The approach of Otto-Reznikoff revisited

In this article we consider a lattice system of unbounded continuos spins. Otto&Reznikoff used the two-scale approach to show that exponential decay of correlations yields a logarithmic Sobolev inequality (LSI) with uniform constant in the system size. We improve their statement by weakening the assumptions. For the proof a more detailed analysis based on two new ingredients is needed. The two new ingredients are a new basic covariance estimate and a uniform moment estimate. We additionally provide a comparison principle for covariances showing that the correlations for the conditioned Gibbs measures are controlled by the correlations of the original Gibbs measure with ferromagnetic interaction. The latter simplifies the application of the main result. As an application, we show how decay of correlations combined with the uniform LSI yields the uniqueness of the infinite-volume Gibbs measure, generalizing a result of Yoshida form finite-range to infinite-range interaction.


Introduction and main results
We consider a lattice system of unbounded and continuous spins on the d-dimensional lattice Z d . The formal Hamiltonian H : R Z d → R of the system is given by We assume that the single-site potentials ψ i : R → R are smooth and perturbed convex. This means that there is a splitting ψ i = ψ c i + ψ b i such that for all i ∈ Z d and z ∈ R (1.2) (ψ c i ) ′′ (z) ≥ 0 and |ψ b i (z)| + |(ψ b i ) ′ (z)| 1.
Here, we used the convention (see Definition 1.16 below for more details) a b :⇔ there is a uniform constant C > 0 such that a ≤ Cb.
Moreover, we assume that • the interaction is symmetric i.e.
• and the matrix M = (M ij ) is strictly diagonal dominant i.e. for some δ > 0 it holds for any i ∈ Z d Definition 1.3 (Finite-volume Gibbs measure). Let Λ be a finite subset of the lattice Z d and let x Z d \Λ be a tempered state. We call the measure µ Λ (dx Λ ) finite-volume Gibbs measure associated to the Hamiltonian H with boundary values x Z d \Λ , if it is a probability measure on the space R Λ given by the density Here, Z µ Λ denotes the normalization constant that turns µ Λ into a probability measure. If there is no ambiguity, we also may write Z to denote the normalization constant of a probability measure. We also used the short notation Note that µ Λ depends on the spin values x Z d \Λ outside of the set Λ.
The main object of study in this article is the question if the finitevolume Gibbs measure µ Λ satisfies a logarithmic Sobolev inequality (LSI). Definition 1.4 (LSI). Let X be a Euclidean space. A Borel probability measure µ on X satisfies the LSI with constant ̺ > 0, if for all smooth functions f ≥ 0 |∇f | 2 f dµ.
Here, ∇ denotes the gradient determined by the Euclidean structure of X.
Definition 1.5 (PI). Let X be a Euclidean space. A Borel probability measure µ on X satisfies the PI with constant ̺ > 0, if for all smooth functions f Here, ∇ denotes the gradient determined by the Euclidean structure of X.
The LSI was originally introduced by Gross [Gro75]. It can be used as a powerful tool for studying spin systems. The LSI implies exponential convergence to equilibrium of the naturally associated conservative diffusion process. The rate of convergence is given by the LSI constant ̺ (cf. [Roy07, Chapter 3.2]). At least in the case of finite-range interaction, independence from the system size of the LSI constant of the local Gibbs state directly yields the uniqueness of the infinite-volume Gibbs state (cf. [Roy07,Yos03,Zit08]).
In the literature, there are several results known that connect the decay of spin-spin correlations to the validity of a LSI uniform in the system size [SZ92a,SZ92b,Zeg96,Yos99,Yos01,BH99]. This means that a static property of the equilibrium state of the system is connected to a dynamic property namely the relaxation to the equilibrium. We refer the reader to the Section 2.2. of the article of Otto & Reznikoff [OR07], which gives a nice overview and discussion on the results in the literature. Otto Assume that the interaction is symmetric i.e. M ij = M ji and has zero diagonal i.e. M ii = 0. Consider a subset Λ tot ⊂ Z d . We assume the uniform control: uniformly in Λ ⊂ Λ tot and i, j ∈ Λ. Here, µ Λ denotes the finite-volume Gibbs measures µ Λ given by (1.4). Then the finite-volume Gibbs measure µ Λtot satisfies the LSI with constant ̺ > 0 depending only on the constant C > 0 in (1.5), (1.6), and (1.7).
