Lectures on

Logarithmic Sobolev Inequalities

by A. Guionnet ^{1} and B. Zegarlinski ^{2}

Introduction

During the last decade some important progress was achieved in the domain of the infinite dimensional analysis involving applications of coercive inequalities. In particular the understanding of the ergodicity problem for stochastic dynamics of large dimensional interacting systems has considerably improved. Besides other things this contributed to better understanding the relations between equilibrium description and out of equilibrium systems described by stochastic dynamics which play a crucial role in statistical mechanics.

The intention of these lecture notes is to present a self consistent and relatively complete introduction to this rapidly developing subject.

One can quickly introduce as follows the problem which we shall consider. Given a Markov semi-group P_{t} ≡ e^{tL} , t > 0, defined on a space C(Ω) of realvalued continuous functions on a Polish space Ω, one can ask the two following questions ;

(I) Is there an invariant measure for P_{t}, that is a probability measure μ on Ω so that for every f ∈ C(Ω)

μ(P_{t}f) = μ(f)

where P_{t}f denotes the image of f by P_{t} and we denoted in short μ(g) = ∫ g(x)dμ(x).

(II) If yes, does P_{t}f converge towards μ and in which sense ? More precisely, can we find a norm || · || on C(Ω) such that

||P_{t}f − μf|| ≤ γ(t)|||f|||

with a semi-norm |||·||| defined on a dense subset of C(Ω) and a rate γ(t) →_{t→∞} 0 independent of f. In general, || · || is either the uniform norm or the norm of L^{p}(μ) for some p ≥ 1.

Frequently, one can introduce P_{t} so that a given measure μ ( or a class of measures ) is P_{t} - invariant and then the question (II) becomes crucial.

1Ecole Normale Suprieure, 45 rue d’Ulm, 75005 Paris, France

2Department of Mathematics, Imperial College, Huxley Building, 180 Queen’s Gate, London SW7 2BZ

One such situation appears in statistical mechanics. In this case the state space Ω one considers is often an infinite product Ω ≡ M^{ZZ^{d}} where M is a finite set or a smooth compact finite dimensional Riemannian manifold. On such a space, one is given an a priori family of conditional expectations E_{X} indexed by the finite subsets X of ZZ^{d}, where E_{X} integrates over the variables (ω_{i}, i ∈ X). Under some mild hypotheses, it can be shown that there exists a probability measure μ on Ω, so-called Gibbs measure, characterized by the Dobrushin-Lanford-Ruelle’s condition

μ(E_{X}f) = μ(f)  (0.0.1)

for every finite subset X of ZZ^{d} and f ∈ C(Ω). One can use these conditional expectations to introduce a class of infinitesimal jump generators formally given by

L^{(X)}f ≡ ∑_{j∈ZZ^{d}}(E_{X+j}f − f),  (0.0.2)

where X + j denotes the set {j + k, k ∈ X} ( see [25, 47, 62] and the references therein ). Naturally the measures μ satisfying (0.0.1) are invariant for the associated Markov semi-groups P_{t}^{(X)} = e^{tL^{(X)}}. However, in general it is not clear that the solutions of (0.0.1) are the only invariant measures of P_{t}^{(X)}.. In [25], [47] and [62], one finds sufficient conditions so that P_{.}^{(X)} is uniformly ergodic and hence has a unique invariant measure. In this set up, the conditional expectations (E_{X}, X ∈ ZZ^{d}) often depend on some parameters (such as the temperature T or magnetic field h). As these parameters vary, the solutions of equation (0.0.1) change as well as their properties. In [23] and later in [19] constructive conditions were given to insure uniqueness of the solutions of (0.0.1) in terms of the kernels E_{{j}} and E_{Y +j} associated with the single point sets ({j}, j ∈ ZZ^{d}) and the cubes with given finite size (Y + j, j ∈ ZZ^{d}), respectively. These conditions are known as the uniqueness conditions of Dobrushin and Dobrushin-Shlosman, respectively. In general, the second is valid in a broader domain of values of the parameters.

Dobrushin-Shlosman’s uniqueness condition was used by Aizenman and Holley in [2] where they showed that a jump dynamics constructed as in (0.0.2) is ergodic in the uniform norm with an exponential rate when the conditional expectations E_{X+j}, j ∈ ZZ^{d}, satisfy this condition. This result in particular implies the ergodicity of the semi-group in L^{2}(μ) and the existence of a spectral gap for the spectrum of the self-adjoint operator L^{(X)} in L^{2}(μ). Moreover, as the quadratic forms associated with the generators L^{(Y)} constructed with other finite sets Y are equivalent to that of L^{(X)}, all the corresponding dynamics are ergodic in L^{2}(μ).

Unfortunately, for many interesting models describing systems in a neighbourhood of a ”critical point”, the size of the cubes for which Dobrushin-Shlosman’s condition is satisfied grows to infinity when one approaches the critical point. In particular, Aizenman-Holley strategy fails even though one can show that there exists a unique Gibbs measure satisfying (0.0.1). (In some specific models such as ferromagnetic systems where the conditional expectations preserve monotonicity properties this strategy was extended in a clever way in [70] (Part I).

To overcome these difficulties, a new clever strategy based on the use of hypercontractivity property was introduced in [55] and [56]. The main idea is to deduce the uniform ergodicity from L^{2}(μ) ergodicity and hypercontractivity. To this end one first approximates the semi-group with another semi-group in finite dimension. In finite dimension, one can bound uniform norms with L^{2} norms by using the ultracontractivity property of the semi-group realized in a unit time. The price to pay for this is a large coefficient growing with the dimension of the space which fortunately can be controlled thanks to the hypercontractivity property.

We recall that the hypercontractivity property allows us to bound the L^{p}(μ) norm of P_{t}f by the L^{q}(μ) norm of f for p = p(t) = 1 + (q − 1)e^{\frac{2}{c} t} and a constant c ∈ (0, ∞). It looks rather unusual since p can be, for large times, much bigger than q. This property was introduced first in the constructive quantum field theory (see [87] and [44]). Similarly to the contractivity property, the hypercontractivity allows an equivalent infinitesimal description given by the following logarithmic Sobolev inequality

μ(f^{2} log |f|) ≤ cμ(f(−Lf)) + μ(f^{2}) log μ(f^{2})  (0.0.3)

with a constant c ∈ (0, ∞) independent of a function f from the quadratic form domain of L. The logarithmic Sobolev inequality was already formally considered in [36], but the equivalence of the hypercontractivity property and of (0.0.3) was first proven in the seminal work [48] opening the door to further progress. In this paper L.Gross proved also that the logarithmic Sobolev inequality has a product property. He showed that this together with (0.0.3) for a uniform measure on the two point set (proven in the same paper) imply the corresponding inequality for a class of Gaussian measures. The first breakthrough which tremendously increased the class of probability measures satisfying (0.0.3) was due to Bakry and Emery who introduced in [6] a very nice sufficient condition. Roughly speaking, this condition, when formulated in the context of a smooth compact finite dimensional Riemannian manifold M, requires the positivity of the Ricci curvature of M. When considered in the setting of a probability measure μ(dx) = Z^{−1}e^{−U(x)}dx on IR^{n}, it necessitates that the smallest eigenvalue of the Hessian of U is uniformly bounded below by a positive constant. One of the first interesting applications of the Bakry - Emery condition was published in [10] where it was used to establish the logarithmic Sobolev inequality for a wide class of Gibbs measures on a product space over the unit sphere of dimension N, N ≥ 2, at high temperatures. Thus [10] provided the first non trivial class of examples of neither product nor Gaussian measures in infinite dimension for which (0.0.3) holds.

Afterwards, a new idea based on the specific structure of Gibbs measures was introduced to deal with Riemannian manifolds with possibly negative Ricci curvature as well as with discrete settings. In particular, in [103] and [104], Sobolev inequality was proved for systems with short range interaction on (S^{1})^{Z^{d}} at high temperature and {−1, +1}^{Z} for any temperature. An extension of these results to systems with long range interactions (of the same type as those studied in the uniqueness theory of Dobrushin) on Riemannian manifolds at high temperature appeared in [105].

Later these results were extended to a variety of directions described in a rich literature including [70], [64], [59], [49] - [50], [11] - [12], [69], …, [92]

- [95], [99], [100] - [101], [100], [3], [4], …, [107] -[109] … Besides the above mentioned control of ergodicity of a hypercontractive Markov semi-group, one of most interesting consequences of the research in this domain is the proof of the equivalence of strong mixing property and the log-Sobolev inequality.

In view of the restricted time and space that we were given to present this area, we will not be able to consider many new and interesting developments of this field, such as the extension to non-compact manifolds M (see [109] and more recent works of Yoshida and Bodineau, Helffer), the study and the applications of other coercivity inequalities (such as Nash’s inequalities (see [7])), the link with isoperimetric’s inequalities, the extension to loop spaces (see [32]…), to non-commutative spaces ([73]…)…

The contents of these notes are as follows

We begin by introducing the objects under study : Markov semi-groups, their infinitesimal construction via infinitesimal generators, invariant and reversible measures.

The second chapter is devoted to L^{2} ergodicity and the simplest coercivity inequality – the spectral gap inequality. After some general discussion, we shall show the stability of this property under perturbations, that is by change of the initial measure by bounded densities, and tensorisation (product property).

In chapter 3, we consider classical Sobolev inequality and its consequences : ultracontractivity property and the classical Nash inequality.

In chapter 4, we study the general properties of log-Sobolev inequalities including stability by perturbation, product property, equivalence with hypercontractivity property and its implications concerning spectral gap inequality. We also present the Bakry-Emery criterion and its equivalent forms in terms of semi-groups. We finally give an example of an infinite dimensional system defined by conditional expectations and discuss its log-Sobolev properties.

Our journey in statistical mechanics begins from chapter 5. The reader willing to learn how to prove log-Sobolev inequality in infinite-dimensional settings can go directly to this chapter. We begin by introducing the statistical mechanics framework of spin systems on a lattice. Then, we describe the general strategy introduced in [104, 92, 93] to prove log-Sobolev inequalities for systems with finite range interactions. It is based on a study of an auxiliary Markov chain constructed as the convolution of conditional expectations on cubes of given finite size as described in 5.2. In section 5.3, we restrict ourselves to one-dimensional lattice whereas in sections 5.4 we consider higher dimensional lattices. Under a strong mixing condition hypothesis, this Markov chain will be shown to converge towards the underlying Gibbs measure, the log-Sobolev inequality being then, in a certain sense, derived as a consequence of the product property. In sections 5.3 and 5.4, we prove the key point of this strategy called sweeping out relations. Our method is constructive in the sense that we always consider finite volume expectations. Our formalism will always cover both discrete and continuous settings, i.e. involving product spaces build upon discrete sets as well as smooth compact connnected Riemannian manifolds.

In chapter 6, we apply the ideas of the previous chapter to a slightly different context. We indeed show that the formalism developped in chapter 5 extends naturally to the case where the measure μ is not given a priori as a Gibbs measure but is described as the stationary measure of a cellular automata. In such a setting, one does not know in general even whether μ is a Gibbs measure for some interaction. Moreover the transition matrix of the cellular automata may not be evnen symmetric for μ.

Another strategy to prove logarithmic Sobolev inequalities is to make a martingale expansion of relative entropy by considering, instead of Markov chains constructed as convolutions of conditional expectations, conditional expectations on an increasing sequences of finite subsets of the lattice. This idea was originally introduced in [64] for finite range interaction systems. We illustrate it in chapter 7 where we study systems with long range interaction (though with a strength decreasing in a sufficiently fast way with the distance between the spins to insure that the potential satisfies Dobrushin’s uniqueness condition). Again, sweeping out relations are the key of this approach.

In chapter 8, we apply the previous log-Sobolev inequalities to deduce ergodic properties for the associated semi-group. We first present in section 8.1 a simple construction of these semi-groups which yields an exponential approximation property. In section 8.2, we study the ergodic properties of these semigroups when log-Sobolev property is satisfied. In section 8.3, we describe the equivalence theorem (see [93]) which establishes the correspondence between properties of the Gibbs measure such as strong mixing, uniform analyticity and other conditions introduced by Dobrushin and Shlosman and properties of the dynamics such as hypercontractivity or L^{2} ergodicity.

In chapter 9, we explore systems with random interactions and apply the ideas of chapters 5-7 to study the ergodic properties of their dynamics at high temperature.

Finally, we describe a few results concerning L^{2} ergodicity of Markov semigroups at low temperature where strong mixing property fails.

We give a long but still very incomplete bibliography of the subject.

These lecture notes represent several attempts to try to understand and organize the diverse evolutions of the subject. Part of this course was already given by B. Zegarlinski in Bochum in 1992, who would like to thank S. Albeverio for giving him the opportunity to spend a long, happy and fruitful period at Ruhr. We could also continue this work during two semesters at Institut Henri Poincaré of Paris where we had the pleasure to give courses. The second of these courses gave birth to these lecture notes originally in French. We are very grateful to the organizers of these two sessions, as well as to Institut Henri Poincaré team. B. Zegarlinski wishes also to thank D. Stroock for their long and fruitful collaboration. Finally, our collaboration was made possible thanks to the financial help of the European Stochastic Analysis Network and EPSRC.

Contents

Chapter 1. Markov semi - groups Section 1.1 Markov semi - groups and Generators Section 1.2 Invariant measures of a semi - group Section 1.3 Markov processes

Chapter 2. Spectral gap inequality and L^{2} ergodicity

Chapter 3. Classical Sobolev inequalities and ultracontractivity

Chapter 4. Logarithmic Sobolev inequalities and hypercontractivity Section 4.1 Properties of logarithmic Sobolev inequality Section 4.2 Logarithmic Sobolev and Spectral Gap inequalities Section 4.3 Bakry - Emery Criterium

Chapter 5. Logarithmic Sobolev inequalities for spins systems on a lattice Section 5.1 Notation and definitions, statistical mechanics Section 5.2 Strategy to demonstrate the logarithmic Sobolev inequality Section 5.3 Logarithmic Sobolev inequality in dimension 1; an example Section 5.4 Logarithmic Sobolev inequalities in dimension ≥ 2

Chapter 6. Logarithmic Sobolev inequalities and cellular automata

Chapter 7. Logarithmic Sobolev inequalities for spin systems with long range interaction. Martingale expansion

Chapter 8. Markov semigroup in infinite volume, ergodic properties Section 8.1. Construction of Markov semi-groups in infinite volume Section 8.2 Uniform ergodicity of Markov semi-groups in infinite volume Section 8.3 Equivalence Theorem

Chapter 9. Disordered systems ; uniform ergodicity in the high temperature regime Section 9.1 Absence of spectral gap for disordered ferromagnetic Ising model Section 9.2 Upper bound for the constant of logarithmic Sobolev inequality In finite volume and uniform ergodicity , d = 2

Chapter 10. Low temperature regime : L^{2} ergodicity in a finite volume Section 10.1 Spectral gap estimate Section 10.2 L^{2} ergodicity in infinite volume

Bibiography

Chapter 1

Markov Semi-groups

This chapter reviews some classical facts from the theory of Markov semi-groups which can be found for instance in [45] or [15]. We first recall the definition of Markov semi-groups and generators, giving examples in exercises. We then introduce the notions of invariant and reversible measures of Markov semi-groups. At the end of the chapter, we briefly sketch the relation between Markov semigroups and Markov processes. For more extended treatment of the related theory of Dirichlet forms and their links with Markov semi-groups the reader is encouraged to look at [40], [65] and [77].

1.1 Markov Semi-groups and Generators

Definition 1.1 A family (P_{t})_{t≥0} of linear operators on a Banach space (B, ||·||) is called a semi-group iff it satisfies the following conditions

(1) P_{0} = I, the identity on B.

(2) The map t → P_{t} is continuous in the sense that for all f ∈ B, t → P_{t}f is a continuous map from IR^{+} into B.

(3) For any f ∈ B and (t, s) ∈ (IR^{+})^{2},

P_{t+s}f = P_{t}P_{s}f.

The space B under consideration in most cases will be the set C(Ω) of real-valued bounded continuous functions on a Polish space Ω equipped with the uniform norm. However, in some important cases one needs to consider a Banach space given by a set of more regular functions, such as for example the set of uniformly bounded continuous functions from a Polish space Ω into IR furnished with the uniform norm. This is one of the reasons why we prefer to introduce a general Banach space B setting. In the sequel, we always consider spaces of (nice) real-valued functions. B will be equipped with a partial order ≥.

Definition 1.2 A semi-group (P_{t})_{t≥0} is Markov iff (4) For any t ∈ IR^{+},

P_{t}1I = 1I.

(5) For any t ∈ IR^{+}, P_{t} preserve the positivity, i.e for any f ∈ B and t ∈ IR^{+},

f ≥ 0  →  P_{t}f ≥ 0.

Properties (4) and (5) imply that, for any t ∈ IR^{+}, P_{t} is contractive, that is

Definition 1.3 P_{t} is contractive iff for any f ∈ B,

||P_{t}f|| ≤ ||f||  (1.1.1)

where || · || denotes the norm on B.

The notion of Markov semi-groups can be illustrated by the following examples

Exercise 1.4 • a) Let A be a non negative linear bounded operator, ||A|| ≤ 1, so that, if 1 is the identity in B, A1 = 1. Let λ > 0. Verify that

P_{t} = e^{tλ(A−I)} = ∑_{n≥0} \frac{(λt)^{n}}{n!} (A − I)^{n}

is a Markov semi-group on B. Here, we have denoted ||A|| = sup_{f∈B,f≠0} ||f||^{−1}||Af|| with || · || the norm on B.

• b) Let μ be a probability measure on a Ω equipped with the σ - algebra F and let m > 0. For any f ∈ C(Ω), we define

P_{t}f(ω) = e^{−mt}f(ω) + (1 − e^{−mt})μ(f).

Verify that (P_{t})_{t≥0} is a Markov semi-group on C(Ω) equipped with the uniform topology.

• c) (d-dimensional Brownian motion) For f ∈ C(IR^{d}) and ω ∈ IR^{d} , we set

P_{t}f(ω) = \frac{1}{(2πt)^{\frac{d}{2}}} ∫ f(y)e^{− \frac{|ω−y|^{2}}{2t}} dy.

Show that (P_{t})_{t≥0} is a Markov semi-group on the space CU(Ω) of uniformly continuous functions equipped with the uniform topology.

• d) (Brownian motion on the circle) If Ω = S^{1}, one can represent any continuous function on Ω by its Fourier expansion

f(x) = ∑_{n∈ZZ} a_{n}e^{inx}.

We then let

P_{t}f(x) = ∑_{n∈ZZ} e^{−tn^{2}}a_{n}e^{inx}.

Verify that (P_{t})_{t≥0} is a Markov semi-group on C(Ω) endowed with the uniform topology.

• e) (Poisson Process) Let λ ∈ IR^{+}, Ω = IR^{d}. For any f ∈ C(IR^{d}) and x ∈ IR^{d} , we define

P_{t}f(x) = e^{−λt} ∑_{k≥0} \frac{(λt)^{k}}{k!} f(x − kλ)

Show that (P_{t})_{t≥0} is a Markov semi-group on the set CU(Ω) endowed with the uniform topology.

• f) (Ornstein-Uhlenbeck’s Process) If Ω = IR, prove that

P_{t}f(x) = ∫ f(e^{−t}x + \sqrt{1 − e^{−2t}}y)\frac{e^{− \frac{y^{2}}{2}}}{\sqrt{2π}} dy

defines a Markov process on C(Ω).

Definition 1.5 The infinitesimal generator L of a semi-group P_{t} is defined by the formula

Lf := lim_{t↓0} \frac{1}{t}(P_{t} − I)f  (1.1.2)

for any function f for which the limit makes sense. The domain D(L) of L is the set of functions of C(Ω) for which the limit (1.1.2) exists.

Exercise 1.6 In the examples given in exercise 1.4, show that the infinitesimal generators are given, in the same order, by

• a) Lf = λ(A − I)f, D(L) = B.

• b) Lf = μf − f, D(L) = C(Ω).

• c) In the case d = 1 to simplify, Lf = (1/2)f′′, D(L) is the set of twice continuously differentiable functions f such that f′ and f′′ are uniformly bounded continuous.

Hints : Show that

g_{λ}(x) := λ ∫_{0}^{∞} e^{−λt}P_{t}f(x)dt = ∫_{−∞}^{∞} \sqrt{\frac{λ}{2}}f(y) exp\{−\sqrt{2λ}|x − y|\}dy

by using the identity

∫_{0}^{∞} exp −\{y^{2} + \frac{c^{2}}{y^{2}} \}dy = \frac{\sqrt{π}}{2} e^{−2c}

for all c > 0. Deduce that 2λ(g_{λ} − f) = g′′_{λ}. On the other hand, prove that g_{λ} ∈ D(L) and

L(g_{λ}) = λ(g_{λ} − f).

Using the above identity, deduce that Lg_{λ} = (1/2)g′′_{λ} and conclude by letting λ go to infinity and using that L is closed (see below).

• d) Lf = (1/2)f′′, D(L) is the set of twice continuously differentiable functions.

• e) For λ ∈ IR^{+}, show that Lf(x) = λ(f(x − λ) − f(x)). D(L) = CU(Ω).

• f) Lf(x) = f′′(x)−xf′(x), D(L) = C^{3}(IR), the set of 3 times continuously differentiable functions on IR with bounded derivatives (see [5]).

The following theorem characterizes the infinitesimal generators :

Theorem 1.7 (Hille-Yoshida’s theorem for Markov semi-groups) A linear operator L is the infinitesimal generator of a Markov semi-group (P_{t}, t ∈ IR^{+}) on B iff

• 1I ∈ D(L) and L1I = 0.

• D(L) is dense in B.

• L is closed.

• For any λ > 0, (λI − L) is invertible. Its inverse (λI − L)^{−1} is bounded

with

sup_{||f||≤1} ||(λI − L)^{−1}f|| ≤ \frac{1}{λ}

and preserves positivity (i.e for all f ≥ 0, (λI − L)^{−1}f ≥ 0) .

\underline{Remark} 1.8: Recall that an operator L is closed iff for any sequence f_{n} of D(L) converging (in the sense of the topology inherited from the norm || · || on B) towards a function f and such that Lf_{n} converges, then

lim_{n→∞} Lf_{n} = Lf.

Proof of theorem 1.7 Necessary condition.

The first point is a direct consequence of (4) of definition (1.2) and of the definition of the generator. To show the second point, we shall see that

D_{0} := \{\frac{1}{t} ∫_{0}^{t} dsP_{s}f, f ∈ C(Ω), t > 0\} ⊂ D(L).  (1.1.3)

Since P_{.}f is continuous for any f ∈ B, D_{0} is dense in B which completes the proof of this point.

