neighborhood set N(*i*, *j*) is determined by hyper-parameters like the kernel size, stride, padding, and dilation . For example, if the kernel size of the convolution filter is 3 × 3, we have N(*i*, *j*) = {(*i* + *s*, *j* + *t*) ∣ − 1 ≤ *s*, *t* ≤ 1}. Readers can refer to for details about how to calculate the neighborhood set N(*i*, *j*) based on these hyper-parameters. Also, the out-of-boundary regions of the input are usually set to zero, which is known as zero-padding.\nLike convolution layer, the pooling layer also scans over the input and applies the same mapping function. Pooling layer generally has no parameter and uses some reduction operations like max, sum, and average for mapping. The formula of the pooling layer is given in the following:\n \nH*k*, *i*, *j* = *g*({X*k*, *s*, *t* ∣ (*s*, *t*) ∈ N(*i*, *j*)}), \nwhere *g* is the element-wise reduction operation.\n**Deconvolution and Unpooling:**The deconvolution and unpooling layer are first proposed to serve as the “inverse” operation of the convolution and the pooling layer. Computing the forward pass of these two layers is equivalent to computing the backward pass of the convolution and pooling layers .\n**Graph Convolution:**Graph convolution generalizes the standard convolution, which is operated over a regular grid topology, to convolution over graph structures . There are two interpretations of graph convolution: the spectral interpretation and the spatial interpretation. In the spectral interpretation , graph convolution is defined by the convolution theorem, i.e., the Fourier transform of convolution is the point-wise product of Fourier transforms:\n \n$$\\label{eq:graph\\_conv}\n\\begin{aligned}\n\\mathbf{x} \\ast\\_G \\mathbf{y} &= F^{-1}(F(\\mathbf{x}) \\circ F(\\mathbf{y})) = \\mathbf{U}(\\mathbf{U}^T \\mathbf{x} \\circ \\mathbf{U}^T \\mathbf{y}).\n\\end{aligned}$$ \nHere, *F*(**x**) = **U***T***x** is the graph Fourier transform. The transformation matrix **U** is defined based on the eigendecomposition of the graph Laplacian. Defferrard et al. defined *U* as the eigenvectors of $\\mathbf{I} - \\mathbf{D}^{-\\frac{1}{2}}\\mathbf{A}\\mathbf{D}^{-\\frac{1}{2}}$, in which **D** is the diagonal degree matrix, **A** is the adjacency matrix, and **I** is the identity matrix.\nIn the spatial interpretation, graph convolution is defined by a localized parameter-sharing operator that aggregates the features of the neighboring nodes, which can also be termed as graph aggregator :\n \n$$\\label{eq:graph\\_conv\\_spatial}\n\\begin{aligned}\n\\mathbf{h}\\_i = \\gamma\\_\\theta(\\mathbf{x}\\_i, \\{\\mathbf{z}\\_{\\mathcal{N}\\_i}\\}),\n\\end{aligned}$$ \nwhere **h***i* is the output feature vector of node *i*, *γ**θ*( ⋅ ) is the mapping function parameterized by *θ*, **x***i* is the input feature of node *i*, and **z**N*i* contains the features of node *i*’s neighborhoods. The mapping function needs to be permutation invariant and dynamically resizing. One example is the pooling aggregator defined in :\n \n$$\\label{eq:graph\\_conv\\_pool}\n\\begin{aligned}\n\\mathbf{h}\\_i &= \\phi\\_o(\\mathbf{x}\\_i \\oplus \\text{pool}\\_{j \\in \\mathcal{N}\\_i}(\\phi\\_v(\\mathbf{z}\\_j))),\n\\end{aligned}$$ \nwhere *ϕ**o* and *ϕ**v* are mapping functions and ⊕ means concatenation. Compared with the spectral interpretation, the spatial interpretation does not require the expensive eigendecomposition and is more suitable for large graphs. Moreover, Defferrard et al. and Kipf & Welling proposed accelerated versions of the spectral graph convolution that could be interpreted from the spatial perspective.\n### Types of DL Models for STSF\nBased on the characteristic of the model formulation, we can categorize DL models for STSF into two major types: *Deep Temporal Generative Models* (DTGMs) and *Feedforward Neural Network* (FNN) and *Recurrent Neural Network* (RNN) based methods. In the following, we will briefly introduce the basics about DTGM, FNN, and RNN.\n**DTGM:**DTGMs are DGMs for temporal sequences. DTGM adopts a generative viewpoint of the sequence and tries to define the probability distribution *p*(**X**1 : *T*). The aforementioned RBM and SBN are building-blocks of DTGM.\n**FNN:**FNNs, also called deep feedforward networks , refer to deep learning models that construct the mapping **y** = *f*(**x**; *θ*) by stacking various of the aforementioned basic blocks such as FC layer, convolution layer, deconvolution layer, graph convolution layer, and activation layer. Common types of FNNs include *Multilayer Perceptron* (MLP) , which stacks multiple FC layers and nonlinear activations, *Convolutional Neural Network* (CNN) , which stacks multiple convolution and pooling layers, and graph CNN , which stacks multiple graph convolution layers. The parameters of FNN are usually estimated by minimizing the loss function plus some regularization terms. Usually, the optimization problem is solved via stochastic gradient-based methods , in which the gradient is computed by backpropagation .\n**RNN:**RNN generalizes the structure of FNN by allowing cyclic connections in the network. These recurrent connections make RNN especially suitable for modeling sequential data. Here, we will only introduce the discrete version of RNN which updates the hidden states using **h***t* = *f*(**h***t* − 1, **x***t*; *θ*) . These hidden states can be further used to compute the output and define the loss function. One example of an RNN model is shown in . The model uses the tanh activation and the fully-connected layer for recurrent connections. It uses another full-connected layer to get the output:\n \n$$\\label{eq:vanilla-rnn}\n \\begin{aligned}\n \\mathbf{h}\\_t &= \\text{tanh}(\\mathbf{W}\\_h \\mathbf{h}\\_{t-1} + \\mathbf{W}\\_x \\mathbf{x}\\_t + b),\\\\\n \\mathbf{o}\\_t &= \\mathbf{W}\\_o \\mathbf{h}\\_t + \\mathbf{c}.\n \\end{aligned}$$ \nSimilar to FNNs, RNNs are also trained by stochastic gradient based methods. To calculate the gradient, we can first unfold an RNN to a FNN and then perform backpropagation on the unfolded computational graph. This algorithm is called *Backpropagation Through Time* (BPTT) .\nDirectly training the RNN shown in with BPTT will cause the vanishing and exploding gradient problems . One way to solve the vanishing gradient problem is to use the *Long-Short Term Memory* (LSTM) , which has a cell unit that is designed to capture the long-term dependencies. The formula of LSTM is given in the following:\n \n$$\\begin{aligned}\n \\mathbf{i}\\_t &= \\sigma(\\mathbf{W}\\_{xi} \\mathbf{x}\\_t + \\mathbf{W}\\_{hi} \\mathbf{h}\\_{t-1} + \\mathbf{b}\\_i),\\\\\n \\mathbf{f}\\_t &= \\sigma(\\mathbf{W}\\_{xf} \\mathbf{x}\\_t + \\mathbf{W}\\_{hf} \\mathbf{h}\\_{t-1} + \\mathbf{b}\\_f ),\\\\\n \\mathbf{c}\\_t &= \\mathbf{f}\\_t \\circ \\mathbf{c}\\_{t-1} +\\\\\n &\\qquad \\mathbf{i}\\_t \\circ \\text{tanh}(\\mathbf{W}\\_{xc} \\mathbf{x}\\_t + \\mathbf{W}\\_{hc} \\mathbf{h}\\_{t-1} + \\mathbf{b}\\_c),\\\\\n \\mathbf{o}\\_t &= \\sigma(\\mathbf{W}\\_{xo} \\mathbf{x}\\_t + \\mathbf{W}\\_{ho} \\mathbf{h}\\_{t-1} + \\mathbf{b}\\_o),\\\\\n \\mathbf{h}\\_t &= \\mathbf{o}\\_t \\circ \\text{tanh}(\\mathbf{c}\\_t).\n \\end{aligned}$$ \nHere, **c***t* is the memory cell and **i***t*, **f***t*, **o***t* are the input gate, the forget gate, and the output gate, respectively. The memory cell is accessed by the next layer when **o***t* is turned on. Also, the cell can be updated or cleared when **i***t* or **f***t* is activated. By using the memory cell and gates to control information flow, the gradient will be trapped in the cell and will not vanish over time, which is also known as the *Constant Error Carousel* (CEC) . LSTM belongs to a broader category of RNNs called gated RNNs . We will not introduce other architectures like *Gated Recurrent Unit* (GRU) in this survey because LSTM is the most representative example. To simplify the notation, we will also denote the transition rule of LSTM as **h***t*, **c***t* = LSTM(**h***t* − 1, **c***t* − 1, **x***t*; **W**).\nDeep Temporal Generative Models\n-------------------------------\nIn this section, we will review DTGMs for STSF. Similar to SSM and GP model, DTGM also adopts a generative viewpoint of the sequences. The characteristics of DTGMs is that models belonging to this class can all be stacked to form a deep architecture, which enables them to model complex system dynamics. We will survey four DTGMs in the following. The general structures of these models are shown in Figure [fig:dtgm-structures].\n**Temporal Restricted Boltzmann Machine:**Sutskever & Hinton proposed *Temporal Restricted Boltzmann Machine* (TRBM), which is a temporal extension of RBM. In TRBM, the bias of the RBM in the current timestamp is determined by the previous *m* RBMs. The conditional probability of TRBM is given in the following:\n \n$$\\begin{aligned}\n p(\\mathbf{v}\\_t, \\mathbf{h}\\_t \\mid \\mathbf{\\hat{v}}\\_t, \\mathbf{\\hat{h}}\\_{t}) &= \\frac{1}{Z}\\exp\\left(-E(\\mathbf{v}\\_t, \\mathbf{h}\\_t \\mid \\mathbf{\\hat{v}}\\_{t}, \\mathbf{\\hat{h}}\\_{t})\\right),\\\\\n E(\\mathbf{v}\\_t, \\mathbf{h}\\_t \\mid \\mathbf{\\hat{v}}\\_{t}, \\mathbf{\\hat{h}}\\_{t}) &=\n -f\\_{b\\_v}(\\mathbf{\\hat{v}}\\_{t})^T \\mathbf{v}\\_t -f\\_{b\\_h}(\\mathbf{\\hat{v}}\\_{t}, \\mathbf{\\hat{h}}\\_{t})^T \\mathbf{h}\\_t \\\\\n &~~~~- \\mathbf{v}\\_t^T\\mathbf{W}\\mathbf{h}\\_t,\\\\\n f\\_{b\\_v}(\\mathbf{\\hat{v}}\\_{t}) &= \\mathbf{a} + \\sum\\_{i=1}^m \\mathbf{A}\\_i^T \\mathbf{v}\\_{t-i},\\\\\n f\\_{b\\_h}(\\mathbf{\\hat{v}}\\_{t}, \\mathbf{\\hat{h}}\\_{t}) =& \\mathbf{b} + \\sum\\_{i=1}^m \\mathbf{B}\\_i^T \\mathbf{v}\\_{t-i} + \\sum\\_{i=1}^m \\mathbf{C}\\_i^T \\mathbf{h}\\_{t-i},\n \\end{aligned}$$ \nwhere $\\mathbf{\\hat{v}}\\_t = \\mathbf{v}\\_{t-m:t-1}$ and $\\mathbf{\\hat{h}}\\_t = \\mathbf{h}\\_{t-m:t-1}$. The joint distribution is modeled as the multiplication of the conditional probabilities, i.e., $p(\\mathbf{v}\\_{1:T}, \\mathbf{h}\\_{1:T}) = \\prod\\_{i=1}^T p(\\mathbf{v}\\_t, \\mathbf{h}\\_t \\mid \\mathbf{\\hat{v}}\\_{t}, \\mathbf{\\hat{h}}\\_{t})$.\nThe inference of TRBM is difficult and involves evaluating the ratio of two partition functions . The original paper adopted a heuristic inference procedure . The results on a synthetic bouncing ball dataset show that a two-layer TRBM outperforms a single-layer TRBM in this task.\n**Recurrent Temporal Restricted Boltzmann Machine:**Later, Sutskever et al. proposed the *Recurrent Temporal Restricted Boltzmann Machine* (RTRBM) that extends the idea of TRBM by conditioning the parameters of the RBM at timestamp *t* on the output of an RNN. The model of RTRBM is described in the following:\n \n$$\\begin{aligned}\n p(\\mathbf{v}\\_t, \\mathbf{h}\\_t \\mid \\mathbf{r}\\_{t-1}) &= \\frac{1}{Z}\\exp\\left(-E(\\mathbf{v}\\_t, \\mathbf{h}\\_t \\mid \\mathbf{r}\\_{t-1})\\right),\\\\\n E(\\mathbf{v}\\_t, \\mathbf{h}\\_t \\mid \\mathbf{r}\\_{t-1}) &= -\\mathbf{h}\\_t^T \\mathbf{W} \\mathbf{v}\\_t - \\mathbf{c}^T \\mathbf{v}\\_t - \\mathbf{b}^T \\mathbf{h}\\_t \\\\\n &~~~~- \\mathbf{h}\\_t^T \\mathbf{U} \\mathbf{r}\\_{t-1},\\\\\n \\mathbf{r}\\_t &= \\begin{cases}\n \\sigma(\\mathbf{W} \\mathbf{v}\\_t + \\mathbf{U} \\mathbf{r}\\_{t-1} + \\mathbf{b}),& t > 1\\\\\n \\sigma(\\mathbf{W} \\mathbf{v}\\_t + \\mathbf{b}\\_{init}),& t = 1\n \\end{cases}.\n \\end{aligned}$$ \nHere, **r***t* encodes the visible states **v**1 : *t*. The inference is simpler in RTRBM than TRBM because the model is the same as the standard RBM once **r***t* − 1 is given. Similar to RBM, RTRBM can also be learned by CD. The author also extended RTRBM to use GRBM for modeling continuous time-series. Qualitative experiments on the synthetic bouncing ball dataset show that RTRBM can generate more realistic sequences than TRBM.\n**Structured Recurrent Temporal Restricted Boltzmann Machine:**Mittelman et al. proposed *Structured Recurrent Temporal Restricted Boltzmann Machine* (SRTRBM) that improves upon RTRBM by learning the connection structure based on the spatiotemporal patterns of the input. SRTRBM uses two block-masking matrices to mask the weights between the hidden and visible nodes. The formula of SRTRBM is similar to RTRBM except for the usage of two masking matrices **M***W* and **M***U*:\n \n$$\\begin{aligned}\n &E(\\mathbf{v}\\_t, \\mathbf{h}\\_t \\mid \\mathbf{r}\\_{t-1}) = -\\mathbf{h}\\_t^T\\mathbf{\\hat{W}} \\mathbf{v}\\_t - \\mathbf{c}^T \\mathbf{v}\\_t\\\\\n &\\qquad\\qquad\\qquad\\quad~~~~- \\mathbf{b}^T \\mathbf{h}\\_t - \\mathbf{h}\\_t^T \\mathbf{\\hat{U}} \\mathbf{r}\\_{t-1},\\\\\n &\\mathbf{r}\\_t = \\begin{cases}\n \\sigma(\\mathbf{\\hat{W}} \\mathbf{v}\\_t + \\mathbf{\\hat{U}} \\mathbf{r}\\_{t-1} + \\mathbf{b}),& t > 1\\\\\n \\sigma(\\mathbf{\\hat{W}} \\mathbf{v}\\_t + \\mathbf{b}\\_{init}),& t = 1\n \\end{cases},\\\\\n &\\mathbf{\\hat{W}} = \\mathbf{W} \\circ \\mathbf{M}\\_W,\\\\\n &\\mathbf{\\hat{U}} = \\mathbf{U} \\circ \\mathbf{M}\\_U.