The most important feature of Theorem 1.6 is that the LSI constant ̺ is independent of the system size |Λ tot | and of the spin values x Z d \Λtot outside of Λ tot . The advantage of Theorem 1.6 over existing results connecting a decay of correlations to a uniform LSI is that it can deal with infinite-range interaction (cf. [SZ92a,SZ92b,Zeg96,Yos99,Yos01,BH99] uniformly in Λ ⊂ Λ tot , and i, j ∈ Λ. Here, µ Λ denote the finite-volume Gibbs measures given by (cf . (1.4)). Then the finite-volume Gibbs measure µ Λtot satisfies the LSI with a constant ̺ > 0 depending only on the constant in (1.2), (1.3), (1.9) and (1.8).
Theorem 1.7 improves Theorem 1.6 in two ways: Note that Theorem 1.6 needs an exponential decay of interaction and spin-spin correlations. However, analyzing the proof of [OR07, Theorem 3] one sees that the exponential decay is only needed to guarantee that certain sums are summable. Therefore this assumption can be weakened to algebraically decaying interaction and spin-spin correlations. Of course now, the order of the algebraic decay depends on the dimension of the underlying lattice to guarantee summability.
The second improvement is more subtle. Theorem 1.6 needs a special structure on the single-site potentials ψ i . Namely, the single-site potentials ψ i have to be perturbed quartic in the sense of (1.5). Analyzing the proof of [OR07,Theorem 3] shows that the argument does not rely on a quartic potential ψ c i . For the argument of Otto & Reznikoff it would be sufficient to have a perturbation of a strictly-superquadratic potential i.e.
(1.10) lim inf The condition (1.10) on the single-site potential ψ i is widespread and accepted in the literature on the uniform LSI (cf. for example [Yos01,Yos03,PS01]  The first new ingredient in the proof of Theorem 1.7 is the covariance estimate of Proposition 3.3. With this estimate it is possible to deduce algebraic decay of correlations, provided the interactions M ij also decay algebraically and the nonconvex perturbation ψ b i is small enough. The second new ingredient in the proof of Theorem 1.7 is a uniform estimate of var µ Λ (x i ) (see Lemma 3.4), which we reduce to a moment estimate due to Robin Nittka (cf. [MN13,Lemma 4.2] and Lemma 3.5). The full proof of Theorem 1.7 is given in Section 3.
However, Theorem 1.7 still calls for further improvements. Note that in the condition (1.9) of Theorem 1.7 one needs to check the decay of correlations for all finite-volume Gibbs measures µ Λ with Λ ⊂ Λ tot . Even if this is a very common assumption (see for example [Yos01, Condition (DS3)]) it may be a bit tedious to verify. Instead of the strong condition (1.9), one would like to have a weak condition like the one used for discrete spins in [MO94]. The main difference between the weak and the strong condition for the decay of correlations is that in the weak condition it suffices to show that for a sufficiently large box Λ the correlations decay nicely. The main advantage of the weak condition is that one does not have to control the decay of correlations for all growing subsets Λ → Z d . Therefore, the weak condition is easier to verify by experiments. Unfortunately, we cannot get rid of the strong decay of correlations condition (1.9) in the Otto-Reznikoff approach. However, we show how verifying the strong decay of correlations condition (1.9) can be simplified by two comparison principles.