To prove (1.1.3), note that for any τ ∈ IR^{+}, we have

\frac{1}{τ} (P_{τ} − I) ∫_{0}^{t} dsP_{s}f = \frac{1}{τ} ∫_{τ}^{τ+t} dsP_{s}f − \frac{1}{τ} ∫_{0}^{t} dsP_{s}f  (1.1.4)
= \frac{1}{τ} ∫_{t}^{τ+t} dsP_{s}f \frac{1}{τ} ∫_{0}^{τ} dsP_{s}f

by property (3) of definition 1.1. Then (2) of definition 1.1 gives the convergence of the right hand side of (1.1.4) so that we obtain

L ∫_{0}^{t} dsP_{s}f = P_{t}f − f,  (1.1.5)

resulting with (1.1.3). Remark as well that from (1.1.4), we also deduce by letting τ going to zero that for any f ∈ D(L),

L ∫_{0}^{t} dsP_{s}f = P_{t}f − f = ∫_{0}^{t} dsP_{s}Lf.  (1.1.6)

To show that L is closed, let us take a sequence f_{n} in D(L) converging to a function f and such that, for some g ∈ B,

lim_{n→∞} Lf_{n} = g.

By (1.1.5), we obtain that

P_{t}f_{n} − f_{n} = L ∫_{0}^{t} dsP_{s}f_{n} = ∫_{0}^{t} dsP_{s}Lf_{n}.  (1.1.7)

Hence, letting n go to infinity,we obtain P_{t}f − f = ∫_{0}^{t} dsP_{s}g. Dividing by t and taking t ↓ 0, we conclude that Lf = g for every f ∈ D(L).

To finish the proof, let us consider the resolvent R(λ, L) of the operator L defined by

R(λ, L) := (λI − L)^{−1}

and show that

R(λ, L)f = ∫_{0}^{∞} dse^{−λs}P_{s}f.  (1.1.8)

In fact, if one considers the semi-group P̃_{s} = e^{−λs}P_{s} with generator (L − λI), we obtain, according to (1.1.5) that

(L − λI) ∫_{0}^{t} dse^{−λs}P_{s}f = e^{−λt}P_{t}f − f

for any bounded continuous function f. Taking the limit as t → ∞ and using that L is closed, we conclude that, for any bounded continuous function f, ∫_{0}^{∞} dse^{−λs}P_{s}f belongs to D(L) and

(L − λI) ∫_{0}^{∞} dse^{−λs}P_{s}f = −f.  (1.1.9)

Consequently, (L − λI) is bijective and (1.1.8) is satisfied. In particular, since P_{s} is contractive,

sup_{||f||=1} ||R(λ, L)f|| ≤ ∫_{0}^{∞} dse^{−λs} = \frac{1}{λ}.

R(λ, L) is thus a bounded operator. Also, it is clear from (1.1.8) that R(λ, L) preserves positivity.

Sufficient condition.

For any λ > 0, we introduce the Yoshida approximation L_{λ} of L

L_{λ} := LλI(λI − L)^{−1} = λ^{2}(λI − L)^{−1} − λI.  (1.1.10)

Since

||(λI − L)^{−1}|| ≤ \frac{1}{λ},  (1.1.11)

L_{λ} is a bounded operator and we can define a family of operators P_{.}^{λ} by

P_{t}^{λ} := e^{tL_{λ}} := ∑_{n=0}^{∞} \frac{t^{n}}{n!} L_{λ}^{n} = e^{−tλI}e^{tλ^{2}(λI−L)^{−1}}.  (1.1.12)

P^{λ} is a Markov semi-group for any λ > 0 ( see exercise 1.4.a)). It remains to show that P_{t}^{λ} converges towards P_{t} as λ goes to infinity and then that P possesses all the properties of a Markov semi- group. To begin with, notice that for any f in the domain D(L) of L,

lim_{λ→∞} L_{λ}f = Lf.  (1.1.13)

Indeed, the identity

λ(λI − L)^{−1} − (λI − L)^{−1}L = I  (1.1.14)

holds on D(L). The second term goes to zero as λ goes to infinity so that we find

lim_{λ→∞} λ(λI − L)^{−1}g = g  (1.1.15)

for every g ∈ D(L), and hence for any g ∈ \overline{D(L)} = B. Since L is closed, we deduce that L_{λ} converges towards L as λ goes to infinity. We can now show that P_{t}^{λ} converges towards P_{t} as λ → ∞. In fact, for any couple (λ_{1}, λ_{2}) of positive real numbers and any f ∈ D(L), we have the interpolation formula

e^{tL_{λ_{1}}} f − e^{tL_{λ_{2}}} f = t ∫_{0}^{1} dse^{t(sL_{λ_{1}} +(1−s)L_{λ_{2}} )}(L_{λ_{1}} − L_{λ_{2}})f  (1.1.16)

since L_{λ_{1}} are L_{λ_{2}} bounded commuting operators. We can see as above that the semi-group Q_{t} := e^{t(sL_{λ_{1}} +(1−s)L_{λ_{2}} )} is contractive as s ∈ [0, 1] and conclude that

||e^{tL_{λ_{1}}} f − e^{tL_{λ_{2}}} f|| ≤ t||(L_{λ_{1}} − L_{λ_{2}})f||.  (1.1.17)

Thus, the convergence of the L_{λ}’s implies that of the P_{t}^{λ}’s. Let P_{t} be the linear operator defined by

P_{t} = lim_{λ→∞} P_{t}^{λ}  (1.1.18)

on D(L). It is clear that the semi-group properties, contractivity property, positivity properties as well as conservation of unit property of the semi-groups P^{λ} can be extended to P. Hence P is a Markov semi-group. Also, for any f ∈ D(L), we obtain by interpolation

P_{t}f − f = lim_{λ→∞}(P_{t}^{λ} − I)f = lim_{λ→∞} ∫_{0}^{t} dsP_{s}^{λ}L_{λ}f = ∫_{0}^{t} dsP_{s}Lf.  (1.1.19)

so that P is continuous with generator L on D(L). By condition (4) of the theorem, the resolvent of the semi-group corresponds to that of L on B so that L is the generator of P. ⋄

Exercise 1.9 Show that the generators L with domains D(L) defined in exercise 1.6 are infinitesimal generators of Markov semi-groups by using Hille-Yoshida’s theorem.

Exercise 1.10 Show that for any integer number n, D(L^{n}) is dense in B. Hint : Generalize (1.1.5).

1.2 Invariant Measures of a semi-group

We recall that

Definition 1.11 Let (P_{t})_{t≥0} be a Markov semi-group. A probability measure μ on (Ω, Σ) is invariant with respect to the semi-group (P_{t})_{t≥0} iff for any f ∈ C(Ω) and any t ≥ 0

μ(P_{t}f) = μ(f).  (1.2.20)

The set of invariant measures for a semi-group (P_{t})_{t≥0} will be denoted hereafter J = J (P).

\underline{Remark} 1.12: Note that a semi-group may have no invariant probability measure (consider for instance the semi-group associated with the Brownian motion). The invariant probability measures are also characterized by

Property 1.13 μ on (Ω, Σ) is invariant with respect to the semi-group (P_{t})_{t≥0} iff for any f ∈ D(L),

μ(Lf) = 0.  (1.2.21)

Proof : By (1.2.20), we obtain for any t

μ[\frac{1}{t}(P_{t} − I)f] = 0.

Taking the limit t → 0, we deduce (1.2.21). Reciprocally, since we already noticed that

(P_{t} − I)f = L ∫_{0}^{t} dsP_{s}f,

by integrating both sides with respect to μ, we get that (1.2.21) implies (1.2.20) for every bounded continuous function since we saw in (1.1.5) that ∫_{0}^{t} dsP_{s}f belongs to D(L). ⋄

The semi-group (P_{t})_{t≥0} has been defined until now on the Banach space B, which a’ priori is a subspace of C(Ω). In fact, we can extend it as shown in

Property 1.14 Let μ ∈ J (P) with respect to a Markov semi-group (P_{t})_{t≥0}. (P_{t})_{t≥0} can be extended to any L^{p}(μ) for p ≥ 1.

Proof : The operators P_{t} being linear, positive and with total mass P_{t}1 = 1, we have by Jensen’s inequality for any p ≥ 1 and any f ∈ B,

|P_{t}f|^{p} ≤ P_{t}|f|^{p}.  (1.2.22)

Integrating both sides with respect to μ ∈ J, we deduce that

μ|P_{t}f|^{p} ≤ μ|f|^{p}.  (1.2.23)

Thus, Hahn-Banach’s theorem (see [83], theorem 5.16) shows that we can extend P_{t} to L^{p}(μ).

⋄

Definition 1.15 A Markov semi-group {P_{t}, t ≥ 0} is L^{q}(μ) ergodic for q ∈ ]1, ∞[ and μ ∈ J (P_{t}) iff for any function f ∈ L^{q}(μ),

lim_{t→∞} ∫ (P_{t}f − μf)^{q}dμ = 0.

A Markov semi-group {P_{t}, t ≥ 0} is uniformly ergodic iff J (P_{t}) is reduced to a unique probability measure and

lim_{t→∞} ||P_{t}f − μf||_{∞} = 0.

In the rest of this course our goal will be to develop tools to study the ergodic properties of Markov semi-groups. We shall often consider semi-groups P and measures μ satisfying the following property stronger than invariance.

Definition 1.16 A probability measure μ on (Ω, Σ) is reversible for a Markov semi-group {P_{t}, t ≥ 0} iff for any (f, g) ∈ B and time t ≥ 0,

μ(gP_{t}f) = μ(fP_{t}g).  (1.2.24)

Equivalently, one says that (P_{t})_{t≥0} satisfies the detailed balance condition for the probability measure μ.

The set of reversible measures of a Markov semi-group {P_{t}, t ≥ 0} will be denoted J_{0}(P_{t}). Clearly

J_{0}(P_{t}) ⊂ J (P_{t})

since we can take g = 1I and that P_{t}1I = 1I. (1.2.24) shows more precisely that the semi-group P̄_{t} obtained as the extension of P_{t} in L^{2}(μ) (as indicated in property 1.14) is self adjoint in L^{2}(μ). We can associate to P̄_{t} a closed infinitesimal generator L̄ (see the proof of Hille-Yoshida’s theorem) which coincides with the infinitesimal generator of P on its domain. Then, it is not hard to see that condition (1.2.24) is equivalent to

μ(gL̄f) = μ(fL̄g)  (1.2.25)

for any f, g in the domain of L̄. In other words, since L̄ is closed, L̄ is self-adjoint in L^{2}(μ).

Exercise 1.17 Verify that the probability measure μ is invariant but not reversible for the generator L in the two following examples

• Let (Ω, Σ, μ) be a probability space and (E_{1}, E_{2}) two different conditional expectations, E_{1} = μ( |A_{1}) et E_{2} = μ( |A_{2}), A_{1} ≠ A_{2}. We define, if I denotes the identity in L^{2}(μ), the operator L in L^{2}(μ) by

L = E_{1}E_{2} − I.

• To U ∈ C^{1}(IR^{d}), we associate the following Gibbs measure on IR

μ(dx) = \frac{1}{Z} exp\{−U(x_{1}, .., x_{d})\}dx_{1}..dx_{d}.

provided that 0 < Z < ∞. For a function α ∈ C^{1}(IR^{d}, IR^{d}), consider the operator on C^{1}(IR^{d}) defined by

Lf(x) = α · ∇f(x) = ∑_{i=1}^{d} α_{i}(x)∂_{x_{i}}f(x).

If

div(α) = α · ∇U,

check that μ is invariant for L. However, it is not reversible in general.

Exercise 1.18 Show that the standard Gaussian law on IR is reversible for the Ornstein-Uhlenbeck semi-group generated by L = Δ − x∂_{x}. More generally, let Λ be a finite subset of ZZ^{d} and Ω = IR^{Λ}. Let U be a continuously differentiable function on Ω diverging to infinity in a sufficiently fast way so that Z_{Λ} = ∫ exp{−U(x)}dx_{Λ} is finite. Let μ be the probability measure

μ(dx) = \frac{1}{Z_{Λ}} exp\{−U(x)\}dx_{Λ}.

Consider the generator on C_{b}^{2}(IR^{|Λ|}) defined by

L = ∑_{i∈Λ}(Δ_{i} − g_{i}∇_{i})

where ∇_{i} (resp. Δ_{i}) denotes the derivation (resp. the Laplacian ) acting on the i-th coordinate. Here the g_{i}’s are bounded Lipschitz functions. Give sufficient conditions on the g_{i}’s so that μ is reversible for the semi-group associated with L.

Exercise 1.19 Let Λ be a finite subset of ZZ^{d} and Ω = {−1, +1}^{Λ}. Let

L = ∑_{i∈Λ} c_{i}(σ)∂_{i}

with ∂_{i}f = f(σ^{i}) − f(σ) if

σ_{j}^{i} = \{ \begin{matrix}+σ_{j}  if j ≠ i \\ −σ_{i}  if j = i. \end{matrix}

Consider the probability measure μ on Ω of the form

μ(f) = \frac{1}{∑_{σ∈Ω} e^{−U(σ)}} ∑_{σ∈Ω} e^{−U(σ)}f(σ).

Give sufficient conditions on the c_{i} so that μ is reversible for the semi-group associated with L.

Exercise 1.20 Let Λ be a finite subset of ZZ^{d}. Consider X ⊂ ZZ^{d} a finite subset of ZZ^{d}. Set

X + i = {j + i; j ∈ X}.

Let μ be a probability measure on a probability space Ω = M^{Λ} for a Polish space M. Let Σ_{X+i}^{Λ} be the sigma-algebra generated by (x_{j})_{j∈(X+i)^{c}∩Λ}. We denote

E_{X+i} = μ( |Σ_{X+i})

the conditional expectation of μ knowing (x_{j})_{j∈(X+i)^{c}∩Λ}. Let L be the generator given, for f ∈ L^{2}(μ), by

Lf(x) = ∑_{i∈Λ}(E_{X+i} − I)f(x).

Show that μ is reversible for the semi-group associated with L.

1.3 Markov Processes

The notion of Markov Semi-group is intimately related with that of Markov Processes. In this section we recall some aspects of such processes and in particular their links with Markov Semi-groups.

We define the Markov property as follows.

Definition 1.21 Let (Ω, F, (F_{t})_{t≥0}, IP) be a probability space; an adapted process X = {X_{t} : Ω → IR^{n}, t ≥ 0, }, n ∈ IN, will be said to be Markov with respect to (F_{t}) iff for all measurable bounded set Γ and any times s, t,

IP(X_{t} ∈ Γ|F_{s}) = IP(X_{t} ∈ Γ|X_{s}).

Exercise 1.22 Show that the Brownian motion, or more generally any process with independent increments, is a Markov process.

In the sequel, we denote for any x ∈ Ω,

IP^{x}(X_{t} ∈ Γ) := IP(X_{t} ∈ Γ|X_{0} = x)

the law of the Markov process with initial condition x.

Later on, homogeneous Markov processes will be of interest to us. They are described as follows

Definition 1.23 Let (Ω, F, (F_{t}), IP) be a probability space. A Markov process X = {X_{t}, t ≥ 0} with respect to (F_{t}) will be said to be homogeneous iff for every bounded measurable set G and any times s, t, u ≥ 0,

IP(X_{t} ∈ Γ|X_{s}) = IP(X_{t+u} ∈ Γ|X_{s+u}).

Exercise 1.24 Show that Brownian motion B_{.} is an homogeneous Markov process with respect to its natural filtration F_{t} = σ(X_{s}, s ≤ t).

If we denote by IP^{x} the law of a homogeneous Markov process starting from x ∈ Ω, we can define a family (P_{t})_{t≥0} of linear operators on the space of bounded measurable functions by

P_{t}f(x) = ∫ f(X_{t})dIP^{x}.

(P_{t})_{t≥0} is then a Markov semi-group.

We can reformulate the homogeneous Markov property as follows

Property 1.25 Let (Ω, F, (F_{t}), IP) be a probability space. A process X = {X_{t}, t ≥ 0} is a homogeneous Markov process iff for any bounded measurable set Γ and any times s, t

IP^{x}(X_{t} ∈ Γ|F_{s}) = (P_{t}1I_{Γ})(X_{s}) IP^{x} − a.s

A proof may be found in [57], p. 75.

Notice that the notion of Markov semi-group is in fact equivalent to the definition of Feller-Markov processes defined below. This point is illustrated in the following exercises.

Exercise 1.26 Let (Ω, F, (F_{t}), IP) be a probability space and X = {X_{t}, t ≥ 0} a Feller-Markov process (i.e a homogeneous Markov process such that for any bounded continuous function f, P_{t}f is bounded continuous). Show that the operator family (P_{t})_{t≥0} is a Markov semi-group on (C(Ω), || · ||_{∞}). Conversely, any Markov semi-group on (C(Ω), ||·||_{∞}) defines a unique Feller-Markov process. Hint : For the second point,

a) Use Riesz ’s theorem to see that for any time t and any x ∈ Ω, there exist a unique probability E(t, x, dy) so that for any f ∈ D(L),

P_{t}(f)(x) = ∫ f(y)E(t, x, dy).

b) Let A be the algebra of cylinder functions on C(IR^{+}, Ω), that is the algebra of functions of the form

F (ξ) = f(ξ_{t_{1}}, .., ξ_{t_{p}})

for some finite integer number p and times (t_{1}, .., t_{p}). Define , for a probability measure μ, a normalized linear functional E^{μ} on A such that, for F as above, E^{μ}(F) is equal to

∫ f(y_{1}, ., y_{p})μ(dy_{0})E(t_{1}, y_{0}, dy_{1})E(t_{2} − t_{1}, y_{1}, dy_{2})..E(t_{p} − t_{p−1}, y_{p−1}, dy_{p}).

The Stone-Weierstrass theorem implies that A is dense in C(IR^{+}, Ω). Conclude by Riesz’s theorem that there is a unique probability measure P^{μ} on this space such that for any F ∈ A,

∫ F (ξ)dP^{μ}(ξ) = E^{μ}(F).

Show that (E^{δ_{x}}, x ∈ Ω) is the law of a Markov process.

Since we saw that the Feller-Markov semi-groups and homogeneous Markov processes are in bijection, the Hille-Yoshida theorem establishes a bijection between Markov processes and infinitesimal generators. This connection may in fact be made more directly by following Stroock and Varadhan [96] who introduced the notion of martingale problems.

To associate a generator to a Markov process, we define

Definition 1.27 Let L be an infinitesimal generator. A probability measure IP on C(IR^{+}, Ω) is said to be solution of the martingale problem for L with initial condition η iff

1.

IP[ξ ∈ C(IR^{+}, Ω) : ξ_{0} = η] = 1.

2. If we denote by x the canonical process under IP, f(x_{t}) − ∫_{0}^{t} Lf(x_{s})ds is a martingale under IP for the canonical filtration

F_{t} = σ(x_{u}, u ≤ t).

We then have the following theorem (see Proposition 4.2 in [57])

Theorem 1.28 Let L be an infinitesimal generator. Let (E^{η}, η ∈ Ω) be the unique Feller-Markov process associated to L. Then, for every (η ∈ Ω), E^{η} is the unique solution of the martingale problem for L with initial condition η.

Finally, let us give as an exercise the following example of Markov process

Exercise 1.29 Let (h_{i})_{1≤i≤d} be continuously differentiable Lipschitz functions on IR^{d}. Show that the stochastic differential system

dx_{t}^{i} = h_{i}(x_{t})dt + dB_{t}^{i} 1 ≤ i ≤ d,

with a d-dimensional Brownian motion (B^{i}, 1 ≤ i ≤ d), admits a unique strong solution which is a Markov process. Describe the generator of its associated Markov semi-group.

For further informations on Markov processes the reader may like to consult the books [57] and [76].

Spectral gap inequalities and L^{2} ergodicity

Before considering Sobolev inequalities and entering the heart of the matter, we study the L^{2} ergodicity of semi-groups satisfying the detailed balance condition with respect to some probability measure μ. As we already mentioned in the last section, this property allows us to consider their infinitesimal generators as self-adjoint operators in L^{2}(μ) so that the study of these operators in that space is rather natural.

Let L be an infinitesimal generator. Let μ be a probability measure on (Ω, Σ) and (P_{t})_{t≥0} a Markov semi-group satisfying the detailed balance condition with respect to μ. We define

Definition 2.1 A probability measure μ ∈ J_{0}(L) satisfies the spectral gap inequality iff there exists a positive real number m such that

mμ(f − μf)^{2} ≤ μ(f(−L)f)  (2.0.1)

for any function f ∈ L^{2}(μ) ∩ D(L) such that the right hand side of (2.0.1) is finite. The largest positive real number m satisfying (2.0.1) is called the spectral gap of the self-adjoint operator L.

The corresponding Dirichlet form is given on elements of the domain of the generator by

E(f, g) = μ(f(−L)g)

It can as well be defined by the carré du champ

Γ_{1}(f, f) := \frac{1}{2} (Lf^{2} − 2fLf) .

Indeed, for any μ ∈ J_{0}(L), we have

E(f, f) = μ(Γ_{1}(f, f)).

2.0 Spectral gap inequalities and L^{2} ergodicity 15

The operator carré du champ is non negative since

Γ_{1}(f, f) = lim_{t↓0} \frac{1}{2t} (P_{t}f^{2} − (P_{t}f)^{2}) ≥ 0

by Cauchy-Schwarz’s inequality. It will play an important role latter.

Example 2.2 Let Λ be a finite subset of ZZ^{d}.

• Let Ω = (S^{d−1})^{Λ} = {x = (x_{i})_{i∈Λ}, x_{i} ∈ S^{d−1} ∀i ∈ Λ} with S^{d−1} the unit sphere in IR^{d}. Consider the Langevin dynamics generator

L = ∑_{i∈Λ} (Δ_{i} − ∇_{i}U · ∇_{i})

for some function U ∈ C^{1}(Ω). Then

Γ_{1}(f, f) = ∑_{i∈Λ}(∇_{i}f)^{2}.

• If Ω = {−1, +1}^{Λ} and we consider the Glauber dynamics generator

L = ∑_{i∈Λ} c_{i}∂_{i}

for some functions c_{i} ≥ 0, then

Γ_{1}(f, f) = ∑_{i∈Λ} c_{i}(∂_{i}f)^{2}.

Exercise 2.3 1) Let (Ω, Σ, μ) be a probability space . Let L be a generator with domain L^{1}(μ) given by Lf = μf − f. Then, show that μ ∈ J_{0}(L) satisfies a spectral gap inequality with m = 1.

2) If Ω = {−1, +1} and μ = pδ_{−1} + (1 − p)δ_{+1} for some p ∈ (0, 1), we define the generator L on the set of measurable functions by Lf = c(σ)(f(−σ) − f(σ)) for some non negative function c. Choose c so that μ ∈ J_{0}(L) and show that μ satisfies a spectral gap inequality.

The spectral gap property is equivalent with the notion of L^{2} ergodicity of the semi-groups

Property 2.4 μ ∈ J_{0}(P_{t}) satisfies a spectral gap inequality with constant m iff for any t ≥ 0 and any function f ∈ L^{2}(μ)

μ(P_{t}f − μf)^{2} ≤ e^{−2mt}μ(f − μf)^{2}.

