The weak limit of Ising models on locally tree-like graphs
aa r X i v : . [ m a t h . P R ] D ec The weak limit of Ising models on locally tree-like graphs
Andrea Montanari ∗ , Elchanan Mossel † and Allan Sly ‡ November 3, 2018
Abstract
We consider the Ising model with inverse temperature β and without external field on sequences ofgraphs G n which converge locally to the k -regular tree. We show that for such graphs the Ising measurelocally weak converges to the symmetric mixture of the Ising model with + boundary conditions and the − boundary conditions on the k -regular tree with inverse temperature β . In the case where the graphs G n are expanders we derive a more detailed understanding by showing convergence of the Ising measurecondition on positive magnetization (sum of spins) to the + measure on the tree. An Ising model on the finite graph G (with vertex set V , and edge set E ) is defined by the followingdistribution over x = { x i : i ∈ V } , with x i ∈ { +1 , − } µ ( x ) = 1 Z ( β, B ) exp n β X ( i,j ) ∈ E x i x j + B X i ∈ V x i o . (1.1)The model is ferromagnetic if β ≥ B ≥
0. Here Z ( β, B ) is anormalizing constant (partition function).The most important feature of the distribution µ ( · ) is the ‘phase transition’ phenomenon. On a varietyof large graphs G , for large enough β and B = 0, the measure decomposes into the convex combinationof two well separated simpler components. This phenomenon has been studied in detail in the case ofgrids [2, 3, 4, 5], and on the complete graph [1]. In this paper we consider sequences of regular graphs G n = ( V n , E n ) with increasing vertex sets V n = [ n ] = { , . . . , n } that converge locally to trees and provea local characterization of the corresponding sequence of measures µ n ( · ), which corresponds to the phasetransition phenomenon.More precisely, consider the case in which G n is a sequence of regular graphs of degree k ≥ B i any vertex i in G n converges to an infinite regular tree of degree k .It is natural to assume that the marginal distribution µ n, B i ( · ) converges to the marginal of a neighborhoodof the root for an Ising Gibbs measure on the infinite tree. For large β , however, there are uncountablymany Gibbs measures on the tree so it is natural to ask which is the limit ∗ Department of Electrical Engineering and Department of Statistics, Stanford University † Faculty of Mathematics and Computer Science, Weizmann Institute and Departments of Statistics and Computer Science,UC Berkeley ‡ Microsoft Research, Redmond, WA ν + ( · ) and ν − ( · ). It was proved in [9] that, for any β , and any B > µ n ( · )converges locally to ν + as n → ∞ and by symmetry when B < µ n ( · ) converges locally to ν − as n → ∞ .In this paper we cover the remaining (and most interesting) case proving that µ n ( · ) −→ n ν + ( · ) + 12 ν − ( · ) for B = 0 and any β ≥ µ n, + ( · ) and µ n, − ( · ) denote the Ising measure (1.1) conditioned to,respectively, P i ∈ V x i > P i ∈ V x i <
0, then we have µ n, ± ( · ) −→ n ν ± ( · ) for B = 0 and any β ≥ µ n = µ + ,n + µ − ,n (exactly for n odd and approximately for even n ), this result implies (1.2). We denote by G n = ( V n , E n ) a graph with vertex set V n ≡ [ n ] = { , . . . , n } . The distance d ( i, j ) between i, j ∈ V n is the length of the shortest path from i to j in G n . Given a vertex i ∈ V n , we denote by B i ( t ) theset of vertices whose distance from i is at most t (and with a slight abuse of notation it will also denotethe subgraph induced by those vertices). We will let I denote a vertex chosen uniformly from the vertices V n , let U n denote the measure induced by I and let J denote a uniformly random neighbor of I .This paper is concerned by sequence of graphs { G n } n ∈ N of diverging size, that converge locally to T k ,the infinite rooted tree of degree k . Let T k ( t ) be the subset of vertices of T k whose distance from theroot ø is at most t (and, by an abuse of notation, the induced subgraph). For a rooted tree T , we write T ≃ T k ( t ) if there is a graph isomorphism between T and T k ( t ) which maps the root of T to that of T k ( t ).The following definition defines what we mean by convergence in the local weak topology. Definition 2.1.
Consider a sequence of graphs { G n } n ∈ N , and let U n be the law of a uniformly randomvertex I in V n . We say that { G n } converges locally to the degree- k regular tree T k if, for any t , lim n →∞ U n { B I ( t ) ≃ T k ( t ) } = 1 . (2.1)Part of our results hold for sequences of expanders (more precisely, edge expanders), whose definitionwe now recall. For a subset of vertices S ⊂ V , we will denote by ∂S the subset of edges ( i, j ) ∈ E havingonly one endpoint in S . Definition 2.2.
The k -regular graph G = ( V, E ) is a ( γ, λ ) (edge) expander if, for any set of vertices S ⊆ V with | S | ≤ nγ , | ∂S | ≥ λS . In analogy with the definition of locally tree-like graph sequences, we introduce local weak convergence forIsing measures. This done in two different ways. First one can look at a random vertex and the randomconfiguration in the neighbourhood of the vertex and examine its limiting measure. Alternatively, we maychoose a random vertex and consider the marginal distribution of the variables in a neighborhood under theIsing model. This induces (via the random choice of the vertex) a distribution over probability measures.We can therefore ask whether this measure converges to a probability measure over Gibbs measures.2ecall that an Ising measure µ on the infinite tree T k may be either defined as a weak limit of Gibbsmeasures on T k ( t ) or in terms of the DLR conditions, see e.g. [11]. An Ising model is in particular aprobability measure over {− , +1 } T k endowed with the σ -algebra generated by cylindrical sets. We let G k denote the space of Ising Gibbs measures on T k and let H k denote the space of all probability measureson { +1 , − } T k . We endow both these spaces with the topology of weak convergence. Since { +1 , − } T k iscompact, G k and H k are also compact in the weak topology by Prohorov’s theorem.We define M k (respectively M G k ) to be the space of probability measures over ( H k , B H ) (resp. ( G k , B H )),with B Ω the Borel σ -algebra. Also M k , M G k are compact in the weak topology.We will use generically µ for Ising measures on G n and ν for Ising measure on T k . For a finite subsetof vertices S ⊆ V n , we let µ S be the marginal of µ on the variables x j , j ∈ S . We the shorthand µ t forwhen S = B i ( t ) is the ball of radius t about i ( i should be clear from the context). For a measure ν ∈ G k we let ν t denote its marginal over the variables x j , j ∈ T k ( t ). In other words ν t is the projection of ν on { +1 , − } T k ( t ) . For a measure m ∈ M k we let m t denote the measure on the space of measures on { +1 , − } T k ( t ) induced by such projection. Definition 2.3.