The first comparison principle (see Lemma 1.9 below) shows that in the case of ferromagnetic interaction (i.e. M ij < 0 for all i, j ∈ Λ tot ) the correlations of a smaller system are controlled by correlations of the larger system. Lemma 1.9. Assume that the formal Hamiltonian H : R Z d → R given by (1.1) satisfies the Assumptions (1.2) -(1.3). Additionally, assume that the interactions are ferromagnetic i.e. M i,j ≤ 0 for i = j. For arbitrary subsets Λ ⊂ Λ tot ⊂ Z d , we consider the finite-volume Gibbs measure µ Λ and µ Λtot with the same tempered state x Z d \Λtot . Then it holds for any i, j ∈ Λ The proof of Lemma 1.9 is given in Section 2. The second comparison principle is rather standard. It states that correlations of a non-ferromagnetic system are controlled by the correlations of the associated ferromagnetic system: Lemma 1.10. Assume that the formal Hamiltonian H : R Z d → R given by (1.1) satisfies the Assumptions (1.2) -(1.3). Let µ Λ denote the finite-volume Gibbs measure given by (1.4). Additionally, consider the corresponding finite-volume Gibbs measure µ Λ,|M | with attractive interaction i.e. the associated formal Hamiltonian is given by Then it holds that for any i, j ∈ Λ We do not state the proof of the last lemma. One can find the proof for example in a recent work by Robin Nittka and the author. The proof follows the argument of [HM79] for discrete spins (see [MN13, Lemma 2.1.]).
This case is also contained in the main results of the article, because the Hamiltonian βH still satisfies the structural Assumptions (1.2) -(1.3). Of course, the LSI constant of Theorem 1.7 would depend on the inverse temperature β.
Remark 1.12. Because we assume that the matrix M = (M ij ) is strictly diagonal dominant (cf. (1.3)), the full single-site potential is perturbed strictly-convex. We want to note that this is the same structural assumption as used in the article [MO13].
Let us turn to an application of Theorem 1.7. We will show how the decay of correlations condition (1.9) combined with the uniform LSI of Theorem 1.7 yields the uniqueness of the infinite-volume Gibbs measure. The statement that a uniform LSI yields the uniqueness of the Gibbs state is already known from the case of finite-range interaction (cf. for example [Yos03], the conditions (DS1), (DS2), and (DS3) in [Yos01]). The related arguments of [Roy07], [Zit08], and [Yos01] are based on semigroup properties of an associated diffusion process. Though the semigroup probably may work in the case of infinite-range interaction, we follow a more straightforward approach to deduce the uniqueness of the Gibbs measure. Before we formulate the precise statement (see Theorem 1.14 below), we specify the notion of an infinitevolume Gibbs measure.
Definition 1.13 (Infinite-Volume Gibbs measure). Let µ be a probability measure on the state space R Z d equipped with the standard product Borel sigma-algebra. For any finite subset Λ ⊂ Z d we decompose the measure µ into the conditional measure µ(dx Λ |x Z d \Λ ) and the marginalμ(dx Z d \Λ ). This means that for any test function f it holds We say that the measure µ is the infinite-volume Gibbs measure associated to the Hamiltonian H, if the conditional measures µ(dx Λ |x Z d \Λ ) are given by the finite-volume Gibbs measures The equations of the last identity are also called Dobrushin-Lanford-Ruelle (DLR) equations.
The precise statement connecting the decay of correlations with the uniqueness of the infinite-volume Gibbs measure is: Theorem 1.14 (Uniqueness of the infinite-volume Gibbs measure). Under the same assumptions as in Theorem 1.7, there is at most one unique Gibbs measure µ associated to the Hamiltonian H satisfying the uniform bound The moment condition (1.11) in Theorem 1.14 is standard in the study of infinite-volume Gibbs measures (see for example [BHK82] and [Roy07, Chapter 4]). It is relatively easy to show that the condition (1.11) is invariant under adding a bounded random field to the Hamiltonian H (cf. Remark 1.8).
Theorem 1.14 is one of the well-known statements for which it is hard to find a proof. Therefore we state the proof in full detail in the Appendix A. The argument does not need that the finite-volume Gibbs measures µ Λ satisfy a uniform LSI. It suffices that the finitevolume Gibbs measures µ Λ satisfy a uniform PI, which is a weaker condition then the LSI (see Definition 1.5).
We also want to note that the main results of this article, namely Theorem 1.7 and Theorem 1.14 were applied in [MN13] to deduce a uniform LSI and the uniqueness of the infinite-volume Gibbs measure of a one-dimensional lattice system with long-range interaction, generalizing Zegarlinsk's result [Zeg96, Theorem 4.1.] to interactions of infinite range.
Remark 1.15. In this article, we do not show the existence of an infinite-volume Gibbs measure. However, the author of this article believes that under the assumption (1.11) the existence should follow by an compactness argument similarly to the one used in [BHK82].