\n \\end{aligned}$$ \nHere, **M***W* and **M***U* are block-masking matrices generated by a graph *G* = (*V*; *E*). To construct the graph, the visible units and the hidden units are divided into $\\abs{V}$ nonintersecting subsets. Each node *ε* in the graph is assigned to a subset of hidden units *C**ε**h* and a subset of visible units *C**ε**v*. The ways to separate and assign the units are based on the spatiotemporal coordinates of the data. If there is an edge between two nodes in the graph, the corresponding hidden and visible subsets are connected in SRTRBM.\nIf the masks are fixed, the learning process of SRTRBM is the same as RTRBM. To learn the mask, the paper proposed to use the sigmoid function *σ*(*ν**i**j*) as a soft mask and learn the parameters *ν**i**j*. Also, a sparsity regularizer is added to *ν**i**j*s to encourage sparse connections. The training process is first to fix the mask while updating the other parameters and then update the mask. The author also gives the ssRBM version of SRTBM and RTRBM by replacing RBM with ssRBM while fixing the part of the energy function related to **r***t* to be unchanged. Experiments on human motion prediction, which is a TF-MPC task, and climate data prediction, which is an STSF-IG task, prove that SRTRBM achieves smaller one-step-ahead prediction error than RTRBM and RNN. This shows that using a masked weight matrix to model the spatial and temporal correlations is useful for STSF problems.\n [fig:dtgm-structures]\n**Temporal Sigmoid Belief Network:**Gan et al. proposed a temporal extension of SBN called *Temporal Sigmoid Belief Network* (TSBN). TSBN connects a sequence of SBNs in a way that the bias of the SBN at timestamp *t* depends on the state of the previous SBNs. For a length-*T* binary sequence **v**1 : *T*, TSBN describes the following joint probability, in which we have omitted the biases for simplicity:\n \n$$\\begin{aligned}\n &p\\_\\theta(\\mathbf{v}\\_{1:T}, \\mathbf{h}\\_{1:T}) = p(\\mathbf{h}\\_1) p(\\mathbf{v}\\_1\\mid \\mathbf{h}\\_1) \\\\\n &\\qquad\\qquad\\quad\\prod\\_{t=2}^T p(\\mathbf{h}\\_t \\mid \\mathbf{h}\\_{t-1}, \\mathbf{v}\\_{t-1}) p(\\mathbf{v}\\_t \\mid \\mathbf{h}\\_t, \\mathbf{v}\\_{t-1}),\\\\\n &p(\\mathbf{h}\\_{t, j} =1 \\mid \\mathbf{h}\\_{t-1}, \\mathbf{v}\\_{t-1}) = \\sigma(\\mathbf{A}\\_j^T \\mathbf{h}\\_{t-1}+ \\mathbf{B}\\_j^T \\mathbf{v}\\_{t-1}),\\\\\n &p(\\mathbf{v}\\_{t, j} =1 \\mid \\mathbf{h}\\_{t}, \\mathbf{v}\\_{t-1}) = \\sigma(\\mathbf{C}\\_j^T \\mathbf{h}\\_{t} + \\mathbf{D}\\_j^T \\mathbf{v}\\_{t-1}).\n \\end{aligned}$$ \nAlso, the author extended TSBN to the continuous domain by using a conditional Gaussian for the visible nodes:\n \n$$\\begin{aligned}\n p(\\mathbf{v}\\_t \\mid \\mathbf{h}\\_t, \\mathbf{v}\\_{t-1}) = \\mathcal{N}(\\bm{\\mu}\\_t, \\text{diag}(\\bm{\\sigma}\\_t^2)),\\\\\n \\bm{\\mu}\\_t = \\mathbf{A}^T \\mathbf{h}\\_{t-1} + \\mathbf{B}^T \\mathbf{v}\\_{t-1} + \\mathbf{e},\\\\\n \\log(\\bm{\\sigma}\\_t^2) = \\mathbf{A^\\prime}^T \\mathbf{h}\\_{t-1} + \\mathbf{B^\\prime}^T \\mathbf{v}\\_{t-1} + \\mathbf{e^\\prime}.\n \\end{aligned}$$ \nMultiple single-layer TSBNs can be stacked to form a deep architecture. For a multi-layer TSBN, the hidden states of the lower layers are conditioned on the states of the higher layers. The connection structure of a multi-layer TSBN is given in Figure [fig:dtgm-structures](e). The author investigated both the stochastic and deterministic conditioning for multi-layer models and showed that the deep model with stochastic conditioning works better. Experiment results on the human motion prediction task show a multi-layer TSBN with stochastic conditioning outperforms variants of SRTRBM and RTRBM.\nFNN and RNN based Models\n------------------------\nOne shortcoming of DTGMs for STSF is that the learning and inference processes are often complicated and time-consuming. Recently, there is a growing trend of using FNN and RNN based methods for STSF. Compared with DTGM, FNN and RNN based methods are simpler in the learning and inference processes. Moreover, these methods have achieved good performance in many STSF problems like video prediction , precipitation nowcasting , traffic speed forecasting , human trajectory prediction , and human motion prediction . In the following, we will review these methods.\n### Encoder-Forecaster Structure\nRecall that the goal of STSF is to predict the length-*L* sequence in the future given the past observations. The *Encoder-Forecaster* (EF) structure first encodes the observations into a finite-dimensional vector or a probability distribution using the encoder and then generates the one-step-ahead prediction or multi-step predictions using the forecaster:\n \n$$\\label{eq:ef-both-deterministic}\n \\begin{aligned}\n \\mathbf{s} &= f(\\mathcal{F}\\_t;\\theta\\_1),\\\\\n \\hat{\\mathbf{X}}\\_{t+1:t+L} &= g(\\mathbf{s};\\theta\\_2).\n \\end{aligned}$$ \nHere, *f* is the encoder and *g* is the forecaster. In our definition, we also allow probabilistic encoder and forecaster, where **s** and $\\hat{\\mathbf{X}}\\_{t+1:t+L}$ turn into random variables:\n \n$$\\label{eq:ef-both-probabilstic}\n \\begin{aligned}\n \\mathbf{s} &\\sim f(\\mathcal{F}\\_t;\\theta\\_1),\\\\\n \\hat{\\mathbf{X}}\\_{t+1:t+L} &\\sim \\pi\\_g(\\mathbf{s};\\theta\\_2).\n \\end{aligned}$$ \nUsing probabilistic encoder and forecaster is a way to handle uncertainty and we will review the details in Section [ch:uncertainty]. All of the surveyed FNN and RNN based methods fit into the EF framework. In the following, we will review how previous works design the encoder and the forecaster for different types of STSF problems, i.e., STSF-RG, STSF-IG, and TF-MPC.\n### Models for STSF-RG\nFor STSF-RG problems, the spatiotemporal sequences lie in a regular grid and can naturally be modeled by CNNs, which have proved effective for extracting features from images and videos . Therefore, researchers have used CNNs to build the encoder and the forecaster. In , two input frames are encoded independently to two state vectors by a 2D CNN. The future frame is generated by linearly extrapolating these two state vectors and then mapping the extrapolated vector back to the image space with a series of 2D deconvolution and unpooling layers. Also, to better capture the spatial features, the author proposed the phase pooling layer, which keeps both the pooled values and pooled indices in the feature maps. In , multi-scale 2D CNNs with ReLU activations were used as the encoder and the forecaster. Rather than encoding the frames independently, the author concatenated the input frames along the channel dimension to generate a single input tensor. The input tensor is rescaled to multiple resolutions and encoded by a multi-scale 2D CNN. To better capture the spatiotemporal correlations, Vondrick et al. proposed to use 3D CNNs as the encoder and forecaster. Also, the author proposed to use a 2D CNN to predict the background image and only use the 3D CNN to generate the foreground. Later, Shi et al. compared the performance of 2D CNN and 3D CNN for precipitation nowcasting and showed that 3D CNN outperformed 2D CNN in the experiments. Kalchbrenner et al. proposed the *Video Pixel Network* (VPN) that uses multiple causal convolution layers in the forecaster. Unlike the previous 2D CNN and 3D CNN models, of which the forecaster directly generates the multi-step predictions, VPN generates the elements in the spatiotemporal sequence one-by-one with the help of causal convolution. Experiments show that VPN outperforms the baseline which uses a 2D CNN to generate the predictions directly. Recently, Xu et al. proposed the PredCNN network that stacks multiple dilated causal convolution layers to encode the input frames and predict the future. Experiments show that PredCNN achieves state-of-the-art performance in video prediction.\nBesides using CNNs, researchers have also used RNNs to build models for STSF-RG. The advantage of using RNN for STSF problems is that it naturally models the temporal correlations within the data. Srivastava et al. proposed to use multi-layer LSTM networks as the encoder and the forecaster. To use the LSTM, the author directly flattened the images into vectors. Oh et al. proposed to use 2D CNNs to encode the image frames before feeding into LSTM layers. However, the state-state connection structure of *Fully-Connected LSTM* (FC-LSTM) is redundant for STSF and is not suitable for capturing the spatiotemporal correlations in the data . To solve the problem, Shi et al. proposed the *Convolutional LSTM* (ConvLSTM). ConvLSTM uses convolution instead of full-connection in both the feed-forward and recurrent connections of LSTM. The formula of ConvLSTM is given in the following:\n \n$$\\label{eq:convlstm}\n \\begin{aligned}\n \\mathcal{I}\\_t &= \\sigma(\\mathcal{W}\\_{xi} \\ast \\mathcal{X}\\_t + \\mathcal{W}\\_{hi} \\ast \\mathcal{H}\\_{t-1} + b\\_i),\\\\\n \\mathcal{F}\\_t &= \\sigma(\\mathcal{W}\\_{xf} \\ast \\mathcal{X}\\_t + \\mathcal{W}\\_{hf} \\ast \\mathcal{H}\\_{t-1} + b\\_f ),\\\\\n \\mathcal{C}\\_t &= \\mathcal{F}\\_t \\circ \\mathcal{C}\\_{t-1} +\\\\\n &\\qquad\\mathcal{I}\\_t \\circ tanh(\\mathcal{W}\\_{xc} \\ast X\\_t + W\\_{hc} \\ast \\mathcal{H}\\_{t-1} + b\\_c),\\\\\n \\mathcal{O}\\_t &= \\sigma(\\mathcal{W}\\_{xo} \\ast \\mathcal{X}\\_t + \\mathcal{W}\\_{ho} \\ast \\mathcal{H}\\_{t-1} + b\\_o),\\\\\n \\mathcal{H}\\_t &= \\mathcal{O}\\_t \\circ tanh(\\mathcal{C}\\_t).\n \\end{aligned}$$ \nHere, all the states, cells, and gates are three-dimensional tensors. The author used two ConvLSTM layers as the encoder and the forecaster and showed that 1) ConvLSTM outperforms FC-LSTM when applied to the precipitation nowcasting problem and 2) the two-layer model performs better than the single-layer model. Similar to the way of extending LSTM to ConvLSTM, other types of RNNs, like GRU, can be extended to *Convolutional RNNs* (ConvRNNs) by using convolution in state-state transitions. Since ConvRNN combines the advantage of CNN and RNN, it is widely adopted in later researches as a basic block for building DL models for STSF-RG. There are also attempts to improve the structure of ConvRNN. Shi et al. proposed the *Trajectory GRU* (TrajGRU) model that improves ConvGRU by actively learning the recurrent connection structure. Because the convolutional recurrence structure adopted in ConvRNNs is location-invariant, it is not suitable for modeling location-variant motions, e.g., rotation and scaling. Thus, TrajGRU uses a learned subnetwork to output the recurrent connection structures:\n \n$$\\begin{aligned}\n \\mathcal{U}\\_t, \\mathcal{V}\\_t &= \\gamma(\\mathcal{X}\\_t, \\mathcal{H}\\_{t-1}),\\\\\n \\mathcal{Z}\\_t &= \\sigma(\\mathcal{W}\\_{xz} \\ast \\mathcal{X}\\_t + \\sum\\_{l=1}^L{\\mathcal{W}^l\\_{hz} \\ast \\mathcal{H}\\_{t-1, \\mathcal{U}\\_{t, l}, \\mathcal{V}\\_{t, l}}}),\\\\\n \\mathcal{R}\\_t &= \\sigma(\\mathcal{W}\\_{xr} \\ast \\mathcal{X}\\_t + \\sum\\_{l=1}^L{\\mathcal{W}^l\\_{hr} \\ast \\mathcal{H}\\_{t-1, \\mathcal{U}\\_{t, l}, \\mathcal{V}\\_{t, l}}}),\\\\\n \\mathcal{H}^\\prime\\_t &= f(\\mathcal{W}\\_{xh} \\ast \\mathcal{X}\\_t +\\\\\n &\\qquad\\mathcal{R}\\_{t} \\circ (\\sum\\_{l=1}^L{\\mathcal{W}^l\\_{hh} \\ast \\mathcal{H}\\_{t-1, \\mathcal{U}\\_{t, l}, \\mathcal{V}\\_{t, l}})}),\\\\\n \\mathcal{H}\\_t &= (1 - \\mathcal{Z}\\_t) \\circ \\mathcal{H}^\\prime\\_t + \\mathcal{Z}\\_t \\circ \\mathcal{H}\\_{t-1}.\n \\end{aligned}$$ \nHere, *γ* is a convolutional subnetwork that outputs *L* couples of flow fields U*t*, V*t* ∈ R*L* × *H* × *W*. H*t* − 1, U*t*, *l*, V*t*, *l* means to shift the elements in H*t* − 1 through the flow field U*t*, *l*, V*t*, *l* by bilinear warping . The connection structures of ConvRNN and TrajRNN are illustrated in Figure [fig:convrnntrajrnn]. Experiments on both the synthetic dataset and the real-world HKO-7 benchmark show that TrajGRU outperforms 2D CNN, 3D CNN, and ConvGRU. Wang et al. proposed the *Predictive RNN* (PredRNN), which extends ConvLSTM by adding a new type of spatiotemporal memory. Unlike ConvLSTM, of which the memory states are constrained inside each LSTM layer, PredRNN maintains another global memory state to store the spatiotemporal information in the lower layers and in the previous timestamps. In PredRNN, memory states are allowed to zigzag across the RNN layers. The structure of PredRNN is illustrated in Figure [fig:predrnn]. Here, M*t**l*s are the global spatiotemporal memory states and are updated in a similar way as the cell states in ConvLSTM except that it is updated in a zigzag order. Based on PredRNN, Wang et al. proposed the PredRNN++ structure which added more nonlinearities when updating M*t**l*s. Experiments show that PredRNN++ outperforms both PredRNN and other baseline models including TrajGRU and variants of ConvLSTM.