Reduction to covariance Step 2: Poincaré inequality for the global Gibbs measure

From the last computation, which holds for all η ∈ Ω τ S and for all t, we finally obtain that c g ap ν τ S ≥ α independently of the size of S, which implies the Poincaré inequality 4.7 with constant c = α −1 = 1 + Oe −cβ . This concludes the proof of Theorem 4.1. ƒ

4.3 Step 2: Poincaré inequality for the global Gibbs measure

With the previous analysis we obtained a like-Poincaré inequality for the marginal of the measure µ i on the level L i see Remark 4.2, which inserted in formula 4.3 provides the bound Var f ≤ c m X i=0 X x∈L i µ ” µ i Var K x g i+1 — . 4.18 Using the same notation as in [26], let us denote the sum in the r.h.s. of 4.18 by Pvar f . The aim of the following analysis is to study Pvar f in order to find an inequality of the kind Pvar f ≤ cD f + ǫPvar f , with ǫ = ǫβ, g, ∆ 1 independent of the size of the system. This would imply that Var f ≤ c · Pvar f ≤ c c 1 − ǫ D f , and then would conclude the proof of Theorem 2.7. In this last part of the section, we will first relate the local variance of g i = µ i f with the local variance of f . This will produce a covariance term that will be analyzed using a recursive argument.

4.3.1 Reduction to covariance

In order to reconstruct the Dirichlet form of f from 4.18, we want to extract the local variance of f from the local variance of g i+1 . Notice that w.r.t. the measure µ K x , the function g i+1 just depends on x. Fixing x ∈ L i and τ ∈ Ω + , and defining pτ := µ τ K x σ x = + and qτ := µ τ K x σ x = −, we can write µ i € Var K x g i+1 Š = X τ µ i τpτqτ ∇ x g i+1 τ 2 . 4.19 Using the martingale property g i+1 = µ i+1 g i+2 , the variance Var K x g i+1 can be split in two terms, stressing the dependence on x of g i+2 and of the conditioned measure µ i+1 . Let us formalize this idea. For a given configuration τ ∈ Ω + we introduce the symbols τ + := ¨ τ + y = τ y if y 6= x τ + y = + if y = x τ − := ¨ τ − y = τ y if y 6= x τ − y = − if y = x and define the density h x σ := µ τ + i+1 σ µ τ − i+1 σ , with µ τ − i+1 h x = 1 . 4.20 1999 Whit this notation and continuing from 4.19, we get µ i Var K x g i+1 = X τ µ i τpτqτ ∇ x µ i+1 g i+2 τ 2 = X τ µ i τpτqτ h µ τ − i+1 g i+2 − µ τ + i+1 g i+2 i 2 = X τ µ i τpτqτ h µ τ + i+1 ∇ x g i+2 − µ τ − i+1 h x , g i+2 i 2 ≤ 2 X τ µ i τpτqτ µ τ + i+1 ∇ x g i+2 2 + µ τ − i+1 h x , g i+2 2 4.21 Consider the first term of 4.21 and notice that µ τ + i+1 ∇ x g i+2 = µ τ + i+1 ∇ x f . To understand this fact, it is enough to observe that the dependence on x of g i+2 = µ i+2 f comes only from f , since the b.c. on B i+1 is fixed equal to τ + . Replacing ∇ x g i+2 by ∇ x f and applying the Jensen inequality, we get X τ µ i τpτqτ µ τ + i+1 ∇ x g i+2 2 ≤ X τ µ i τpτqτ µ τ + i+1 ∇ x f 2 . 4.22 Now, notice that pτµ τ + i+1 σ + = µ τ K x σ + = µ σ x σ x = + µ τ K x σ + + µ τ K x σ − . This, together with the fact that ∇ x f does not depend on x, means that the expression on the r.h.s. of 4.22 equals to X τ µ i τqτ X σ µ τ K x σµ σ x σ x = + ∇ x f σ 2 ≤ X τ µ i τ inf σ∈Ω τ Kx ¨ qτ µ σ x σ x = − « . 