Proof : Note that if f is a centered function, ∫ fdμ = 0, then f_{t} = P_{t}f is also centered since μ is invariant. Hence, (2.0.1), applied to f_{t} = P_{t}f, gives

−∂_{t}μ(P_{t}f)^{2} ≥ 2mμ(P_{t}f)^{2}

16 2.Spectral gap inequalities and L^{2} ergodicity

and so, as P_{0} = I,

μ(P_{t}f)^{2} ≤ e^{−2mt}μ(f^{2}).  (2.0.2)

Conversely, if f ∈ D(L) satisfies (2.0.2), then t → e^{mt}μ(P_{t}f −μf)^{2} is decreasing. Hence,

\frac{d}{dt}e^{mt}μ(P_{t}f − μf)^{2} ≤ 0.

In t = 0, we deduce the spectral gap inequality.

⋄

The notion of spectral gap inequality can easily be extended in infinite dimension. In fact, we have the following product property

Theorem 2.5 Let (μ_{i})_{i=1,2} be two probability measures on (Ω_{i}, Σ_{i})_{i=1,2} satisfying the spectral gap inequality with coefficients (m_{i})_{i=1,2} for some generators (L_{i})_{i=1,2}. Let Ω = Ω_{1} × Ω_{2} be equipped with the product σ-algebra. For any function f on Ω, we note

f_{1,ω}(x) = f(x, ω) f_{2,ω}(x) = f(ω, x)

and extend (L_{i})_{i=1,2} to functions on Ω by

L_{1}f(x_{1}, x_{2}) = L_{1}f_{1,x_{2}}(x_{1}) L_{2}f(x_{1}, x_{2}) = L_{2}f_{2,x_{1}}(x_{2}).

Then, if L = L_{1} + L_{2} and m = min(m_{1}, m_{2}), the product law μ_{1} ⊗ μ_{2} satisfies the spectral gap inequality with constant m

mμ_{1} ⊗ μ_{2}(f − μ_{1} ⊗ μ_{2}f)^{2} ≤ μ_{1} ⊗ μ_{2}(f(−L)f))  (2.0.3)

for any measurable function f for which the right hand side is finite.

Proof : Integrating with respect to one variable, we obtain

μ_{1} ⊗ μ_{2}(f − μ_{1} ⊗ μ_{2}f)^{2} = μ_{2} [μ_{1}(f − μ_{1}f)^{2}] + μ_{2}(μ_{1}f − μ_{1} ⊗ μ_{2}f)^{2}.

Applying spectral gap inequality for μ_{1} and μ_{2}, we deduce

μ_{1} ⊗ μ_{2}(f − μ_{1} ⊗ μ_{2}f)^{2} ≤ m^{−1}μ_{1} ⊗ μ_{2} (f(−L_{1})f + μ_{1}f(−L_{2})μ_{1}f) .

On the other hand, f → Γ_{1}(f, f) is convex since

Γ_{1}(f, f)(x) = lim_{t↓0} \frac{1}{2t}P_{t}(f^{2} − P_{t}f(x))^{2}(x).

Thus,

μ_{2} (μ_{1}f(−L_{2})μ_{1}f) = μ_{2} (Γ_{1}^{L_{2}}(μ_{1}f, μ_{1}f)) ≤ μ_{1} ⊗ μ_{2} (Γ_{1}^{L_{2}}(f, f))

which finishes the proof.

⋄

Further, the spectral gap property is preserved when we perturb a measure by a bounded density.

Property 2.6 Let μ be a probability measure on a space (Ω, Σ) satisfying the spectral gap inequality for the generator L with constant m. Let U be a bounded measurable function and consider the probability measure ν given by

ν = \frac{1}{Z} e^{−U}dμ

with

Z = ∫ e^{−U}dμ.

Then ν satisfies the spectral gap inequality for the generator L with constant me^{−2osc(U)}.

Proof : This property is clear once one notices that, if osc(U) = sup U − inf U, for every measurable set A,

e^{−osc(U)}μ(A) ≤ ν(A) ≤ e^{osc(U)}μ(A).  (2.0.4)

Indeed, one then obtains, with the additional observation that for every constant C, ν(f − νf)^{2} ≤ ν(f − C)^{2},

ν(f − νf)^{2} ≤ ν(f − μf)^{2}
≤ e^{osc(U)}μ(f − μf)^{2}
≤ \frac{e^{osc(U)}}{m} μΓ_{1}(f, f)
≤ \frac{e^{2osc(U)}}{m} νΓ_{1}(f, f)

⋄

In fact, this property can be improved when one considers finite volume Gibbs measures with short range interaction. We then have (see [86] or [106])

Property 2.7 Let (Ω, Σ, μ) be a probability space and assume that μ satisfies a spectral gap inequality with constant m for the carré du champ Γ_{1}. Let Λ = [−L, L] × [−l, l]^{d−1} for some (l, L) ∈ (IN^{∗})^{2}, l ≤ L. Let U be a real valued bounded continuous function on Ω^{2}. We denote by ω ∈ Ω^{|∂Λ|} the boundary condition . Let

H_{Λ}^{ω}(x) = ∑_{i,j∈Λ,|i−j|=1} U(x_{i}, x_{j}) + ∑_{i∈Λ,j∈Λ^{c},|i−j|=1} U(x_{i}, ω_{j})

and set

dμ_{Λ}^{ω}(x) = \frac{1}{Z_{Λ}^{ω}} e^{−βH_{Λ}^{ω}(x)}dμ^{⊗|Λ|}(x).

Then, if Γ̄_{1} = ∑_{i∈Λ} Γ_{1}^{i}, there exists a finite constant c (depending on U) so that

μ_{Λ}^{ω}(f − μ_{Λ}^{ω}f)^{2} ≤ |Λ|e^{cl^{d−1}}μ_{Λ}^{ω}(Γ̄_{1}(f, f))

for any measurable function f such that the above right hand side is finite.

Remark that the previous property would only yield in this example

μ_{Λ}^{ω}(f − μ_{Λ}^{ω}f)^{2} ≤ e^{cl^{d}L}μ_{Λ}^{ω}(Γ̄_{1}(f, f)).

Hence, property 2.7 is a real improvement when L is large. It already points out that in dimension 1, the spectral gap is bounded below by the inverse of the volume of Λ. In fact, we shall see later that it is of order one, a fact linked with absence of phase transition for systems with finite range interactions on one dimensional lattice. Property 2.7 is optimal (modulo the multiplication by the volume of Λ in the right hand side and the choice of the constant) in such a generality (see the lower bound proved by Thomas [98] at low temperatures).

Proof : Let us choose a lexicographic order in Λ and denote by {i_{1}, .., i_{k}, ...} the points in Λ in this order. We choose it so that (i_{k}^{1})_{1≤k≤|Λ|}, the first coordinate of (i_{k})_{1≤k≤|Λ|}, increases slowly, that is that we fill in the faces of volume l^{d−1} of Λ with coordinate i^{1} one after the other. Set I_{k} = {i_{1}, .., i_{k}}.

We shall proceed by interpolation : note first that

μ_{Λ}^{ω}(f − μ_{Λ}^{ω}f)^{2} = \frac{1}{2} ∫ ∫ (f(x_{i_{1}}, .., x_{i_{|Λ|}}) − f(x̃_{i_{1}}, .., x̃_{i_{|Λ|}}))^{2} dμ_{Λ}^{ω}(x)dμ_{Λ}^{ω}(x̃).

Writing, with Δ_{p}f(x, x̃) ≡ f(x_{i_{1}}, ., x_{i_{p}}, x̃_{i_{p+1}}., x̃_{i_{|Λ|}})−f(x_{i_{1}}, ., x_{i_{p+1}}, x̃_{i_{p+2}}., x̃_{i_{|Λ|}}),

f(x_{i_{1}}, .., x_{i_{|Λ|}}) − f(x̃_{i_{1}}, .., x̃_{i_{|Λ|}}) = ∑_{p=0}^{|Λ|−1} Δ_{p}f(x, x̃),

and using Jensen’s inequality we obtain

μ_{Λ}^{ω}(f − μ_{Λ}^{ω}f)^{2} ≤ |Λ| ∑_{p=0}^{|Λ|−1}(μ_{Λ}^{ω})^{⊗2} (Δ_{p}f(x, x̃))^{2} .  (2.0.5)

In each term of the right hand side, only the variable at the i_{p+1}-th site can differ. Notice as well that the points of J_{p}^{+} = {i_{m}, m ≥ p + c_{d}l^{d−1}} and of J_{l}^{−} = {i_{m}, m ≤ p − c_{d}l^{d−1}} are at distance larger than one from i_{p+1} for some well chosen finite constant c_{d}. Consequently, bounding the density of μ_{Λ}^{ω} uniformly on the {x_{i_{k}}, |k−p| ≤ c_{d}l^{d−1}}, we get, if S_{p} = {i_{k} ∈ Λ; |k−p| ≤ c_{d}l^{d−1}}, that for any function F ≥ 0,

μ_{Λ}^{ω}F ≤ e^{4c_{d}l^{d−1}||U||_{∞}}μ_{J_{p}^{−}}^{0} ⊗ μ_{S_{p}} ⊗ μ_{J_{p}^{+}}^{0}F.

Heredμ_{S_{p}}(x_{i}, i ∈ S_{p}) = dμ^{⊗|S_{p}|}(x_{i}, i ∈ S_{p}) and

dμ_{J_{p}^{ϵ}}^{0} = \frac{1}{Z_{J_{p}^{ϵ}}} e^{−βH_{J^{ϵ}_{p}}} dμ^{⊗|J_{p}^{ϵ}|}

with the convention that for ϵ = + or −,

H_{J_{p}^{ϵ}} = ∑_{i,j∈J_{p}^{ϵ},|i−j|=1} U(x_{i}, x_{j}).

From the above bound, we obtain that

(μ_{Λ}^{ω})^{⊗2} (Δ_{p}f(x, x̃))^{2} ≤ e^{4c_{d}l^{d−1}||U||_{∞}}(μ_{J_{p}^{−}}^{0} ⊗ μ_{S_{p}} ⊗ μ_{J_{p}^{+}}^{0})^{⊗2} (Δ_{p}f(x, x̃))^{2} . (2.0.6)

Using the spectral gap inequality for μ, we deduce that

(μ_{Λ}^{ω})^{⊗2} (Δ_{p}f(x, x̃))^{2} ≤ \frac{1}{m}e^{4c_{d}l^{d−1}||U||_{∞}}μ_{J_{p}^{−}}^{0} ⊗ μ_{S_{p}} ⊗ μ_{J_{p}^{+}}^{0} (Γ_{1}^{p+1}(f, f)) (2.0.7)

with Γ_{1}^{p+1} acting on the i_{p+1}-th variable.

To find back our initial probability measure, we input by force the potential to find

(μ_{Λ}^{ω})^{⊗2} (Δ_{p}f(x, x̃))^{2} ≤ \frac{e^{6c_{d}l^{d−1}||U||_{∞}}}{m} μ_{Λ}^{ω} (Γ_{1}^{p+1}(f, f)) .  (2.0.8)

Thanks to (2.0.5) and (2.0.8), we can conclude that

μ_{Λ}^{ω}(f − μ_{Λ}^{ω}f)^{2} ≤ \frac{|Λ|}{m}e^{6c_{d}l^{d−1}||U||_{∞}} ∑_{l=0}^{|Λ|−1} μ_{Λ}^{ω} (Γ_{1}^{l+1}(f, f)) .  (2.0.9)
⋄

Exercise 2.8 Generalize the last property to the case where the interaction has finite range, (but not necessarily of nearest neighbours type), and the interaction inhomogeneous, that is that the Hamiltonian is given by

H_{Λ}^{ω}(x) = ∑_{X∈X:X∩Λ≠∅} U_{X}(x)

for a family X of finite subsets of ZZ^{d} with uniformly bounded diameters and a family (U_{X})_{X∈X} of local functions so that, for any X ∈ X, U_{X}(x) only depends on (x_{i}, i ∈ X).

The next exercise gives a link between the spectral gap inequality and the concentration of measure phenomenon

Exercise 2.9 ([52]) If μ satisfies a spectral gap inequality with constant m on IR^{d} for a carré du champ Γ_{1} satisfying the Leibnitz rule

Γ_{1}(f, gh) = Γ_{1}(f, g)h + Γ_{1}(f, h)g,

then for all f ∈ D(L), and t sufficiently small so that

\frac{t^{2}}{2m}||Γ_{1}(f, f)||_{∞} ≤ 1

we have

μe^{tf} ≤ const.e^{\frac{t^{2}}{m} ||Γ_{1}(f,f)||_{∞}}e^{tμf}.

Hint : Write

μ(e^{tf}) = μ(e^{\frac{tf}{2}} , e^{\frac{tf}{2}} ) + μ(e^{\frac{tf}{2}} )^{2}.

and apply the spectral gap inequality to μ(e^{\frac{tf}{2}} , e^{\frac{tf}{2}} ). Observe that when Γ_{1} satisfies the Leibnitz rule,

Γ_{1}(e^{\frac{tf}{2}} , e^{\frac{tf}{2}} ) = \frac{t^{2}}{4} e^{tf}Γ_{1}(f, f).

Proceed inductively.

We shall continue our discussion on the carré du champ and Leibnitz rule when studying Bakry-Emery criterion.

In the next exercise, we suggest proofs of the spectral gap inequality in a few simple cases.

Exercise 2.10 • Ω = [0, a] for some positive real number a. For f ∈ C^{2}(Ω) such that f(a) = f(0) = 0, we set

L = \frac{d^{2}}{dx^{2}} .

If μ is the normalized Lebesgue measure on Ω, then L is self-adjoint in L^{2}(μ) and the spectral gap inequality is satisfied with m ≥ (1/a^{2}). Hint : Use integration by parts formula.

• Let Ω = {−1, +1} and μ be the Bernouilli law pδ_{−1} + (1 − p)δ_{+1}. Let Lf(x) = f(−x) − ν(f). Show that the spectral gap inequality is satisfied with m = 1.

For further general informations about spectral gap inequality, the reader can read [51] and references therein. Even though one most often try to bound the spectral gap constant from below, it can be useful also to obtain upper bounds. We then recommend to read [14]. Finally, we remind the reader that in forthcoming chapter 5 we shall be interested in the logarithmic Sobolev inequalities which imply spectral gap inequalities. Nevertheless, in most cases we shall consider, one can obtain directly (and by similar methods) lower bounds for the spectral gap constant.

To obtain a stronger control on the asymptotic behaviour of P_{t}f than the L^{2} ergodicity studied above, we shall need contractivity inequalities. It is the role played by Sobolev inequalities.

Classical Sobolev inequalities and Ultracontractivity

In this chapter, we study classical Sobolev inequalities and their links with uniform ergodicity of Markov semi-groups. We shall see that classical Sobolev inequalities is equivalent to Nash’s inequalities, which are themselves equivalent to the ultracontractivity of the associated semi-groups. This last property entails the uniform ergodicity of the semi-groups. Unfortunately, we shall see that in general these inequalities cannot be true in infinite dimension ( see exercise 3.7). It is one of the main motivations to study log-Sobolev inequalities which satisfy a product property and therefore can hold in infinite dimension.

Definition 3.1 A probability measure μ ∈ J_{0}(P_{t}) satisfies a classical Sobolev inequality iff for some p ∈]2, ∞[ and two finite constants (a, b) ∈ [0, ∞[^{2}, we have

||f||_{p}^{2} ≤ aμΓ_{1}(f, f) + bμf^{2}  (3.0.1)

for any measurable function f such that μΓ_{1}(f, f) and ||f||_{2} are finite and, in the case where b = 0, so that ∫ fdμ = 0.

In the next exercise, we describe some classes of probability measures for which a Sobolev inequality holds.

Exercise 3.2 • Let O ⊂ IR^{d} be a bounded convex open subset of IR^{d} and μ be the normalized Lebesgue measure on O. Show that μ satisfies a classical Sobolev inequality restricted to the functions which vanish on the boundary of O. Hint : For d > 2 use Taylor’s formula to write

|f(x)| = ∏_{i=1}^{d} | ∫_{x_{0}^{i}}^{x_{i}} ∂_{i}f(x_{i^{c}} ∘ y_{i})dy_{i}|^{\frac{1}{d}}

for any function f which vanishes outside O and any point (x_{0}^{i}, 1 ≤ i ≤ d) with coordinates outside of O. Multiply all Taylor’s formulas and rise to the power p = (1/d − 1). Integrate with respect to the d dimensional Lebesgue measure, use inductively Hölder inequality and the geometric - arithmetic mean inequality to arrive at an useful inequality. By substituting to this inequality a power of the absolute value of a sufficiently smooth local function one can obtain a family of useful inequalities including the desired one (with optimal exponent).

• Show that, if μ satisfies a Sobolev inequality, any probability measure ν ≪ μ such that there exists λ ∈ [1, ∞[ for which

\frac{1}{λ} ≤ \frac{dν}{dμ} ≤ λ

also satisfies a Sobolev inequality.

Note that when one considers a probability measure μ on the entire real line IR, it was shown in [79] that if a probability measure μ has a tail like e^{−ax^{2+ϵ}} for a and ϵ strictly positive real numbers, then μ satisfies a Sobolev inequality. This result is false when ϵ = 0 since the heat semi-group on IR is not ultracontractive; ( the latter as we shall see is equivalent to a Sobolev inequality ).

Setting

||f||_{∞} = sup{μ(fg), ||g||_{1} = 1},

we shall see that

Theorem 3.3 If μ ∈ J_{0}(P_{t}) satisfies a Sobolev inequality, there exists a positive real number γ and a finite constant c such that for any f ∈ L^{1}(μ)

||P_{t}f − μf||_{∞} ≤ c(\frac{a}{2et} + b)^{γ}||f − μf||_{1}  (3.0.2)

In particular, if b = 0, P_{t} converges to μ with a polynomial rate.

Moreover, if μ satisfies both the spectral gap inequality and the Sobolev inequality, there exists a finite constant c′ so that for any f ∈ L^{2}(μ)

||P_{t}f − μf||_{∞} ≤ c′e^{−mt}||f − μf||_{2}.  (3.0.3)

Let us point out that the above control is not really uniform on the entire space but with probability one with respect to μ. However, if μ is compactly supported and is absolutely continuous with respect to the uniform measure on the underlying space, the theorem implies uniform ergodicity.

Theorem 3.3 relates Classical Sobolev inequalities with a contractivity property of the associated semi-groups as follows. To prove it, let us recall the standard notation

Definition 3.4 For (p, q) ∈ [1, ∞[^{2},

||P_{t}||_{q,p} = sup{||P_{t}f||_{p} : f ∈ C(Ω), ||f||_{q} = 1, μ(f) = 0}

where ||f||_{q} = (∫ f^{q}dμ)^{\frac{1}{q}} . This definition extends to the case p = ∞.

With this definition, the first point of the theorem is equivalent to

Property 3.5 Under the hypotheses of theorem 3.3, there exists a positive constant γ and a finite constant c so that

||P_{t}||_{1,∞} ≤ c(\frac{a}{2et} + b)^{γ}

This property is itself equivalent to

Lemma 3.6 Under the hypotheses of theorem 3.3, there exists a positive constant γ̃ and a finite constant c̃ so that

||P_{t}||_{1,2} ≤ c̃(\frac{a}{4et} + b)_{γ}

Indeed, it is clear that property 3.5 implies

||P_{t}||_{1,2} ≤ ||P_{t}||_{1,∞} ≤ c(\frac{a}{2et} + b)^{γ}

so that the lemma is verified. Conversely, the semi-group property implies that

||P_{t}||_{1,∞} ≤ ||P_{\frac{t}{2}} ||_{1,2}||P_{\frac{t}{2}} ||_{2,∞}.  (3.0.4)

But, by duality,

||P_{\frac{t}{2}} ||_{2,∞} = sup\{μ(f_{\frac{t}{2}} g), ||f||_{2} = 1, ||g||_{1} = 1\}
= sup\{μ(g_{\frac{t}{2}} f), ||f||_{2} = 1, ||g||_{1} = 1\}
= ||P_{\frac{t}{2}} ||_{1,2}

which gives with (3.0.4) the desired equivalence. Proof of Lemma 3.6 [106]

To prove the lemma, we first remark that the Sobolev inequality implies the following classical Nash inequality

||f||_{2} ≤ (aμΓ_{1}(f, f) + bμf^{2})^{\frac{α}{2}} ||f||_{1}^{β}  (3.0.5)

with α = \frac{1}{2} \frac{p}{p−1} and β = 1 − α. This inequality is in fact directly derived via Hölder ’s inequality

||f||_{2} ≤ ||f||1^{\frac{1}{2u}} ||f||v+1^{\frac{v+1}{2v}}

for every couple (u, v) of conjugate exponents between one and infinity, by taking v + 1 = p. Using (3.0.5) with f_{t} we get

||f_{t}||_{2} ≤ (aμΓ_{1}(f_{t}, f_{t}) + bμf^{2})^{\frac{α}{2}} ||f_{t}||_{1}^{β}  (3.0.6)

To estimate Γ_{1}(f_{t}, f_{t}), note first that the spectral theorem allows us to write

P_{t} = ∫_{0}^{∞} e^{−λt}dE_{λ}^{μ}

with {E_{λ}^{μ}, λ ≥ 0} the projections on the eigenspaces of L in L^{2}(μ) (see Bourbaki, théorie spectrale or [78] ). With these notations

μΓ_{1}(f_{t}, f_{t}) = −μ(P_{t}fLP_{t}f)
= ∫ λe^{−2λt}d(E_{λ}^{μ}f, f)_{L^{2}(μ)}.

Noticing that for any real number u

ue^{−2u} ≤ \frac{1}{2e}

we deduce

μΓ_{1}(f_{t}, f_{t}) ≤ \frac{1}{2et} ∫ d(E_{λ}^{μ}f, f)_{L^{2}(μ)} = \frac{1}{2et}μ(f^{2}).  (3.0.7)

Plugging this result into (3.0.6) and noticing that μ|P_{t}f|^{q} ≤ μ|f|^{q} for q ≥ 1, we deduce that for any t ≥ 0,

||f_{t}||_{2} ≤ (\frac{a}{2et} + b)^{\frac{α}{2}} ||f||_{2}^{α}||f||_{1}^{β}.  (3.0.8)

Choosing t_{0} = t/2, we obtain by induction

||f_{t}||_{2} ≤ ∏_{n=1}^{∞}(\frac{a}{2e^{\frac{t}{2^{n}}}} + b)^{\frac{α^{n}}{2}} ||f||_{1}^{β ∑_{n≥0} α^{n}}.  (3.0.9)

Since

(\frac{a}{2e\frac{t}{2^{n}}} + b) ≤ 2^{n}(\frac{a}{2et} + b)

we deduce that

||f_{t}||_{2}^{2} ≤ 2^{∑_{n≥1} nα^{n}}(\frac{a}{2et} + b)^{\frac{α}{1−α}} ||f||1^{\frac{2β}{1−α}} .  (3.0.10)

The lemma is established.

⋄

To complete the proof of theorem 3.3, it is enough to notice that, when the spectral gap inequality holds,

||P_{t}f − μf||_{∞} ≤ ||P_{1}||_{2,∞}||P_{t−1}f − μf||_{2} ≤ ce^{−mt}||f − μf||_{2}.
⋄

Unfortunately, classical Sobolev inequality is not satisfied in infinite dimension in the sense that it does not satisfies a product property as the spectral gap inequality.