\n [fig:convrnntrajrnn]\nApart from studying the basic building blocks like CNNs and RNNs, another line of research tries to disentangle the motion and content of the spatiotemporal sequence to better forecast the future. Jia et al. and Finn et al. proposed to predict how the frames will transform instead of directly predicting the future frames. Finn et al. proposed two transformation methods named *Convolutional Dynamic Neural Advection* (CDNA) and *Spatial Transformer Predictors* (STP). To predict the frame at timestamp *t* + 1, CDNA generates *N* 2D convolutional filters {**W***t*1, **W***t*2, ..., **W***t**N*} and a masking tensor M ∈ R*N* × *H* × *W* where ∑*c* = 1*N*M*c*, *i*, *j* = 1. The previous frame X*t* is transformed *N* times by these filters and these *N* transformed frames are linearly combined by the masking tensor to generate the final prediction. STP is similar to CDNA except that the **W***t**n*s become the parameters of different affine transformations . Similar to CDNA and STP, Jia et al. proposed the *Dynamic Filter Network* (DFN) that directly outputs the parameters of a *K* × *K* local filter for every location. The prediction is generated by applying these filters to the previous frame. Actually, DFN can be viewed as a special case of CDNA. If we manually fix the 2-dimensional convolutional filters in CDNA to have *K*2 elements, of which only a single element is non-zero, CDNA turns into DFN. Villegas et al. proposed *Motion-Content Network* (MCnet) which uses two separated networks to encode the motion and the content. In MCnet, the input of the motion encoder is the sequence of difference images {X2 − X1, ..., X*t* − X*t* − 1} and the input of the content encoder is the last observed image X*t*. The encoded motion features and content features are fused together to generate the prediction. Experiments show that disentangling the motion and the content helps improve the performance. This architecture has been used in other papers to enhance the prediction performance.\n [fig:stackconvrnnpredrnn]\n### Models for STSF-IG\nUnlike STSF-RG problems, the measurements in STSF-IG problems are observed in some sparsely distributed stations which are more challenging to deal with. The simple strategy is to model the measurements observed at each station independently, which has been adopted in Yu et al. . However, this strategy has not considered the correlation among different stations. A recent trend in solving this problem is to convert these sparsely distributed stations to a graph and utilize graph convolution to build the encoder and the forecaster. We will mainly introduce these graph convolution-based methods in this section.\nLi et al. proposed the *Diffusion Convolutional Recurrent Neural Network* (DCRNN) for traffic speed forecasting. To convert the traffic stations into a graph, the author generated the adjacency matrix based on the thresholded pairwise road network distances. After that, DCRNN is used to encode the sequence of graphical data and predict the future. The formula of DCRNN is similar to ConvRNN except that the *Diffusion Convolution* (DC) is used in the input-state and state-state transitions. DC is a type of graph convolution that considers the direction information in the graph, which has the following formula:\n \n$$\\label{eq:dcrnn}\n\\begin{aligned}\n\\mathbf{Y} = \\sum\\_{k=0}^{K - 1} (\\theta\\_{k, 1} (\\mathbf{D}\\_O^{-1} \\mathbf{A})^k) + \\theta\\_{k, 2}(\\mathbf{D}\\_I^{-1} \\mathbf{A}^T)^k) \\mathbf{X},\n\\end{aligned}$$ \nwhere **X** ∈ R*N* × *C* is the input features, **A** is the adjacency matrix, **D***O* is the diagonal out-degree matrix, **D***I* is the diagonal in-degree matrix, *θ**k*, 1, *θ**k*, 2 are the parameters, *K* is the number of diffusion steps, *N* is the number of nodes in the graph, and *C* is the number of the features. Unlike other graph convolution operators like ChebNet which operates on undirected graphs, DC operates on a directed graph and is more suitable for real-life scenarios. Experiments on two real-world traffic datasets show that DCRNN outperforms FC-LSTM and the ChebNet-based *Graph Convolutional Recurrent Neural Network* (GCRNN) . Later, Zhang et al. proposed the *Graph GRU* (GGRU) network that generalizes the idea of DCRNN. Instead of proposing a single model, the author proposed a unified method for constructing an RNN based on an arbitrary graph convolution operator. The formula of GGRU is given in the following:\n \n$$\\label{eq:graph\\_gru}\n \\begin{aligned}\n \\mathbf{U}\\_t = &\\sigma(\\Gamma\\_{\\Theta\\_{xu}}(\\mathbf{X}\\_t, \\mathbf{X}\\_t; \\mathcal{G}) \\\\\n &\\quad + \\Gamma\\_{\\Theta\\_{hu}}(\\mathbf{X}\\_t \\oplus \\mathbf{H}\\_{t-1}, \\mathbf{H}\\_{t-1}; \\mathcal{G})),\\\\\n \\mathbf{R}\\_t = &\\sigma(\\Gamma\\_{\\Theta\\_{xr}}(\\mathbf{X}\\_t, \\mathbf{X}\\_t; \\mathcal{G})\\\\\n &\\quad + \\Gamma\\_{\\Theta\\_{hr}}(\\mathbf{X}\\_t \\oplus \\mathbf{H}\\_{t-1}, \\mathbf{H}\\_{t-1}; \\mathcal{G})),\\\\\n \\mathbf{H}^\\prime\\_t =& h(\\Gamma\\_{\\Theta\\_{xh}}(\\mathbf{X}\\_t, \\mathbf{X}\\_t; \\mathcal{G})\\\\\n &\\quad + \\mathbf{R}\\_t \\circ \\Gamma\\_{\\Theta\\_{hh}}(\\mathbf{X}\\_t \\oplus \\mathbf{H}\\_{t-1}, \\mathbf{H}\\_{t-1}; \\mathcal{G})),\\\\\n \\mathbf{H}\\_t =& (1 - \\mathbf{U}\\_t) \\circ \\mathbf{H}^\\prime\\_t + \\mathbf{U}\\_t \\circ \\mathbf{H}\\_{t-1}.\n \\end{aligned}$$ \nHere, **U***t*, **R***t* are the update gate and reset gate that controls the memory flow, **H***t* is the hidden state, **X***t* is the input, *h*( ⋅ ) is the activation function, and Γ( ⋅ , ⋅ ) is an arbitrary graph convolution operator. The author also proposed a new multi-head gated graph attention aggregator that uses neural attention to aggregate the neighboring features. Experiments show that GGRU outperforms DCRNN if the gated attention aggregator is used. Yu et al. proposed the *Spatio-Temporal Graph Convolutional Network* (STGCN) that applies graph convolution in a different manner. The author directly stacked multiple *Spatio-Temporal Convolution* (ST-Conv) blocks to build the network. The ST-Conv block is a concatenation of two temporal convolution layers and one graph convolution layer. For the graph convolution operator, the paper tested the ChebNet and its first-order approximation and found that ChebNet performed better. The major advantage of STGCN over RNN-based methods is it is faster in the training phase.\n### Models for TF-MPC\nThe goal of TF-MPC is to predict the future trajectories of all objects in the moving point cloud. Fragkiadaki et al. proposed a basic RNN based method for this problem. In the model, the coordinate sequence of each human joint is encoded by the same LSTM. The drawback of this basic method is that it does not model the correlation among the individuals. To solve the problem, Alahi et al. proposed the SocialLSTM model that uses a set of interacted LSTMs to model the trajectory of the crowd. In SocialLSTM, the position of each person is modeled by an LSTM and a social pooling layer interconnects these LSTMs. Social pooling aggregates the hidden states of the nearby LSTMs at the previous timestamp and use this aggregated state to help decide the current state. The transition rule of the SocialLSTM is given in the following:\n \n$$\\label{eq:social-lstm}\n \\begin{aligned}\n \\mathcal{N}^i\\_t &= \\{j \\mid j\\in \\mathcal{N}\\_i, \\abs{x\\_t^j - x\\_t^i} \\leq m, \\abs{y\\_t^j - y\\_t^i} \\leq n\\},\\\\\n \\mathbf{H}\\_t^i &= \\sum\\_{j\\in \\mathcal{N}^i\\_t} \\mathbf{h}\\_{t-1}^j,\\\\\n \\mathbf{e}\\_t^i &= \\phi\\_1(x\\_t^i, y\\_t^i; \\mathbf{W}\\_e),\\\\\n \\mathbf{a}\\_t^i &= \\phi\\_2(\\mathbf{H}\\_t^i; \\mathbf{W}\\_l),\\\\\n \\mathbf{h}\\_t^i, \\mathbf{c}\\_t^i &= \\text{LSTM}(\\mathbf{h}\\_{t-1}^i, \\mathbf{c}\\_{t-1}^i, \\text{vec}(\\mathbf{e}\\_t^i, \\mathbf{a}\\_t^i); \\mathbf{W}\\_l).\n \\end{aligned}$$ \nHere, **H***t**i* is the aggregated state vector in the social pooling step, (*x**t**j*, *y**t**j*) is the coordinate of the *j*th person at timestamp *t*, *ϕ*1, *ϕ*2 are mapping functions, and *m*, *n* are distance thresholds. Experiments show that adding the social pooling layer is helpful for the TF-MPC task. In , the author proposed an improved version of social pooling called *View Frustum Of Attention* (VFOA) social pooling. This method defines the neighborhood set in the social pooling layer based on the calculated VFOA interest region. Jain et al. proposed the *Structural-RNN* (S-RNN) network to solve the human motion prediction problem. S-RNN imposes structures in the recurrent connections based on a given spatiotemporal graph. For human motion prediction, this spatiotemporal graph is constructed by the relationship between different human joints. S-RNN maintains two types of RNNs, nodeRNN and edgeRNN. The outputs of edgeRNNs are fed to the corresponding nodeRNNs. Experiments show that the S-RNN achieves the state-of-the-art result in the human motion prediction problem.\n### Methods for Handling Uncertainty\nMost real-world dynamical systems, e.g., atmosphere, are inherently stochastic and unpredictable. Simply assuming the outcome is deterministic, which is common in FNN and RNN based models, will generate blurry predictions. Thus, recent studies have proposed ways for handling uncertainty. In , the author proposed to use a probabilistic forecaster that outputs the parameters of a *Gaussian Mixture Model* (GMM). This enables the network to generate stochastic predictions. The author found that the smallest forecasting error was always produced by the most probable sample, which was similar to the output of a model trained by the Euclidean loss. Apart from using a probabilistic forecaster, the author also proposed a curriculum learning strategy which gradually adds noise to the input during training. This can be viewed as regularizing the model not to be overconfident about its predictions. This technique has also been adopted in later works . Goroshin et al. proposed to add a latent variable *δ* in the network to represent the uncertainty of the prediction. The latent variable *δ**i* is not determined by the input X*i* and can be adjusted in the learning process to minimize the loss. Before updating the parameters of the network, the algorithm first updates *δ**i* for *k* steps where *k* is a hyperparameter. At test time, the *δ**i*s is sampled based on its distribution on the training set.\nAnother approach to handle uncertainty is to utilize the recently popularized conditional *Generative Adversarial Network* (GAN) . Mathieu et al. proposed to use conditional GAN as the loss function to train the multi-scale 2D CNN. The GAN loss used in the paper is given as follows:\n \n$$\\label{eq:gan}\n\\begin{aligned}\n&\\min\\_{\\theta\\_G} \\max\\_{\\theta\\_D} \\mathbb{E}\\_{\\mathcal{Y} \\sim p\\_{\\text{data}}(\\mathcal{Y} \\mid \\mathcal{X})}[\\log D\\_{\\theta\\_D}(\\mathcal{X}, \\mathcal{Y})]\\\\\n&\\quad+\\mathbb{E}\\_{z \\sim p\\_z} [\\log(1 - D\\_{\\theta\\_D}(\\mathcal{X}, G\\_{\\theta\\_G}(\\mathcal{X}, z)))].\n\\end{aligned}$$ \nHere, *D**θ**D* is the discriminator, *G**θ**G* is the forecasting model, X is the observed sequence, Y is the ground-truth sequence, and *p*(*z*) is the noise distribution. The discriminator is trained to differentiate between the ground-truth output and the prediction generated by the forecaster. On the other hand, the forecasting model is trained to fool the discriminator. The final loss function in the paper combined the GAN loss, the L2 loss, and the image gradient difference loss. Experiments show that the model trained with GAN generates sharper predictions. Vondrick et al. used a similar GAN loss to train the 3D CNN. Jang et al. proposed to use two discriminators to respectively discriminate the motion and appearance part of the generated samples.\nBesides GAN, the *Variational Auto-encoder* (VAE) has also been adopted for dealing with the uncertainty in STSF. Babaeizadeh et al. proposed *Stochastic Variational Video Prediction* (SV2P) which uses the following objective function:\n \n$$\\label{eq:sv2p}\n\\begin{aligned}\nl &= - \\mathbb{E}\\_{q\\_\\phi(\\mathbf{Z} \\mid \\mathbf{X}\\_{1:T})}[\\log p\\_\\theta(\\mathbf{X}\\_{t:T}\\mid \\mathbf{X}\\_{1:t}, \\mathbf{Z})] \\\\\n&\\quad\\quad+ D\\_{\\text{KL}}(q\\_\\phi(\\mathbf{Z} \\mid \\mathbf{X}\\_{1:T}) || p(\\mathbf{Z})),\n\\end{aligned}$$ \nwhere *q**ϕ* is the variational inference network that approximates the posterior of the latent variable **Z** and *p**θ* is an RNN generator. The experiments show that adding VAE greatly alleviates the blurry problem of the predictions. Later, Denton & Fergus proposed the *Stochastic Video Generator with Learned Prior* (SVG-LP) which uses a learned time-varying prior *p**ψ*(**Z***t* ∣ **X**1 : *t* − 1) to replace the fixed prior *p*(**Z**) in . The extension in their paper is similar to the *Variational RNN* (VRNN) . Learning a time-variant prior can be interpreted as learning a predictive model of uncertainty. The author showed in the experiments that using a learned prior improves the forecasting performance.