In order to avoid confusion, let us make the notation a b from above precise.
Definition 1.16. We will use the notation a b for quantities a and b to indicate that there is a constant C ≥ 0 which depends only on a lower bound for δ and upper bounds for |ψ b i |, |(ψ b i ) ′ |, and sup i j∈Z d |M ij | such that a ≤ Cb. In the same manner, if we assert the existence of certain constants, they may freely depend on the above mentioned quantities, whereas all other dependencies will be pointed out.
We close the introduction by giving an outline of the article.
• In Section 2, we prove Lemma 1.9. This contains the comparison principle for covariances of smaller systems to larger systems. • In Section 3, we consider the generalization of Theorem 1.6 and give the proof of Theorem 1.7. • In the Appendix A, we consider the uniqueness of the infinitevolume Gibbs measure and give the proof of Theorem 1.14. • In the Appendix B we state some well-known facts about the LSI and the PI.
2. Comparing covariances of a smaller system to covariances to a bigger system: Proof of Lemma 1.9 The proof of Lemma 1.9 uses an idea of Sylvester of expanding the exponential function [Syl76]. Sylvester used this idea to give a simple unified derivation of a bunch of correlation inequalities for ferromagnets.
Proof of Lemma 1.9. We fix the spin values m i , i ∈ Λ tot \Λ. Recall that in our notations µ Λ coincides with the conditional measure We introduce the auxiliary Hamiltonian H α , α > 0, by the formula We denote by µ α the associated Gibbs measure active on the sites Λ tot . The measure µ α is given by the density Note that the measure µ α interpolates between the measure µ Λ and µ Λtot in the sense that µ 0 = µ Λtot and for any integrable function f : This yields by the fundamental theorem of calculus that We will now show that which yields the statement of Lemma 1.9. Indeed, direct calculation shows that We will show now that For this purpose, we follow the method by Sylvester [Syl76] of expanding the interaction term. Recall that this method is also used to show for example that provided the interactions are ferromagnetic. By doubling the variables we get Because the partition function Z > 0 is positive, the sign of the covariance is determined by the integral on the right hand side of the last identity. We change variables according to x i = (p i + q i ) and x i = (p i − q i ) and get where C > 0 is the constant from the transformation. Straightforward calculation reveals For convenience, we only consider the first summand on the right hand side of (2.2). The second summand can be estimated in the same way. Due to symmetry ofH α (p, q) in the q l variables it holds q j l∈Λtot\Λ ((p l + q l ) − m l ) 2 exp(−H α (p, q))dpdq = 0 Therefore we get by doubling the variable p first and then changing of variables p = r +q andp = r −q that where the HamiltonianH α (r,q, q) is given bỹ As we have seen in (2.3) from above, the HamiltonianH α (r,q, q) contains no mixed terms in the variables r,q and q. More precisely, H α (r,q, q) has three interaction terms i.e. 4r · Mr, 4q · Mq, and 2q · Mq.
So we can rewriteH α (r,q, q) as where the function F is of the form for some single-site potentialsψ l that are symmetric in the variablesq l and q l . Expanding the term on the right hand side of (2.4) yields a sum of terms of the form − M mn q n 1 l 1 · · · q n k l kqñ Because the functionsψ l are symmetric in the variablesq l and q l any term with an odd exponent vanishes. Hence, the exponents n 1 , . . . , n k , andñ 1 , . . . ,ñ k , are all even. Because −M mn ≥ 0 due to the fact that the interaction is ferromagnetic we get − M mn q n 1 l 1 · · · q n k l kqñ All in all, the last inequality yields the desired estimate (2.1) and therefore completes the proof.