Exercise 3.7 Let P_{t} = e^{tL} be a Markov semi-group on C(Ω). Let μ ∈ J_{0}(P_{t}) satisfying the Sobolev inequality

μ(f^{p})^{\frac{2}{p}} ≤ aμΓ_{1}(f, f) + bμf^{2}.

For any integer number n, we define a semi-group P_{t}^{n} = e^{tL^{n}} with

L^{n} = ∑_{i=1}^{n} L_{i}

if L_{i} acts on the i-th variable. Show that for sufficiently large integer number n, one cannot have

μ^{⊗n}(f^{p})^{\frac{2}{p}} ≤ aμ^{⊗n}Γ_{1}^{n}(f, f) + bμ^{⊗n}f^{2}

for all functions f for which the right hand side is finite.

Hint : Assume the above equality true and obtain a contradiction by taking for some g ∈ D(L),

f(ω) = ∏_{i=1}^{n} g(ω_{i}).

We saw that if μ ∈ J_{0}(P_{t}) satisfies a Sobolev inequality, P_{t} is ultracontractive, that is that for sufficiently large times t

||P_{t}||_{2,∞} < ∞.

This property was the key point to obtain uniform ergodicity. In fact, such a property can be obtained directly when the state space is a compact Riemaniann manifold as one can see by doing the following exercise.

Exercise 3.8 Let S^{1} be the unit circle in IR^{2}. Let Λ ⊂⊂ ZZ^{d} and U_{Λ} be a twice continuously bounded differentiable function on S^{1} × S^{1}. Set

H_{Λ}(x) = ∑_{i,j∈Λ,|i−j|=1} U_{Λ}(x_{i}, x_{j}).

We consider the semi-group associated with

L = ∑_{i∈Λ}(Δ_{i} − ∇_{i}H_{Λ}(x).∇_{i})

where Δ_{i} = (1/2)\frac{d^{2}}{dx_{i}^{2}} et ∇_{i} = \frac{d}{dx_{i}} on S^{1} for i ∈ Λ.

Show by use of Girsanov formula ( see theorem 5.1 in [57] ) that there exists a finite constant c which only depends on U_{Λ} so that for any function f ≥ 0,

P_{t}f ≤ e^{ct|Λ|} ∫ f(y) ∏_{i∈Λ} p_{t}(x_{i}, dy_{i}).

Here, p_{t} is the semi-group associated with the Laplacian on the circle. Show that there exists, for any t ≥ 0, a finite constant c_{t} such that

sup_{x_{i}} p_{t}(x_{i}, dy_{i}) ≤ c_{t}λ(dy_{i})

with λ the uniform measure on the circle. Conclude that there exists a finite constant c′ so that

||P_{t}||_{1,∞} ≤ (c_{t})^{|Λ|}e^{c′t|Λ|}.

Finally, let us point out that Nash’s inequality is equivalent to the control we obtained for ||P_{t}||_{1,2}. Indeed, we have

Exercise 3.9 Let μ ∈ J_{0}(P_{t}) and assume that there exists γ, a, b > 0 such that

||P_{t}||_{1,2} ≤ (\frac{a}{2et} + b)^{γ}.

Then, there exists A, B > 0 such that

||f||_{2} ≤ (AμΓ_{1}(f, f) + Bμf^{2})^{\frac{1}{2} α}||f||_{1}^{β}

with α = γ(γ + 1) and β = 1 − α. Hint : Note that for all t ≥ 0,

||f_{t}||_{2}^{2} ≥ ||f||_{2}^{2} − 2tμΓ_{1}(f, f)

and deduce from the hypothesis that

||f||_{2}^{2} ≤ 2tμΓ_{1}(f, f) + (\frac{a}{2et} + b)^{2γ}||f||_{1}^{2}

Conclude by optimization over the time parameter t.

In the set up of finite Markov chains, Diaconis and Saloff-Coste [21] gave simple proofs of Nash’s inequalities. In [7], generalized log-Nash and Nash inequalities were introduced.

Logarithmic Sobolev inequalities and Hypercontractivity

We begin this chapter by describing the equivalence theorem of Gross contained in his seminal work [48] which inspired intensive interest and development in logarithmic Sobolev Inequalities. Then, we study the properties of logarithmic Sobolev Inequalities, such as its stability by product (see property 4.4) and its stability by perturbation (see property 4.6 and exercise 4.7). We then compare logarithmic Sobolev inequality with spectral gap inequality, showing in particular that logarithmic Sobolev inequality implies spectral gap inequality (see theorem 4.9). Finally, we discuss Bakry-Emery criterion, giving the examples of probability measures satisfying logarithmic Sobolev inequality.

Let P_{t} = e^{tL} be a Markov semi-group and μ ∈ J_{0}(P_{t}). Given p ∈]1, ∞[ and a constant c ∈]0, ∞[, we define

q(t) := q(t, p, c) = 1 + (p − 1)e^{\frac{2t}{c}} , t ∈ IR^{+}.

Theorem 4.1 [Gross’ Integration Lemma] Let c ∈]0, ∞[ and d ∈]0, ∞[. The following statements are equivalent : (i)

μ(f^{2} log \frac{f}{||f||_{2}} ) ≤ cμΓ_{1}(f, f) + d||f||_{2}^{2}  (4.0.1)

for all non negative functions for which the right hand side is finite. (ii) For any q ∈ [2, ∞[,

μ(f^{q} log \frac{f}{||f||_{q}}) ≤ \frac{2c}{q}μΓ_{1}(f^{\frac{q}{2}} , f^{\frac{q}{2}} ) + \frac{2d}{q}||f||_{q}^{q}  (4.0.2)

for all non negative functions for which the right hand side is finite.

28 4. Logarithmic Sobolev inequalities and Hypercontractivity

(iii) Inequality (4.0.2) for some q ≥ 1. (iv) For all p ∈]1, ∞[ and p ≤ q ≤ q(t, p, c), we have

||P_{t}||_{p,q} ≤ exp(2d(\frac{1}{p} − \frac{1}{q}))  (4.0.3)

(v) For all q ∈ [2, ∞[, t ∈ IR^{+},

||P_{t}||_{2,q} ≤ exp(2d(\frac{1}{2} − \frac{1}{q}))  (4.0.4)

Properties (i)-(v) imply (vi) For all t ∈ IR^{+} and any non negative function f such that ||f||_{1} = 1 we have

S_{f}(t) = μf_{t} log f_{t} ≤ e^{\frac{−2t}{c}} S_{f}(0) + 2d(1 − e^{\frac{−2t}{c}})  (4.0.5)

for all t ∈ IR^{+}.

Inequality (4.0.1) is called the logarithmic Sobolev (or in short log-Sobolev) inequality.

Constants (c, d), c ∈ (0, ∞) and d ∈ [0, ∞), such that (i) is satisfied will be called log-Sobolev coefficients.

\underline{Remark} 4.2: If additionally

μ ((Lf) log f) = −4μΓ_{1}(f^{\frac{1}{2}} , f^{\frac{1}{2}} )  (4.0.6)

then (vi) is equivalent to (i)-(v). (4.0.6) is in particular satisfied when Γ_{1} satisfies the Leibnitz rule (see 4.13).

Exercise 4.3

• Show that Sobolev inequality implies logarithmic Sobolev inequality.

Hint : write

μ[f^{2} log \frac{f^{2}}{||f||_{2}^{2}} ] = ||f||_{2}^{2}μ[\frac{f^{2}}{||f||_{2}^{2}} log \frac{f^{2}}{||f||_{2}^{2}} ],

and use Jensen’s inequality together with Sobolev inequality .

• Prove the logarithmic Sobolev inequality for a Bernoulli law μ = \frac{1}{2}δ_{a}+ \frac{1}{2}δ_{b}, with (a, b) ∈ IR^{2}, by a direct computation (see [48]). The generalization to μ = pδ_{a} + (1 − p)δ_{b} for p ∈ (0, 1) can be found for instance in [85].

Note that the theorem shows that the logarithmic Sobolev inequality with d = 0 implies that the associated semi-group is hypercontractive. In this situation one shows that the spectral gap inequality is satisfied, see theorem 4.9 below. Proof : The equivalence between (i), (ii) and (iii) is easily obtained by substitution (For (i)→(ii), we take f = g^{\frac{q}{2}} , (ii) can be clearly reduced to (iii) and (iii)→(i) by taking f = g^{\frac{2}{q}}).

4.0 Logarithmic Sobolev inequalities and Hypercontractivity 29

Let us show that (ii) implies (iv). To do so, we remark that

∂_{t} log ||f_{t}||_{q(t)} = −\frac{∂_{t}q(t)}{q(t)^{2}} log μf_{t}^{q(t)} + \frac{1}{q(t)μf_{t}^{q(t)}} μ(f_{t}^{q(t)}∂_{t}q(t) log f_{t})
= −\frac{∂_{t}q(t)}{q(t)^{2}} log μf_{t}^{q(t)} + \frac{1}{q(t)μf_{t}^{q(t)}} μ((∂_{t}q(t))f_{t}^{q(t)} log f_{t}
+q(t)f_{t}^{q(t)−1}Lf_{t})  (4.0.7)
= \frac{∂_{t}q(t)}{q(t)μf_{t}^{q(t)}} (μ(f_{t}^{q(t)} log \frac{f_{t}}{||f_{t}||_{q(t)}} ) + \frac{cq(t)}{2(q(t) − 1)}μ(f_{t}^{q(t)−1}Lf_{t}))

where in the last line we used that

∂_{t}q(t) = \frac{2}{c}(q(t) − 1).

Consider now more closely the last term in the bracket on the right hand side of (4.0.7). By definition, for any q ≥ 1 and any non negative function g,

μ(g^{q−1}Lg) = lim_{τ→0} \frac{1}{τ} (g^{q−1}(P_{τ} − 1)g)
= − lim_{τ→0} \frac{1}{2τ} ∫ μ(dω)P_{τ} (ω, dω̃)(g^{q−1}(ω) − g^{q−1}(ω̃))(g(ω) − g(ω̃))
≤ −\frac{4(q − 1)}{q^{2}} lim_{τ→0} \frac{1}{2τ} ∫ μ(dω)P_{τ} (ω, dω̃)(g^{\frac{q}{2}} (ω) − g^{\frac{q}{2}} (ω̃))^{2} (4.0.8)
= −\frac{4(q − 1)}{q^{2}}μΓ_{1}(g^{\frac{q}{2}} , g^{\frac{q}{2}} ).

In (4.0.8), we used the following inequality satisfied for any non negative (x, y)

(x^{\frac{q}{2}} − y^{\frac{q}{2}} )^{2} ≤ \frac{q^{2}}{4(q − 1)}(x^{q} − y^{q})(x − y).

(4.0.7) and (4.0.8) yield

∂_{t} log ||f_{t}||_{q(t)}

≤ \frac{∂_{t}q(t)}{q(t)μf_{t}^{q(t)}} (μ(f_{t}^{q(t)} log \frac{f_{t}}{||f_{t}||_{q(t)}} ) − 2\frac{c}{q(t)}μΓ_{1}(ft^{\frac{q(t)}{2}} , ft^{\frac{q(t)}{2}} )) (4.0.9)

Under hypothesis (ii), we deduce that

∂_{t} log ||f_{t}||_{q(t)} ≤ 2d\frac{∂_{t}q(t)}{q^{2}(t)}

and hence, since q(0) = p,

||f_{t}||_{q(t)} ≤ ||f_{t}||_{p} exp\{2d(\frac{1}{p} − \frac{1}{q(t)})\}.

Finally, for any given q ≤ q(t), we can find a t_{0} ≤ t such that q = q(t_{0}). Using the definition of the semi-group and its contractivity property in any L^{p}, p ≥ 1, we obtain

||f_{t}||_{q} = ||f_{t}||_{q(t_{0})} = ||P_{t_{0}}f_{t−t_{0}}||_{q(t_{0})} ≤ ||f_{t−t_{0}}||_{p} exp\{2d(\frac{1}{p} − \frac{1}{q})\}
≤ ||f||_{p} exp\{2d(\frac{1}{p} − \frac{1}{q})\}

which extends the property to any q ≤ q(t).

To show that (iv) implies (i), we note first that (iv) induces that for any non negative function

ϕ_{f}(t) : t → exp\{−2d(\frac{1}{p} − \frac{1}{q(t)})\}||f_{t}||_{q(t)}

is decreasing. Indeed, it implies that for all s, t ≥ 0,

ϕ_{f}(t + s) ≤ exp\{−2d(\frac{1}{p} − \frac{1}{q(t + s)})\}||P_{t}||_{q(s),q(t+s)}||f||_{q(s)}
≤ exp\{−2d(\frac{1}{p} − \frac{1}{q(s)})\}||f||_{q(s)}.

Consequently, log ϕ_{f}(t) is differentiable and also decreasing. We thus obtain

∂_{t} log ||f_{t}||_{q(t)} ≤ ∂_{t}\{2d(\frac{1}{p} − \frac{1}{q(t)})\} = 2d\frac{∂_{t}q}{q^{2}} (t).

Going back to (4.0.7), we deduce

μ(f_{t}^{q} log \frac{f_{t}}{||f_{t}||_{q}}) + \frac{cq}{2(q − 1)}μ(f_{t}^{q−1}Lf_{t}) ≤ 2d\frac{μf_{t}^{q}}{q} (t).

Taking t = 0 and p = 2, we have

μ(f^{2} log \frac{f}{||f_{t}||_{2}} ) ≤ cμΓ_{1}(f, f) + \frac{2d}{q}||f||_{2}^{2}.

Finally, to show that (iv) implies (vi), we differentiate with respect to the time variable the following function

ψ_{f}(t) = e^{\frac{2t}{c}} S_{f}(t)

defined with a non negative f from the domain of the generator. Assuming that ||f_{t}||_{1} = ||f||_{1} = 1, we find

∂_{t}e^{\frac{2t}{c}} S_{f}(t) = \frac{2}{c}e^{\frac{2t}{c}} S_{f}(t) + e^{\frac{2t}{c}} μ(Lf_{t} log f_{t}).  (4.0.10)

To estimate the last term, we note that for any function F ∈ D(L),

μ[(LF) log F] = lim_{τ→0} \frac{−1}{2τ} ∫ μ(dω)P_{τ} (ω, dω̃)(F (ω) − F (ω̃))(log F (ω) − log F (ω̃)).

Since for any non negative x, y,

(log x − log y)(x − y) ≥ (x^{\frac{1}{2}} − y^{\frac{1}{2}} )^{2},

gives

μ[(LF) log F] ≤ −4μΓ_{1}(F^{\frac{1}{2}} , F^{\frac{1}{2}} ).  (4.0.11)

We deduce from (4.0.10) and (4.0.11) that

∂_{t}e^{\frac{2t}{c}} S_{f}(t) ≤ \frac{2}{c}e^{\frac{2t}{c}} (S_{f}(t) − 2cμ(Γ_{1}(ft^{\frac{1}{2}} , ft^{\frac{1}{2}} ))  (4.0.12)

and, if logarithmic Sobolev inequality is satisfied,

∂_{t}e^{\frac{2t}{c}} S_{f}(t) ≤ \frac{4d}{c}e^{\frac{2t}{c}} .  (4.0.13)

(vi) follows from the integration with respect to the time variable.

⋄

4.1 Properties of logarithmic Sobolev inequality

We begin by showing that, similarly to the spectral gap inequality, the logarithmic Sobolev inequality has the product property.

Theorem 4.4 Let (μ_{i})_{i=1,2} be two probability measures on probability spaces satisfying the logarithmic Sobolev inequality with coefficients (c_{i}, d_{i})_{i=1,2} for the infinitesimal generators (L_{i}, i = 1, 2). Let L be the generator on the product space Ω_{1} × Ω_{2} defined as in property 2.5 and Γ_{1} its carré du champ. Then, the product probability measure μ_{1} ⊗ μ_{2} satisfies

μ_{1} ⊗ μ_{2}(f^{2} log \frac{f}{||f||_{2}} ) ≤ cμ_{1} ⊗ μ_{2}(Γ_{1}(f, f) + dμ_{1} ⊗ μ_{2}f^{2} (4.1.14)
with  c = max(c_{1}, c_{2})  (4.1.15)
d = d_{1} + d_{2}  (4.1.16)

for any non negative function f for which the right hand side of (4.1.14) is finite. Consequently, if μ is a probability measure satisfying a logarithmic Sobolev inequality with a coefficient c < ∞ and d = 0, then the product probability measure μ^{⊗n} satisfies the logarithmic Sobolev inequality with the same coefficient c for any integer number n.

Exercise 4.5 (Gross) Deduce from exercise 4.3 and property 4.4 that the Gaussian law satisfies a logarithmic Sobolev inequality (see [48]). Hint : Use the central limit theorem.

Proof : Let f be a bounded measurable function on Ω_{1} × Ω_{2} in the domain of L. Then, using the logarithmic Sobolev inequality for the probability measure μ_{1}, we obtain

μ_{1} ⊗ μ_{2}(f^{2} log \frac{f^{2}}{||f||_{2}^{2}} ) = μ_{2} (μ_{1}(f^{2} log \frac{f^{2}}{μ_{1}(f^{2})}) + μ_{1}(f^{2}) log \frac{μ_{1}(f^{2})}{μ_{1} ⊗ μ_{2}(f^{2})})
≤ μ_{2} (2c_{1}μ_{1}Γ_{1,1}(f, f) + 2d_{1}μ_{1}(f^{2}) + μ_{1}(f^{2}) log \frac{μ_{1}(f^{2})}{μ_{1} ⊗ μ_{2}(f^{2})})

Applying the logarithmic Sobolev inequality to g = \sqrt{μ_{1}(f^{2})} and Cauchy-Schwarz’s inequality, we deduce

μ_{1} ⊗ μ_{2}(f^{2} log \frac{f^{2}}{||f||_{2}^{2}} ) ≤ 2c_{1}μ_{2} ⊗ μ_{1}Γ_{1,1}(f, f) + 2c_{2}μ_{2}Γ_{1,2}(g, g)
+2(d_{1} + d_{2})μ_{2} ⊗ μ_{1}(f^{2}).

We can estimate the last term of the above inequality by noticing that

μ_{2}Γ_{1,2}(g, g) = μ_{2}Γ_{1,2}(\sqrt{μ_{1}(f^{2})}, \sqrt{μ_{1}(f^{2})})
= lim_{τ↓0} \frac{1}{2τ} ∫ μ_{2}(dω)P_{τ,2}(ω, dω̃) (\sqrt{μ_{1}(f^{2})}(ω) − \sqrt{μ_{1}(f^{2})}(ω̃))^{2}
≤ lim_{τ↓0} \frac{1}{2τ} ∫ μ_{2}(dω)P_{τ,2}(ω, dω̃)dμ_{1}(η) (f(ω, η) − f(ω̃, η))^{2}
= μ_{1} ⊗ μ_{2} (Γ_{1,2}(f, f))

where the last inequality is due to Cauchy-Schwarz’s inequality. We hence have shown that

μ_{1} ⊗ μ_{2}(f^{2} log \frac{f^{2}}{||f||_{2}^{2}} )

≤ 2c_{1}μ_{2} ⊗ μ_{1}Γ_{1,1}(f, f) + 2c_{2}μ_{2} ⊗ μ_{1}(Γ_{1,2}(f, f)) + 2(d_{1} + d_{2})μ_{2} ⊗ μ_{1}(f^{2})
≤ 2 max(c_{1}, c_{2})μ_{2} ⊗ μ_{1}Γ_{1}(f, f) + 2(d_{1} + d_{2})μ_{2} ⊗ μ_{1}(f^{2})

which was the announced statement.

⋄

Logarithmic Sobolev inequality is also stable under perturbation.

Property 4.6 Let U be a bounded measurable function on a probability space (Ω, Σ, μ). Assume that μ satisfies the logarithmic Sobolev inequality with coefficients (c, d) for an infinitesimal generator L. Then

dμ_{U} := \frac{1}{Z_{U}} e^{−U}dμ with Z_{U} = ∫ e^{−U}dμ

satisfies the logarithmic Sobolev inequality for the same generator L with coefficients bounded above by (ce2osc(U), de2osc(U)) with osc(U) = sup U − inf U.

Proof : The proof is based on the formula

μ(f^{2} log \frac{f^{2}}{μf^{2}} ) = inf_{t≥0} μ (f^{2} log(\frac{f^{2}}{t}) − f^{2} + t)

which is easily checked. Moreover,

ϕ(t, x) = f^{2}(x) log(\frac{f^{2}(x)}{t}) − f^{2}(x) + t = t (z(t, x) log z(t, x) − z(t, x) + 1) ,

with z(t, x) = (f^{2}(x)/t), is clearly non negative. Hence, we can use (2.0.4) to deduce

μ_{U}(f^{2} log \frac{f^{2}}{μ_{U}f^{2}} ) ≤ e^{osc(U)}μ(f^{2} log \frac{f^{2}}{μf^{2}} ) ≤ e^{osc(U)} (cμ(Γ_{1}(f, f) + dμ(f^{2})) .

(4.1.17)

By further use of (2.0.4), we finish the proof of the lemma.

⋄

We recall that for a finite volume Gibbs measure with finite range interaction we could improve the estimate of property 2.6 for the spectral gap inequality . In the next exercise, we propose to generalize this result to logarithmic Sobolev inequality in the discrete case.

Exercise 4.7 Let ν be the uniform measure on {−1, +1} and Λ = [−L, L] × [−l, l]^{d−1} for (l, L) ∈ (IN^{∗})^{2}, l ≤ L. Let U(x, y) be a function on {−1, +1}^{2} and set

H_{Λ}^{ω}(x) = ∑_{i,j∈Λ,|i−j|=1} U(x_{i}, x_{j}) + ∑_{i∈Λ,j∈Λ^{c},|i−j|=1} U(x_{i}, ω_{j}).

Let

dμ_{Λ}^{ω}(x) = \frac{1}{Z_{Λ}^{ω}} e^{−βH_{Λ}^{ω}(x)}dμ^{⊗|Λ|}(x).

Then, with Γ̄_{1}(f, f) = ∑_{i∈Λ}(∂_{i}f)^{2} if ∂_{i}f(σ) = f(σ_{1}, ..σ_{i−1}, −σ_{i}, σ_{i+1}, ..)−f(σ), prove that there exist two constants (c, C) ∈ (0, ∞) depending only on U so that for every ω ∈ Ω, μ_{Λ}^{ω} satisfies logarithmic Sobolev inequality for the carré du champ Γ̄_{1} with coefficient bounded by C|Λ|^{2}e^{cl^{d−1}}.

Hint : Choose a lexicographic order (i_{j}, j ∈ {1, .., |Λ|}) of Λ as in property 2.7. Denoting Λ_{k} = {i_{j}, j ≤ k} and, for a non negative bounded function f, f_{k}(ω) = μ_{Λ_{k}}^{ω}f, f_{0} = f, write

μ_{Λ}^{ω}(f log \frac{f}{μ_{Λ}^{ω}(f)}) = ∑_{k=0}^{|Λ|−1} μ_{Λ}^{ω}μ_{Λ_{k+1}} (f_{k} log \frac{f_{k}}{f_{k+1}} ) .