Consider a sequence of graphs/Ising measures pairs { ( G n , µ n ) } n ∈ N and let P tn ( i ) denotethe law of the pair ( B i ( t ) , x B i ( t ) ) when x is drawn with distribution µ n and i ∈ [ n ] is vertex in the graph. Let U n denote the uniform measure over a random vertex I ∈ [ n ] . Let P tn = E U n ( P tn ( I )) denote the average of P tn ( I ) .A. The first mode of convergence concerns picking a random vertex I and a random local configurationin the neighbourhood of I . Formally, for ¯ ν ∈ G k we say that { µ n } n ∈ N converges locally on averageto ¯ ν if for any t and any ǫ > it holds that lim n →∞ d TV (cid:0) P tn , δ T k ( t ) × ¯ ν t (cid:1) = 0 . (2.2) B. A stronger form of convergence involves picking a random vertex I and the associated random localmeasure P tn ( I ) and asking if this distribution of distributions converges. Formally, we say that thelocal distributions of { µ n } n ∈ N converge locally to m ∈ M G k if it holds that the law of P tn ( I ) convergesweakly to δ T k ( t ) × m t for all t .C. If m is a point mass on ¯ ν ∈ G k and if the local distributions of { µ n } n ∈ N converge locally to m thenwe say that { µ n } n ∈ N converges in probability locally to ¯ ν . Equivalently convergence in probabilitylocally to ¯ ν says that for any t and any ǫ > it holds that lim n →∞ U n (cid:0) d TV ( P tn ( I ) , δ T k ( t ) × ν t (cid:1) > ǫ ) = 0 . (2.3) It is easy to verify that C ⇒ B ⇒ A . Similar notions of the convergence has been studied before under the name metastates for Gibbsmeasures. Aizenman and Wehr [6], while investigating the quenched behaviour of lattice random fieldmodels, introduced the notion of a metastate which is a probability measures over Gibbs measures as afunction of the disorder (the random field). Here, rather than taking a finite graph and choosing a randomvertex they take a fixed random environment in Z d , and study the measure over increasing finite volumes.Rather than prove convergence (which depending on the model may not hold) they take subsequentiallimits and study the properties of these limiting distributions of Gibbs measures (metastates). Another,similar notion of convergence to metastates was developed by Newman and Stein [16] where they took theempirical measure over Gibbs measures at over increasing volumes to study spin-glasses. More referencesand discussions can be found in [13]. 3n order to state our main result formally, we recall that an Ising measure on T k is Gibbs if, for anyinteger t ≥ µ T k ( t ) | T c k ( t ) ( x T k ( t ) | x T c k ( t ) ) = 1 Z t,x ( β ) exp β X ( i,j ) ∈ E ( T k ( t +1)) x i x j , (2.4)where Z t ( β ) is a normalization function that depends on the conditioning, namely on x T k ( t +1) \ T k ( t ) .It is well known that if ( k −
1) tanh β ≤
1, there exist only one Gibbs measure on a k -regular treewhile for ( k −
1) tanh β > ν + and ν − play a special role in the following. The ‘plus-boundary conditions’ measure ν + is defined as the monotone decreasing limit (with respect to the natural partial ordering on the spaceof configurations { +1 , − } T k ) of ν t + as t → ∞ , where ν t + is the measure on x T k ( t ) defined by ν t + ( x T k ( t ) ) = 1 Z + ,t ( β ) exp β X ( i,j ) ∈ E ( T k ( t )) x i x j Y i ∈ T k ( t ) \ T k ( t − I ( x i = +1) . (2.5)The measure ν − is defined analogously, by forcing spins on the boundary to take value − ν we have ν − (cid:22) ν (cid:22) ν + (with (cid:22) the stochastic ordering induced by the partialordering on { +1 , − } configurations, see e.g. [14]). Our main result may be now stated as follows Theorem 2.4.
Let { G n } n ∈ N be a sequence of k -regular graphs that converge locally to the tree T k . For ( k −
1) tanh β > , define the sequence { µ n } n ∈ N , { µ n, + } n ∈ N by µ n, + ( x ) = 1 Z n, + ( β ) exp n β X ( i,j ) ∈ E n x i x j o I n X i ∈ V n x i > o , (2.6) µ n ( x ) = 1 Z n ( β ) exp n β X ( i,j ) ∈ E n x i x j o . (2.7) ThenI. µ n converges locally in probability to ( ν + + ν − ) II. If the graphs { G n } are (1 / , λ ) edge expanders for some λ > , then µ n, + converges locally inprobability to the plus-boundary Gibbs measure on the infinite tree ν + . This characterization has a number of useful consequences. In particular, ‘spatial’ averages of localfunctions are roughly constant under the conditional measure µ n, + . To be more precise, for each i ∈ V n ,let f i,n : { +1 , − } B i ( ℓ ) → [ − , , be a function of its neighborhood B i ( ℓ ). Theorem 2.5.