3. The Logarithmic Sobolev inequality: proof of Theorem 1.7 This section is devoted to the proof of Theorem 1.7. We adapt the strategy of Otto & Reznikoff [OR07, Theorem 3] to our situation. Recall that compared to Theorem 1.7, we work with weaker assumptions: • The single-site potentials ψ i are only quadratic and not superquadratic (cf. (1.2) vs. (1.5)). Also note that in Theorem 1.6 it is assumed that M ii = 0, whereas in Theorem 1.7 it is assumed that M ii ≥ c > 0 (cf. (1.3)). In order to compare both statements it makes sense to think of the single-site potentials in Theorem 1.7 as Let us turn to the first auxiliary Lemma (cf. [OR07, Lemma 3] or Lemma 3.1 from below). It states that the single-site conditional measures satisfy a LSI uniformly in the in the system size and the conditioned spin-values. The argument of [OR07, Lemma 3] by Otto & Reznikoff is heavily based on the assumption that the single-site potential ψ is super-quadratic. At this point we provide a new, different, and more elaborated argument showing that the statement of [OR07, Lemma 3] remains valid if the single-site potential ψ is only perturbed quadratic. One could say that the proof of Lemma 3.1 represents the main new ingredient compared to the argument of [OR07].
Lemma 3.1 (Generalization of [OR07, Lemma 3]). We assume the same conditions as in Theorem 1.7. We consider for an arbitrary subset S ⊂ Λ tot and site i ∈ S the single-site conditional measurē Then the single-site conditional measureμ(dx i |x S ) satisfies a LSI with constant ̺ > 0 (cf. Definition 1.4) that is uniform in Λ tot , S and the conditioned spins x S .
We state the proof of Lemma 3.1 in Section 3.1.
Let us turn to the second auxiliary Lemma (cf. [OR07, Lemma 4] or Lemma 3.2 from below). For some fixed but large enough integer K let us consider the K-sublattice Λ K given by Let S an arbitrary subset satisfying Λ K ⊂ S ⊂ Λ tot . The second auxiliary lemma states that measure on Λ K , which is conditioned on the spins in S\Λ K and averaged over the spins in Λ tot \S, satisfies a LSI with constant ̺ > 0 uniformly in S and the conditioned spins: Lemma 3.2 (Generalization of [OR07, Lemma 4]). We assume the same conditions as in Theorem 1.7. Let S be an arbitrary set with Λ K ⊂ S ⊂ Λ tot . Consider the conditional measurē Then there is some integer K such that the conditional measureμ(dx Λ K |x S\Λ K ) satisfies a LSI with constant ̺ > 0 (cf. Definition 1.4) that is uniform in Λ tot , S and the conditioned spins x S\Λ K .
3.1. Proof of Lemma 3.1 and Lemma 3.2. Let us first turn to the proof of Lemma 3.1. For the argument we need the two new ingredients. The first one is the covariance estimate of Proposition 3.3 from below. The second one is that the variances of our kind of Gibbs measure are uniformly bounded (see Lemma 3.4 from below).
Let us now state the covariance estimate of Proposition 3.3.
Proposition 3.3. Let Λ ⊂ Z d an arbitrary finite subset of the ddimensional lattice Z d . We consider a probability measure dµ := Here, | · | denotes the operator norm of a bilinear form.
• the numbers κ ij decay algebraically in the sense of is strictly diagonally dominant i.e. for some δ > 0 it holds for any i ∈ Λ j∈Λ,j =i Then for all functions f = f (x i ) and g = g(x j ), i, j ∈ Λ, and for any i, j ∈ Λ For the proof of Proposition 3.3 we refer the reader to the article [Men14].
Proof of Lemma 3.1. The strategy is to show that the HamiltonianH i (x i ) of the single-site conditional measureμ(dx i |x S ) is perturbed strictlyconvex in the sense that there exists a splittinḡ uniformly in x i ∈ R, i ∈ S, Λ tot and S. Once The aim is to decomposeH i such that (3.4) and (3.5) is satisfied. For that purpose, let us define the auxiliary Hamiltonian H aux (x), x ∈ R Λtot , as Note that H aux is strictly convex, if restricted to spins x j with |i − j| ≤ R. For convenience, let us introduce the notation S c := Λ tot \S. The HamiltonianH i is then written as . Now, let us check that the functionsψ c i (x i ) andψ b i (x i ) defined by the last identity satisfy the structural condition (3.5).