Using the previous proof and the product property 4.4, show that there exists a finite constant C such that

μ_{Λ_{k+1}}^{ω} (f_{k} log \frac{f_{k}}{f_{k+1}} ) ≤ e^{Cl^{d−1}}μ_{Λ_{k+1}}^{ω}(∂_{i_{k+1}}f_{k})^{2}

Remark that there exists a finite constant D such that

(∂_{i_{k+1}}f_{k})^{2}(ω) ≤ 2μ_{Λ_{k}}^{ω}|∂_{i_{k+1}}f|^{2} + Dμ_{Λ_{k}}^{ω}(f − E_{Λ_{k}}f)^{2}

and conclude by property 2.7.

The next exercise links the logarithmic Sobolev inequality with a property of concentration of measure

Exercise 4.8 (see e.g. [61], [1]) Let (Ω, Σ, μ) be a probability space with a probability measure μ satisfying the logarithmic Sobolev inequality with coefficients (c, 0) for a carré du champ Γ_{1} satisfying Leibnitz rule. Show that for any z ≥ 0, any measurable function f satisfying ||Γ_{1}(f, f)||_{∞} < ∞,

μ(e^{z(f−μ(f))}) ≤ e^{\frac{c||Γ_{1}(f,f)||∞}{4} z^{2}}.

Hint : Show that

∂_{z} \frac{1}{z} log ∫ e^{zf}dμ ≤ \frac{c}{4}||Γ_{1}(f, f)||_{∞}

by applying the logarithmic Sobolev inequality to g = e^{zf}.

For other connections between logarithmic Sobolev inequalities and concentration of measure phenomenon, we refer the reader to [8] and [61].

4.2 Logarithmic Sobolev and Spectral Gap inequalities

We have the following relationship between logarithmic Sobolev inequality and spectral gap inequality.

Theorem 4.9 If μ is a probability measure satisfying the logarithmic Sobolev inequality with coefficients (c, 0), then μ satisfies the spectral gap inequality with a coefficient

m ≥ \frac{1}{c}.

Conversely, if μ satisfies logarithmic Sobolev inequality with coefficients (c, d) as well as a spectral gap inequality with coefficient m, then μ satisfies logarithmic Sobolev inequality with coefficients (c′, 0) with

c′ ≤ c + \frac{d + 1}{m}.

This result can be found in [81], (see also [20] or [106]). Proof : The first result is obtained as follows. Let f ∈ D(L) be a bounded measurable function centered with respect to μ. For ϵ sufficiently small, 1 + ϵf is a positive bounded function. Hence, applying logarithmic Sobolev inequality, we obtain

μ(1 + ϵf)^{2} log(1 + ϵf)^{2} ≤ 2cϵ^{2}μ(Γ_{1}(f, f)) + μ(1 + ϵf)^{2} log μ(1 + ϵf)^{2}. (4.2.18)

By Taylor’s expansion and identifying the first order terms (in ϵ^{2}), we get

μ(1 + ϵf)^{2} log(1 + ϵf)^{2} = 3ϵ^{2}μf^{2} + O(ϵ^{3})

and

μ(1 + ϵf)^{2} log μ(1 + ϵf)^{2} ≈ ϵ^{2}μf^{2} + O(ϵ^{3})

so that

μf^{2} ≤ cμ(Γ_{1}(f, f)).

To prove the second point, it is enough to notice that for any measurable function f,

μ(f^{2} log \frac{f^{2}}{||f||_{2}^{2}} ) ≤ μ((f − μf)^{2} log \frac{|f − μf|^{2}}{||f − μf||_{2}^{2}} ) + 2||f − μf||_{2}^{2}.  (4.2.19)

The statement is then obtained by using both the logarithmic Sobolev inequality and the spectral gap inequality in the right hand side. To show (4.2.19), note first that for μf ≠ 0 it is equivalent to the following inequality

μ((1 + tϕ)^{2} log \frac{(1 + tϕ)^{2}}{1 + t^{2}} ) ≤ t^{2}μϕ^{2} log ϕ^{2} + 2t^{2}  (4.2.20)

with

t = \frac{||f − μf||_{2}}{μf} ϕ = \frac{f − μf}{||f − μf||_{2}} .

We shall use the following observations concerning ϕ ;

μϕ = 0, μ(ϕ^{2}) = 1.  (4.2.21)

Up to replacement ϕ by −ϕ, we can and do assume that t ≥ 0. We shall bound the second derivative of

F (t) = μ((1 + tϕ)^{2} log \frac{(1 + tϕ)^{2}}{1 + t^{2}} ).

Using (4.2.21), we find

F′(t) = μ(2ϕ(1 + tϕ) log \frac{(1 + tϕ)^{2}}{1 + t^{2}} )

and then

F′′(t) = μ(2ϕ^{2} log \frac{(1 + tϕ)^{2}}{1 + t^{2}} ) + 4 − \frac{4t}{1 + t^{2}} .

Using Jensen’s inequality applied to the probability measure μ(ϕ^{2}.), we remark that

μ (ϕ^{2} log \frac{(1 + tϕ)^{2}}{ϕ^{2}(1 + t^{2})}) ≤ log μ (ϕ^{2} \frac{(1 + tϕ)^{2}}{ϕ^{2}(1 + t^{2})}) = 0,

and since we assumed t ≥ 0, we get

F′′(t) ≤ 2μϕ^{2} log ϕ^{2} + 4.

Integrating twice with the initial conditions F′(0) = F (0) = 0 gives the result.

Note here that F may not be differentiable. Applying the same arguments with

F_{δ}(t) = μ([δ^{2} + (1 + tϕ)^{2}] log \frac{[δ^{2} + (1 + tϕ)^{2}]}{1 + t^{2} + δ^{2}})

for some non negative real number δ, one can show that

F′′_{δ}(t) ≤ 2μϕ^{2} log ϕ^{2} + 4(1 + δ^{2})

and conclude as above with further use of dominated convergence theorem. ⋄

Exercise 4.10 In the set up of property 4.6 and with additional assumption that the carré du champ Γ_{1}(f, f) is equal to |∇f|^{2}, show that if μ_{U} satisfies a spectral gap inequality with coefficient m, μ_{U} satisfies a logarithmic Sobolev inequality with coefficients (c’,0) with

c′ ≤ c(1 + \frac{1}{4}||∇U||_{∞}) + \frac{1}{m} ((c||∇U||_{∞}^{2} + 1)^{2} + d + ||U||_{∞}) .

Compare this result with the one obtained by property 4.6. Hint : Denoting ρ_{U} = \frac{e^{−U}}{Z_{U}}, and being given a function f ∈ D(L) centered with respect to μ, apply logarithmic Sobolev inequality under μ to the function g = ρ^{\frac{1}{2}} f to find

μ_{U} (f^{2} log \frac{ρ_{U}f^{2}}{μ_{U}(f^{2})}) ≤ (c + \frac{1}{4}|| log ρ_{U}||_{∞}) μ_{U} (Γ_{1}(f, f)) + (c||\frac{|∇ρU^{\frac{1}{2}}|^{2}}{ρ_{U}} ||_{∞}

+\frac{c}{2}||∇ log ρU^{\frac{1}{2}}||_{∞} + d) μ_{U} (f^{2}) .

4.3 Bakry-Emery Criterion

We shall assume in this section that the carré du champ Γ_{1} of the generator under study satisfies the Leibnitz rule

Γ_{1}(f, gh) = Γ_{1}(f, g)h + Γ_{1}(f, h)g.

This rule was already encountered previously but we shall now describe it more precisely. First, let us consider the following examples.

Example 4.11 Consider the Brownian motion on the real line IR or on the unit circle S^{1}, L = Δ, and show that

Γ_{1}(f, f) = |∇f|^{2}

satisfies Leibnitz rule. More generally, consider the operator

L = ∑_{1≤i,j≤d} a_{ij}∂_{i}∂_{j} − ∑_{1≤j≤d} β_{j}∂_{j}

with a positive definite matrix A = (a_{ij})_{ij} with real valued entries and some vector (β_{j})_{j} and show that

Γ_{1}(f, f) = ∑_{i,j} a_{ij}(∂_{i}f)(∂_{j}f)

satisfies Leibnitz rule.

On the other hand, Leibnitz rule may not be satisfied if the state space Ω is discrete ; take Ω = {−1, +1} and L = ∂ to be the discrete derivative ; ∂f(σ) = f(−σ) − f(σ). Then, show that

Γ_{1}(f, f) = |f(−σ) − f(σ)|^{2}

does not satisfy Leibnitz rule.

More generally, considering a measurable space (Ω, Σ), a family of probability measures (ν^{ω})_{ω∈Ω} on (Ω, Σ), and the generators Lf(ω) = ν^{ω}f −f(ω), show that Leibnitz rule fails.

Necessary and sufficient conditions for a semi-group to have a generator satisfying Leibnitz rule can be found in [58] and [80].

One of the consequence of Leibnitz rule is the following fact.

Lemma 4.12 Assume that Γ_{1} satisfies the Leibnitz rule. Then, for any entire function v on IR, we have

Γ_{1}(g, v ∘ f) = Γ_{1}(g, f)v′ ∘ f

for any functions (f, g) ∈ D(L).

Proof : Since for any monomial function x^{n}, n ∈ IN, one easily check

Γ_{1}(g, f^{n}) = nΓ_{1}(g, f)f^{n−1}

the result is straightforward for entire functions. ⋄ We notice also the following interesting fact.

Property 4.13 Assume that Γ_{1} satisfies the Leibnitz rule. Then, the last condition (vi) of Gross’s theorem [48] is equivalent to the others.

Proof : It is sufficient to show that, for any non negative function f, we have, thanks to the previous lemma,

μ[(Lf) log f] = −μ(Γ_{1}(f, log f)) = −4μΓ_{1}(f^{\frac{1}{2}} , f^{\frac{1}{2}} ).

Plugging this equality in the proof of Gross’s theorem [48] yields the result. ⋄

Hereafter, we shall consider so called the carré du champ itéré , (or simply Gamma two), given, for f in the domain of L, by

Γ_{2}(f, f) = \frac{1}{2} \frac{d}{dt} (P_{t}(Γ_{1}(f, f) − Γ_{1}(P_{t}f, P_{t}f)) |_{t=0}
= \frac{1}{2}\{LΓ_{1}(f, f) − 2Γ_{1}(f, Lf)\}.

We shall define the Bakry-Emery condition by

Definition 4.14 We say that Bakry-Emery’s condition ( denoted (BE) ) is satisfied if there exists a positive constant c > 0 such that

Γ_{2}(f, f) ≥ \frac{1}{c}Γ_{1}(f, f)  (4.3.22)

for any function f for which Γ_{1}(f, f) and Γ_{2}(f, f) are well defined.

We have the following characterization of Bakry-Emery’s criterion (see [106])

Theorem 4.15 Condition (BE) is satisfied iff for any t ≥ 0,

Γ_{1}(P_{t}f, P_{t}f) ≤ e^{− \frac{2}{c} t}P_{t}Γ_{1}(f, f)  (4.3.23)

for any function f so that Γ_{1}(f, f) and Γ_{2}(f, f) are well defined.

Proof : Let us assume that (4.3.23) is satisfied. Then, for any t > 0, we have

0 ≤ \frac{1}{t} (e^{− \frac{2}{c} t}P_{t}Γ_{1}(f, f) − Γ_{1}(P_{t}f, P_{t}f))
= \frac{e^{− \frac{2}{c} t} − 1}{t}P_{t}Γ_{1}(f, f) + \frac{1}{t} (P_{t}Γ_{1}(f, f) − Γ_{1}(P_{t}f, P_{t}f)) (4.3.24)

Taking the limit t ↓ 0, we deduce, according to the definition of Γ_{2} that

0 ≤ −\frac{2}{c}Γ_{1}(f, f) + 2Γ_{2}(f, f)

that is condition (BE).

Conversely, if condition (BE) is satisfied, for any t ≥ 0 and s ∈ [0, t], the function

F : s → e^{\frac{2}{c} s}P_{t−s}Γ_{1}(P_{s}f, P_{s}f)

is decreasing. In fact,

\frac{d}{ds}F (s) = e^{\frac{2}{c} s}P_{t−s} (\frac{2}{c}Γ_{1}(P_{s}f, P_{s}f) − LΓ_{1}(P_{s}f, P_{s}f) + 2Γ_{1}(LP_{s}f, P_{s}f))
= 2e^{\frac{2}{c} s}P_{t−s} (\frac{1}{c}Γ_{1}(P_{s}f, P_{s}f) − Γ_{2}(P_{s}f, P_{s}f)) ≤ 0.

In particular, F (t) ≤ F (0), that is

e^{\frac{2}{c} t}Γ_{1}(P_{t}f, P_{t}f) ≤ P_{t}Γ_{1}(f, f)

that is (4.3.23).

⋄

The main application of Bakry-Emery’s criterion is the following

Theorem 4.16 Let L be the generator of a Markov semi-group P_{t}, t ∈ IR^{+} with carré du champ Γ_{1} satisfying Leibnitz rule. Let μ ∈ J_{0}(P_{t}) so that P_{t} is weakly ergodic, i.e

lim_{t→∞} P_{t}f(ω) = μf  μ − a.s.  (4.3.25)

for any bounded continuous function f. Then, Bakry-Emery’s criterion implies that μ satisfies logarithmic Sobolev inequality.

Proof : Let f be a positive bounded continuous function so that μf = 1. We set f_{t} = P_{t}f and let

S_{f}(t) = μ(f_{t} log f_{t}).

Under the weak ergodic hypothesis, we have

lim_{t→∞} S_{f}(t) = 0.

Hence,

S_{f}(0) = − ∫_{0}^{∞} dt\frac{d}{dt}S_{f}(t) = ∫_{0}^{∞} dtμΓ_{1}(f_{t}, log f_{t}).  (4.3.26)

But, with Leibnitz rule, we have

μΓ_{1}(f_{t}, log f_{t}) = μΓ_{1}(ft^{\frac{1}{2}} , ft^{\frac{1}{2}} ))

so that the characterization of theorem 4.15 of (BE) criterion implies

S_{f}(0) ≤ ∫_{0}^{∞} e^{− \frac{2t}{c}} dtμΓ_{1}(f^{\frac{1}{2}} , f^{\frac{1}{2}} )) = 2cμΓ_{1}(f^{\frac{1}{2}} , f^{\frac{1}{2}} )

which completes the proof. ⋄ One can also show that

Proposition 4.17 If (BE) is satisfied,

P_{t}(f log f) ≤ 2c(1 − e^{− \frac{2}{c} t})P_{t}Γ_{1}(f^{\frac{1}{2}} , f^{\frac{1}{2}} ) + P_{t}f log P_{t}f

for any function f for which the right hand side is well defined.

The proof of this proposition is very similar to the previous one. It is given by Bakry and Emery, [6], Proposition 5.

Let us give a few examples where (BE) is fulfilled.

Exercise 4.18 • Let U be a twice continuously differentiable function so that Z = ∫ e^{−U(x)}dx is well define and finite. Set

μ(dx) = Z^{−1}e^{−U(x)}dx.

Let L = Δ−U′(x).∂_{x}. Show that, if U′′ is uniformly bounded below by a positive constant, Bakry-Emery criterion holds.

• Let μ_{G} be the Gaussian measure on IR^{n} with covariance G = {G_{ij}}_{1≤i,j≤n}. Given U ∈ C^{2}(IR^{n}) such that

0 < μ_{G}(e^{−U}) < ∞,

we define

μ_{U}(f) = \frac{μ_{G}(fe^{−U})}{μ_{G}(e^{−U})}.

Let L be the Markov generator given by

Lf = Δf − ∑_{i} β_{i}∂_{x_{i}}f

and, with M = G^{−1},

β_{i} = ∂_{x_{i}}U + ∑_{j} M_{ij}x_{j}.

Then, μ_{U} is reversible for L. Show that

Γ_{1}(f, f) = |∇f|^{2}

and

Γ_{2}(f, f) = ∑_{i,j} (|∂_{x_{i}}∂_{x_{j}}f|^{2} + (∂_{x_{j}}β_{i})∂_{x_{i}}f · ∂_{x_{j}}f) .

Prove that, if we denote N(x) the matrix with entries ∂_{x_{i}}∂_{x_{j}}U, and if for all x, the smallest eigenvalue of the symmetric matrix M + N(x) is bounded below by (1/c), then Γ_{1} satisfies (BE) with coefficient c. In particular, μ satisfies logarithmic Sobolev inequality with coefficient c for the carré du champ Γ_{1}.

The interest of Bakry-Emery criterion is also to obtain log-Sobolev inequalities in smooth compact Riemaniann manifolds. We send the reader to the original article [6], as well as to the more recent paper [13], for such applications.

Exercise 4.19 Ω = IR^{ZZ^{d}}. For a finite subset Λ of ZZ^{d}, a constant m ≠ 0, we denote μ_{G}^{∂Λ} the Gaussian measure on IR^{Λ} with covariance

G = (−Δ_{δ}^{∂Λ} + m^{2})^{−1}

where Δ_{δ}^{∂Λ} is the discrete Laplacian with Dirichlet boundary conditions. In other words, if we denote x_{Λ} = (x_{i})_{i∈Λ} and i ≈ j for |i − j| = 1, ω ∈ Ω,

μ_{G}^{∂Λ,ω}(dx_{Λ}) = \frac{1}{Z_{G}^{∂Λ}} exp\{−\frac{1}{2} ∑_{i∈Λ,i≈j}(x_{i} − x_{j})^{2} − \frac{m^{2}}{2} ∑_{i∈Λ} x_{i}^{2}\}dx_{Λ}

with x|_{Λ^{c}} = ω|_{Λ^{c}}. Let V be a bounded below C^{2}(IR^{Λ}) function and set

U_{Λ}(x_{Λ}) = ∑_{i∈Λ} V (x_{i}).

For instance, for λ > 0, V (x) = λx^{4} + ax^{2}. We can then define

μ_{U}^{∂Λ,ω}(dx_{Λ}) = \frac{1}{Z_{Λ}^{∂Λ,ω}} e^{−U_{Λ}}μ_{G}^{∂Λ,ω}(dx_{Λ}).

Following the previous example, see that if, for some c > 0,

−Δ_{δ}^{∂Λ} + m^{2} + D(x) ≥ \frac{1}{c}I

with D the diagonal matrix D_{ij}(x) = δ_{ij}V′′(x_{i}) and the inequality is understood in the sense of quadratic forms, then μ_{U}^{∂Λ,ω} satisfies the logarithmic Sobolev inequality with coefficient bounded by c. Moreover, if the following limit exists

μ = lim_{Λ↑ZZ^{d}} μ_{Λ}^{∂Λ,ω} ,

μ satisfies also logarithmic Sobolev inequality with coefficient bounded by c.

Let us remark that such a measure may not be unique and that then the logarithmic Sobolev inequality is satisfied by any such limit. The proof of the existence of a limit is in general not easy. We note that for any Λ ⊂ ZZ^{d} and Λ_{0} ⊂ Λ, the conditional expectation μ_{Λ}^{∂Λ,ω} knowing Σ_{Λ_{0}^{c}}, the σ-algebra generated by {x_{i}, i ∈ Λ_{0}^{c}}, is independent of Λ and is defined by δ_{ω}(μ_{Λ_{0}}^{∂Λ_{0},ω}(·)) (exercise).

One can use this idea to define the infinite volume measure as follows.

Let a family of conditional expectations (E_{Λ}^{ω}, ω ∈ Ω, Λ ⊂ ZZ^{d}) be given so that

(1) E_{Λ}^{ω}1 = 1 ∀ω ∈ Ω, ∀Λ ⊂ ZZ^{d}

(2) ω → E_{Λ}^{ω}(f) is Σ_{Λ^{c}} measurable for any bounded measurable function f.

(3)If Λ_{1} ⊂ Λ_{2}, then Σ_{Λ_{2}^{c}} ⊂ Σ_{Λ_{1}^{c}} and

E_{Λ_{2}}^{ω}E_{Λ_{1}}^{.}f = E_{Λ_{2}}^{ω}f.

The Gibbs measures in infinite volume μ associated with the specification (E_{Λ}^{ω}, ω ∈ Ω, Λ ⊂ ZZ^{d}) are described as the solutions of the equation (DLR)

μE_{Λ}^{.} = μ.

This equation may have several solutions.

To be more precise, let us consider the case where U = 0. Denote E_{μ_{G}}^{ω}(f|Σ_{Λ^{c}}) the conditional expectation of μ_{G} with respect to Σ_{Λ^{c}}. The family of these conditional expectation is called local specification. We can consider the Gibbs measure associated with this local specification. It is not hard to check that

E_{μ_{G}}^{ω}(f|Σ_{Λ^{c}}) = μ_{G}^{∂Λ} (f(. + ϕ_{ω}^{∂Λ}))  (4.3.27)

where

\{ \begin{matrix}G^{−1}ϕ_{ω}^{∂Λ}(i) = 0  ∀i ∈ Λ \\ ϕ_{ω}^{∂Λ}(j) = ω_{j}  ∀j ∈ Λ^{c} \end{matrix}

Consequently, if ϕ is a global solution of

G^{−1}ϕ = 0,

any probability measure

μ_{G,ϕ}(f) = μ_{G}(f(. + ϕ))

have the same conditional expectations. Hence, they define Gibbs measures in infinite volume with the same local specification. All these measures satisfy the logarithmic Sobolev inequality with the same coefficient.

Logarithmic Sobolev inequalities for spins systems on a lattice

This chapter will be concerned with logarithmic Sobolev inequalities for Gibbs measures. We begin by describing the statistical mechanics context in which we shall work. In particular, we define local specifications and the associated Gibbs measures. We introduce as well some Markov semi-groups, (via their generators), satisfying the detailed balance condition with respect to a Gibbs measure. Then, we present a general strategy to prove logarithmic Sobolev inequalities for a given Gibbs measure with Dirichlet forms of relevant generators. The application of this strategy requires to check four general conditions. In section 5.3 we show that they are satisfied for all one dimensional systems with bounded finite range potential. For ”geometric” reasons, this case is slightly easier to handle than the higher dimensional case. In section 5.4, assuming some mixing condition, we show that the requirements of the general stratedy listed in section 5.2 hold for systems with bounded finite range potential in dimension d ≥ 2. The proof is similar to that of section 5.3, except that one needs an extra argument taking into account the mixing condition (based on the so-called sweeping out relations defined in paragraph 5.4.1).