Let { G n } n ∈ N be a sequence of k -regular (1 / , λ ) edge expanders, for some λ > , thatconverge locally to the tree T k . For each n , let { f i,n } ni =1 be a collection of local functions as above. Then,for any ε > n →∞ µ n, + n(cid:12)(cid:12)(cid:12) n X i ∈ V n [ f i,n ( x B i ( ℓ ) ) − µ n, + n X i ∈ V n f i,n ( x B i ( ℓ ) )) ! ] (cid:12)(cid:12)(cid:12) ≥ ε o = 0 . (2.8)The proof can be found in Section 5. 4 .3 Examples and remarks Notice that, for ( k −
1) tanh β ≤
1, the set of Ising Gibbs measures on T k contains a unique element, thatcan be obtained as limit of free boundary measures. Therefore, the local limits of { µ n } n ∈ N , { µ n, + } n ∈ N coincide trivially with this unique Gibbs measure.Therefore, the claim I is proved under the weakest possible, hypothesis, namely local convergence ofthe graphs to T k . An important class of graphs for which Theorem 2.4 is applicable are random k -regulargraphs. These are known to converge locally to T k [17].The expansion condition (or an analogous ‘connectedness’ condition) is needed to obtain the convergenceof the conditional measures µ n, + . For example consider r identical but disjoint graphs on n/r vertices.Then conditioning on the sum of the spins being positive the probability that the sum of spins in a specificcomponent is positive is of order r − / . Therefore in this case we have: µ n, + → (1 − q ) ν + + qν − , with q = 1 / − O ( r − / ). A similar construction may be repeated with a small number of edges connectingdifferent components, e.g., when the components are connected in a cyclic fashion.In order to identify the limit for µ n and obtain our results, there are a number of challenges thatneed to be overcome. First, while soft compactness arguments imply that subsequential limits exist, sucharguments do not imply the existence of a proper limit. Second, recalling that there are uncountably manyextremal Gibbs measures for T k , it is remarkable we are able to identify precisely those that appear in thelimit. Finally, for conditional measures such as µ n, + it is not even a priori clear that (subsequential) limitsare in fact Gibbs measures. The basic idea of the proof is the following. Look at a ball of radius t around a vertex i in G n . Since G n istree like, the ball is with high probability a tree. The measure µ n restricted to the ball is clearly a Gibbsmeasure on a tree of radius t . The same is true (although less obvious) for µ n, + .In order to characterize the limit of this measure as n → ∞ ,1. The probability of agreement between neighboring spins in the ball is asymptotically the same as inthe measure ν + on the infinite tree.2. We further show that ν + maximizes the probability of agreement between neighboring spins amongall Gibbs measures on the tree. These two facts together imply that any local limit must convergeto a convex combination of ν + and ν − .3. By symmetry this already implies converges of µ n to ( ν + + ν − ). Note that this step does not requireexpansion, just the local weak convergence of the tree.4. In order to deal with the conditional measure, we use expansion to show that it is unlikely thatsimultaneously a positive fraction of the vertices have their neighborhood “in the + state” andanother positive fraction “in the − state”. We now proceed with the proof. For each of claims I and II we break the proof into 3 steps:( i ) We consider a subsequence of sizes { n ( m ) } m ∈ N along which µ n ( m ) or µ n ( m ) , + converge locally inaverage to a limit ¯ ν or ¯ ν + (respectively). 5 ii ) We prove that any such limit is in fact always the same and is ¯ ν = (1 / ν + + ν − ) for µ n ( m ) and(using expansion) ¯ ν + = ν + for µ n ( m ) , + . As a consequence the sequences themselves converge.( iii ) Finally we show how is it possible to deduce local convergence from convergence in average. The construction of subsequential weak limits is based on a standard diagonal argument, for similar resultssee [8]. For the sake of simplicity we refer to the measures µ n, + , and construct the subsequential limit ¯ ν + ,but the same procedure works for µ n with limit ¯ ν . Let B I ( t ) be the ball of radius t centered at a uniformlyrandom vertex I in V n , and x be an Ising configuration with distribution µ n, + . If P n denotes the jointdistribution of ( B I ( t ) , x B I ( t ) ), we let µ t + ,n ( x ∗ T k ( t ) ) ≡ P n (cid:8) ( B I ( t ) , x B I ( t ) ) ≃ ( T k ( t ) , x ∗ T k ( t ) ) (cid:9) . (3.1)Since this is a sequence of measures over a finite state space, it converges over some subsequence { n t ( m ) } m ≥ .Further, since by hypothesis P n { B i ( t ) ≃ T k ( t ) } →
1, the limits of µ t + ,n t ( m ) and µ tn t ( m ) are in fact probabilitymeasures. We call the limit ¯ ν t + .Fix one of these subsequences { n t ( m ) } m ≥ for t = t , leading to the limit ¯ ν t + , and recursively refine itto { n t ( m ) } m ≥ ⊇ { n t +1 ( m ) } m ≥ ⊇ { n t +2 ( m ) } m ≥ ⊇ . . . leading to limits ¯ ν t + for all t ≥ t . Notice that,for any graph G n , any vertex i and any t we have µ tn, + ( x B i ( t ) ) = X x B i ( t +1) \ B i ( t ) µ t +1 n, + ( x B i ( t +1) ) . (3.2)As a consequence, for any t , the measures limit ¯ ν ( t )+ measure satisfies¯ ν t + ( x T k ( t ) ) = X x T k ( t +1) \ T k ( t ) ¯ ν t +1+ ( x T k ( t +1) ) . (3.3)By Kolmogorov extension theorem, there exist measures ¯ ν + over { +1 , − } T k such that ¯ ν t + are the marginalsof ¯ ν + over the variables in the subtree T k ( t ). By taking the diagonal subsequence n ( m ) = n m ( m ) we obtainthe desired subsequence { n ( m ) } m ∈ N such that µ n ( m ) , + converges locally on average to ¯ ν + . ¯ ν = ( ν + + ν − ) In this section we carry out our program in the case of the unconditional measures µ n . It is immediate that,since each of the measures µ tn is a Gibbs measure on T k (although with a complicate boundary condition),the limit measure ¯ ν is also a Gibbs measure on T k (i.e. ¯ ν ∈ G k ).For proving convergence of the unconditional measure we need two lemmas. The first one establishesthat the + (equivalently − ) Gibbs measure ν + has the correct expected number of edge disagreements (inphysics terms, the correct energy density). Lemma 3.1.