\n[tb!] [tbl:overall-summary]\n| Category | Subcategory | Methods | STSF-RG | STSF-IG | TF-MPC | Unc. |\n| --- | --- | --- | --- | --- | --- | --- |\n| | | Spatiotemporal indicator | | $\\surd$ | | |\n| | | Multiple predictors | | $\\surd$ | | |\n| | | Group-based | | | $\\surd$ | $\\surd$ |\n| | | STARIMA | $\\surd$ | $\\surd$ | | $\\surd$ |\n| | | Low-rankness | $\\surd$ | $\\surd$ | | $\\surd$ |\n| | GP | GP | $\\surd$ | $\\surd$ | $\\surd$ | $\\surd$ |\n| | | TRBM | $\\surd$ | | $\\surd$ | $\\surd$ |\n| | | RTRBM | $\\surd$ | | $\\surd$ | $\\surd$ |\n| | | SRTRBM | $\\surd$ | $\\surd$ | $\\surd$ | $\\surd$ |\n| | | TSBN | $\\surd$ | | $\\surd$ | $\\surd$ |\n| | | FC-LSTM | $\\surd$ | $\\surd$ | $\\surd$ | $\\surd$ |\n| | | 2D CNN & 3D CNN | $\\surd$ | | | $\\surd$ |\n| | | ConvLSTM & TrajGRU | $\\surd$ | | | |\n| | | PredRNN | $\\surd$ | | | |\n| | | VPN & PredCNN | $\\surd$ | | | |\n| | | Learn transformation | $\\surd$ | | | |\n| | | Motion + Content | $\\surd$ | | | $\\surd$ |\n| | | DCRNN & GGRU | | $\\surd$ | | |\n| | | STGCN | | $\\surd$ | | |\n| | | Social pooling | | | $\\surd$ | $\\surd$ |\n| | | S-RNN | | | $\\surd$ | $\\surd$ |\n| | | *δ*-gradient | $\\surd$ | | | $\\surd$ |\n| | | SV2P & SVG-LP | $\\surd$ | | | $\\surd$ |\nRemarks\n-------\nIn this section, we reviewed two types of DL based methods for STSF: DTGMs and FNN and RNN based methods. The majority of the reviewed DTGMs treat STSF as a general multivariate sequence forecasting problem and SRTRBM is the only model that gives special treatment of the spatiotemporal sequences. Also, learning a DTGM requires approximation and sampling techniques and is more complicated compared with the FNN and RNN based methods. However, DTGM can well capture the uncertainty in the data due to its stochastic generative process. On the other hand, the FNN and RNN based models are easy to train and fast for prediction. We reviewed the representative methods for STSF-RG, STSF-IG, and TF-MPC. Although these three types of STSF problems involve different tasks, the proposed methods for these problems have strong relationships. For example, ConvLSTM, DCRNN, and SocialLSTM, which are respectively proposed for STSF-RG, STSF-IG, and TF-MPC, improve upon RNN by introducing structured recurrent connections. To be more specific, ConvLSTM uses convolution in recurrent transitions, DCRNN uses the graph convolution, and SocialLSTM uses the social pooling layer. Therefore, the success of these models greatly relies on the appropriate design of the basic network architecture. Also, FNN and RNN based methods are usually deterministic and are not well-designed for capturing the uncertainty in the data. We thus reviewed some recently popularized techniques for handling uncertainty including GAN and VAE, which have improved the forecasting performance.\nSummary and Future Works\n========================\nIn this survey, we reviewed the machine learning based methods for STSF. We first examined the general strategies for multi-step sequence forecasting, which contain the IMS, DMS, boosting strategy, and scheduled sampling. We showed that IMS and DMS have their advantages and disadvantages. The boosting strategy and scheduled sampling can be viewed as the mid-grounds between these two approaches. Next, we reviewed the specific methods for STSF. We divided these methods into two major categories: classical methods and DL based methods. As mentioned in [ch:classical-remark] and [ch:deep-remark], the existing classical methods and deep learning methods have their own trade-offs when solving the STSF problem. The overall summarization of the reviewed methods is given in Table [tbl:overall-summary].\nThere are many future research directions for machine learning for STSF. 1) We can enhance the strategies for the multi-step sequence forecasting. One direction is to design new training curriculum to shift from IMS to DMS. For example, we can extend the forecasting horizon exponentially from 1 to *N* to better model longer sequences, which is common in TF-MPC problems. Also, we can generalize the boosting method described in Section [ch:boosting] to the general STSF problems. In addition, we can reduce the complexity of DMS from *O*(*h*) to *O*(log*h*) by training *O*(log*h*) models that generate the predictions of horizons 20, 21, ..., 2log*h* − 1. We can use these models to make prediction for any horizon up to *h* in the spirit of binary coding. Moreover, the objective function mentioned in Section [ch:scheduled-sampling] will not be differentiable due to the sampling process and the original paper has just ignored this problem. This problem can be solved by approximating the gradient through techniques in reinforcement learning . 2) We can propose new FNN and RNN based methods for STSF-IG problems. Compared with STSF-RG, DL for STSF-IG is less mature and we can bring the ideas in STSF-RG, like decomposing motion and content, adding a global memory structure, and using GAN/VAE to handle uncertainty, to STSF-IG. 3) We can combine DTGMs with FNN and RNN based methods to get the best of both worlds. The generation flavor of DTGMs has its role to play in FNN and RNN based models particularly for handling uncertainty. One feasible approach is to use the idea of *Bayesian Deep Learning* (BDL) , in which FNN and RNN based models are used for perception and DTGMs are used for inference. 4) We can study the type of TF-MPC problem where both the measurements and coordinates are changing. We can consider to solve the “video object prediction” problem where we have the bounding boxes of the objects at the previous frames and the task is to predict both the objects’ future appearances and bounding boxes. 5) We can study new learning scenarios like online learning and transfer learning for STSF. For the STSF problem, the online learning scenario is essential because spatiotemporal data, like regional rainfall, usually arrive in an online fashion . The transfer learning scenario is also essential because STSF can act as a pretraining method. The model trained by STSF can be transferred to solve other tasks like classification and control . Designing a better transfer learning method for STSF will improve the performance of these tasks.\nXingjian Shi received his BEng degree in information security from Shanghai Jiao Tong University, China. He is currently a Ph.D student at the Department of Computer Science and Engineering of Hong Kong University of Science and Technology. His research interests are in deep learning and spatiotemporal analysis. He received the Hong Kong PhD Fellowship in 2014.\nDit-Yan Yeung received his BEng degree in electrical engineering and MPhil degree in computer science from the University of Hong Kong, and PhD degree in computer science from the University of Southern California. He started his academic career as an assistant professor at the Illinois Institute of Technology in Chicago. He then joined the Hong Kong University of Science and Technology where he is now a full professor and the Acting Head of the Department of Computer Science and Engineering. His research interests are in computational and statistical approaches to machine learning and artificial intelligence.\n---\n1. \n\nMachine Learning for Spatiotemporal Sequence Forecasting: A Survey\n==================================================================\nreal-world phenomena are spatiotemporal, such as the traffic flow, the diffusion of air pollutants and the regional rainfall. Correctly predicting the future of these spatiotemporal systems based on the past observations is essential for a wide range of scientific studies and real-life applications like traffic management, precipitation nowcasting, and typhoon alert systems.\nIf there exists an accurate numerical model of the dynamical system, which usually happen in the case that we have fully understood its rules, forecasting can be achieved by first finding the initial condition of the model and then simulate it to get the future predictions . However, for some complex spatiotemporal dynamical systems like atmosphere, crowd, and natural videos, we are still not entirely clear about their inner mechanisms. In these situations, machine learning based methods, by which we can try to learn the systems’ inner “laws” based on the historical data, have proven helpful for making accurate predictions . Moreover, recent studies have shown that the “laws” inferred by machine learning algorithms can further be used to guide other tasks like classification and control. For example, controlling a robotic arm will be much easier once you can anticipate how it will move after performing specific actions .\n[tb!] [tbl:type-spatiotemporal-forecasting]\n| Problem Name | Coordinates | Measurements |\n| --- | --- | --- |\n| Trajectory Forecasting of Moving Point Cloud | Changing | Fixed/Changing |\n| Spatiotemporal Forecasting on Regular Grid | Fixed regular grid | Changing |\n| Spatiotemporal Forecasting on Irregular Grid | Fixed irregular grid | Changing |\nIn this paper, we review these machine learning based methods for spatiotemporal sequence forecasting. We define a length-*T* spatiotemporal sequence as a series of matrices **X**1 : *T* = [**X**1, **X**2, ...**X***T*]. Each **X***t* ∈ R*K* × (*D* + *E*) contains a set of coordinates and their corresponding measurements. Here, *K* is the number of coordinates, *D* is the number of measurements, and *E* is the dimension of the coordinate. We further denote the measurement part and coordinate part of **X***t* as **M***t* ∈ R*K* × *D* and **C***t* ∈ R*K* × *E*. For many forecasting problems, we can use auxiliary information to enhance the prediction accuracy. For example, regarding wind speed forecasting, the latitudes and past temperatures are also helpful for predicting the future wind speeds. We denote the auxiliary information available at timestamp *t* as A*t*. The *Spatiotemporal Sequence Forecasting* (STSF) problem is to predict the length-*L* (*L* > 1) sequence in the future given the previous observations plus the auxiliary information that is allowed to be empty. The mathematical form is given in .\n \n$$\\label{eq:stsf}\n\\begin{aligned}\n \\mathbf{\\hat{X}}\\_{t+1:t+L} = \\operatorname\\*{arg\\,max}\\_{\\mathbf{X}\\_{t+1:t+L}} p(\\mathbf{X}\\_{t+1:t+L} \\mid \\mathbf{X}\\_{1:t}, \\mathcal{A}\\_{t}).\n\\end{aligned}$$ \nHere, we limit STSF to only contain problems where the input and output are both sequences with multiple elements. According to this definition, problems like video generation using a static image and dense optical flow prediction , in which the input or the output contain a single element, will not be counted.\nBased on the characteristics of the coordinate sequence **C**1 : *T* and the measurement sequence **M**1 : *T*, we classify STSF into three categories as shown in Table [tbl:type-spatiotemporal-forecasting]. If the coordinates are changing, the STSF problem is called *Trajectory Forecasting of Moving Point Cloud* (TF-MPC). Problems like human trajectory prediction and human dynamics prediction fall into this category. We need to emphasize here that the entities in moving point cloud can not only be human but also be general moving objects. If the coordinates are fixed, we only need to predict the future measurements. Based on whether the coordinates lie in a regular grid, we can get the other two categories, called *STSF on Regular Grid* (STSF-RG) and *STSF on Irregular Grid* (STSF-IG). Problems like precipitation nowcasting , crowd density prediction and video prediction , in which we have dense observations, are STSF-RG problems. Problems like air quality forecasting , influenza prediction and traffic speed forecasting , in which the monitoring stations (or sensors) spread sparsely across the city, are STSF-IG problems. We need to emphasize that although the measurements in TF-MPC can also be time-variant due to factors like appearance change, most models for TF-MPC deal with the fixed-measurement case and only predict the coordinates. We thus focus on the fixed-measurement case in this survey.