5.1 Notation and definitions, statistical mechanics

In this chapter, we shall consider random variables, (in applications representing spins or particles and therefore frequently called by these names), with values in a Polish space M. We shall assume that either M is a finite set, (frequently being simply given as a two point set M = {+1, −1}), or a finite dimensional smooth (compact) connected Riemaniann manifold. These two cases will be called discrete and continuous settings respectively. In most of the proofs, we will consider in more detail the discrete case which is usually more complicated to deal with, (mainly because the discrete derivative used there does not satisfy the Leibnitz rule). The spins will be ”located” on the lattice ZZ^{d}, for some positive integer number d ∈ IN^{∗}. We equipp ZZ^{d} with a distance d(x, y) = max_{1≤i≤d} |x_{i} − y_{i}| for (x, y) in ZZ^{d}. The notation Λ ⊂⊂ ZZ^{d} will be used to say that Λ is a finite subset of ZZ^{d}. Let Ω = M^{ZZ^{d}} be the state space. For Λ ⊂⊂ ZZ^{d}, we shall denote by

σ_{Λ} : Ω → M^{Λ}

the projection defined by

σ_{Λ} : Ω → M^{Λ}
ω = (ω_{j})_{j∈ZZ^{d}} → σ_{Λ}(ω) = (ω_{j})_{j∈Λ}

In particular, a projection σ_{i} corresponding to single point set {i} is called a spin at site i. For Λ ⊂ ZZ^{d}, we shall denote by Σ_{Λ} the smallest σ-algebra for which all the spins {σ_{i}, i ∈ Λ} are Σ_{Λ}-measurable. We shall say that a function f on Ω is localized in Λ if it is Σ_{Λ}-measurable. We denote by Λ_{f} (or alternatively Λ(f)) the smallest subset of ZZ^{d} for which f is localized in Λ_{f}. Later on, a function f for which the cardinality of Λ_{f} is finite will be called a local function.

Let ϕ = (ϕ_{X})_{X⊂ZZ^{d}} be a potential with finite range R, that is a family (ϕ_{X})_{X⊂ZZ^{d}} of continuous functions on Ω so that for all X ⊂ ZZ^{d}, ϕ_{X} is localized in X and ϕ_{X} ≡ 0 if the diameter of X is greater than R. We shall assume hereafter that

||ϕ|| = sup_{i∈ZZ} ∑_{X:i∈X} ||ϕ_{X}||_{∞} < ∞.  (5.1.1)

The energy U_{Λ} in a finite volume Λ ⊂⊂ ZZ^{d} is then well defined by

U_{Λ} = ∑_{X∩Λ≠∅} ϕ_{X}.  (5.1.2)

Let ν be the uniform measure on M and let

ν_{Λ}(dσ_{Λ}) = ⊗_{i∈Λ}ν(dσ_{i}).

We can define a local Gibbs measure in a finite volume Λ and with boundary conditions ω ∈ Ω by

E_{Λ}^{ω}f = \frac{δ_{σ_{Λ^{c}} =ω} ⊗ ν_{Λ} (e^{−U_{Λ}}f)}{δ_{σ_{Λ^{c}} =ω} ⊗ ν_{Λ} (e^{−U_{Λ}})}.  (5.1.3)

Frequently it will be convenient to use the following notation

E_{Λ}f(ω) = E_{Λ}^{ω}f.

We shall call local specification the family of local Gibbs measures {E_{Λ}^{ω}}_{Λ⊂⊂ZZ^{d},ω∈Ω}.

By μ, we shall denote a Gibbs measure in infinite volume associated to the local specification {E_{Λ}^{ω}}_{Λ⊂⊂ZZ^{d},ω∈Ω}, that is a solution of the (DLR) equation, (DLR being a shorthand for Dobrushin, Landford and Ruelle), which is given by

μE_{Λ}^{.}f = μf  (5.1.4)

for all local bounded measurable function f and all Λ ⊂⊂ ZZ^{d}.

For an introduction to the theory of Gibbs measures the reader may like to look at the classical reference [75]. Concerning the uniqueness versus non uniqueness problem (a phase transition phenomenon) the literature is very wide; the reader may consult [90] and [41] for more detailed discussion and further guide to the literature. We note that, except in the last chapter of these notes, we will only be concerned with Gibbs measures for which log-Sobolev inequalities hold for all local Gibbs measures with uniformly bounded coefficients. This entails the uniqueness of the infinite volume Gibbs measure.

The Markov generators under study will be defined as follows. In the continuous setting where we consider a smooth connected Riemaniann manifold M equipped with the Laplace-Baltrami operator Δ and a gradient ∇, for any i ∈ ZZ^{d}, we set Δ_{i} and ∇_{i} to be the corresponding operators acting on the i^{th} variable ω_{i}. We shall then consider the operator defined on the set of local twice continuously differentiable functions by

Lf = ∑_{i∈ZZ^{d}}(Δ_{i} − ∇_{i}U_{Λ}.∇_{i})f

with Λ any finite subset of ZZ^{d} so that d(Λ^{c}, Λ_{f}) > R.

In the discrete setting, for any finite subset X ⊂ ZZ^{d}, we introduce a generator by means of the local Gibbs measures as follows

L^{(X)} = ∑_{j∈ZZ^{d}} L_{X+j}f

with

L_{X+j}f(ω) = E_{X+j}^{ω}f − f(ω)

for any j ∈ ZZ^{d} and f any integrable local function.

It is not hard to see that μ is reversible for such operators. Moreover, L (resp. L^{(X)}) is non positive and with dense domain in L^{2}(μ) for any Gibbs measure μ satisfying (5.1.4) for the specification (E_{Λ}^{.}, Λ ⊂⊂ ZZ^{d}). Consequently, the semi-group P_{t} = e^{tL} (or P_{t} = e^{tL^{(X)}}) is symmetric and leaves μ invariant. We shall call in the sequel the standard Dirichlet form the quantity

E(f, f) = μ|∇f|^{2}

with

|∇f|^{2} = ∑_{j∈ZZ^{d}} |∇_{j}f|^{2}.

where in the discrete setting, we denote ∇f = (∂_{σ_{j}}f)_{j∈ZZ^{d}} with

∂_{σ_{j}}f = ∫ f(σ_{k}, k ≠ j, σ̄)dν(σ̄) − f(σ).

We shall denote, whenever it makes sense,

|||f||| = ( ∑_{i∈ZZ^{d}} ||∇_{i}f||_{∞}^{2})^{\frac{1}{2}}

and say that a function f is of class C^{1} iff |||f||| is well defined and finite. E(f, f) is clearly well defined for any function f of class C^{1}.

Exercise 5.1 Show that if Φ is a bounded potential with finite range, for any X ⊂⊂ ZZ^{d}, the operator L^{(X)} has a quadratic form μΓ_{1}^{(X)}(f, f) equivalent to the standard Dirichlet form, that is that there exists two constants c_{X} and C_{X}, 0 < c_{X} ≤ C_{X} < ∞, so that

c_{X}E(f, f) ≤ μΓ_{1}^{(X)}(f, f) ≤ C_{X}E(f, f).

5.2 Strategy to prove the logarithmic Sobolev inequality

To prove logarithmic Sobolev inequality for the Gibbs measure μ, the idea is to use the local Gibbs measures to define an auxiliary Markov chain on (Ω, Σ) with transition matrix E satisfying the following conditions

(Ci)

μEf = μf

for any bounded measurable function f.

(Cii) There exists a positive finite constant c̄ so that

μ (E(f log f) − (Ef) log(Ef)) ≤ 2c̄μ|∇f^{\frac{1}{2}} |^{2}.

(Ciii) There exists λ ∈ (0, 1) so that

μ|∇(Ef)^{\frac{1}{2}} |^{2} ≤ λμ|∇f^{\frac{1}{2}} |^{2}.

(Civ) Denoting for any bounded measurable function f, (f_{n})_{n∈IN} the sequence of measurable functions given by f_{0} = f and f_{n} = E(f_{n−1}),

lim_{n→∞} f_{n} = μ(f) μ − a.s.

Before proving that a construction of such transition matrix E is possible and giving a sufficient condition for this, we first show that conditions (C) imply that the Gibbs measure μ satisfies a logarithmic Sobolev inequality .

To this end for a fixed non negative function f we consider the sequence (f_{n}) given by

f_{0} = f,  f_{n} = Ef_{n−1}.

We notice that by using (Ci), we have

μ(f log \frac{f}{μf}) = μ (E(f log f) − (Ef) log(Ef)) + μ ((Ef) log \frac{(Ef)}{μf}) . (5.2.5)

and hence by induction

μ(f log \frac{f}{μf}) = ∑_{n=0}^{N−1} μ (E (f_{n} log f_{n}) − (Ef_{n}) log(Ef_{n})) + μ(f_{N} log \frac{f_{N}}{μf}) (5.2.6)

On the other hand, by (Cii),

μ (E (f_{n} log f_{n}) − (Ef_{n}) log(Ef_{n})) ≤ 2c̄μ|∇fn^{\frac{1}{2}} |^{2}.  (5.2.7)

Applying (Ciii), we obtain by induction that

μ|∇fn^{\frac{1}{2}} |^{2} ≤ λμ|∇fn−1^{\frac{1}{2}}|^{2} ≤ λ^{n}μ|∇f^{\frac{1}{2}} |^{2}.  (5.2.8)

Combining (5.2.6), (5.2.7) and (5.2.8), we get for any N ∈ IN,

μ(f log \frac{f}{μf}) ≤ \frac{2c̄}{1 − λ}μ|∇f^{\frac{1}{2}} |^{2} + μ(f_{N} log \frac{f_{N}}{μf}).  (5.2.9)

Finally, hypothesis (Civ) implies that the last term on the right hand side of the inequality (5.2.9) goes to zero as N goes to infinity. Indeed, the convexity of x → x log x together with Jensen’s inequality imply that μ(f_{N} log \frac{f_{N}}{μf}) ≥ 0. Since we also have that for any ϵ > 0, any f ≥ 0 which is not identically null, f_{N} log \frac{f_{N} +ϵ}{μf} is uniformly bounded, the monotone convergence theorem implies that

lim sup_{N→∞} μ(f_{N} log \frac{f_{N}}{μf}) ≤ lim sup_{N→∞} μ(f_{N} log \frac{f_{N} + ϵ}{μf}) = μ(f) log(\frac{μf + ϵ}{μf})

which allows us to conclude by letting ϵ going to zero.

Hence, we proved that μ satisfies a logarithmic Sobolev inequality with coefficient bounded by c = \frac{c̄}{1−λ}.

5.3 Logarithmic Sobolev inequality in dimension 1 ; an example

Log-Sobolev inequalities on the one dimensional lattice were first studied for discrete spins in [56]. In that paper a bound on logarithmic Sobolev inequality coefficients (logarithmically growing in the dimension of the configuration space bounds) were obtained and used to prove a form of ergodicity of the associated semi-group (although in general not an exponential decay in the uniform norm ). A first proof of logarithmic Sobolev inequality on infinite the one dimensional lattice was obtained in [104].

In this section we give a new proof of that result (with improved estimates on logarithmic Sobolev inequality coefficients).

Theorem 5.2 Assume d = 1 and consider the local Gibbs measures on Ω = {−1, 1}ZZ constructed with a potential Φ with finite range R as in (5.1.3). Then, the unique Gibbs measure μ on Ω solving the corresponding (5.1.4) satisfies the logarithmic Sobolev inequality .

Consequently, logarithmic Sobolev inequality is satisfied (with possibly different coefficients) for any quadratic form associated with the generator L^{(X)} for any finite subset X ⊂⊂ ZZ (see exercise 5.1). The semi-groups P_{t}^{(X)} = e^{tL^{(X)}} are thus hypercontractive (and therefore, as we shall see later, uniformly ergodic).

In dimension greater or equal to 2, such a result is obtained in general only under some additional mixing conditions that will be considered in the next section.

5.3.1 Construction of the auxiliary Markov chain

Let

Λ_{0} = [0, 2(L + R)]

with R the range of the interaction. L is an integer number the value of which will be properly chosen later. For k ∈ ZZ, set

Λ_{k} = Λ_{0} + 2k(L + 2R).

With such a choice, we have

dist(Λ_{k}, Λ_{k+1}) = 2R.  (5.3.10)

For l = 0 or 1, we shall denote

Γ_{l} = ∪_{k∈ZZ} {Λ_{k} + l(L + 2R)} .

In this manner, we have constructed two sets (each composed with disjoint sets at a distance greater or equal to 2R), the union of which covers the whole lattice.

ZZ = Γ_{0} ∪ Γ_{1}.  (5.3.11)

The goal of this construction is to compare the coefficient of the log-Sobolev inequality satisfied by μ with the maximum of those satisfied by (E_{Γ_{l}}^{ω}, l = 0, 1)

E_{l}^{ω} = E_{Γ_{l}}^{ω} = ⊗_{k}E_{Λ_{k}+l(L+2R)}^{ω}, l = 0, 1.

with properly chosen L (sufficiently large).

The coefficient in the inequality corresponding to E_{l}^{ω} can easily be estimated thanks to the product property and the estimates of the log-Sobolev coefficients for local Gibbs measures.

We define a Markov chain on (Ω, Σ) by the transition matrix

E^{ω} = E_{1}^{ω}E_{0}^{.}

By definition E preserves the unit and positivity. Moreover, E preserves the set of cylindrical functions.

5.3.2 Checking of conditions (C)

Condition (Ci) is clearly satisfied by E since, by property of local Gibbs measures,

μE_{1}^{.}E_{0}^{.}f = μE_{0}^{.}f = μf.

Conditions (Cii), (Ciii) and (Civ) result from conditions (a) and (b) of the following auxiliary lemma.

Lemma 5.3

(a) For any finite subset Λ of ZZ, there exist non negative constants B_{1}(Λ) and B_{2}(Λ) so that for any i ∈ ZZ, we have

μ|∇_{i}(E_{Λ}f)^{\frac{1}{2}} |^{2} ≤ B_{1}(Λ)μ|∇_{i}f^{\frac{1}{2}} |^{2} + B_{2}(Λ)μ|∇_{Λ}f^{\frac{1}{2}} |^{2}

for any non negative function f for which the right hand side is finite. Moreover, for any l ∈ IN,

B_{1}(l) = sup_{Λ:|Λ|≤l} B_{1}(Λ) < ∞, B_{2}(l) = sup_{Λ:|Λ|≤l} B_{2}(Λ) < ∞.

(b) There exists L_{0} ∈ IN and a constant λ̄ ∈ (0, 1), satisfying

λ̄ max{RB_{1}(l), R^{2}B_{2}(l)} < 1

for l ≡ 2(L + R), with L ≥ L_{0}, such that for any finite Λ ⊂⊂ ZZ of size l and any Λ̃ ⊂ Λ for which dist(Λ̃, Λ^{c}) ≥ L − R, for any i ∈ ZZ, we have

μ|∇_{i}(E_{Λ}f)^{\frac{1}{2}} |^{2} ≤ λ̄μ|∇_{Λ}f^{\frac{1}{2}} |^{2}

for any differentiable Σ_{Λ^{c}∪_{Λ}}-measurable function f ≥ 0

We shall first show that Proof of (Cii) assuming lemma 5.3 (a)

Remark that

μ[E(f log f) − (Ef) log(Ef)] = μ[E_{0}(f log \frac{f}{E_{0}f})] + μ[E_{0}(f) log \frac{E_{0}(f)}{E_{1}(E_{0}f)}]

(5.3.12)

Since, for any ω ∈ Ω, E_{0}^{ω} (resp. E_{1}^{ω}) is a product measure, the product property of logarithmic Sobolev inequality , (theorem 4.4), shows that E_{0}^{ω} (resp. E_{1}^{ω}) satisfies a logarithmic Sobolev inequality with coefficient bounded above by

c_{0} = sup_{ω∈Ω} sup_{Λ_{k}∈Γ_{0}∪Γ_{1}} c(E_{Λ_{k}}^{ω}).

In particular,

E_{0}(f log \frac{f}{E_{0}f}) ≤ 2c_{0}E_{0}|∇_{Γ_{0}}f^{\frac{1}{2}} |^{2}  (5.3.13)

and

E_{1} [E_{0}(f) log \frac{E_{0}(f)}{E_{1}(E_{0}f)}] ≤ 2c_{0}E_{1}|∇_{Γ_{1}}(E_{0}f)^{\frac{1}{2}} |^{2}.  (5.3.14)

Moreover, by definition of E_{0}, we have

|∇_{Γ_{1}}(E_{0}f)^{\frac{1}{2}} |^{2} = ∑_{i∈Γ_{1}} |∇_{i}(E_{0}f)^{\frac{1}{2}} |^{2} = ∑_{i∈Γ_{1}\Γ_{0}} |∇_{i}(E_{0}f)^{\frac{1}{2}} |^{2} (5.3.15)

Since the range of the interaction is finite, it is not hard to see that for any i ∈ Γ_{1}\Γ_{0}, there exists a subset Λ^{(i)} of Γ_{0} with length larger or equal to l = 2(L + R) so that |∇_{i}E_{0}f| = |∇_{i}E_{0}E_{Λ^{(i)}}f| ≤ E_{0}|∇_{i}E_{Λ^{(i)}}f|. Then, we have

μ|∇_{i}(E_{0}f)^{\frac{1}{2}} |^{2} ≤ μ|∇_{i}(E_{Λ^{(i)}}f)^{\frac{1}{2}} |^{2}.  (5.3.16)

Assuming (a) of the lemma and setting for short B_{1} = B_{1}(l) and B_{2} = B_{2}(l), we deduce from (5.3.14) the following bound

μ [E_{0}(f) log \frac{E_{0}(f)}{E_{1}(E_{0}f)})]

≤ 2c_{0} ∑_{i∈Γ_{1}\Γ_{0}} (B_{1}μ|∇_{i}(f)^{\frac{1}{2}} |^{2} + B_{2}μ|∇_{Λ^{(i)}}(f)^{\frac{1}{2}} |^{2})
= 2c_{0}B_{1} ∑_{i∈Γ_{1}\Γ_{0}} μ|∇_{i}(f)^{\frac{1}{2}} |^{2} + 2c_{0}RB_{2} ∑_{i∈Γ_{0}} μ|∇_{i}(f)^{\frac{1}{2}} |^{2}. (5.3.17)

Plugging (5.3.13) and (5.3.17) in (5.3.12), we conclude that

μ[E(f log f) − (Ef) log(Ef)] ≤ 2c_{0}B_{1} ∑_{i∈Γ_{1}\Γ_{0}} μ|∇_{i}(f)^{\frac{1}{2}} |^{2}

+2c_{0}(1 + RB_{2}) ∑_{i∈Γ_{0}} μ|∇_{i}(f)^{\frac{1}{2}} |^{2}

(5.3.18)

which gives (Cii) with c̄ = c_{0} max{B_{1}, 1 + RB_{2}}.

Proof of (Ciii) assuming lemma 5.3 (a) and (b)

Let us notice first that E_{0}f is Σ_{Γ_{1}\Γ_{0}} measurable and that Ef = E_{1}(E_{0}f) is Σ_{Γ_{0}\Γ_{1}} measurable. By our choice of Γ_{0} and Γ_{1},

dist(Γ_{1}\Γ_{0}, Γ_{0}\Γ_{1}) = L

and

μ|∇_{Γ_{0}}(Ef)^{\frac{1}{2}} |^{2} = ∑_{i∈Γ_{0}\Γ_{1}} μ|∇_{i}(E_{1}E_{0}f)^{\frac{1}{2}} |^{2}.  (5.3.19)

For any i ∈ Γ_{0}\Γ_{1}, following the arguments of (5.3.16) and denoting by Λ^{(i)} the corresponding subset of Γ_{1} with diameter l = 2(L + R), we deduce that

μ|∇_{i}(E_{1}E_{0}f)^{\frac{1}{2}} |^{2} ≤ μ|∇_{i}(EΛ̄^{(i)}E_{0}f)^{\frac{1}{2}} |^{2}.  (5.3.20)

Since Λ^{(i)} is at a distance smaller or equal to R of i ∈ Γ_{0}\Γ_{1}, the function E_{Γ_{0}}f is localized in a subset of Γ_{1}\Γ_{0} at distance greater or equal to L of i. Thus, if L ≥ L_{0} for suitable L_{0} ≥ 0, we can apply lemma (b) to obtain

μ|∇_{i}(E_{1}E_{0}f)^{\frac{1}{2}} |^{2} ≤ λ̄μ|∇_{Λ^{(i)}}(E_{0}f)^{\frac{1}{2}} |^{2}.  (5.3.21)

Introducing for any j ∈ Λ^{(i)} the sets Λ^{(j)} ⊂ Γ_{0}, (in a similar way as Λ^{(i)}), we have

μ|∇_{j}(E_{0}f)^{\frac{1}{2}} |^{2} ≤ μ|∇_{j}(E_{Λ^{(j)}}f)^{\frac{1}{2}} |^{2}.

We deduce from (5.3.21) by lemma (a) that

μ|∇_{i}(E_{1}E_{0}f)^{\frac{1}{2}} |^{2} ≤ λ̄B_{1} ∑_{j∈Λ^{(i)}} μ|∇_{j}(f)^{\frac{1}{2}} |^{2}+λ̄RB_{2} ∑_{j∈Λ^{(i)}} μ|∇_{Λ^{(j)}}(f)^{\frac{1}{2}} |^{2}. (5.3.22)

By (5.3.20) and (5.3.22), and since the diameters of Λ^{(j)} and Λ^{(i)} are equal to l = 2(L + R), we conclude that

μ|∇_{0}(Ef)^{\frac{1}{2}} |^{2} ≤ λ̄RB_{1} ∑_{i∈Γ_{1}\Γ_{0}} μ|∇_{i}(f)^{\frac{1}{2}} |^{2} + λ̄R^{2}B_{2} ∑_{i∈Γ_{0}} μ|∇_{i}(f)^{\frac{1}{2}} |^{2}. (5.3.23)

Hence, (Ciii) holds with

λ ≤ λ̄R max{B_{1}, RB_{2}}.

Proof of (Civ) assuming lemma 5.3 (a) and (b)

According to theorem 4.9, E_{0} and E_{1} satisfy a spectral gap inequality with coefficient bounded below by (1/c_{0}) so that

μ(f − Ef)^{2} ≤ 2μE_{Γ_{1}}(E_{Γ_{0}}f − E_{Γ_{1}}E_{Γ_{0}}f)^{2} + 2μE_{Γ_{0}}(f − E_{Γ_{0}}f)^{2}
≤ 2c_{0} (μ(|∇_{Γ_{1}}E_{Γ_{0}}f|^{2}) + |∇_{Γ_{0}}f|^{2}) ≤ Kμ|∇f|^{2} (5.3.24)

with the constant K obtained from lemma 5.3 (a). Now, if f_{0} = f and f_{n+1} = Ef_{n}, we deduce from (Ciii) that

μ|∇fn^{\frac{1}{2}} |^{2} ≤ λμ|∇fn−1^{\frac{1}{2}}|^{2} ≤ λ^{n}μ|∇f^{\frac{1}{2}} |^{2}

which converges towards zero as n goes to infinity. Hence, (5.3.24) implies that the sequences (f_{n} −μf_{n})_{n∈IN} and (|∇fn^{\frac{1}{2}} |)_{n∈IN} converges μ-almost surely by the Borel-Cantelli lemma. The limit of f_{n} −μ(f_{n}) = f_{n} −μ(f) is therefore constant, and hence identically zero.

Proof of Lemma 5.3(a)

Let us recall first that, if ν is the uniform Bernoulli law on {−1, 1},

∇_{j}F = ν_{j}F − F = \frac{1}{2}(T_{j}F − F)

with T_{j}F := F (ω^{(j)}) where

ω^{(j)} := \{ \begin{matrix}−ω_{k}  for k = j, \\ +ω_{k}  otherwise. \end{matrix}

We have the following discrete analogue of Leibnitz rule

∇_{j}(F G) = F ∇_{j}G + (∇_{j}F)T_{j}G.  (5.3.25)

In particular,

∇_{j}(F^{2}) = 2A_{j}(F)∇_{j}F  (5.3.26)

with

A_{j}(F) := \frac{1}{2}(F + T_{j}F).