4.23 Since σ agrees with τ on K c x , then qτ µ σ x σ x = − = µ τ K x σ x = − µ τ K x σ x = −|σ = µ τ K x σ µ τ K x σ|σ x = − ≤ h x σ , 4.24 and hence, inf σ § qτ µ σ x σ x =− ª ≤ kh x k ∞ . To bound kh x k ∞ , we first write h x σ = µ τ k x σ|σ x = + µ τ k x σ|σ x = − = µ τ k x σ x = +|σ µ τ k x σ x = −|σ · µ τ k x σ x = − µ τ k x σ x = + . 4.25 Taking the supremum over σ of h x , we then have kh x k ∞ ≤ 1 µ τ − i+1 σ y = +; ∀ y ∈ N x ∩ L i+1 ≤ 1 1 − P y∈N x ∩L i+1 µ − i+1 σ y = − , 4.26 where we denoted by µ − i+1 the measure conditioned on having all minus spins in B i and plus spins in ∂ V B. Inequality 3.9, applied to U = F i+1 , implies that µ − i+1 σ y = − ≤ ce −β ′ , 4.27 2000 with β ′ = 2gβ − δ as in Proposition 3.1. Combining 4.26 and 4.27, we get that for all β ≥ δ 2g , kh x k ∞ ≤ 1 + ce −β ′ . 4.28 Altogether, inequalities 4.22-4.28 imply that X τ µ i τpτqτ µ τ + i+1 ∇ x g i+2 2 ≤ c 1 µ i Var x f , 4.29 with c 1 = c 1 β, ∆, g = 1 + Oe −cβ . Thus, summing both sides of 4.21 over x ∈ L i and i ∈ {0, . . . , m}, and applying inequality 4.29, we obtain Pvar f ≤ 2c 1 D f + 2 m X i=0 X x∈L i µ – X τ µ i τpτqτ µ τ − i+1 h x , g i+2 2 ™ . 4.30 Notice that since g m+2 ≡ f is constant w.r.t. µ m+1 , then µ τ − m+1 h x , g m+2 ≡ 0 and the value m can be removed from the summation over i in the r.h.s. of 4.30. It now remains to analyze the covariance µ τ − i+1 h x , g i+2 . 4.3.2 Recursive argument Before going on with the proof, we need some more definitions and notation. For every x ∈ L i , let D x denote the set of nearest neighbors of x in the level L i+1 descendants of x. Given x ∈ L i and ℓ ∈ N, let us define the following objects: i D x,ℓ := { y ∈ L i+1 : d y, D x ≤ ℓ} is the ℓ-neighborhood of D x in L i+1 ; ii F x,ℓ := σ € σ y : y ∈ B i+1 \ D x,ℓ Š is the σ-algebra generated by the spins on B i+1 \ D x,ℓ ; iii µ x,ℓ · := µ · |F x,ℓ is the Gibbs measure conditioned on the σ-algebra F x,ℓ . We remark that D x,0 = D x , and that there exists some ℓ ≤ 2i + 1 such that, for all integers ℓ ≥ ℓ , D x,ℓ = L i+1 and µ x,ℓ = µ i+1 . We also remark that for any function f ∈ L 1 Ω, F i+1 , µ, the set of variables {µ x,ℓ f } ℓ∈N is a Martingale with respect to the filtration {F x,ℓ } ℓ=0,1,...,ℓ . Let us now come back to our proof and recall the following property of the covariance. For all subsets D ⊆ C ⊆ B, µ η C f , g = µ η C µ D f , g + µ η C µ D f , µ D g . 4.31 Since the support of µ i+1 strictly contains the support of µ x,0 , we can apply the property 4.31 to the square covariance µ τ − i+1 h x , g i+2 2 appearing in 4.30, in order to get µ τ − i+1 h x , g i+2 2 ≤ 2µ τ − i+1 µ x,0 h x , g i+2 2 + 2µ τ − i+1 µ x,0 h x , µ x,0 g i+2 2 . 4.32 The first term in the r.h.s. of 4.32 can be bounded, by the Schwartz inequality, as µ τ − i+1 µ x,0 h x , g i+2 2 ≤ µ τ − i+1 Var x,0 h x · µ τ − i+1 Var x,0 g i+2 . 