\nHaving both the characteristics of spatial data like images and temporal data like sentences and audios, spatiotemporal sequences contain information about “what”, “when”, and “where” and provide a comprehensive view of the underlying dynamical system. However, due to the complex spatial and temporal relationships within the data and the potential long forecasting horizon, spatiotemporal sequence forecasting imposes new challenges to the machine learning community. The first challenge is how to learn a model for multi-step forecasting. Early researches in time-series forecasting have investigated two approaches: *Direct Multi-step* (DMS) estimation and *Iterated Multi-step* (IMS) estimation. The DMS approach directly optimizes the multi-step forecasting objective while the IMS approach learns a one-step-ahead forecaster and iteratively applies it to generate multi-step predictions. However, choosing between DMS and IMS involves a trade-off among forecasting bias, estimation variance, the length of the prediction horizon and the model’s nonlinearity . Recent studies have tried to find the mid-ground between these two approaches . The second challenge is how to model the spatial and temporal structures within the data adequately. In fact, the number of free parameters of a length-*T* spatiotemporal sequence can be up to *O*(*K**T**D**T*) where *K* is usually above thousands for STSF-RG problems like video prediction. Forecasting in such a high dimensional space is impossible if the model cannot well capture the data’s underlying structure. Therefore, machine learning based methods for STSF, including both classical methods and deep learning based methods, have special designs in their model architectures for capturing these spatiotemporal correlations.\nIn this paper, we organize the literature about STSF following these two challenges. Section [ch:multistep] covers the general learning strategies for multi-step forecasting including IMS, DMS, boosting strategy and scheduled sampling. Section [ch:method-classical] reviews the classical methods for STSF including the feature-based methods, state-space models, and Gaussian process models. Section [ch:dl] reviews the deep learning based methods for STSF including deep temporal generative models and feed-forward and recurrent neural network-based methods. Section [ch:discussion] summarizes the survey and discusses some future research directions.\nLearning Strategies for Multi-step Forecasting\n==============================================\nCompared with single-step forecasting, learning a model for multi-step forecasting is more challenging because the forecasting output $\\mathbf{\\hat{X}}\\_{t+1:t+L}$ is a sequence with non-i.i.d elements. In this section, we first introduce and compare two basic strategies called IMS and DMS and then review two extensions that bridge the gap between these two strategies. To simplify the notation, we keep the following notation rules in this and the following sections. We denote the information available at timestamp *t* as F*t* = {**X**1 : *t*, A*t*}. Learning an STSF model is thus to train a model, which can be either probabilistic or deterministic, to make the *L*-step-ahead prediction $\\mathbf{\\hat{X}}\\_{t+1:t+L}$ based on F*t*. We also denote the ground-truth at timestamp *t* as $\\mathbf{\\tilde{X}}\\_t$ and recursively applying the basic function *f*(*x*) for *h* times as *f*(*h*)(*x*). Setting the *k*th subscript to be “:” means to select all elements at the *k*th dimension. Setting the *k*th subscript to a letter *i* means to select the *i*th element at the *k*th dimension. For example, for matrix **X**, **X***i*, : means the *i*th row, **X**: , *j* means the *j*th column, and **X***i*, *j* means the (*i*, *j*)th element. The flattened version of matrices **A**, **B** and **C** is denoted as vec(**A**, **B**, **C**).\nIterative Multi-step Estimation\n-------------------------------\nThe IMS strategy trains a one-step-ahead forecasting model and iteratively feeds the generated samples to the one-step-ahead forecaster to get the multi-step-ahead prediction. There are two types of IMS models, the deterministic one-step-ahead forecasting model **X***t* = *m*(F*t*; *ϕ*) and the probabilistic one-step-ahead forecasting model *p*(**X***t* ∣ F*t*; *ϕ*). For the deterministic forecasting model, the forecasting process and the training objective of the deterministic forecaster are shown in and respectively where *d*(*x̃*, *x*) measures the distance between *x̃* and *x*.\n \n$$\\label{eq:ims-testing}\n \\begin{aligned}\n \\mathbf{\\hat{X}}\\_{t+h} &= m^{(h)}(\\mathcal{F}\\_t;\\phi),~~\\forall 1 \\leq h \\leq L.\\\\\n \\end{aligned}$$ \n \n$$\\label{eq:ims-training-deterministic}\n \\phi^\\star = \\operatorname\\*{arg\\,min}\\_\\phi \\mathbb{E}\\left[d(\\mathbf{\\tilde{X}}\\_{t+1}, m(\\mathcal{F}\\_t;\\phi))\\right].$$ \nFor the probabilistic forecasting model, we use the chain-rule to get the probability of the length-*L* prediction, which is given in , and estimate the parameters by *Maximum Likelihood Estimation* (MLE).\n \n$$\\label{eq:ims-testing-probabilistic}\n \\begin{aligned}\n &p(\\mathbf{X}\\_{t+1:t+L}\\mid \\mathcal{F}\\_t;\\phi) = p(\\mathbf{X}\\_{t+1} \\mid \\mathcal{F}\\_t;\\phi) \\\\\n &\\qquad\\prod\\_{i=2}^L p(\\mathbf{X}\\_{t+i} \\mid \\mathcal{F}\\_t, \\mathbf{X}\\_{t+1:t+i-1};\\phi).\\\\\n \\end{aligned}$$ \nWhen the one-step-ahead forecaster is probabilistic and nonlinear, the MLE of is generally intractable. In this case, we can draw samples from *p*(**X***t* + 1 : *t* + *L* ∣ F*t*; *ϕ*) by techniques like *Sequential Monte-Carlo* (SMC) and predict based on these samples.\nThere are two advantages of the IMS approach: 1) The one-step-ahead forecaster is easy to train because it only needs to consider the one-step-ahead forecasting error and 2) we can generate predictions of an arbitrary length by recursively applying the basic forecaster. However, there is a discrepancy between training and testing in IMS. In the training process, we use the ground-truths of the previous *L* − 1 steps to generate the *L*th-step prediction. While in the testing process, we feed back the model predictions instead of the ground-truths to the forecaster. This makes the model prone to accumulative errors in the forecasting process . For the deterministic forecaster example above, the optimal muti-step-ahead forecaster, which can be obtained by minimizing $\\mathbb{E}\\left[\\sum\\_{h=1}^L d(\\mathbf{\\tilde{X}}\\_{t+h}, m^{(h)}(\\mathcal{F}\\_t;\\phi))\\right]$, is not the same as recursively applying the optimal one-step-ahead forecaster defined in when *m* is nonlinear. This is because the forecasting error at earlier timestamps will propagate to later timestamps .\nDirect Multi-step Estimation\n----------------------------\nThe main motivation behind DMS is to avoid the error drifting problem in IMS by directly minimizing the long-term prediction error. Instead of training a single model, DMS trains a different model *m**h* for each forecasting horizon *h*. There can thus be *L* models in the DMS approach. When the forecasters are deterministic, the forecasting process and training objective are shown in and respectively where *d*( ⋅ , ⋅ ) is the distance measure.\n \n$$\\label{eq:dms-testing}\n \\begin{aligned}\n \\mathbf{\\hat{X}}\\_{t+h} &= m\\_h(\\mathcal{F}\\_t;\\phi\\_L), \\forall 1 \\leq h \\leq L.\\\\\n \\end{aligned}$$ \n \n$$\\label{eq:dms-training}\n \\begin{aligned}\n \\phi\\_1^\\star,..., \\phi\\_L^\\star &= \\operatorname\\*{arg\\,min}\\_{\\phi\\_1,...,\\phi\\_L} \\mathbb{E}\\left[d(\\mathbf{\\tilde{X}}\\_{t+1:t+L}, \\mathbf{\\hat{X}}\\_{t+1:t+L})\\right].\n \\end{aligned}$$ \n \n$$\\label{eq:dms-training-recursive}\n \\begin{aligned}\n \\phi^\\star = \\operatorname\\*{arg\\,min}\\_{\\phi} \\mathbb{E}\\left[\\sum\\_{h=1}^L d(\\mathbf{\\tilde{X}}\\_{t+h}, m^{(h)}(\\mathcal{F}\\_t;\\phi))\\right].\n \\end{aligned}$$ \nTo disentangle the model size from the number of forecasting steps *L*, we can also construct *m**h* by recursively applying the one-step-ahead forecaster *m*1. In this case, the model parameters {*ϕ*1, ..., *ϕ**L*} are shared and the optimization problem turns into . We need to emphasize here that , which optimizes the multi-step-ahead forecasting error, is different from , which only minimizes the one-step-ahead forecasting error. This construction method is widely adopted in deep learning based methods .\nComparison of IMS and DMS\n-------------------------\nChevillon compared the IMS strategy and DMS strategy. According to the paper, DMS can lead to more accurate forecasts when 1) the model is misspecified, 2) the sequences are non-stationary, or 3) the training set is too small. However, DMS is more computationally expensive than IMS. For DMS, if the *ϕ**h*s are not shared, we need to store and train *L* models. If the *ϕ**h*s are shared, we need to recursively apply *m*1 for *O*(*L*) steps . Both cases require larger memory storage or longer running time than solving the IMS objective. On the other hand, the training process of IMS is easy to parallelize since each forecasting horizon can be trained in isolation from the others . Moreover, directly optimizing a *L*-step-ahead forecasting loss may fail when the forecasting model is highly nonlinear, or the parameters are not well-initialized .\nDue to these trade-offs, models for STSF problems choose the strategies that best match the problems’ characteristics. Generally speaking, DMS is preferred for short-term prediction while IMS is preferred for long-term forecast. For example, the IMS strategy is widely adopted in solving TF-MPC problems where the forecasting horizons are long, which are generally above 12 and can be as long as 100 . The DMS strategy is used in STSF-RG problems where the dimensionality of data is usually very large and only short-term predictions are required . For STSF-IG problems, some works adopt DMS when the overall dimensionality of the forecasting sequence is acceptable for the direct approach .\nOverall, IMS is easier to train but less accurate for multi-step forecasting while DMS is more difficult to train but more accurate. Since IMS and DMS are somewhat complementary, later studies have tried to bridge the gap between these two approaches. We will introduce two representative strategies in the following two sections.\nBoosting Strategy\n-----------------\nThe boosting strategy proposed in combines the IMS and DMS for univariate time series forecasting. The strategy boosts the linear forecaster trained by IMS with several small and nonlinear adjustments trained by DMS. The model assumption is shown in .\n \n*x̃**t* + *h* = *m*(*h*)(F*t*; *ϕ*) + ∑*i* = 1*I**h**ν**l**i*(*x̃**t* − *j*, *x̃**t* − *k*; *ψ**i*) + *e**t*, *h*. \nHere, *m*(F*t*; *ϕ*) is the basic one-step-ahead forecasting model, *l**i*(*x̃**t* − *j*, *x̃**t* − *k*; *ψ**i*) is the chosen weak learner at step *i*, *I**h* is the boosting iteration number, *ν* is the shrinkage factor and *e**t*, *h* is the error term. The author set the base linear learner to be the auto-regressive model and set the weak nonlinear learners to be the penalized regression splines. Also, the weak learners were designed to only account for the interaction between two elements in the observation history. The training method is similar to the gradient boosting algorithm . *m*(F*t*; *ϕ*) is first trained by minimizing the IMS objective . After that, the weak-learners are fit iteratively to the gradient residuals.\nThe author also performed a theoretical analysis of the bias and variance of different models trained with DMS, IMS and the boosting strategy for 2-step-ahead prediction. The result shows that the estimation bias of DMS can be arbitrarily small while that of the IMS will be amplified for larger forecasting horizon. The model estimated by the boosting strategy also has a bias, but it is much smaller than the IMS approach due to the compensation of the weak learners trained by DMS. Also, the variance of the DMS approach depends on the variability of the input history F*t* and the model size which will be very large for complex and nonlinear models. The variance of the boosting strategy is smaller because the basic model is linear and the weak-learners only consider the interaction between two elements.\nAlthough the author only proposed the boosting strategy for the univariate case, the method, which lies in the mid-ground between IMS and DMS, should apply to the STSF problems with multiple measurements, which leads to potential future works as discussed in Section [ch:discussion].\nScheduled Sampling\n------------------\nThe idea of *Scheduled Sampling* (SS) is first to train the model with IMS and then gradually replaces the ground-truths in the objective function with samples generated by the model. When all ground-truth samples are replaced with model-generated samples, the training objective becomes the DMS objective. The generation process of SS is described in :\n \n$$\\label{eq:ss-training}\n \\begin{aligned}\n &\\forall 1 \\leq h \\leq L,\\\\\n &\\mathbf{\\hat{X}}\\_{t+h} \\sim p(\\mathbf{X} \\mid \\mathcal{F}\\_t, \\mathbf{\\bar{X}}\\_{t+1:t+h-1};\\phi),\\\\\n &\\mathbf{\\bar{X}}\\_{t+h} = (1 - \\tau\\_{t+h}) \\mathbf{\\hat{X}}\\_{t+h} + \\tau\\_{t+h} \\mathbf{\\tilde{X}}\\_{t+h},\\\\\n &\\tau\\_{t+h} \\sim B(1, \\epsilon\\_k).