Let us consider first the functionψ b i (x i ). We introduce the auxiliary measure µ aux by Then it follows from the definition (3.6) of H aux that It is now left to show thatψ c i (x i ) is uniformly strictly convex. Direct calculation yields We decompose the measure µ aux into Here, µ aux (dx j ) j∈S c ,|j−i|≤R | (x j ) j∈S c ,|j−i|>R denotes the conditional measure given by whereasμ aux ((dx j ) j∈S c ,|j−i|>R ) denotes the marginal measure given bȳ For convenience, we write µ aux,c instead of the conditional measure Applying the decomposition to (3.7) yields The first term on the right hand side of the last identity is controlled easily. Note that the Hamiltonian H aux is strictly-convex, if restricted to spins x j with |j − i| ≤ R. So it follows from a standard argument based on the Brascamp lieb inequality that (for details see for example [Diz07, Chapter 3]) uniformly in R and therefore also Let us now turn to the second term in (3.8). Straightforward calculation yields Because the measuresμ aux and µ aux,c live on a subset of S c , i ∈ S, and the variance is invariant under adding constants, we have The first summand on the right hand side of the last identity is estimated in a straightforward manner i.e. varμ aux Here we have used one of the new ingredients, namely the uniform estimate (3.10) stated in Lemma 3.4 from below. Note that Lemma 3.4 also applies to the measure µ aux because µ aux satisfies the same structural assumptions as the measure µ Λ . Let us consider now the second summand on the right hand side of (3.9).
By doubling the variables we get Without loss of generality we may assume that the interaction is ferromagnetic i.e. M kl ≤ 0 for all k = l (else use M kl ≤ |M kl | and Lemma 1.10). Note that the measure µ aux,c has strictly convex singlesite potentials. Therefore the single-site conditional measures µ(dx 1 |x) satisfy a LSI with constant 1 2 M ii by the Bakry-Émery criterion (see Theorem B.1). Because the interaction is strictly-diagonally dominant in the sense of (1.3), an application of Proposition 3.3 yields that the covariance can be estimated as where the matrix M is given by the elements M ln for l, n ∈ S c , |l − i| ≤ R, |k − i| ≤ R or l = n = i.
We want to note that by an simple standard result (see for example [OR07, Lemma 5]) or [MN13, Lemma 4.3]) it holds (M −1 ) kl ≥ 0 for all k, l. Using this information, we get by an application of Jensen's inequality that Rα 2 Note that here we also used the second ingredient, namely the covariance estimates (3.2) and (3.3). Hence, both terms on the right hand side of (3.9) are arbitrarily small, if we choose R big enough. Overall this leads to the desired statement (cf. (3.8) ff.) which completes the argument.
In the proof of Lemma 3.1, we needed the following auxiliary statement.
Lemma 3.4. Under the same assumptions as in Lemma 3.1, it holds that for all i ∈ Λ var µ Λ (x i ) ≤ C, (3.10) where the bound is uniform in Λ and only depends on the constants appearing in (1.2) and in (1.3).
The proof of Lemma 3.4 is a simple and straightforward application of a exponential moment bound due to Robin Nittka. Additionally, we assume that for all i ∈ Z d the convex part ψ c i of the single-site potentials ψ i has a global minimum in x i = 0. Let δ > 0 be given by (1.3). Then for every 0 ≤ a ≤ δ 2 and any subset Λ ⊂ Z d it holds E µ Λ e ap 2 i 1.
In particular, for any k ∈ N 0 this yields i ] k!. The statement of Lemma 3.5 is a slight improvement of [BHK82, Section 3], because the assumptions are slightly weaker compared to [BHK82]. More precisely, ψ ′′ i may change sign outside every compact set and there is no condition on the signs of the interaction. Even if [MN13,Lemma 4.3] is formulated in [MN13] for systems on an one-dimensional lattice, a simple analysis of the proof shows that the statement is also true on lattices of any dimension.
Proof of Lemma 3.4. By doubling the variables we get By the change of coordinates x k = q k +p k and y k = q k −p k for all k ∈ Λ, the last identity yields by using the definition (1.4) of the finite-volume Gibbs measure µ Λ that By conditioning on the values q Λ it directly follows from the definition (1.1) of H that Here, the conditional measure µ Λ,q is given by the density Because of symmetry in the variable p k ,the convex part of the single-site potential ψ c k,q (p k ) has a global minimum at p k = 0 for any k. Therefore, an application of Lemma 3.5 yields the desired statement.