4.33 2001 The second term can be rearranged and bounded as follows: [µ τ − i+1 µ x,0 h x , µ x,0 g i+2 ] 2 = h µ τ − i+1 € µ x,0 h x − µ i+1 h x , g i+2 Ši 2 =  µ τ − i+1   ℓ X ℓ=1 € µ x,ℓ−1 h x − µ x,ℓ h x Š , g i+2     2 ≤ ℓ X ℓ=1 ℓ 2 h µ τ − i+1 µ x,ℓ−1 h x − µ x,ℓ h x , g i+2 i 2 = ℓ X ℓ=1 ℓ 2 h µ τ − i+1 € µ x,ℓ µ x,ℓ−1 h x , g i+2 Ši 2 , 4.34 where in the second line, due to the fact that µ x,ℓ = µ i+1 for some ℓ , we substituted µ x,0 h x − µ i+1 h x by the telescopic sum P ℓ ℓ=1 µ x,ℓ−1 h x − µ x,ℓ h x . Applying again the Cauchy-Schwartz inequality to the last term in 4.34, we get [µ τ − i+1 µ x,0 h x , µ x,0 g i+2 ] 2 ≤ ≤ ℓ X ℓ=1 ℓ 2 µ τ − i+1 € Var x,ℓ µ x,ℓ−1 h x Š · µ τ − i+1 € Var x,ℓ g i+2 Š 4.35 To conclude the estimate on the covariance, it remains to analyze the three quantities appearing in 4.33 and 4.35: i µ τ − i+1 € Var x,ℓ g i+2 Š , for all ℓ = 0, 1, . . . , ℓ ; ii µ τ − i+1 Var x,0 h x ; iii µ τ − i+1 € Var x,ℓ µ x,ℓ−1 h x Š , for all ℓ = 1, . . . , ℓ , We proceed estimating separately these three terms. First term: Poincaré inequality for the marginal measure on D x,ℓ . Let us consider the variance Var x,ℓ g i+2 appearing in i. By definition, the function g i+2 depends on the spin configuration on B i+1 . Since the measure µ η x,ℓ fixes the configuration on B i+1 \ D x,ℓ , it follows that µ η x,ℓ g i+2 = µ η x,ℓ| Dx,ℓ g i+2 . Thus, for every configuration η ∈ Ω + , we can apply the Poincaré inequality stated in Theorem 4.1 to Var η x,ℓ g i+2 , and obtain the inequality µ τ − i+1 € Var x,ℓ g i+2 Š ≤ c X y∈D x,ℓ µ τ − i+1 Var K y g i+2 , 4.36 with c = 1 + Oe −cβ independent of the size of system. 2002 Second term: computation of the variance of h x . Notice that, from definition 4.20, it turns out that h x only depends on the spin configuration on D x . In particular, for all η which agrees with τ − on B i , µ η x,0 h x = 0 and Var η x,0 h x ≤ kh x k 2 ∞ − 1 . 4.37 Together with inequality 4.28, this yields the bound: Var η x,0 h x ≤ ce −β ′ =: k β . 4.38 Third term: the variance of µ x,ℓ−1 h x . We now consider the variance Var η x,ℓ µ x,ℓ−1 h x , with η ∈ Ω + and ℓ ≥ 1. Applying the result of Theorem 4.1, we obtain Var η x,ℓ µ x,ℓ−1 h x ≤ c X z∈D x,ℓ µ η x,ℓ Var K z µ x,ℓ−1 h x = c X z∈D x,ℓ \D x,ℓ−1 µ η x,ℓ Var K z µ x,ℓ−1 h x , 4.39 where in the last line we used that the function µ x,ℓ−1 h x does not depend on the spin configuration on D x,ℓ−1 . Let z ∈ D x,ℓ \ D x,ℓ−1 , and for any configuration ζ ∈ Ω η D x,ℓ , let us denote by ζ + and ζ − the configura- tions that agree with ζ in all sites but z, and have respectively a +-spin and a −-spin on z. The summand in 4.39 can be trivially bounded as µ η x,ℓ Var K z µ x,ℓ−1 h x ≤ 1 2 sup ζ∈ Ω η x,ℓ µ ζ + x,ℓ−1 h x − µ ζ − x,ℓ−1 h x 2 . 