\n \\end{aligned}$$ \nHere, $\\mathbf{\\hat{X}}\\_{t+h}$ and $\\mathbf{\\tilde{X}}\\_{t+h}$ are the generated sample and the ground-truth at forecasting horizon *h*, correspondingly. $p(\\mathbf{X} \\mid \\mathcal{F}\\_t, \\mathbf{\\bar{X}}\\_{t+1:t+h-1};\\phi)$ is the basic one-step-ahead forecasting model. *τ**t* + *h* is a random variable following the binomial distribution, which controls whether to use the ground-truth or the generated sample. *ε**k* is the probability of choosing the ground-truth at the *k*th iteration. In the training phase, SS minimizes the distance between $\\mathbf{\\hat{X}}\\_{t+1:t+L}$ and $\\mathbf{\\tilde{X}}\\_{t+1:t+L}$, which is shown in . In the testing phase, *τ* is fixed to 0, meaning that the model-generated samples are always used.\n \n$$\\label{eq:ss-objective}\n\\begin{aligned}\n&\\mathbb{E}\\_{p(\\mathbf{\\hat{X}}\\_{t+1:t+L}\\mid \\mathcal{F}\\_t;\\phi)}\\big[d(\\mathbf{\\hat{X}}\\_{t+1:t+L}, \\mathbf{\\tilde{X}}\\_{t+1:t+L}) \\big] \\\\\n&= \\mathbb{E}\\_{p(\\tau\\_{t+1:t+L-1})} \\big[d(\\mathbf{\\hat{X}}\\_{t+1:t+L}, \\mathbf{\\tilde{X}}\\_{t+1:t+L})\\\\\n&\\qquad\\qquad\\prod\\_{h=1}^L p(\\mathbf{\\hat{X}}\\_{t+1} \\mid \\mathcal{F}\\_t, \\mathbf{\\bar{X}}\\_{t+1:t+h-1};\\phi)\\big].\n\\end{aligned}$$ \nIf *ε**k* equals to 1, the ground-truth will always be chosen and the objective function will be the same as in the IMS strategy. If *ε**k* is 0, the generated sample will always be chosen and the optimization objective will be the same as in the DMS strategy. Based on this observation, the author proposed to gradually decay *ε**k* during training to make the optimization objective shift smoothly from IMS to DMS, which is a type of *curriculum learning* . The paper also provided some ways to change the sampling ratio. One issue of SS is that the expectation over *τ**t* + *h*s in makes the loss function non-differentiable. The original paper obviates the problem by treating *X̂**t* + *h*s as constants, which does not optimize the true objective .\nThe SS strategy is widely adopted in DL models for STSF. In , the author applied SS to video prediction and showed that the model trained via SS outperformed the model trained via IMS. In , the author proposed a training strategy similar to SS except that the training curriculum is adapted from IMS to DMS by controlling the number of the prediction steps instead of controlling the frequency of sampling the ground-truth. In , the author used SS to train models for traffic speed forecasting and showed it outperformed IMS.\nClassical Methods for STSF\n==========================\nIn this section, we will review the classical methods for STSF, including feature-based methods, state space models, and Gaussian process models. These methods are generally based on the shallow models or spatiotemporal kernels. Since a spatiotemporal sequence can be viewed as a multivariate sequence if we flatten the **X***t*s into vectors or treat the observations at every location independently, methods designed for the general multivariate time-series forecasting are naturally applicable to STSF. However, directly doing so ignores the spatiotemporal structure within the data. Therefore, most algorithms introduced in this section have extended this basic approach by utilizing the domain knowledge of the specific task or considering the interactions among different measurements and locations.\nFeature-based Methods\n---------------------\nFeature-based methods solve the STSF by training a regression model based on some human-engineered spatiotemporal features. To reduce the problem size, most feature-based methods treat the measurements at the *K* locations to be independent given the extracted features. The high-level IMS and DMS models of the feature-based methods are given in and respectively.\n \n$$\\begin{aligned}\n \\mathbf{\\hat{X}}\\_{t+1, k} &= f\\_{\\text{IMS}}(\\Psi\\_k(\\mathcal{F}\\_t);\\phi) \\label{eq:feature-based-ims},\\\\\n \\mathbf{\\hat{X}}\\_{t+1:t+L, k} &= f\\_{\\text{DMS}}(\\Psi\\_k(\\mathcal{F}\\_t;\\phi) \\label{eq:feature-based-dms}.\\end{aligned}$$ \nHere, Ψ*k*(F*t*) represents the features at location *k*. Once the feature extraction method Ψ*k* is defined, any regression model *f* can be applied. The feature-based methods are usually adopted in solving STSF-IG problems. Because the observations are sparsely and irregularly distributed in STSF-IG, it is sometimes more straightforward to extract human-engineered features than to develop a purely machine learning based model. Despite their straightforwardness, the feature-based methods have a major drawback. The model’s performance relies heavily on the quality of the human-engineered features. We will briefly introduce two representative feature-based methods for two STSF-IG problems to illustrate the approach.\nOhashi and Torgo proposed a feature-based method for wind-speed forecasting. The author designed a set of features called spatiotemporal indicators and applied several regression algorithms including regression tree, support vector regression, and random forest. There are two kinds of spatiotemporal indicators defined in the paper: the average or weighted average of the observed historical values within a space-time neighborhood and the ratio of two average values calculated with different distance thresholds. The regression model takes the spatiotemporal indicators as the input to generate the one-step-ahead prediction. Experiment results show that the spatiotemporal indicators are helpful for enhancing the forecasting accuracy.\nZheng et al. proposed a feature-based method for air quality forecasting. The author adopted the DMS approach and combined multiple predictors, including the temporal predictor, the spatial predictor, and the inflection predictor, to generate the features at location *k* and timestamp *t*. The temporal predictor focuses on the temporal side of the data and uses the measurements within a temporal sliding window to generate the features. The spatial predictor focuses on the spatial side of the data and uses the previous measurements within a nearby region to generate the features. Also, to deal with the sparsity and irregularity of the coordinates, the author partitioned the neighborhood region into several groups and aggregated the inner measurements within a group. The inflection predictor extracts the sudden changes within the observations. For the regression model, the temporal predictor uses the linear regression model, the spatial predictor uses a 2-hidden-layer neural network and the inflection predictor used the regression tree model. A regression tree model further combines these predictors.\nState Space Models\n------------------\nThe *State Space Model* (SSM) adopts a generative viewpoint of the sequence. For the observed sequence **X**1 : *T*, SSM assumes that each element **X***t* is generated by a hidden state **H***t* and these hidden states, which are allowed to be discrete in our definition, form a Markov process. The general form of the state space model is given in . Here, *g* is the transition model, *f* is the observation model, **H***t* is the hidden state, $\\bm{\\epsilon}\\_t$ is the noise of the transition model, and $\\bm{\\sigma}\\_t$ is the noise of the observation model. The SSM has a long history in forecasting and many classical time-series models like *Markov Model* (MM), *Hidden Markov Model* (HMM), *Vector Autoregression* (VAR), and *Autoregressive Integrated Moving Average Model* (ARIMA) can be written as SSMs.\n \n$$\\begin{aligned}\n \\label{eq:ssm}\n \\mathbf{H}\\_t &= g(\\mathbf{H}\\_{t-1}, \\bm{\\epsilon}\\_t),\\\\\\nonumber\n \\mathbf{X}\\_t &= f(\\mathbf{H}\\_t, \\bm{\\sigma}\\_t).\\end{aligned}$$ \nGiven the underlying hidden state **H***t*, the observation **X***t* is independent concerning the historical information F*t*. Thus, the posterior distribution of the sequence for the next *L* steps can be written as follows:\n \n$$\\label{eq:ssm:forecasting-posterior}\n \\begin{"},"data_id":{"kind":"string","value":"arxiv_0002056"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":4,"numItemsPerPage":100,"numTotalItems":46503,"offset":400,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sInJlcG9JZHMiOlsiNjdmNjBjOGNiMmRmNzFkZDhjOWZkYmIzIl0sImlhdCI6MTc5MTY1ODYwNSwianRpIjoiODI3YWRjOWEtNTI3OC00ZGU2LWFmZDktMzNlN2ZkNWVhMzAzIiwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3OTE2NjIyMDUsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.bjyjri1p5vU0bTT3-TdKCmWe6ElWl-DN4ModYKlVWZQlwzCjWCiQsmqDYz9WvofSO_oZTW6F9K3z_EC9_n9iCQ","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
content stringlengths 34.5k 188k | data_id stringlengths 13 13 |
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Byzantine Agreement with Unknown Participants and Failures
==========================================================
A set of mutually distrusting participants that want to agree on a common opinion must solve an instance of a Byzantine agreement problem. These problems have been extensively studied in the literature.... | arxiv_0001611 |
are the brightest in our sample which the S11 sample excludes. The S11 work fits each galaxy with three models and uses an *F* − test to determine whether a bulge + disk model is needed. A galaxy is fit with a bulge + disk if there is less than 32% chance that a single Sérsic profile is a better fit. This threshold is... | arxiv_0001617 |
of directly to stellar population models and then compare the reddened models with the observed SEDs to derive SFRs and other parameters. Thus we use the full UV/optical SED to constrain the dust attenuation. This procedure is equivalent to an implicit use of the correlation between the UV slope (i.e., UV color) and ... | arxiv_0001620 |
*Have Large Language Models Developed a Personality?*: Applicability of Self-Assessment Tests in Measuring Personality in LLMs
==============================================================================================================================
Have Large Language Models (LLMs) developed a personality? The sho... | arxiv_0001626 |
events that could have failed the S11 event selection due to finite source effects did not occur in the two years of the S11 sample.
The number of events predicted to be found in the $0.5 < t\_{\rm E}/{\rm day} < 2$ range has also changed for reasons relating to our light curve analysis, but changes to our Galactic mo... | arxiv_0001635 |
\_i}) + \textrm{diag}((\boldsymbol{\sigma^1\_i})^2)\right|^2}>0 \nonumber$$
for any element $(\boldsymbol{\sigma^1\_i})^2\_j$ of $(\boldsymbol{\sigma^1\_i})^2$; hence Ψ**n\_i**(*n*) is increasing in any such element and its positivity follows. This completes the proof of the lemma.
Now suppose that some *n* ∈ {1, …, ... | arxiv_0001636 |
Phase transitions of composition schemes:Mittag-Leffler and mixed Poisson distributions
=======================================================================================
,
,
Multitudinous combinatorial structures are counted by generating functions satisfying a composition scheme *F*(*z*) = *G*(*H*(*z*)). The cor... | arxiv_0001637 |
A general framework for estimation and inference from clusters of features
==========================================================================
Applied statistical problems often come with pre-specified groupings to predictors. It is natural to test for the presence of simultaneous group-wide signal for groups in... | arxiv_0001640 |
Deep Chandra, HST-COS, and Megacam Observations of the Phoenix Cluster:Extreme Star Formation and AGN Feedback on Hundred Kiloparsec Scales
===========================================================================================================================================
We present new ultraviolet, optical, and... | arxiv_0001643 |
Dissipative dynamics in a quantum bistable system: Crossover from weak to strong damping
========================================================================================
The dissipative dynamics of a quantum bistable system coupled to a Ohmic heat bath is investigated beyond the spin-boson approximation. Within... | arxiv_0001647 |
Vector Magnetic Fields and Electric Currents from the Imaging Vector Magnetograph
=================================================================================
First, we describe a general procedure to produce high quality vector magnetograms using the Imaging Vector Magnetograph (IVM) at Mees Solar Observatory. Tw... | arxiv_0001649 |
Irreducible decompositions and stationary states of quantum channels
====================================================================
**Abstract.** For a quantum channel (completely positive, trace-preserving map), we prove a generalization to the infinite dimensional case of a result by Baumgartner and Narnhofer (... | arxiv_0001653 |
)} \in \cS^{-1}$ is compact and $\Phi\_j \rightharpoonup 0$, it follows that $\la {{\rm Op}^W\!\left(r\_{-1}\right)} \Phi\_j, \Phi\_j \ra \to 0$ and therefore it suffices to show that $\la {{\rm Op}^W\!\left((v-\lambda)^2\right)} \Phi\_j, \Phi\_j \ra \stackrel{j \to \infty}{\to} 0$. To estimate this last term we shall ... | arxiv_0001658 |
concealed algebras) by Zwara.