Let { G n } n ∈ N be a sequence of k -regular graphs converging locally to T k , let I be a uniformlyrandom vertex in G n , and J be chosen uniformly among its k neighbors. Then lim n →∞ E U n µ n, + ( x I · x J ) = lim n →∞ E U n µ n ( x I · x J ) = ν + ( x ø · x ) = ν − ( x ø · x ) , (3.4) where is one of the neighbors of the root in T k , and E U n denotes the expectation over the random edge ( I, J ) in G n . ν + and ν − have the same expectationof the product x ø x by symmetry under inversion { x i } → {− x i } . The probability that the spins at ø and1 agree is simply (1 + ν ( x ø · x )) /
2. The second Lemma shows that ν + , ν − are uniquely characterized bythis agreement probability among all Ising Gibbs measures on T k . Lemma 3.2.
Let ν be a Gibbs measure for the Ising model on T k . Then ν ( x ø · x ) ≤ ν + ( x ø · x ) = ν − ( x ø · x ) , (3.5) and the inequality is strict unless ν is a convex combination of ν + and ν − . The proof of this Lemma can be found in Section 4.3. We can now prove the following:
Proposition 3.3.
Let { G n } n ∈ N be a sequence of k -regular graphs that converge locally to the tree T k .Then for ( k −
1) tanh β > , it holds that µ n converges locally in average to (1 / ν + + ν − ) .Proof. By Lemma 3.1 and weak convergence, we have ¯ ν ( x ø · x ) = ν + ( x ø · x ). By Lemma 3.2, ¯ ν =(1 − q ) ν + + q ¯ ν − for some q ∈ [0 , µ n, + is symmetric under spin inversion for each n ,and therefore ¯ ν must be symmetric as well, whence q = 1 / (cid:3) We can now prove the first part of our main result.
Proof (Theorem 2.4, part I).
By a similar construction to the one recalled in Section 3.1, and compactnessof M k , we can construct a subsequence { n ( m ) } m ∈ N such that µ n ( m ) converges locally (not only in average)to a distribution m over H k . By the arguments above, m is in fact a measure over the space Ising Gibbsmeasures G k .We claim that any such subsequential weak limit m is in fact a point mass at (1 / ν + + ν − ). Since ν ν ( x ø · x ) is continuous in the weak topology it follows thatlim m →∞ E U n µ n ( m ) ( x I · x J ) = Z ν ( x ø · x ) m (d ν ) . (3.6)By Lemma 3.1, this implies Z ν ( x ø · x ) d m ( ν ) = ν + ( x ø · x ) , (3.7)and therefore, by Lemma 3.2, m is supported on Ising Gibbs measures ν that are convex combinationsof ν + and ν − . Finally, µ n is almost surely symmetric for any n . Here ‘symmetric’ means that, for anyconfiguration x B i ( t ) , µ tn ( x B i ( t ) ) = µ tn ( − x B i ( t ) ). Therefore m is supported on Ising Gibbs measures that aresymmetric.There is only one Ising Gibbs measure that is a convex combination of ν + and ν − and is symmetric,namely ν = (1 / ν + + ν − ). Hence m is a point mass on this distribution. (cid:3) ¯ ν + = ν + We now turn to the subsequence of conditional measures { µ n ( m ) , + } m ∈ N converging locally in average to¯ ν + . The goal of this subsection is to show that ¯ ν + is equal to ν + .For this we repeat the previous proof with two additional ingredients. First we need to show that ¯ ν + is a Gibbs measure on the tree T k . This requires proof since the conditioning on { P i ∈ V n x i > } impliesthat the measures µ tn, + are not Gibbs measures. The Gibbs property is only recovered in the limit.7econd even after we have established that ¯ ν + is a Gibbs measure, this measure is not symmetric withrespect to spin flip. Therefore the argument above only implies that ¯ ν + = (1 − q ) ν + + qν − . It remainsto show that q = 0. This is where the expansion assumption is used. The first lemma we prove is thefollowing: Lemma 3.4.
Any subsequential limit ¯ ν + constructed as above is an Ising-Gibbs measure on T k . We defer the proof to Section 4.1. Given Lemma 3.4 the following lemma follows immediately fromLemmas 3.1 and 3.2.
Lemma 3.5.
For any subsequential limit ¯ ν + there exists a q ∈ [0 , such that ¯ ν + = (1 − q ) ν + + q ν − . (3.8) Proof.
By Lemma 3.4 the measure ¯ ν + is an Ising Gibbs measure on T k . If it was not a convex combinationof ν + and ν − a contradiction to Lemma 3.2 would be derived. (cid:3) .The last step consists of arguing that q = 0. Given a vertex i (either in a graph G n of the sequence orof T k ), an integer ℓ ≥ x , let F i ( ℓ, δ, x ) ≡ I n X j ∈ B i ( ℓ ) x j ≤ − δ | B i ( ℓ ) | o , (3.9)where δ ∈ (0 ,
1) will be chosen below. Roughly speaking F i indicates which vertices are in the “ − state”.We will drop reference to δ and to the configuration x when clear from the context. The following lemmaswill be proven in Section 4.4. Lemma 3.6.
Let { G n } be a sequence of graphs converging locally to T k , and, for each n , x = x ( n ) be aconfiguration in the support of µ n, + . Then there exists n , depending on δ , ℓ and the graph sequence, butnot on x , such that, for all n ≥ n , E U n ( F I ( ℓ, δ, x )) ≤
11 + δ/ , (3.10) where E U n denotes expectation with respect to the uniformly random vertex I in V n . The following lemma is an immediate consequence of the definition of local weak convergence.
Lemma 3.7.