Consequently, to estimate ∇_{j}(E_{Λ}f)^{\frac{1}{2}} , we have to find a bound of the form

|∇_{j}(E_{Λ}f)| ≤ 2A_{j} ((E_{Λ}f)^{\frac{1}{2}} ) × desired terms.

Let us recall that

E_{Λ}f = ν_{Λ}(ρ_{Λ}f) with ρ_{Λ} = \frac{e^{−U_{Λ}}}{ν_{Λ}(e^{−U_{Λ}})}.

Hence, we have that for any j ∈ Λ^{c},

∇_{j}E_{Λ}f = ν_{Λ} [∇_{j}ρ_{Λ}f]
= ν_{Λ} [(∇_{j}f)T_{j}ρ_{Λ} + f∇_{j}ρ_{Λ}]
= (T_{j}E_{Λ})[∇_{j}f] + E_{Λ}[f(ρ_{Λ}^{−1}∇_{j}ρ_{Λ})]  (5.3.27)

In the first term of (5.3.27), one of the boundary conditions is inverted. To estimate this term, we use the discrete Leibnitz rule ∇_{j}(f) = 2A_{j}(f^{\frac{1}{2}} )∇_{j}f^{\frac{1}{2}} to obtain by Cauchy-Schwarz ’s inequality

|(T_{j}E_{Λ})(∇_{j}f)| ≤ 2 ((T_{j}E_{Λ})(A_{j}(f^{\frac{1}{2}} ))^{2})^{\frac{1}{2}} ((T_{j}E_{Λ})(|∇_{j}f^{\frac{1}{2}} |^{2}))^{\frac{1}{2}} . (5.3.28)

We note that

ρ_{Λ}^{−1}T_{j}ρ_{Λ} = \frac{e^{−2∇_{j}U_{Λ}}}{E_{Λ}[e^{−2∇_{j}U_{Λ}}]} ≤ e^{4||∇_{j}U_{Λ}||_{∞}},  (5.3.29)

for any non negative measurable function F

(T_{j}E_{Λ})(F) ≤ e^{4||∇_{j}U_{Λ}||_{∞}}E_{Λ}F.

Thus, we can bound the first term on the right hand side of (5.3.28) by

2 ((T_{j}E_{Λ})(A_{j}(f^{\frac{1}{2}} )^{2}))^{\frac{1}{2}} ≤ ((T_{j}E_{Λ})(f))^{\frac{1}{2}} + ((T_{j}E_{Λ})(T_{j}f))^{\frac{1}{2}}
≤ 2e^{2||∇_{j}U_{Λ}||_{∞}}A_{j}[(E_{Λ}f)^{\frac{1}{2}} ]  (5.3.30)

This last inequality together with (5.3.28) yields

|(T_{j}E_{Λ})(∇_{j}f)| ≤ 2e^{4||∇_{j}U_{Λ}||_{∞}}A_{j}[(E_{Λ}f)^{\frac{1}{2}} ] (E_{Λ}|∇_{j}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}} .  (5.3.31)

To estimate the second term in the right hand side of (5.3.27), we first notice that E_{Λ}(ρ_{Λ}^{−1}∇_{j}ρ_{Λ}) = 0 and henceforth

E_{Λ} (fρ_{Λ}^{−1}∇_{j}ρ_{Λ}) = E_{Λ} [(f − E_{Λ}f)ρ_{Λ}^{−1}∇_{j}ρ_{Λ}]  (5.3.32)
= \frac{1}{2} ∫ ∫ (f(σ) − f(σ̃))(ρ_{Λ}^{−1}∇_{j}ρ_{Λ}(σ) − ρ_{Λ}^{−1}∇_{j}ρ_{Λ}(σ̃))dE_{Λ}(σ)dE_{Λ}(σ̃).

We immediately obtain the following bound

|E_{Λ}(fρ_{Λ}^{−1}∇_{j}ρ_{Λ})| ≤  (5.3.33)

≤ sup_{σ,_{σ}} |ρ_{Λ}^{−1}∇_{j}ρ_{Λ}(σ) − ρ_{Λ}^{−1}∇_{j}ρ_{Λ}(σ̃)|\frac{1}{2} ∫ ∫ |f(σ) − f(σ̃)|dE_{Λ}(σ)dE_{Λ}(σ̃).

Moreover,

sup_{σ,_{σ}} |ρ_{Λ}^{−1}∇_{j}ρ_{Λ}(σ) − ρ_{Λ}^{−1}∇_{j}ρ_{Λ}(σ̃)| = \frac{1}{2} sup_{σ,_{σ}} |\frac{e^{−2∇_{j}U_{Λ}(σ)}}{E_{Λ}[e^{−2∇_{j}U_{Λ}(σ)}]} − \frac{e^{−2∇_{j}U_{Λ}(_{σ})}}{E_{Λ}[e^{−2∇_{j}U_{Λ}(_{σ})}]}|
≤ e^{4||∇_{j}U_{Λ}||_{∞}}.  (5.3.34)

Furthermore Cauchy-Schwarz’s inequality shows that for any f ≥ 0

(∫ ∫ |f(σ) − f(σ̃)|dE_{Λ}(σ)dE_{Λ}(σ̃))^{2}
≤ E_{Λ}^{⊗2}[(\sqrt{f(σ)} + \sqrt{f(σ̃)})^{2}]E_{Λ}^{⊗2}[(\sqrt{f(σ)} − \sqrt{f(σ̃)})^{2}]
≤ 4E_{Λ}[f]E_{Λ}[(\sqrt{f} − E_{Λ}\sqrt{f})^{2}]
≤ 16(A_{j}\sqrt{E_{Λ}[f]})^{2}E_{Λ}[(\sqrt{f} − E_{Λ}\sqrt{f})^{2}]  (5.3.35)

where the last inequality is trivial. Finally, we saw in property 2.7 that there exists a finite constant C so that for any Λ ⊂⊂ ZZ,

E_{Λ}[(\sqrt{f(σ)} − E_{Λ}\sqrt{f})^{2}] ≤ C|Λ|E_{Λ}[|∇_{Λ}f^{\frac{1}{2}} |^{2}].  (5.3.36)

From (5.3.33)-(5.3.36), we deduce

|E_{Λ}(fρ_{Λ}^{−1}∇_{j}ρ_{Λ})| ≤
≤ 2\sqrt{C|Λ|}e^{4||∇_{j}U_{Λ}||}(A_{j}E_{Λ}[f]^{\frac{1}{2}} )E_{Λ}[|∇_{Λ}f^{\frac{1}{2}} |^{2}]^{\frac{1}{2}}  (5.3.37)

Plugging this estimate into (5.3.27) and using (5.3.31), we obtain

|∇_{j}(E_{Λ}f)^{\frac{1}{2}} | = \frac{∇_{j}E_{Λ}f}{2A_{j}(E_{Λ}f)^{\frac{1}{2}}}

≤ e^{4||∇_{j}U_{Λ}||} (E_{Λ}|∇_{j}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}} + \sqrt{C|Λ|}e^{4||∇_{j}U_{Λ}||} (E_{Λ}|∇_{Λ}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}}(5.3.38)

This ends the proof of lemma 5.3 (a) with the constants

B_{1}(Λ) = 2 sup_{j} e^{8||∇_{j}ϕ||} B_{2}(Λ) = 2C|Λ|sup_{j} e^{8||∇_{j}ϕ||}.

⋄

Proof of Lemma 5.3 (b)

The proof of this second property relies on several facts already developed in the course of the preceding proof. In fact, going back to equation (5.3.27) of the previous proof but with f being Σ_{Λ}-measurable with a set Λ̃ ⊂ Λ such that

dist(Λ̃, Λ^{c}) ≥ L > R,

we find out that the first term on the right hand side of (5.3.27) is equal to zero. Hence, we need only to improve the estimate of the second term.

To this end, let us remark that, if R is the range of the interaction, for j at positive distance smaller than R from Λ, ρ_{Λ}^{−1}∇_{j}ρ_{Λ} is Σ_{Λ_{1}}-measurable for

Λ_{1} = {i; dist(i, ∂Λ) ≤ R}.

Thus, we can replace the estimates (5.3.33)-(5.3.35) by the following bound

|E_{Λ}[fρ_{Λ}^{−1}∇_{j}ρ_{Λ}] | = |E_{Λ}[(f − E_{Λ}f)E_{Λ\_{Λ}}ρ_{Λ}^{−1}∇_{j}ρ_{Λ}]|
≤ 4 (A_{j}(E_{Λ}f)^{\frac{1}{2}} ) Var_{Λ} (E_{Λ\_{Λ}}ρ_{Λ}^{−1}∇_{j}ρ_{Λ}) (E_{Λ}(\sqrt{f} − E_{Λ}\sqrt{f})^{2})^{\frac{1}{2}}

with the notation Var_{X}G = sup_{ω_{X^{c}} =ω_{X^{c}}} |G(ω) − G(ω̃)|.

Moreover, if f is localized in Λ̃, denoting by E_{Λ,_{Λ}} the restriction of E_{Λ} to Σ_{Λ}, we have

E_{Λ}(f^{\frac{1}{2}} − E_{Λ}f^{\frac{1}{2}} )^{2} = E_{Λ,_{Λ}}(f^{\frac{1}{2}} − E_{Λ,_{Λ}}f^{\frac{1}{2}} )^{2} ≤ m(Λ̃)^{−1}E_{Λ}|∇_{Λ}f^{\frac{1}{2}} |^{2} (5.3.39)

with m(Λ̃)^{−1} ≤ C|Λ̃|, according to property 2.7. Hence, we obtain that

|∇_{j}(E_{Λ}f)^{\frac{1}{2}} |^{2} ≤ λ̄E_{Λ}|∇_{Λ}f^{\frac{1}{2}} |^{2}  (5.3.40)

where

λ̄ = \frac{1}{8} (Var_{Λ}(E_{Λ\_{Λ}}(ρ_{Λ}^{−1}∇_{j}ρ_{Λ})))^{2} · m(Λ̃)^{−1}.

In dimension d = 1, we have the following estimate

Var_{Λ}(E_{Λ\_{Λ}}(ρ_{Λ}^{−1}∇_{j}ρ_{Λ})) ≤ e^{−M(d(_{Λ},∂Λ)−R)}C(Φ),  (5.3.41)

for a finite constant C(Φ) which only depends on the potential Φ. This result is standard (see [41] and [84]) and is given as an exercise below.

As a consequence, since B_{1}(l) and B_{2}(l) grow at most polynomially with l = 2(L + R) according to the previous estimates, we can choose L large enough so that

\frac{1}{8} max\{RB_{1}(l), R^{2}B_{2}(l)\} sup \{ (Var_{Λ}(E_{Λ\_{Λ}}(ρ_{Λ}^{−1}∇_{j}ρ_{Λ})))^{2} · m(Λ̃)^{−1}\} < 1

where the supremum runs over all the sets (Λ, Λ̃) such that Λ̃ ⊂ Λ, |Λ| ≤ 2(L+R) and d(Λ̃, Λ^{c}) ≥ L − R. The proof of the second point of lemma 5.3 is thus complete. ⋄

Exercise 5.4 Proof of estimate (5.3.41). Hints : Assume that f is localized in Λ_{0} ≡ [a_{0}, b_{0}] and let Λ ≡ [a, b] be so that Λ_{0} ⊂ Λ. Note that, by anti-symmetry properties, if Λ ∩ Λ(f) = ∅,

E_{Λ}^{ω}f − E_{ω} ; Λf = ∫ E_{Λ}^{ω}(dx) ⊗ E_{ω} ; Λ(dy)(f(x) − f(y))

= ∫ E_{Λ}^{ω}(dx) ⊗ E_{ω} ; Λ(dy) (G_{Λ}^{ω,_{ω}}(x, y)(f(x) − f(y)))

with

G_{Λ}^{ω,_{ω}}(x, y) ≡ \frac{e^{−W_{Λ}(x_{Λ}•ω_{Λ^{c}})−W_{Λ}(y_{Λ}•ω_{Λ^{c}})} − e^{−W_{Λ}(y_{Λ}•ω_{Λ^{c}})−W_{Λ}(x_{Λ}•ω_{Λ^{c}})}}{e^{−W_{Λ}(x_{Λ}•ω_{Λ^{c}})−W_{Λ}(y_{Λ}•ω_{Λ^{c}})} + e^{−W_{Λ}(y_{Λ}•ω_{Λ^{c}})−W_{Λ}(x_{Λ}•ω_{Λ^{c}})}}

where W_{Λ} = ∑_{X∩Λ≠∅,X∩Λ^{c}≠∅} Φ_{X},

Consider an increasing sequence of subsets Λ_{l} ≡ [a, b_{l}], l = 1, ⋯, k so that b_{l+1} = b_{l} + R + 1 < b_{0}. Denote f_{l} ≡ E_{Λ_{l}}f, G_{l} = G_{Λ_{l}} and show by induction that

E_{Λ_{k}}^{ω}f − E_{ω} ; Λ_{k}f = E_{Λ_{k}}^{ω} ⊗ E_{ω} ; Λ_{k} ((∏_{l=1}^{k} G_{l}) · (f − f̃))

to deduce

||E_{Λ_{k}}^{ω}(f) − E_{ω} ; Λ_{k}(f)||_{∞} ≤ ∏_{l=1}^{k} ||G_{l}||_{∞}Var(f).

Show that when the state space is finite and the potential Φ uniformly bounded, sup_{l} ||G_{l}||_{∞} ≤ e^{−MR} for a constant M > 0.

5.4 Logarithmic Sobolev inequalities in dimension d ≥ 2

In dimension d ≥ 2, log-Sobolev inequalities for local Gibbs measures with constant bounded independently of the volume can be obtained when some mixing conditions are satisfied. This last property gives a spatial decorrelation which roughly allows to approximate the system by a system of particles which are interacting only in cubes of finite size for which one can prove logarithmic Sobolev inequality .

This rough idea was developed in different ways in the literature ; the reader may look at [64], [69], [92] and [104]. Here we present a formalism quite close to the one developed in [92] and which is based on the strategy described in the beginning of this chapter. It will clearly rely on a key intermediate property characterized by what we shall call sweeping out relations. (One should realize that our strategy does not rely on an a priori spectral gap property of the Gibbs measure as described in [94].)

In order to illustrate the general considerations above, we will begin by describing what are the so-called sweeping out relations and show how they lead to log-Sobolev inequalities. Later we shall show how these relations can be deduced from strong mixing hypotheses.

We shall again restrict ourselves to the setting introduced at the beginning of this chapter where we are given a local specification (E_{Λ}^{.}, Λ ⊂⊂ ZZ^{d}) described by a finite range potential Φ. Later we also consider the case where the range of the interaction is infinite.

5.4 Log-Sobolev inequalities in dimension d ≥ 2 57

5.4.1 Sweeping out relations

We shall say in the following that sweeping out relations are satisfied for a finite subset X_{0} of ZZ^{d} if For any set Λ = {j + X_{0}} for some j ∈ ZZ^{d}, there exist constants α_{ij}^{(Λ)} ∈ (0, ∞), such that for any i ∈ ZZ^{d}\Λ so that dist(i, Λ) ≤ R,

(q_{i}|∇_{i}(E_{Λ}f)^{\frac{1}{2}} |^{2})^{\frac{1}{2}} ≤ α_{ii}^{(Λ)} (E_{Λ}q_{i}|∇_{i}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}} + ∑_{j∈Λ∪\{i\}} α_{ij}^{(Λ)} (E_{i}E_{Λ}q_{j}|∇_{j}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}}

(5.4.42)

with

α_{ij}^{(Λ)} ≤ De^{−ε|i−j|}  (5.4.43)

where D, ε ∈ (0, ∞) are finite constants which are independent of i, j and Λ. Here, in the discrete setting q_{i} = ν_{i}(x_{i}) where ν_{i} is an isomorphic copy of the uniform measure on the finite set M (acting on the particle x_{i} at site i), whereas in the continuous setting q_{i} equal to the identity.

Let us note that in the following, we shall always restrict ourselves to cubes X_{0} and shall subdivide large subsets Λ ⊂ ZZ^{d} (or even ZZ^{d} itself) in small cells of the type {j + X_{0}, j ∈ ZZ^{d}}. In this way, we shall obtain bounds on log-Sobolev coefficients for E_{Λ}^{.} (or μ) in terms of the log-Sobolev coefficients of the local Gibbs measures {E_{X_{0}+j}^{.}, j ∈ ZZ^{d}} when the sweeping out relations are satisfied for X_{0}. The strategy is then to optimize on the choice of X_{0}. The fact that we choose homogeneous partitions is of course irrelevant, except that it helps to write the formulae more easily.

5.4.2 Proof of logarithmic Sobolev inequalities assuming sweeping out relations

We assume in this part that the sweeping out relations are satisfied. M shall be either a connected compact and smooth Riemannian manifold of finite dimension or a finite set.

To obtain log-Sobolev inequalities, we shall follow the strategy of section 5.2 and first define a suitable auxiliary Markov chain Π, analogous to the chain E studied in the last section.

To this end, for an integer number L ∈ IN, for k ∈ ZZ^{d}, we denote by X_{k} ≡ k + ([0, 2(L + R)])^{d}, the translation by the vector k of the cube X_{0} = [0, 2(L + R)]^{d} containing the origin. Later on it will be convenient to represent s ∈ IN as follows s = ∑_{l=1,...,d} v_{s}^{(l)} 2^{l−1} with a family {v_{s}}_{s=0,...,2^{d}−1} ⊂ {0, 1}^{d}.

We denote T_{s} = {k ∈ ZZ^{d}; k ∈ (L + 2R)v_{s} + (2(L + 2R)Z)^{d}} and T = ∪_{s=0}^{2^{d}−1}T_{s}. Set

Γ_{s} ≡ ⋃ {X_{k} : k ∈ T_{s}} .

Γ_{s} is the union of disjoint cubes of shape X_{0} at distance greater to 2R of each other. Let us also remark that ∪_{1≤s≤2^{d}−1}Γ_{s} covers ZZ^{d}. It is important to note that, by construction, for any i ∈ ZZ^{d}, there exists an s ∈ {0, …, 2^{d} − 1} and a cube in X_{k} ⊂ Γ_{s} such that i ∈ X_{k} and d(i, X_{k}^{c}) ≥ (L/2).

Setting IE_{s} ≡ ⊗_{X_{k}⊂Γ_{s}}E_{X_{k}}, we shall consider in the following

Π(f)(ω) ≡ IE_{2^{d}−1}^{ω}(⋯ IE_{1}^{·}(IE_{0}^{·}f)).  (5.4.44)

We shall assume hereafter that the local Gibbs measures E_{X_{k}}^{ω} satisfy a log-Sobolev inequality with coefficient c̃ ∈ (0, ∞) independent of k ∈ ZZ^{d},

E_{X_{k}}^{ω}f log f − E_{X_{k}}^{ω}f log E_{X_{k}}^{ω}f ≤ 2c̃ E_{X_{k}}^{ω}|∇f^{\frac{1}{2}} |^{2}  (5.4.45)

for any non negative function f of class C^{1}. This assumption is naturally satisfied when the local specification is defined by a finite range potential Φ according to property 4.6; the constant c̃ is then bounded by c_{0}e^{4||Φ||(2(L+R))^{d}} if c_{0} is the log-Sobolev constant for the uniform measure ν (see also exercise after 4.6).

Our goal is to prove that when sweeping out relations hold, Π satisfies the conditions (Ci)-(Civ) of section 5.2. By definition of the local Gibbs measures E_{X_{k}}, it is clear that (Ci) is true. Hence, we need to consider conditions (Cii) and (Ciii); Proof of properties (Cii) and (Ciii) assuming the sweeping out relations In this part, we shall prove that

Lemma 5.5 If X_{0} satisfies the sweeping out relations (5.4.42), there exist finite constants λ ∈ (0, ∞) and c̄ ∈ (0, ∞) such that

|∇ (Πf)^{\frac{1}{2}} |^{2} ≤ λ Π |∇f^{\frac{1}{2}} |^{2}  (5.4.46)

and

Πf log f − (Πf) log(Πf) ≤ 2c̄ Π |∇f^{\frac{1}{2}} |^{2}.  (5.4.47)

Moreover, if there exists L_{0} ≥ 0 such that for any L ≥ L_{0}, any v ∈ T ,

α_{ij}^{(X_{v})} ≤ De^{−ε|i−j|}  (5.4.48)

with constants D, ε ∈ (0, ∞) independent of L, v ∈ T , i ∈ Y^{c} so that d(i, Y) ≤ R and of j ∈ X ∪ {i}, then we can choose L sufficiently large so that (5.4.46) is satisfied with λ ∈ (0, 1). Consequently, if for n ∈ IN, we define inductively Π^{n,ω}f ≡ Π^{n}f(ω) = Π^{n−1}Πf(ω), Π^{0}f = f, we have for any ω ∈ Ω,

Π^{n,ω}f log f − (Π^{n,ω}f) log(Π^{n,ω}f) ≤ 2\frac{c̄}{1 − λ}Π^{n,ω}|∇f^{\frac{1}{2}} |^{2}  (5.4.49)

\underline{Remark} 5.6: The constants (λ, c̄) can be chosen as follows. Setting

α ≡ sup_{i} (α_{ii}^{(X_{k(i)})} + ∑_{j∈X_{k(i)}∪{i}} α_{ij}^{(X_{k(i)}} ) , η^{(s)} = sup_{j} ∑_{i∈Z^{d}} η_{ij}^{(s)}

were for s ∈ {0, …, 2^{d} − 1}

η_{ij}^{(s)} ≡ ∑_{j_{1}∈X_{k(i)}∪i\Γ_{s−2}} ⋯ ∑_{j_{s−2}∈X_{k(j_{s−3})}∪j_{s−3}\Γ_{1}} α_{ij_{1}}^{(X_{k(i)})}α_{j_{1}j_{2}}^{(X_{k(j_{1})})} ⋯ α_{j_{s−2} j}^{(X_{k(j_{s−2})})}

with X_{k(j_{l})} being the cube of Γ_{s−l} at distance less than or equal to R from j_{l} for l = 0, …, s − 1. Then we can choose

c̄ ≡ 2^{d}c̃ sup_{s∈{0,...,2^{d}−1}} λ^{(s)}

with

λ^{(s)} ≡ b^{2}e^{2^{d+1}||Φ||}α^{s}η^{(s)},

with appropriately chosen constant b dependent only on M, and choose λ ≡ λ^{(2^{d}−1)} In a few cases encountered for instance in random media, (considered in Chapter 9), the decay (5.4.48) of the coefficients α_{ij}^{(Λ)} appearing in the sweeping out relations for the local Gibbs measure E_{Λ} is satisfied only for points (i, j) at a distance greater or equal to some typical length l(Λ) depending on Λ. For the study of such situations in Chapter 9, it will be useful to make the following exercise.