Another remarkable characterization due to Zwara, building on work of Riedtmann, is that *M* ≤ *d**e**g**N* exactly when there exists another representation *Z* (of unknown dimension) and an exact sequence of the form
0 → *N* → *M* ⊕ *Z* → *Z* → 0,
and that in this case the degener... | arxiv_0001667 |
, but also on the orbital period, *P*, on *e*, and on the periastron epoch, *T*0, so the model is finally based on 12 unknowns: the 5 parameters of the single star solution, and *A*, *B*, *F*, *G*, *P*, *e*, *T*0; the partial derivatives of *w* with respect to the TI elements are given in Eq. [eq:wAO], and those with r... | arxiv_0001673 |
The Southern HIIRegion Discovery Survey (SHRDS): Pilot Survey
=============================================================
The Southern HIIRegion Discovery Survey (SHRDS) is a survey of the third and fourth quadrants of the Galactic plane that will detect radio recombination line and continuum emission at cm-wavelengt... | arxiv_0001678 |
M}}$ and $\Delta {\rm Sp}^{\mathrm{M}}$, by varying the factors within ranges of plausible values.
The fixed values and variation intervals were chosen according to studies concerning *C. Trachtomatis* or infectious diseases such as tuberculosis or pneumonia (see and references therein). Hence, we admit the following ... | arxiv_0001684 |
Maximum Marginal Likelihood Estimation of Phase Connections in Power Distribution Systems
=========================================================================================
Accurate phase connectivity information is essential for advanced monitoring and control applications in power distribution systems. The exi... | arxiv_0001686 |
IFIC/21-31
FTUV-21-0901.5855
**Multiple soft radiation at one-loop order
and the emission of a soft quark–antiquark pair**
**Stefano Catani** (*a*) and **Leandro Cieri** (*a*)(*b*)
(*a*)INFN, Sezione di Firenze and Dipartimento di Fisica e Astronomia,
Università di Firenze, I-50019 Sesto Fiorentino, Florence, It... | arxiv_0001700 |
ula around S61.
As shown in Section [sec:analysis], there are hints of inhomogeneities in the nebula. Following, who described the radio spectrum of a clumped stellar wind, we can assume discrete gas clumps of relatively higher density (*n**H*), embedded in a lower-density medium (*n**L*). Both the clumps and the inter... | arxiv_0001704 |
Distributional Results for Thresholding Estimators in High-Dimensional Gaussian Regression Models [1](#fn1)
===========================================================================================================
### June 2011Revised November 2011
We study the distribution of hard-, soft-, and adaptive soft-threshol... | arxiv_0001713 |
pinning center $j\_{\rm p}$, we can confirm the existence of such Peierls-Nabarro barriers by numerically computing the ground-state energy as a function of the position of the soliton $ E\_{\rm g}(j\_{\rm p})$ (see Fig. [fig:loclengthbeta]**(a**)). Due to the Peierls’ dimerization, we find a periodic arrangement of e... | arxiv_0001719 |
that, in particular, for *p* = 1 and *r* ≥ 2 (which implies *q* = 2 and $\widetilde{p}=2$), and taking into account that *w* = 2 and $s=\frac{1}{2}$, the error rate reads
$$\norm{\bbE[u(\cdot,t)] - E^L[U(\cdot,t)]}\_{\mathrm{L}^2(\Omega;\mathrm{L}^1(D))} \leq C
\left(W\_L^{\text{MLMC}}\right)^{-\frac{1}{5}}\mper... | arxiv_0001720 |
Quantum Optimization: Potential, Challenges, and the Path Forward
=================================================================
Recent advances in quantum computers are demonstrating the ability to solve problems at a scale beyond brute force classical simulation. As such, a widespread interest in quantum algorithm... | arxiv_0001721 |
5, 0.5); (-0.5, -0.5, 0.5) – ( 0.5, -0.5, 0.5); ( 0.5, -0.5, 0.5) – ( 0.5, 0.5, 0.5); ( 0.5, 0.5, -0.5) – ( 0.5, -0.5, -0.5); ( 0.5, -0.5, 0.5) – ( 0.5, -0.5, -0.5); ( 0,0,0) circle (0.2cm); ( -0.5, 0.5, -0.5) circle (0.2cm);
[shift=(1.5, 0)] (-0.5, 0.5, -0.5) – (-0.5, -0.5, -0.5); (-0.5, -0.5, -0.5) – ( 0.5, -0.5, -0.... | arxiv_0001725 |
600){\usebox{\tow}}
\end{picture}$$
The first system does not admit a set of colors Δ(*τ*), so there is no ambiguity.
We assume now that no color moved by *α**l* is zero on both *α**l* − 1 and *α**l* + 1. If *D**l* − 1− is free, then we have the following.
$$\begin{picture}(14700,2400)(-300,-1200)
\put(0,0){\thic... | arxiv_0001735 |
& 0.0042 & 0.0041
0.7 & 0.1203 & 0.005 & 0.0048
0.75 & 0.1531 & 0.0064 & 0.0061
0.8 & 0.1921 & 0.008 & 0.0077
0.85 & 0.2589 & 0.0119 & 0.0143
0.881 & 0.3568 & 0.0186 & 0.012
0.9 & 0.4647 & 0.025 & 0.0186
0.916 & 0.8113 & 0.0428 & 0.0325
A priori distribution of unknown parameters
----------------------... | arxiv_0001737 |
A Two-Level Block Preconditioned Jacobi-Davidson Method for Multiple and Clustered Eigenvalues of Elliptic Operators
====================================================================================================================
**Abstract:** In this paper, we propose a two-level block preconditioned Jacobi-David... | arxiv_0001738 |
Mean-Field Approximation of Cooperative Constrained Multi-Agent Reinforcement Learning (CMARL)
==============================================================================================
Mean-Field Control (MFC) has recently been proven to be a scalable tool to approximately solve large-scale multi-agent reinforceme... | arxiv_0001745 |
. ADP accepts a (typically very small) error *δ* that characterizes the mass that violates the pure *ɛ*-differential privacy guarantees. However, prior work does not provide a generic method for finding noise distributions with strong utility-privacy trade-offs for ADP under sequential composition.
* We present a **gra... | arxiv_0001749 |
Evaluation of Digital Micromirror Devices for use in space-based Multi-Object Spectrometer application
======================================================================================================
The astronomical community continues to be interested in suitable programmable slit masks for use in multi-object ... | arxiv_0001751 |
Time-Bandwidth Product Perspective for Multi-Soliton Phase Modulation
=====================================================================
Multi-soliton pulses are potential candidates for fiber optical transmission where the information is modulated and recovered in the so-called nonlinear Fourier domain. While this ... | arxiv_0001752 |
photoabsorption characteristics can be influenced by solvation effects in two ways. The ground-state geometries and electronic structures are influenced by the solvent, which, in turn, affects the optical spectrum via its dielectric constant. Additional changes are directly related to the interaction between the dynam... | arxiv_0001753 |
/*T*3 vs *T* plot showed a maximum near 27 *K* for all the compounds indicating the presence of low frequency Einstein modes of vibrations. Resistivity measurements showed that all the samples behave as normal Fermi liquid type compounds and *ρ*(*T*) due to electron-phonon scattering follows Bloch-Gr$\ddot{\textrm u}$n... | arxiv_0001756 |
Power-law corrections to entanglement entropy of horizons
=========================================================
We re-examine the idea that the origin of black-hole entropy may lie in the entanglement of quantum fields between inside and outside of the horizon. Motivated by the observation that certain modes of gra... | arxiv_0001758 |
over r^2}}+{3\over 2}\tau \right].
\label{eqt}$$
The optical depth *τ* in eq. ([eqt]) is found from the differential relation,
$${d\tau\over dr}=-\rho k{R^2\over r^2}
\label{eqtau}$$
with the limiting condition that *τ* → 0 for *r* → ∞.
The wind model, determined via integration of eqs. ([eqv])–([eqtau]), is cal... | arxiv_0001762 |
Search for neutrino-induced particle showers with IceCube-**40**
================================================================
[1](#fn1)
[2](#fn2)
[3](#fn3)
[4](#fn4)
We report on the search for neutrino-induced particle-showers, so-called cascades, in the IceCube-40 detector. The data for this search was collected ... | arxiv_0001771 |
Classical Lagrangians and Finsler structuresfor the nonminimal fermion sector of the Standard-Model Extension
=============================================================================================================
This article is devoted to finding classical point-particle equivalents for the fermion sector of th... | arxiv_0001779 |
ries this model does not possess any Goldstone boson, which is expected in the absence of a global symmetry. There are of course special points as for the previous model, such as *μ**s*4 = − *c*1*c*2*m*12*m*22 or *μ̄**s*4 = − *c*1*m*12*κ*/*c*3*m*32, for which massless bosons enter the model. However, these massless b... | arxiv_0001790 |
Sloppy model analysis identifies bifurcation parameters without normal form analysis
====================================================================================
Bifurcation phenomena are common in multi-dimensional multi-parameter dynamical systems. Normal form theory suggests that bifurcations are driven by r... | arxiv_0001792 |
).. controls (80,64.21) and (78.21,66).. (76,66).. controls (73.79,66) and (72,64.21).. (72,62) – cycle ; (72,102).. controls (72,99.79) and (73.79,98).. (76,98).. controls (78.21,98) and (80,99.79).. (80,102).. controls (80,104.21) and (78.21,106).. (76,106).. controls (73.79,106) and (72,104.21).. (72,102) – cycle ; ... | arxiv_0001793 |
*next, *σ*) *// Schedule *σ* for next update*
---
### Numerical Experiments
We first present a simple numerical example demonstrating the good energy behavior of our asynchronous integrator. The AVI was used to integrate in time over a 2-D rectangular cavity with perfectly electrically conducting (PEC) boundaries, so t... | arxiv_0001797 |
* = 0 and *j* ∈ Ω*m* → *ϕ**j* = *ϕ*0.
The non-dimensional parameters, i.e., the inertial number, $ I\_i={|\dot{\gamma}|\_i}{d\_g}\sqrt{\rho\_g/{p\_g}\_i} $, the viscous number, $ {I\_\nu}\_i=\mu\_w{|\dot{\gamma}|\_i}/{p\_g}\_i $, and the mixed number, $ I\_m=\sqrt{I^2+2I\_\nu} $ govern the visco-inertial rheology model... | arxiv_0001801 |
…, *N* − 1) above, yields
$${\mbox{\boldmath $\hat{h}$}\_{}^{(r)}}=
\left[\sum\_{n=0}^{N-1}{\mathrm{diag}}({\mbox{\boldmath $c$}\_{n}^{\*}}\odot{\mbox{\boldmath $c$}\_{n}^{}})\right]^{-1}
\sum\_{n=0}^{N-1}{\mathrm{diag}}({\mbox{\boldmath $c$}\_{n}^{\*}}){\mbox{\boldmath $y$}\_{n}^{(r)}},$$
or equivalently ([eq:Hr... | arxiv_0001805 |
Neglected Clouds in T and Y Dwarf Atmospheres
=============================================
As brown dwarfs cool, a variety of species condense in their atmospheres, forming clouds. Iron and silicate clouds shape the emergent spectra of L dwarfs, but these clouds dissipate at the L/T transition. A variety of other cond... | arxiv_0001806 |
-the-art video understanding methods, which almost always involve deep-neural networks, on large datasets. Recent systems adapt prior work in visual data management for modern tasks that involve applying machine learning techniques to extract insights from large video datasets potentially comprising of millions of hou... | arxiv_0001814 |
Minimax Estimation of Neural Net Distance
==========================================
An important class of distance metrics proposed for training generative adversarial networks (GANs) is the integral probability metric (IPM), in which the neural net distance captures the practical GAN training via two neural networks... | arxiv_0001815 |
≥ *v**x*. By Lemma [lem:vallazy], an acceptable value of *v**x* is ${\text{\rm min}}(0, {\text{\rm val}}\_p({\texttt{\rm x}\xspace}(0)))$ (or more generally ${\text{\rm min}}(i, {\text{\rm val}}\_p({\texttt{\rm x}\xspace}(i)))$ for any value of *i*). A lazy *p*-adic number whose value if *x**y* is then given by the fol... | arxiv_0001819 |
Exploiting dynamic bifurcation in elastic ribbons for mode skipping and selection
=================================================================================
In this paper, we systematically study the dynamic snap-through behavior of a pre-deformed elastic ribbon by combining theoretical analysis, discrete numeri... | arxiv_0001831 |
from the digital main-memory of the accelerator to the digital cache-memory array; and a matrix read time and energy of
*T*RM*d* ≈ 250microseconds and *E*RM*d* ≈ 12000micro-Joules,
respectively. The vector read time and energy on the purely digital accelerator should be comparable to the vector read time and ... | arxiv_0001833 |
Multiwavelength modeling the SED of Luminous Supersoft X-ray Sources in Large Magellanic Cloud and Small Magellanic Cloud
=========================================================================================================================
Classical supersoft X-ray sources (SSSs) are understood as close binary syst... | arxiv_0001835 |
Excursion set peaks in energy as a model for haloes
===================================================
[firstpage]
The simplest models of dark matter halo formation rely on the heuristic assumption, motivated by spherical collapse, that virialized haloes originate from initial regions that are maxima of the smoothed m... | arxiv_0001836 |
9 3 - 11286 4 + 4536 5)) 3 & =
& (107520 8 (1 - 3 )8 (1 - )12 - 61440 6 (1 - 3 )10 (1 - )14
& + 23040 4 (1 - 3 )12 (1 - )16 - 5120 2 (1 - 3 )14 (1 - )18 + 512 (1 - 3 )16 (-1 + )20
& + 20 (512 - 48603 + 471708 2 - 1645251 3 + 2623320 4 - 1957284 5 + 555660 6)
& + 18 (1 - )2 (-5120 - 639867 + 20457090 2 - 1779354... | arxiv_0001843 |
Coding Methods in ComputabilityTheory and Complexity Theory
===========================================================
### January 1998
Preface
=======
A major part of computability theory focuses on the analysis of a few structures of central importance. As a tool, the method of coding with first-order formulas has b... | arxiv_0001845 |
not locally equivalent.