Consider a uniformly random vertex I in G n , let J be one of its neighbors (again uniformlyrandom), and let { n ( m ) } m ∈ N a subsequence of graph sizes along which µ n ( m ) , + converges locally on averageto ¯ ν + . Then we have lim m →∞ E U n ( m ) µ n ( m ) , + ( F I ( ℓ )) = ¯ ν + ( F ø ( ℓ )) , (3.11)lim m →∞ E U n ( m ) µ n ( m ) , + ( F I ( ℓ ) = F J ( ℓ )) = ¯ ν + ( F ø ( ℓ ) = F ( ℓ )) , (3.12) with E denoting expectation with respect to the law U n ( m ) of vertices I and J , and one of the neighborsof ø . Now the limit quantities can be estimated as follows.
Lemma 3.8.
Assume ( k −
1) tanh β > and let ν = (1 − q ) ν + + qν − be a mixture of the plus and minusmeasures for the Ising model on T k . Then there exist δ = δ ( β ) > such that, letting F i ( ℓ ) = F i ( ℓ, δ ; x ) , lim ℓ →∞ ν ( F ø ( ℓ ) = 1) = q , (3.13)lim ℓ →∞ ν ( F ø ( ℓ ) = F ( ℓ )) = 0 . (3.14)8e can now prove the following: Proposition 3.9.
Let { G n } n ∈ N be a sequence of k -regular graphs that are (1 / , λ ) expanders for some λ > and converge locally to the tree T k . Then for ( k −
1) tanh β > , it holds that µ n, + converges locallyon average to ν + Proof.
Let n ( m ) be a subsequence along which µ n, + converges locally on average to some ¯ ν + . By Lemma 3.5we can write this in the form ¯ ν + = (1 − q ) ν + + q ν − . Then by Eqs. (3.11), (3.12), for any ε >
0, thereexists ℓ , such that for large enough n ( m ), E µ n ( m ) , + ( F I ( ℓ )) ≥ q − ε , (3.15) E µ n ( m ) , + ( I { F I ( ℓ ) = F J ( ℓ ) } ) ≤ ε . (3.16)On the other hand, since G n is a (1 / , λ ) expander, and using Eq. (3.10), we have X ( i,j ) ∈ E n I { F i ( ℓ ) = F j ( ℓ ) } ≥ λ min( X i ∈ V n F i ( ℓ ) , X i ∈ V n (1 − F i ( ℓ ))) (3.17) ≥ λ min( X i ∈ V n F i ( ℓ ) , nδ/ (2 + δ )) (3.18) ≥ λδ δ X i ∈ V n F i ( ℓ ) . (3.19)Recalling (3.15), (3.16), taking expectation of both sides with respect to µ n, + and representing the sumsover E n , V n as expectations, we get k ε ≥ k E µ n ( m ) , + ( I { F I ( ℓ ) = F J ( ℓ ) } ) ≥ λδ δ E µ n ( m ) , + ( F I ( ℓ )) ≥ λδ δ ( q − ε ) . (3.20)Since ε > q = 0. The proof follows. (cid:3) We can now complete the proof of Theorem 2.4.
Proof (Theorem 2.4, part II).
Let n ( m ) be a subsequence along which the local distributions of µ n, + con-verge locally to some m (by the same compactness arguments used in the previous section, one alwaysexists). Now by Proposition 3.9 it follows that ν + = R G k ν m (d ν ) which implies that m is a point measureon ν + since it is extremal. This implies local convergence in probability to ν + , which completes the proof. (cid:3) We start from a very general remark, which is implicit in [10] holding for a general Markov random fieldon a graph G = ( V, E ) µ ( x ) = 1 Z Y ( i,j ) ∈ E ψ i,j ( x i , x j ) (4.1)where x = { x i } i ∈ V ∈ X V for a finite spin alphabet X , and ψ ij : X × X → R is a collection of potentials.Recall that a subset S of the vertices of G is an independent set if, for any i, j ∈ S , ( i, j ) E .9 emma 4.1. Assume < ψ min ≤ ψ ij ( x i , x j ) ≤ ψ max , let k be the maximum degree of G , and I ( G ) themaximum size of an independent set of G . Then there exists a constant C = C ( k, ψ max /ψ min ) > suchthat, for any x ∈ X and any ℓ ∈ N , µ (cid:16) X i ∈ V I x i = x = ℓ (cid:17) ≤ C p I ( G ) . (4.2) Proof.