Let us turn to the verification of Lemma 3.2. We also need an auxiliary statement, namely Lemma 3.6 from below. It is a generalization of [OR07, Lemma 2] and states that the interactions of the Hamiltonian H((x i ) i∈S ) given by (3.1) decay sufficiently fast.
Lemma 3.6 (Generalization of [OR07, Lemma 2]). In the same situation as in Lemma 3.1, the interactions ofH(x S ) decay algebraically i.e. there are constants 0, ε, C < ∞ such that Proof of Lemma 3.6. Direct calculation as in [OR07,Lemma 2] shows that The last identity immediately yields the estimate (cf. [OR07,(52) Using the decay of interactions (1.6) and the decay of correlations (1.7) we get Now we use the same kind of argument as used in in the proof of Proposition 3.3 to estimate the term T k . This means that for any multi-indexes i, k, l, j ∈ Λ tot it holds either  Let dµ := Z −1 exp(−H(x)) dx be a probability measure on a direct product of Euclidean spaces X = X 1 × · · · × X N . We assume that • the conditional measures µ(dx i |x i ), 1 ≤ i ≤ N, satisfy a uniform LSI(̺ i ). • the numbers κ ij , 1 ≤ i = j ≤ N, satisfy uniformly in x ∈ X. Here, | · | denotes the operator norm of a bilinear form.
• the symmetric matrix A = (A ij ) N ×N defined by satisfies in the sense of quadratic forms Then µ satisfies LSI(̺).
Proof of Lemma 3.2. We want to apply Theorem 3.7. By an application of Lemma 3.1, we know that the single-site measures conditional measuresμ(dx i |x Λ K x S\,Λ K ), i ∈ Λ k satisfy a LSI with uniform constant ̺ > 0.
For the mixed derivatives of the Hamiltonian, we have according to Hence, in order to apply Theorem 1 we have to consider the symmetric We will argue that A is strict positive-definite if we choose the integer K large enough. We have Let us estimate the second term of the right hand side. We have where the last inequality holds if we choose K large enough. So we get overall that which yields the desired statement of Lemma 3.2 by an application of Theorem 3.7.
Appendix A. Uniqueness of the infinite-volume Gibbs measure: proof of Theorem 1.14 The proof of Theorem 1.14 is straightforward and only needs four ingredients: • A sufficient decay of interactions (cf. (1.6)).
• The uniform PI for the finite-volume Gibbs measures µ Λ . This is provided by Theorem 1.7 and the fact that the LSI yields a PI with the same constant. • The fact that the variances of the infinite-volume Gibbs measure µ are uniformly bounded (cf. (1.11)).
Proof of Theorem 1.14. Let us assume that there are two infinite-volume Gibbs measures µ andμ. It suffices to show that for a function f with bounded support Let B R denote a ball with radius R and center in the root of the lattice Z d . We decompose the measures µ andμ w.r.t. B R into where µ(dx B R |ω Z d \B R ) andμ(dx B R |ω Z d \B R ) denote the conditional measures andμ(dω Z d \B R ) andμ(dω Z d \B R ) denote the corresponding marginals. For convenience we will write x and ω instead of x B R and ω Z d \B R .
This decomposition of µ(dxdz|ω t ) yields the following decomposition of the covariance, namely For, convenience here and from now on, the indexes i, l, and j are always belonging to the set i ∈ supp(f ) l ∈ B R \ supp(f ), and j ∈ B R .
We start with the estimation of the term T 1 . We get Because the measure µ(dxdz|ω t ) satisfies a uniform LSI by Theorem 1.7, it also satisfies a uniform PI (see Definition 1.5). Then also the marginal µ(dz|ω t ) satisfies a uniform PI. Hence we can continue the estimation of T 1 according to where we used the decay of interaction (1.8). Because i ∈ supp(f ) and j / ∈ B R we get |i − j| ≥ dist (supp(f ), B R ) .
Hence, we continue the estimation of T 1 as Using the decay of interaction (1.8) and the decay of correlations (1.7) we get the estimate Plugging this to the estimation of T 2 we get where we used in the last step that ω 2 jμ (dω) + ω 2 jμ (dω) = ω 2 j µ(dω) + ω 2 jμ (dω) (1.11) ≤ C.
So, we have deduced the desired estimate (A.1), which closes the argument.