Let us emphasize that no real solution for the AME states of four subsystems with six levels each was found. It was then conjectured that no real 2-unitary orthogonal matrices of size 36 exist. From a geometric perspective it is thus interesting to analyze differences between the structures of... | arxiv_0001849 |
Faster Relative Entropy Coding withGreedy Rejection Coding
==========================================================
Relative entropy coding (REC) algorithms encode a sample from a target distribution *Q* using a proposal distribution *P* using as few bits as possible. Unlike entropy coding, REC does not assume discre... | arxiv_0001857 |
bar \omega\_C}\sim \frac{m e^{3/2}}{\epsilon \hbar^{3/2}}(Bn)^{-1/2},$$
which turns out to be nothing other than the usual dimensionless coupling constant
$$r\_s=\frac{me^2}{\epsilon \hbar^2}n\_{el}^{-1}$$
for the 2D Coulomb gas in Fermi liquid theory. The last expression is obtained by identifying the Fermi e... | arxiv_0001858 |
Edge theory approach to topological entanglement entropy, mutual informationand entanglement negativity in Chern-Simons theories
================================================================================================================================
We develop an approach based on edge theories to calculate the... | arxiv_0001861 |
AMiBA: Cluster Sunyaev-Zel’dovich Effect Observations with the Expanded 13-Element Array
========================================================================================
The Yuan-Tseh Lee Array for Microwave Background Anisotropy (AMiBA) is a co-planar interferometer array operating at a wavelength of 3mm to me... | arxiv_0001872 |
t*
Introduction
============
than 100 countries have committed to achieving carbon neutrality in the 21st century to mitigate global climate change. Reducing greenhouse gas emissions is one of the key steps. CO2 is the main greenhouse gas whose emissions exceeded 37.9 Gt in 2021. Meanwhile, a large part of CO2 emission... | arxiv_0001873 |
76:083512, 2007, astro-ph/0511041.
J. Martin, C. Ringeval, and V. Vennin.., 2014, 1303.3787.
A. Albrecht and R. H. Brandenberger.., D31:1225, 1985.
I. Moss.., B154:120, 1985.
B. Hu and D. O’Connor.., D34:2535, 1986.
A. Vilenkin.., 72:3137–3140, 1994, hep-th/9402085.
D. Cormier and R. Holman.., D60:041301, 1999, hep-ph/... | arxiv_0001882 |
An algebraic convergence rate for the optimal control of McKean-Vlasov dynamics
===============================================================================
-0.5in
We establish an algebraic rate of convergence in the large number of particles limit of the value functions of *N*-particle stochastic control problems t... | arxiv_0001885 |
Non-Malleable Coding Against Bit-wise and Split-State Tampering[3](#fn3)
========================================================================
Non-malleable coding, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), aims for protecting the integrity of information against tampering attacks in situations where... | arxiv_0001905 |
Edge effects in graphene nanostructures:II. Semiclassical theory of spectral fluctuations and quantum transport
================================================================================================================
We investigate the effect of different edge types on the statistical properties of both the en... | arxiv_0001907 |
. Combining this bound with inequalities and completes the proof of inequality.
Finally, we prove Lemma [CorMixingTimeRestWalk]:
[Proof of Lemma [CorMixingTimeRestWalk]] Translating the main result of into our discrete-time setting, we have n,k Cd n-1 -, for some constant *C**d* that depends only on *d*. By Lemma [Lem... | arxiv_0001912 |
{j\}}
& \ge& m{\sum\_{{\mathbf{i}}}}\nu\_{{\mathbf{i}},\{j\}}(1-2^{r+1}{\epsilon})\\
& = & m\nu\_{\{j\}}\bigl(1+O({\epsilon})\bigr).\end{aligned}$$
Therefore,
$$\frac{{\mathbb{E}}\_{{\mathrm{RE}}}( {\widetilde}{\operatorname{Var}}\_{{\mathrm{PW}}}(\bar X^\*))}{{\operatorname{Var}}(\bar X)}
=\frac{{\sum\_{{\mathbf{... | arxiv_0001921 |
*in-vivo* procedures.
Regarding the thalamus, a recent hypothesis argues that its nuclei are distinguished in their characteristic fiber orientation. Indeed, based on this hypothesis, division of the thalamus into several nuclei was made using Diffusion Tensor Imaging (DTI). DTI is a novel MRI-based technique, which e... | arxiv_0001923 |
General relations for quantum gases in two and threedimensions: Two-component fermions
======================================================================================
We derive exact general relations between various observables for *N* spin-1/2 fermions with zero-range or short-range interactions, in continuous... | arxiv_0001926 |
Self-duality protected multi-criticality in deconfined quantum phase transitions
================================================================================
Duality places an important constraint on the renormalization group flows and the phase diagrams. For self-dual theories, the self-duality can be promoted as ... | arxiv_0001930 |
rank 1269, which seems out of reach for any practical computation.
* Bell inequalities.
Bell inequalities related to AME(4,3) are unknown.
The small corner of the Hilbert space formed by AME states may well be more complex than expected.
**Acknowledgements**. It is a pleasure to thank J. Bielawski, M. Grassl and S. Pa... | arxiv_0001934 |
The extension problem in free harmonic analysis
===============================================
This paper studies certain aspects of harmonic analysis on nonabelian free groups. We focus on the concept of a positive definite function on the free group and our primary goal is to understand how such functions can be ext... | arxiv_0001937 |
How Auto-Encoders Could Provide Credit Assignment in Deep Networks via Target Propagation
=========================================================================================
We propose to exploit *reconstruction* as a layer-local training signal for deep learning. Reconstructions can be propagated in a form of ta... | arxiv_0001946 |
Conservative finite volume schemes for the quasi-geostrophic equation on coastal-conforming unstructured primal-dual meshes
===========================================================================================================================
In this paper we propose finite volume schemes for solving the inviscid ... | arxiv_0001951 |
The circularization timescales of late–type binary stars
========================================================
### Accepted XXX. Received YYY; in original form ZZZ
[firstpage]
We examine the consequences of, and apply, the formalism developed in Terquem (2021) for calculating the rate *D**R* at which energy is excha... | arxiv_0001952 |
Understanding the effect of sheared flow on microinstabilities
==============================================================
The competition between the drive and stabilization of plasma microinstabilities by sheared flow is investigated, focusing on the ion temperature gradient mode. Using a twisting mode representat... | arxiv_0001960 |
Harmonic analysis and spherical functions for multiplicity-free induced representations of finite groups
========================================================================================================
In this work, we study multiplicity-free induced representations of finite groups. We analyze in great detail ... | arxiv_0001967 |
The impact of feedback and the hot halo on the rates of gas accretion onto galaxies
===================================================================================
We investigate the physics that drives the gas accretion rates onto galaxies at the centers of dark matter haloes using the EAGLE suite of hydrodynamica... | arxiv_0001973 |
Nonperturbative quantization à la Heisenbergand thermodynamics of monopole configurations
=========================================================================================
For field theories in which no small parameter is available, we propose a definition of nonperturbative quantum states in terms of the compl... | arxiv_0001975 |
Fine-Grained Analysis of Stability and Generalization for Stochastic Gradient Descent[2](#fn2)
==============================================================================================
Recently there are a considerable amount of work devoted to the study of the algorithmic stability and generalization for stochast... | arxiv_0001976 |
*N*1 + *N*2} − *N*1)*T*1 equations, and receiver 2 will acquire (min{*A*1, *N*1 + *N*2} − *N*1)*T*1 equations as well by $\underline{\textbf{y}}\_2^{a + b} - \underline{\textbf{H}}\_2^{a + b}\textbf{x}\_a^1 \in \mathbb{C}^{(\min\{A\_1,N\_1 + N\_2\}-N\_1)T\_1}$, due to min{*A*1, *N*1 + *N*2} − *N*1 ≤ *N*2. If we provid... | arxiv_0001987 |
23 published galaxy-scale lens candidates from our Pan-STARRS CNN search are shown in Fig. [fig:cnncand] and in Appendix A, together with $p\_{\rm CNN}$ and visual inspection grades. Given that some visual identifications rely on Legacy imaging, which is sometimes deeper than PS1, we also show Legacy 3-color images of ... | arxiv_0001988 |
On the microstructure of higher-dimensional Reissner-Nordström black holes in quantum regime
============================================================================================
Thermodynamic Riemannian geometry provides great insights into the microscopic structure of black holes (BHs). One such example is the... | arxiv_0001995 |
3}$ corresponding to the brightest X-ray sources, and down to ∼ (0.04 − 0.1) × 10− 3 for the faintest ones. For a pulsar interpretation, this probes the conversion ranges of spin-down powers to (non-thermal or thermal) X-rays for possible neutron star candidates.
In contrast to the thermal X-ray sources discussed abov... | arxiv_0002002 |
**On Properties of the Support of Capacity-Achieving Distributions for Additive Noise Channel Models with Input Cost Constraints[1](#fn1)**
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We study the classical problem of charac... | arxiv_0002006 |
, a general formula for any theory involving arbitrary contractions of the Riemann tensor L(*R**μ**ν**ρ**σ*), which seems to satisfy several consistency checks, has been recently proposed by Dong (see also, e.g. )). The corresponding expressions would contain a Wald-like term as well as additional terms involving cont... | arxiv_0002007 |
Equations for point configurations to lie on a rational normal curve
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The parameter space of *n* ordered points in projective *d*-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in (P*d*)*n*. Th... | arxiv_0002012 |
Simple Homogeneous Structures and Indiscernible Sequence Invariants
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We introduce some properties describing dependence in indiscernible sequences: *F**i**n**d* and its dual *F**M**b*, the definable Morley property, and *n*-resolvability. Applying these... | arxiv_0002015 |
when considering the hopping along the z bond between l_1 and l_1'. [fig-9]](Figure9)Site and plaquette labeling conventions around the sites *l*1 (white dot) and *l*1ʹ (black dot) when considering the hopping along the *z* bond between *l*1 and *l*1ʹ. [fig-9]
For a generic open string connecting the sites 1 and 1ʹ, w... | arxiv_0002021 |
ref. and reported in this chapter) are quite naturally understood, such as:
* the asymmetry of the phase diagram around the total filling *n* = 6, which clearly in the presently exposed rationale does not represent a special point, all physical quantities evolving monotonously through it, in the paramagnetic phase;
*... | arxiv_0002022 |
X*) can be neglected in the propagator within our approximations (see the discussion below Eq. ([D0i])).
Now, the integral in *p*+ is Eq. ([D0ff]) can be performed analytically. We first remove the *θ*-function by splitting the integral from − ∞ to + ∞ into two integrals, one from − ∞ to 0 and the other from 0 to +... | arxiv_0002025 |
∅ respectively. Another easy calculation shows that case (B1) corresponds to II.1 (where there is no *Ẽ*7 singularity present) or III.1 and case (B2) corresponds to II.2 or III.2 in Table [eqtable1]. In subcase (B3), and Lemma [1cyclelem] (in the Type II and III cases respectively) show that, prior to contracting th... | arxiv_0002027 |
v v\_1 \cdots v\_m \bar v = v\.$$
But then we have
$$vf = v(v\_1 \cdots v\_m \bar v)^\omega f =
v(v\_1 \cdots v\_m \bar v)^\omega v\_{n+1} \cdots v\_m \bar v (v\_1
\cdots v\_m \bar v)^\omega f =
v(v\_1 \cdots v\_m \bar v)^\omega g = vg\.$$
The second equality follows from ([eq:da]).
Empty forests
========... | arxiv_0002030 |
Sparse random graphs with clustering
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### October 21, 2009
In 2007 we introduced a general model of sparse random graphs with (conditional) independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and ... | arxiv_0002031 |
All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces
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We demonstrate that the linear quotient singularity for the exceptional subgroup *G* in $\Sp(4,{\mathbb{C}})$ of order 32 is isomorphic to an affine qui... | arxiv_0002032 |
Electrostatic stability of insulating surfaces: Theory and applications
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We analyze the electrostatic stability of insulating surfaces in the framework of the bulk modern theory of polarization. We show that heuristic arguments based on a fully ioni... | arxiv_0002037 |
Bayes correlated equilibria and no-regret dynamics
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This paper explores equilibrium concepts for Bayesian games, which are fundamental models of games with incomplete information. We aim at three desirable properties of equilibria. First, equilibria can be naturally rea... | arxiv_0002050 |
a straight vortex has zero interaction with itself. On the other hand a curved vortex acts on itself with a divergent force (made finite by cutting off the integral at a distance on the order of the core radius). This force can be described by the ``local induction approximation", Eq. ([eq:LIA]). This force points tow... | arxiv_0002053 |
Γ-Convergence involving nonlocal gradients with varying horizon: Recovery of local and fractional models
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This work revolves around the rigorous asymptotic analysis of models in nonlocal hyperelasticity. The correspo... | arxiv_0002054 |
The Schrödinger Cat: Physics, Myth, and Philosophy
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The most discussed of “live” metaphors in physical sciences is that of the Schrödinger cat. Introduced in the early, but mature days of the new quantum theory, in 1935, the parable of the cat has provoked and enlivened... | arxiv_0002055 |
Machine Learning for Spatiotemporal Sequence Forecasting: A Survey
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real-world phenomena are spatiotemporal, such as the traffic flow, the diffusion of air pollutants and the regional rainfall. Correctly predicting the future of these spatiotemporal syst... | arxiv_0002056 |
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