Let S be a maximum size independent set and S c = V \ S its complement. Further, let Y U ≡ P i ∈ U I x i = x for U ⊆ V . Conditioning on x S c = { x i : i ∈ S c } µ (cid:16) X i ∈ V I x i = x = ℓ (cid:17) = E µ n µ (cid:16) Y S = ℓ − Y S c | x S c (cid:17)o . (4.3)Conditional on x S c , the variables { x i } i ∈ S are independent with δ ≤ µ ( x i = x | x S c ) ≤ − δ for some δ > k and ψ max /ψ min . As a consequence Y S is the sum of | S | = I ( G ) independent Bernoullirandom variables with expectation bounded away from 0 and 1. By the Berry-Esseen Theorem µ (cid:16) Y S = ℓ − Y S c | x S c (cid:17) ≤ C p I ( G ) , (4.4)which implies the thesis. (cid:3) Proof. (Lemma 3.4) Recall that for T k , the infinite rooted k -regular tree, we denote by T k ( t ) the subtreeinduced by nodes with distance at most t from the root ø. Also, denote T k ( t, t + ) = T k ( t + ) \ T k ( t ), thesubgraph induced by nodes i with distance t + 1 ≤ d ( i, ø) ≤ t + . Let ¯ ν + denote a subsequential limitof the measures µ n, + constructed as in Section 2.4. For any t ≥ t + > t we will prove that theconditional distribution of x T k ( t ) given x T k ( t,t + ) is given by (here and below we adopt the convention ofwriting p ( x | y ) ∼ = f ( x, y ) for a conditional distribution p , whenever p ( x | y ) = f ( x, y ) / P x ′ f ( x ′ , y )):¯ ν T k ( t ) | T k ( t,t +)+ ( x T k ( t ) | x T k ( t,t + ) ) ∼ = exp β X ( i,j ) ∈ E ( T k ( t +1)) x i x j . (4.5)This establishes the DLR conditions and implies that ¯ ν + is a Gibbs measure as required.In analogy with the notation introduced above (and recalling that B i ( t ) is the ball of radius t aroundvertex i in G n ), we let B i ( t, t + ) = B i ( t + ) \ B i ( t ) be the subgraph induced by vertices j such that t + 1 ≤ d ( i, j ) ≤ t + . Also E i ( t ) will be the set of edges in B i ( t ), and E c i ( t ) = E n \ E c i ( t ). The marginal distributionof x B i ( t + ) under µ n, + is given by µ t + n, + ( x B i ( t + ) ) ∼ = F B i ( t + ) ( x B i ( t + ) ) Z B i ( t + ) ( x B i ( t + ) ) (4.6) F B i ( t + ) ( x B i ( t + ) ) ≡ exp n β X ( l,j ) ∈ E i ( t + ) x l x j o , (4.7) Z B i ( t + ) ( x B i ( t + ) ) ≡ X x Vn \ B i ( t +) exp n β X ( l,j ) ∈ E c i ( t + ) x l x j o I (cid:16) X j ∈ B c i ( t + ) x j > − X j ∈ B i ( t + ) x j (cid:17) . (4.8)We, therefore, have the following expression for the conditional distribution of x B i ( t ) , given x B i ( t,t + ) : µ B i ( t + ) | B i ( t,t + ) n, + ( x B i ( t + ) | x B i ( t,t + ) ) = F B i ( t + ) ( x B i ( t + ) ) Z B i ( t + ) ( x B i ( t + ) ) P x B i ( t ) F B i ( t + ) ( x B i ( t + ) ) Z B i ( t + ) ( x B i ( t + ) ) . (4.9)10n the other hand we have Z − B i ( t + ) ( x B i ( t + ) ) ≤ Z B i ( t + ) ( x B i ( t + ) ) ≤ Z + B i ( t + ) ( x B i ( t + ) ) where we define Z ± B i ( t + ) ( x B i ( t + ) ) ≡ X x Vn \ B i ( t +) exp n β X ( l,j ) ∈ E c i ( t + ) x i x j o I (cid:16) X j ∈ B c i ( t + ) x j > ∓| B i ( t + ) | (cid:17) . (4.10)Notice that Z ± B i ( t + ) ( x B i ( t + ) ) depend on x B i ( t + ) only through x B i ( t,t + ) . Using the expression (4.9) for theconditional probability (and dropping subscripts on µ to lighten the notation), we have µ n, + ( x B i ( t + ) | x B i ( t,t + ) ) ≤ µ ∗ ( x B i ( t + ) | x B i ( t,t + ) ) max x ∈{ +1 , − } T k ( t,t +) Z + B i ( t + ) ( x ) Z − B i ( t + ) ( x ) , (4.11) µ n, + ( x B i ( t + ) | x B i ( t,t + ) ) ≥ µ ∗ ( x B i ( t + ) | x B i ( t,t + ) ) min x ∈{ +1 , − } T k ( t,t +) Z − B i ( t + ) ( x ) Z + B i ( t + ) ( x ) , (4.12)with µ ∗ ( x B i ( t + ) | x B i ( t,t + ) ) ∼ = exp n β X ( l,j ) ∈ E i ( t +1) x l x j o . (4.13)The claim (4.5) thus follows from the fact that B i ( t + ) ≃ T k ( t + ) with probability going to 1 as n → ∞ , ifwe can show that Z − B i ( t + ) ( x ) Z + B i ( t + ) ( x ) → x ∈ { +1 , − } T k ( t,t + ) as n → ∞ .Let ˆ µ denote the Ising measure on x B c i ( t + ) with boundary conditions x B i ( t + ) ˆ µ ( x B c i ( t + ) ) = 1ˆ Z ( x B i ( t + ) ) exp n β X ( l,j ) ∈ E c i ( t + ) x i x j o . (4.15)Now 1 − Z − B i ( t + ) ( x ) Z + B i ( t + ) ( x ) = ˆ µ (cid:16) P j ∈ B c i ( t + ) x j > −| B i ( t + ) | (cid:17) − ˆ µ (cid:16) P j ∈ B c i ( t + ) x j > | B i ( t + ) | (cid:17) ˆ µ (cid:16) P j ∈ B c i ( t + ) x j > −| B i ( t + ) | (cid:17) . Observe that by the Gibbs construction of µ for any x B c i ( t + ) , we have thatˆ µ ( x B c i ( t + ) ) ≥ exp( − βk | B i ( t + ) | ) µ n ( x B c i ( t + ) )as this is the maximum affect that conditioning on a set of size | B i ( t + ) | can have on the measure µ . Bysymmetry of the measure µ n with respect to the sign of x ,ˆ µ (cid:16) X j ∈ B c i ( t + ) x j > −| B i ( t + ) | (cid:17) ≥ ˆ µ (cid:16) X j ∈ B c i ( t + ) x j ≥ (cid:17) ≥ exp( − βk | B i ( t + ) | ) µ n (cid:16) X j ∈ B c i ( t + ) x j ≥ (cid:17) ≥
12 exp( − βk | B i ( t + ) | ) . (4.16)11ow applying Lemma 4.1 to the measure ˆ µ we have thatˆ µ (cid:16) X j ∈ B c i ( t + ) x j > −| B i ( t + ) | (cid:17) − ˆ µ (cid:16) X j ∈ B c i ( t + ) x j > | B i ( t + ) | (cid:17) = ˆ µ (cid:16)(cid:12)(cid:12) X j ∈ B c i ( t + ) x j (cid:12)(cid:12) ≤ | B i ( t + ) | (cid:17) ≤ C | B i ( t + ) | p n − | B i ( t + ) | → C = C ( k, β ) as n → ∞ . Combining equations (4.17) and (4.16) we establish equation (4.14)which completes the proof. (cid:3) For the convenience of the reader, we restate the main result of [9] in the case of k -regular graphs, with nomagnetic field B . This provides an asymptotic estimate of the partition function Z n ( β ) = X x exp n β X ( i,j ) ∈ E x i x j + X i ∈ V x i o . (4.18) Theorem 4.2.