4.40 Moreover, by stochastic domination, the inequality µ ζ + x,ℓ−1 h x ≥ µ ζ − x,ℓ−1 h x holds. Let νσ, σ ′ denote a monotone coupling with marginal measures µ ζ + x,ℓ−1 and µ ζ − x,ℓ−1 . We then have µ ζ + x,ℓ−1 h x − µ ζ − x,ℓ−1 h x = X σ,σ ′ νσ, σ ′ h x σ − h x σ ′ ≤ kh x k ∞ νσ y 6= σ ′ y , y ∈ D x ≤ ∆kh x k ∞ max y∈D x νσ y = + − νσ ′ y = + = ∆kh x k ∞ max y∈D x µ ζ + x,ℓ−1 σ y = + − µ ζ − x,ℓ−1 σ y = +, 4.41 where we used that the function h x only depends on the spins on D x . From Proposition 3.1, with U = F i+2 ∪ D x,ℓ , and since dz, y ≥ dz, D x ≥ ℓ, the probability of disagreement appearing in 4.41 can be bounded as µ ζ + x,ℓ−1 σ y = + − µ ζ − x,ℓ−1 σ y = + ≤ c e −β ′ ℓ . 4.42 2003 Putting together formulas 4.39-4.42, and applying inequality 4.28, we obtain that for all η ∈ Ω + , Var η x,ℓ µ x,ℓ−1 h x ≤ k ′ β e −2β ′ ℓ 4.43 with k ′ β = c1 + Oe −cβ . Conclusion. Let us go back to inequalities 4.33 and 4.35. Applying bounds 4.36,4.38 and 4.43, we get respectively • µ τ − i+1 µ x,0 h x , g i+2 2 ≤ k β X y∈D x µ τ − i+1 Var K y g i+2 , • [µ τ − i+1 µ x,0 h x , µ x,0 g i+2 ] 2 ≤ k ′ β ℓ X ℓ=1 ℓ 2 e −2β ′ ℓ X y∈D x,ℓ µ τ − i+1 Var K y g i+2 , where we included in k β and k ′ β all constants non depending on β. For all β ≫ 1, there exists a constant ǫ ≡ ǫβ, g, ∆ = Oe −cβ such that k β ≤ ǫ and k ′ β ℓ 2 e −β ′ ℓ ≤ k ′ β e −β ′ ≤ ǫ. Substituting ǫ in the inequalities above and summing the two terms as in 4.32, we obtain µ τ − i+1 h x , g i+2 2 ≤ ǫ ℓ X ℓ=0 e −β ′ ℓ X y∈D x,ℓ µ τ − i+1 Var K y g i+2 . Inserting this result in the second term of formula 4.30 and rearranging the summation, we get m−1 X i=0 X x∈L i µ – X τ µ i τpτqτ µ τ − i+1 h x , g i+2 2 ™ ≤ ǫ m−1 X i=0 X x∈L i ℓ X ℓ=0 X y∈D x,ℓ e −β ′ ℓ µVar K y g i+2 ≤ ǫ m−1 X i=0 X y∈L i+1 µVar K y g i+2 ℓ X ℓ=0 e −β ′ ℓ nℓ , 4.44 where in the last line we denoted by nℓ the factor which bounds the number of vertices x such that a fixed vertex y belongs to D x,ℓ . Since nℓ grows at most like ∆ ℓ , the product e −β ′ ℓ nℓ decays exponentially with ℓ for all β ≫ 1. Thus the sum over ℓ ∈ {0, . . . , ℓ } can be bounded by a finite constant c which will be included in the factor ǫ in front of the summations. Continuing from 4.44, we get m−1 X i=0 X x∈L i µ – X τ µ i τpτqτ µ τ − i+1 h x , g i+2 2 ™ ≤ ǫ m X i=1 X y∈L i µVar K y g i+1 ≤ ǫ Pvar f . 4.45 2004 Inserting this result in 4.30 and noticing that ǫ = Oe −cβ 1 for β large enough, we obtain Pvar f ≤ 2c 1 D f + ǫPvar f =⇒ Pvar f ≤ 2c 1 1 − ǫ D f . Together with inequality 4.18, this implies that Var f ≤ c Pvar f ≤ c 2 D f , that is the desired Poincaré inequality with c 2 = c 2 β, g, ∆ = 21 + Oe −cβ independent of the size of the system . Notice also that the lower bound on the spectral gap, 1c 2 , increases with β and converges to 12 when β ↑ ∞. This concludes the proof of Theorem 2.7. ƒ 5 Influence of boundary conditions on the mixing time In this section we discuss two examples of influence of the boundary condition on the mixing time derived from Theorem 2.7. In particular, we first prove Lemma 2.3, which implies the applicability of Theorem 2.7 to hyperbolic graphs with sufficiently high degree. Then we prove Theorem 2.8 providing an explicit example of growing graph which exhibits the behavior stated in the theorem.

5.1 Hyperbolic graphs