Let { G n } n ∈ N be a sequence of graphs that converges locally to the k -regular tree T k . For β > , let h be the largest solution of h = ( k −
1) tanh[tanh( β ) tanh( h )] . (4.19) Then lim n →∞ n log Z n = φ ( β ) , where φ ( β ) ≡ k β ) − k { β ) tanh( h ) } + log n [1 + tanh( β ) tanh( h )] k + [1 − tanh( β ) tanh( h )] k o , (4.20)For the proof of Lemma 3.1 we start by noticing that, by symmetry under change of sign of the x i ’s,we have µ n, + ( x i · x j ) = µ n ( x i · x j ). Simple calculus yields1 n ∂∂β log Z n ( β ) = 1 n X ( i,j ) ∈ E n µ n ( x i · x j ) = k E µ n ( x I · x J ) , (4.21)where the expectation E is taken with respect to I uniformly random vertex, and J one of its neighborstaken uniformly at random.On the other hand, differentiating Eq. (4.20) with respect to β , and using the fixed point condition(4.19), we get after some algebraic manipulations ∂∂β φ ( β ) = k β + (tanh h ) β (tanh h ) = k ν + ( x ø · x ) . (4.22)The last identification comes from the fact that the joint distribution of x ø and x on a k -regular treeunder the plus-boundary Gibbs measure is ν + ( x ø , x ) ∝ exp { βx ø x + hx ø + hx } (see [9]).Further β n log Z n ( β ) is convex because its second derivative is proportional to the variance of P ( i,j ) x i x j with respect to the measure µ n . Therefore, its derivative ( k/ E µ n ( x i · x j ) converges to( k/ ν + ( x ø · x ) for a dense subset of values of β . Since the limit β ν + ( x ø · x ) is continuous, con-vergence takes place for every β . 12 .3 Proof of Lemma 3.2 Recalling that T k denotes the infinite k -regular tree rooted at ø let T ø and T be the subtrees obtained byremoving the edge (ø ,
1) where 1 is a neighbor of ø. It is sufficient to prove the claim when ν is an extremalGibbs measure on T k since of course we may decompose any Gibbs measure into a mixture of extremalmeasures. For i ∈ { ø , } define m νi = lim ℓ →∞ E T i ( x i | x B ci ( ℓ ) ∩ T i )where E T i denotes expectation with respect to the Ising model on the tree T i and the boundary condition x B i ( ℓ ) ∩ T i is chosen according to ν . The limit exists by the Backward Martingale Convergence Theorem.Further it is a constant almost surely, because it is measurable with respect to the tail σ -field, and ν isextremal.By the monotonicity of the Ising model if ν (cid:22) ν ′ , then m νi ≤ m ν ′ i . Furthermore ν ( x ø ) = m ν ø + tanh( β ) m ν β ) m ν ø m ν . (4.23)Now if ν = ν + then ν ( x ø = 1) < ν + ( x ø = 1). Under the plus measure m ν + ø = m ν + = m + which by themonotonicity of the system is the maximal such value. Since the right hand side of Eq. (4.23) is increasingin m ø , m it follows that m ν ø = m ν = m + if and only if ν = ν + .An easy tree calculation shows that the expectation of x ø · x is ν ( x ø · x ) = tanh( β ) + m ν ø m ν β ) m ν ø m ν . which is strictly increasing in m ν ø when m ν >
0. By symmetry it is also strictly increasing in m ν when m ν ø >
0. Hence amongst measures ν with m ν ø ≥
0, the expectation ν ( x ø · x ) is uniquely maximizedwhen m ν ø = m ν = m + , that is when ν = ν + . Similarly amongst measures ν with m ν ø ≤ ν − , which completes the proof. Observe first by the local weak convergence of the graphs { G n } that all but o ( n ) vertices appear in | B i ( ℓ ) | balls B i ( ℓ ). Hence given a configuration x with P i x i ≥
0, we have X i ∈ V n | B i ( ℓ ) | X j ∈ B i ( ℓ ) x j ≥ − o ( n ) . (4.24)By Markov’s inequality (applied to the uniform choice of i ∈ V n ) we have1 n X i ∈ V n F i ( ℓ ) ≤
11 + δ + o n (1) ≤
11 + δ/ , (4.25)where the second inequality holds for all n large enough Setting ρ = ν + ( x ø ) note that by invariance of ν + under graph homomorphisms of T k , we have ν + (cid:18) X j ∈ B i ( ℓ ) x j (cid:19) = ρ | B i ( ℓ ) | . ν + , along any path of vertices in T k the states are distributed as a 2-state homogenousMarkov chain and hence ν + ( x j · x j ′ ) − ν + ( x j ) ν + ( x ′ j ) = A b d ( j,j ′ ) where d ( j, j ′ ) is the graph distance between vertices i and j , and b ∈ (0 ,
1) is a constant depending on β .This in particular implies that Var ν + (cid:18) X j ∈ B i ( ℓ ) x j (cid:19) = o (cid:18)(cid:12)(cid:12)(cid:12) B i ( ℓ ) (cid:12)(cid:12)(cid:12) (cid:19) , and therefore, using Chebychev inequality, B i ( ℓ ) P j ∈ B i ( ℓ ) x j converges in probability to ρ as ℓ → ∞ . Sim-ilarly under the measure ν − we have that B i ( ℓ ) P j ∈ B i ( ℓ ) x j converges in probability to − ρ . Now taking0 < δ < ρ we have that lim ℓ →∞ ν + ( F ø ( ℓ ) = 1) = 0 , lim ℓ →∞ ν − ( F ø ( ℓ ) = 1) = 1 . Therefore, for ν = (1 − q ) ν + + qν − , we have ν + ( F ø ( ℓ ) = 1) → q .Moreover, by translation invariance ν + ( F ø ( ℓ ) = F ( ℓ )) = 2 ν + ( F ø ( ℓ ) = 1 , F ( ℓ ) = 0) ≤ ν + ( F ø ( ℓ ) = 1) → . By applying the same argument to ν − , we deduce that the probability that F ø ( ℓ ) and F ( ℓ ) differ goes to0 under any mixture of ν + and ν − . Since ν is a mixture of ν + and ν − this completes the lemma. To simplify notation we will write f i or f i ( x ) for f i,n ( x B i ( ℓ ) ). We will prove that, denoting by Var n, + ,Cov n, + variance and covariance under µ n, + ,lim n →∞ Var n, + (cid:16) n X i ∈ V n f i ( x B i ( ℓ ) ) (cid:17) = lim n →∞ E U n Cov n, + ( f I ( x B I ( ℓ ) ) , f L ( x B L ( ℓ ) )) = 0 . Here E U n denotes expectation with respect to two independent and uniformly random vertices I, L in V n .The thesis then follows by Chebyshev inequality.Since the f i ’s are bounded, we have for r > ℓ , E U n Cov n, + ( f I , f L ) ≤ P U n ( d ( I, L ) ≤ r ) + E U n n Cov n, + ( f I , f L ); d ( I, L ) > r o . Since { G n } n ∈ N are k -regular, the probability d ( I, L ) ≤ r vanishes as n → ∞ . It therefore suffices to showthat lim r →∞ lim n →∞ E U n n Cov n, + ( f U , f V ); d ( U, V ) > r o = 0 . Define ˆ f + i ( r )( x ) = E n, + { f ( x B i ( ℓ ) ) | x V n \ B i ( r ) } , the conditional expectation being taken with respect to µ n, + . Then we have for all i, j that I ( d ( i, j ) > r ) Cov n, + ( f i , f j ) = I ( d ( i, j ) > r ) Cov n, + ( ˆ f + i ( r ) , f j ) ≤ q Var n, + ( ˆ f + i ( r ))14nd thereforelim r →∞ lim n →∞ E U n n Cov n, + ( f I , f L ); d ( I, L ) > r o ≤ lim r →∞ lim n →∞ E U n q Var n, + ( ˆ f + I ( r )) (5.1) ≤ lim r →∞ lim n →∞ q E U n Var n, + ( ˆ f + U ( r )) . (5.2)Define the modified function ˆ f i ( r )( x ) = E n { f ( x B i ( ℓ ) ) | x V n \ B i ( r ) } , (5.3)where the expectation is taken with respect to the measure µ n . Since the latter is a Gibbs measure ˆ f i ( r )depends on x only through the variables x j , j ∈ B i ( r ) \ B i ( r − f + i ( r ) and ˆ f i ( r ) differ only if | P j ∈ V n \ B i ( r ) x j | ≤ | B i ( r ) | . ThereforeVar n, + ( ˆ f + I ( r )) ≤ n, + ( ˆ f I ( r )) + 2Var n, + ( ˆ f + I ( r ) − ˆ f I ( r )) ≤ n, + ( ˆ f I ( r )) + 8 µ n, + (cid:16)(cid:12)(cid:12)(cid:12) X j ∈ V n \ B I ( r ) x j (cid:12)(cid:12)(cid:12) ≤ | B I ( r ) | (cid:17) . The last term vanishes as n → ∞ by Lemma 4.1.We are therefore left with the task of showing that lim r →∞ lim n →∞ E U n Var n, + ( ˆ f I ( r )) = 0. For afunction f : {− , } T k ( ℓ ) → [ − , f ( r )( x ) = E ν + { f ( x T k ( ℓ ) ) | x T k \ T k ( r ) } . For all functions whose domain is not {− , } T k ( ℓ ) we let ¯ f ( r ) = 0 by convention. Also, with an abuse ofnotation, we define ¯ f i ( r ) = ¯ g ( r ) for g = ˆ f i . Since ˆ f I ( r ) depends on x only through x B I ( r ) , we obtain byTheorem 2.4 for every ε > n →∞ E U n | Var n, + ( ˆ f I ( r )) − Var ν + ( ¯ f I ( r )) | ≤ ε + lim n →∞ U n (cid:0) d TV (cid:0) P tn ( I ) , δ T k ( t ) × ν t + (cid:1) > ε (cid:1) = 2 ε , and thereforelim r →∞ lim n →∞ E U n Var n, + ( ˆ f I ( r )) ≤ lim r →∞ sup n Var ν + ( ¯ f ( r )) | f : {− , } T k ( ℓ ) → [ − , o . By extremality of ν + , for each f : {− , } T k ( ℓ ) → [ − , f ( r ) converges to an almost sure constant as r → ∞ and since f is bounded, lim r →∞ Var ν + ( ¯ f ( r )) = 0. For each r , the map f → ¯ f ( r ) is a contraction in L and therefore the map f → q Var ν + ( ¯ f ( r )) is a Lipchitz map with constant 1. Since the set of functions f : {− , } T k ( ℓ ) → [ − ,
1] is compact in L and for each f we have lim r →∞ Var ν + ( ¯ f ( r )) = 0 we concludethat lim r →∞ sup n Var ν + ( ¯ f ( r )) | f : {− , } T k ( ℓ ) → [ − , o = 0 , as needed. Acknowledgements
A.M. was partially supported by by a Terman fellowship, the NSF CAREER award CCF-0743978 and theNSF grant DMS-0806211. E.M. was partially supported by the NSF CAREER award grant DMS-0548249,by DOD ONR grant (N0014-07-1-05-06), by ISF grant 1300/08 and by EU grant PIRG04-GA-2008-239317.Part of this work was carried out while two of the authors (A.M. and E.M.) were visiting MicrosoftResearch. 15 eferences [1] R. S. Ellis and C. M. Newman,
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