Exercise 5.7 Let us assume that for any cube Λ_{L} = [−L, L]^{d}, the decay (5.4.48) of the coefficients α_{ij}^{(Λ)} appearing in the sweeping out relations for the local Gibbs measure is satisfied only for cubes Y_{0} ⊂ Λ_{L} of side greater or equal to a function d(L) of L. Show that E_{Λ_{L}} satisfies a logarithmic Sobolev inequality with coefficient bounded by Cd(L)^{2d}e^{cd(L)^{d−1}} for two finite constants (c, C) ∈ (0, ∞). Show in particular that if d(L) = A log L for some finite constant A, the log-Sobolev constant increases at most polynomially with the length of the side of the cube Λ in dimension 2. Hints : Use cubes of sides of length of order d(L) to construct Π, the bound on log-Sobolev constants obtained in exercise 4.7 for the local Gibbs measures of these cubes and the above controls.

Proof of lemma 5.5.

Let f be a differentiable function of L^{1}(μ). Denoting f_{−1} = f and f_{k} = E_{Γ_{k}}E_{Γ_{k−1}} ⋯ E_{Γ_{0}}f for k = 0, …, 2^{d} − 1, we first notice that

Π^{ω}f log f − (Π^{ω}f) log(Π^{ω}f) = ∑_{s=0,...,2^{d}−1} IE_{2^{d}−1}^{ω} ⋯ [IE_{s}f_{s−1} log \frac{f_{s−1}}{IE_{s}f_{s−1}} ] .

(5.4.50)

According to the hypothesis (5.4.45) and the product property of theorem 4.4, E_{Γ_{k}} satisfies, for k = {0, …, 2^{d} − 1}, a logarithmic Sobolev inequality with coefficient bounded above by c̃. Consequently, we deduce from (5.4.50) that

Π^{ω}f log f − Π^{ω}f log Π^{ω}f ≤ 2c̃ ∑_{s=0,...,2^{d}−1} (IE_{2^{d}−1} ⋯ (IE_{s}|∇_{Γ_{s}}fs−1^{\frac{1}{2}}|^{2}))

(5.4.51)

The sweeping out relations are now going to be useful to control the gradient terms in the right hand side of (5.4.51). To this end, let us observe that for any i ∈ Γ_{s}, we have the following two possibilities ; either i ∈ Γ_{s−1} and then ∇_{i}f_{s−1} is null, or there exists a unique X_{k} ≡ X_{k(i)} ⊂ Γ_{s−1} so that i ∈ X_{k(i)}^{c} and d(i, X_{k(i)}) ≤ R. In this second case, we use lemma 5.10 (see the next section) which tells us that

|∇_{i}fs−1^{\frac{1}{2}}|^{2} = |∇_{i} (IE_{Γ_{s−1}\X_{k(i)}}IE_{X_{k(i)}}f_{s−2})^{\frac{1}{2}} |^{2}  (5.4.52)
≤ b^{2} IE_{Γ_{s−1}\X_{k(i)}}q_{i}|∇_{i} (IE_{X_{k(i)}}f_{s−2})^{\frac{1}{2}} |^{2}

with a finite constant b, equal to one in the continuous setting or when |M| = 2, and otherwise bounded by \sqrt{|M|}. We can now deduce from the sweeping out relations and from Hölder’s inequality that

q_{i}|∇_{i}(E_{X_{k(i)}}f)^{\frac{1}{2}} |^{2} ≤ α · (α_{ii}^{(X_{k(i)})}E_{X_{k(i)}}q_{i}|∇_{i}f^{\frac{1}{2}} |^{2}  (5.4.53)

+ ∑_{j∈X_{k(i)}∪\{i\}} α_{ij}^{(X_{k(i)})}E_{i}E_{X_{k(i)}}q_{j}|∇_{j}f^{\frac{1}{2}} |^{2})

where

α ≡ sup_{i} (α_{ii}^{(X_{k(i)})} + ∑_{j∈X∪{i}} α_{ij}^{(X_{k(i)})}) .

With (5.4.52), we thus obtain that

|∇_{i}fs−1^{\frac{1}{2}}|^{2} ≤ b^{2}α IE_{Γ_{s−1}\X_{k(i)}}(α_{ii}^{(X_{k(i)})}E_{X_{k(i)}}q_{i}|∇_{i}fs−2^{\frac{1}{2}}|^{2}+  (5.4.54)

+ ∑_{j∈X_{k(i)}∪\{i\}} α_{ij}^{(X_{k(i)})}E_{i}E_{X_{k(i)}}q_{j}|∇_{j}fs−2^{\frac{1}{2}}|^{2})

for any s ≥ 1. We can reinterpret this last result in the following form

|∇_{i}fs−1^{\frac{1}{2}}|^{2} ≤ b^{2}α ( ∑_{j∈X_{k(i)}∪i} ((1 − δ_{ij})E_{j} + δ_{ij}) α_{ij}^{(X_{k(i)})}IE_{s−1}q_{j}|∇_{j}fs−2^{\frac{1}{2}}|^{2}) .

(5.4.55)

Repeating inductively the above arguments, we find for s ≥ 2

|∇_{i}fs−1^{\frac{1}{2}}|^{2} ≤ b^{2}α^{2} ∑_{j_{1}∈X_{k(i)}∪i\Γ_{s−2}}  ∑_{j_{2}∈X_{k(j_{1})}∪j_{1}\Γ_{s−3}} α_{ij_{1}}^{(X_{k(i)})}α_{j_{1}j_{2}}^{(X_{k(j_{1})})} (5.4.56)

((1 − δ_{ij_{1}})E_{j_{1}} + δ_{ij_{1}}) IE_{s−1}IE_{s−2}q_{j_{2}}|∇_{j_{2}}fs−3^{\frac{1}{2}}|^{2}

where, since j_{1} ∈ X_{k(i)} ∪ {i}, we did not need to write the term (1 − δ_{j_{1}j_{2}})E_{j_{1}} + δ_{j_{1}j_{2}}. By induction, we arrive at the bound

IE_{2^{d}−1} ⋯ IE_{s}|∇_{i}fs−1^{\frac{1}{2}}|^{2} ≤ b^{2}α^{s−1} ∑_{j∈Z^{d}} η_{ij}^{(s)}Πq_{j}|∇_{j}f^{\frac{1}{2}} |^{2}  (5.4.57)

with

η_{ij}^{(s)} ≡ ∑_{j_{1}∈X_{k(i)}∪i\Γ_{s−2}} ⋯ ∑_{j_{s−2}∈X_{k(j_{s−3})}∪j_{s−3}\Γ_{1}} α_{ij_{1}}^{(X_{k(i)})}α_{j_{1}j_{2}}^{(X_{k(j_{1})})} ⋯ α_{j_{s−2} j}^{(X_{k(j_{s−2})})}

(5.4.58)

and where X_{k(j_{l})} ⊂ Γ_{s−l} for l = 0, …, s − 1.

Noticing that

Πq_{j}|∇_{j}f^{\frac{1}{2}} |^{2} ≤ e^{2^{d}||Φ||}Π|∇_{j}f^{\frac{1}{2}} |^{2}  (5.4.59)

and summing over all the i ∈ Γ_{s} in (5.4.57), we conclude that for 1 ≤ s ≤ 2^{d} −1,

IE_{2^{d}−1} ⋯ IE_{s}|∇_{Γ_{s}}fs−1^{\frac{1}{2}}|^{2} ≤ b^{2}α^{s−1}η^{(s)}e^{2^{d}||Φ||} Π|∇f^{\frac{1}{2}} |^{2}  (5.4.60)

with

η^{(s)} ≡ sup_{j} ∑_{i∈Z^{d}} η_{ij}^{(s)}.

With (5.4.50) and (5.4.51), we obtain (5.4.47).

To prove (5.4.46), let us come back to (5.4.55) and note that for s = 2^{d} − 1, we inductively obtain

|∇_{i}fs−1^{\frac{1}{2}}|^{2} ≤ b^{2}α^{2^{d}} ∑_{j_{1}∈X_{k(i)}∪i\Γ_{s−2}} ∑_{j_{2}∈X_{k(j_{1})}∪j_{1}\Γ_{s−3}} ⋯ ∑_{j_{s−2}∈X_{k(j_{s−3})}∪j_{s−3}\Γ_{1}}

(5.4.61)

α_{ij_{1}}^{(X_{k(i)})}α_{j_{1}j_{2}}^{(X_{k(j_{1})})} ⋯ α_{j_{s−2}j}^{(X_{k(j_{s−2})})} ((1 − δ_{ij_{1}})E_{j_{1}} + δ_{ij_{1}}) Πq_{j}|∇_{j}f^{\frac{1}{2}} |^{2}.

Since ∪_{s=0}^{2^{d}}Γ_{s} = ZZ^{d}, for any j we can find an s ∈ {0, …, 2^{d} − 1} so that j ∈ Γ_{s}

E_{i}Πq_{j}|∇_{j}f^{\frac{1}{2}} |^{2} ≤ e^{2^{d}||Φ||}Πq_{j}|∇_{j}f^{\frac{1}{2}} |^{2} ≤ e^{2^{d+1}||Φ||}Π|∇_{j}f^{\frac{1}{2}} |^{2}  (5.4.62)

where we used (5.4.59) to obtain the last bound. Summing up over all the i ∈ ZZ^{d}, we deduce (5.4.46) from (5.4.61).

Finally, to prove the second part of the lemma, we remark that under the additional assumption (5.4.43), η can be chosen as small as one wishes as long as L is chosen large enough. Indeed, by construction, for any j ∈ ZZ^{d}, there exists r ∈ {0, …, 2^{d} − 1} and a cube X_{k} ⊂ Γ_{r} so that for any j ∈ X_{k}, d(j, X_{k}^{c}) ≥ L/2. Henceforth, in any path W_{ij} ≡ {i ≡ j_{0}, j_{1} ∈ X_{k(i)}∪i\Γ_{s}, …, j_{l} ∈ X_{k(j_{l−1})}∪j_{l−1}\ Γ_{s−l}, ⋯, j_{s} ∈ X_{k(j_{s−1})} ∪ j_{s−1} \ Γ_{0}, j ≡ j_{s}}, there is at least one couple of points (j_{l−1}, j_{l}) at distance greater or equal to L/2. This in particular implies that in the sum defining of η_{ij}^{2^{d}−1}, there is at least one term α_{j_{l−1},j_{l}}^{(X_{k(j_{l−1}))}} ≤ De^{−ε·L/2}. We hence obtain the bound

∑_{i} η_{ij}^{(2^{d}−1)} ≤ α^{2^{d}−2}De^{−ε·L/2}  (5.4.63)

Since, under hypothesis (5.4.48), α is clearly bounded independently of L, we can choose L sufficiently large so that λ < 1.

Finally, the last part of the lemma can be proved following (5.2.6)-(5.2.9) but without bothering about the limiting Gibbs measure μ.

⋄

Proof of condition (Civ) when the sweeping out relations are satisfied

To show that Π^{n} converges, as n goes to infinity, towards a measure μ, we are going to prove that (Π^{n,ω}f, n ∈ IN) is a Cauchy sequence for any continuously differentiable function f localized in a finite subset of ZZ^{d}, uniformly in ω ∈ Ω. This in turn implies the weak convergence of Π^{n} towards a unique measure μ. Indeed, given a finite subset Λ of ZZ^{d} and considering the set F_{Λ} of continuous functions f localized in Λ, we note that F_{Λ} is separable as M^{|Λ|} is compact since M is. Thus, we can consider a countable subset (f_{i}, i ∈ IN) of F_{Λ} dense in F_{Λ}. By the standard diagonalization procedure, we see that if (Π^{n,ω}f_{i}, n ∈ IN) is Cauchy for any i ∈ IN, uniformly in ω ∈ Ω, (Π^{n,ω}f_{i}, i ∈ IN) converges simultaneously along some subsequence. The limit then defines a probability measure. By the property of local Gibbs measures, μ is independent of the choice of the finite set Λ so that we can define μ on the set Ω of all configurations. Finally, uniqueness of μ can be deduced from the uniformity of the convergence with respect to boundary conditions. Hence, we need to prove that, in view of (5.4.46 ), (Π^{n,ω}f, n ∈ IN) is a Cauchy sequence for any f ∈ F_{Λ}, Λ ⊂⊂ ZZ^{d}. We can of course restrict ourselves to non negative functions f. Let n, m ∈ IN, n < m and a non negative function f ∈ F_{Λ} be given. Then we have

|Π^{n}f(ω) − Π^{m}f(ω)| ≤ ∫ Π^{m−n,ω}(dω̃)|Π^{n}f(ω) − Π^{n}f(ω̃)| (5.4.64)
≤ sup_{ω,_{ω}} |Π^{n}f(ω) − Π^{n}f(ω̃)| ≤ |||Π^{n}f|||.

Moreover, by construction of Π, Π^{n}f is localized in a set Λ_{n} so that |Λ_{n}| ≤ [diam(Λ) + n2(L + R)]^{d}. Thus, we deduce that

|||Π^{n}f||| ≤ |Λ_{n}|·||f^{\frac{1}{2}} ||_{∞}· sup_{i∈Λn} ||∇_{i} (Π^{n}f)^{\frac{1}{2}} ||_{∞}

(5.4.65)

≤ |Λ_{n}| · ||f^{\frac{1}{2}} ||_{∞} · || ∑_{i∈Λ_{n}} |∇_{i} (Π^{n}f)^{\frac{1}{2}} |^{2}||∞^{\frac{1}{2}} ≤ λ^{n/2}|Λ_{n}| · ||f^{\frac{1}{2}} ||_{∞} · |||∇f^{\frac{1}{2}} |^{2}||∞^{\frac{1}{2}}

where in the last line we made an inductive use of (5.4.46). Since the volume of Λ_{n} increases at most polynomially in n, we conclude that, when λ < 1, |||Π^{n}f||| goes to zero when n goes to infinity, which gives with the help of (5.4.64) the desired result.

⋄

5.4.3 Proof of sweeping out relations

In this section, we relate the sweeping out relations, (used in the previous part to prove logarithmic Sobolev inequality ), to strong mixing conditions satisfied by local Gibbs measures. We first describe these mixing conditions and show that they are equivalent to decay of correlations properties. Then, we prove that they imply that sweeping out relations hold. At the end of this section, we prove that mixing conditions are necessary to obtain logarithmic Sobolev inequality for local Gibbs measures with uniformly bounded constants. Thus, this last result holds iff mixing conditions are satisfied. We shall finally see that mixing conditions are satisfied in some situations such as for example a high temperature regime.

Strong mixing conditions

For a subset Y ⊂ Z^{d}, we denote by |||f|||_{Y} a semi-norm of f given by

|||f|||_{Y} ≡ ∑_{i∈Y} ||∇_{i}f||_{∞}.

Here, when M is a connected smooth Riemannian manifold, ∇_{i}f denotes as usual the gradient operator acting on the variable at i, and, in case M is a finite set, ∇_{i}f ≡ ν_{i}f − f. If Y = Z^{d}, we use a short notation |||f|||_{Z^{d}} ≡ |||f|||. In the following, we shall measure the variations of a function f with respect to the coordinates ω_{Y} ≡ (ω_{i} : i ∈ Y) by

Var_{Y} (f) ≡ sup_{ω_{Y ^{c}} =ω_{Y ^{c}}} |f(ω_{Y}) − f(ω̃_{Y})|.

If Y = {j}, we simplify the notation as Var_{Y} (f) ≡ Var_{j}(f). We remark that Var_{Y} (f) ≤ 2||f||_{∞} and that the variation of f is related to the triple semi-norm by

Var_{Y} (f) ≤ a|||f|||_{Y}  (5.4.66)

with, in the discrete setting, a ≤ 2 and in the continuous setting, a ≤ s_{M} ≡ sup_{x,y∈M} inf l_{x,y}, with l_{x,y} the length of a geodesic containing x and y and the infimum being taken on the set of these geodesics.

Exercise 5.8 Prove (5.4.66). Hints : Proceed by interpolation between ω and ω̃ by a sequence ω^{(j_{k})} (for j_{k} ∈ Y, k = 1, …, |Y | a lexicographical order in Y) such that each term of this sequence only differs from the previous by one coordinate to deduce that

Var_{Y} (f) ≤ ∑_{k=1,...,|Y |} Var_{j_{k}}(f).

Prove the result for Y = {j} and conclude.

With the above notations, we consider the following mixing conditions :

Strong Mixing Condition (SMC):

There exists a constant ε ∈ (0, ∞) such that for any X ⊂ ZZ^{d}, and any cube Y ⊂ X^{c},

Var_{Y} (E_{X}f) ≤ RC(Y)e^{−εd(Y, Λ_{f} ∩X)}|||f|||  (5.4.67)

with a finite constant C(Y) depending only on the size of Y.

Let us point out that we shall only use in the sequel (SMC)for sets X defined as the union of a finite number of sets obtained by translation of a given cube Y_{0}. As we will show below this condition is equivalent to the following strong decay of correlations property

Strong Decay of Correlations (SDC):

There exists a constant ε ∈ (0, ∞) such that for any functions f, g with finite triple norm on ZZ^{d} and, for any boundary conditions ω ∈ Ω, we have

|E_{X}^{ω}(g, f)| ≤ Ae^{−εd(Λ_{g},Λ_{f})}|||g||| · |||f|||  (5.4.68)

for a constant A ∈ (0, ∞) depending only on min(|X ∩ Λ_{f}|, |X ∩ Λ_{g}|). We have the following property.

Theorem 5.9 The strong mixing condition (SMC) is equivalent to the strong decay of correlations (SDC).

Proof : Let us first prove that (SDC)⟹ (SMC). Fix Y ⊂ ZZ^{d}, a set X ⊂ Y^{c}, two elements (ω, ω̃) ∈ Ω so that ω_{Y^{c}} = ω_{Y^{c}} and a function f localized in Λ_{f} ⊂ X. Since, by definition, the local Gibbs measures (E_{X}^{ω}, ω ∈ Ω) are all equivalent to ν_{X}, we can write

E_{X}^{ω}(f) − E_{ω} ; X(f) = E_{ω} ; X(ξ_{ω,_{ω}}; f)  (5.4.69)

with

ξ_{ω,_{ω}} = \frac{dE_{X}^{ω}}{dE_{ω} ; X}

and E_{ω} ; X(g; f) = E_{ω} ; X(gf) − E_{ω} ; X(f)E_{ω} ; X(g). In the case where the range of the interaction is finite (say R), one easily sees that d(Y, Λ(ξ_{ω,_{ω}})^{c}) ≤ R so that the triangle inequality implies

d(Λ_{f}, Λ(ξ_{ω,_{ω}})) ≥ d(Λ_{f}, Y) − R.

Applying (SDC) to (5.4.69) and using the last remark, we deduce that

|E_{X}^{ω}(f) − E_{ω} ; X(f)| ≤ Ce^{−εd(Y,Λ_{f})}|||f|||  (5.4.70)

with

C = Ae^{εR}|||ξ_{ω,_{ω}}|||.

(SMC)⟹ (SDC) Conversely, for any functions (f, g) on Ω such that Λ_{f} ∩ X and Λ_{g} ∩ X are disjoint, we have

E_{X}^{ω}(g, f) = E_{X}^{ω}(E_{X\Λ_{f}}^{·} (g), f) = \frac{1}{2}E_{X}^{ω} ⊗ Ẽ_{X}^{ω} ((E_{X\Λ_{f}}^{·} (g) − Ẽ_{X\Λ_{f}}^{·} (g))(f − f̃)) .

Consequently, if (SMC) is satisfied, we obtain

|E_{X}^{ω}(g, f)| ≤ Var_{Λ_{f} ∩X}(E_{X\Λ_{f}}^{·} (g)) · Var_{Λ_{f} ∩X}(f)  (5.4.71)
≤ Ae^{−εd(Λ_{g}∩X, Λ_{f} ∩X)}|||g||| · |||f|||

with A ≤ 2C. ⋄

Study of sweeping out relations

We shall in this paragraph show that (SMC) results with sweeping out relations. In fact, we show in lemma 5.11 that inequality (5.4.42) holds with the coefficients (α_{ij}^{(Λ)}) described in (5.4.80) and (5.4.81) respectively. We estimate these coefficients at the end of the paragraph and show that mixing conditions result with the decay (5.4.43) of these coefficients, second condition for sweeping out relations to hold. To prove lemma 5.11, a key ingredient is the following estimate.

Lemma 5.10 Let (Θ, Λ) be two subsets of ZZ^{d} such that dist(Θ, Λ) > R. Set E_{Θ∪Λ} ≡ E_{Θ} ⊗ E_{Λ}. Then, for any i ∈ ZZ^{d} such that dist(i, Λ) ≤ R, we have

|∇_{i}(E_{Θ∪Λ}f)^{\frac{1}{2}} | ≤ b (E_{Θ}q_{i}|∇_{i}(E_{Λ}f)^{\frac{1}{2}} |^{2})^{\frac{1}{2}}  (5.4.72)

with b ≤ |M|^{\frac{1}{2}} if |M| < ∞ and b = 1 in the case |M| = 2 or if M is a smooth connected Riemannian manifold.

Proof : Let f be a measurable function of L^{1}(E_{Λ}) and F ≡ E_{Λ}f. Let us first consider the discrete setting. We then have

|∇_{i}(E_{Θ∪Λ}f)^{\frac{1}{2}} | = |∇_{i}(E_{Θ}F)^{\frac{1}{2}} | = |ν_{i}(E_{Θ}F)^{\frac{1}{2}} − (E_{Θ}F)^{\frac{1}{2}} |  (5.4.73)

and so, as ν_{i} is the uniform measure on M,

|∇_{i}(E_{Θ∪Λ}f)^{\frac{1}{2}} | ≤ |M|^{\frac{1}{2}} (ν_{i}|ν_{i}(E_{Θ}F)^{\frac{1}{2}} − (E_{Θ}F)^{\frac{1}{2}} |^{2})^{\frac{1}{2}} .  (5.4.74)

Moreover, Minkowski’s and Cauchy-Schwarz’s inequalities imply

ν_{i}|ν_{i}(E_{Θ}F)^{\frac{1}{2}} − (E_{Θ}F)^{\frac{1}{2}} |^{2} = \frac{1}{2}ν_{i} ⊗ ν̃_{i} ((E_{Θ}F)^{\frac{1}{2}} − (E_{Θ}F̃)^{\frac{1}{2}} )^{2} (5.4.75)
≤ \frac{1}{2}ν_{i} ⊗ ν̃_{i} (E_{Θ} (F^{\frac{1}{2}} − F̃^{\frac{1}{2}} )^{2}) .

Since dist(i, Θ) > R, E_{Θ} is independent of the coordinate ω_{i}, and we have

\frac{1}{2}ν_{i} ⊗ ν̃_{i} (E_{Θ} (F^{\frac{1}{2}} − F̃^{\frac{1}{2}} )^{2}) = E_{Θ}ν_{i}(F^{\frac{1}{2}} − ν_{i}F^{\frac{1}{2}} )^{2} = E_{Θ}ν_{i}|∇_{i}F^{\frac{1}{2}} |^{2}.

Putting (5.4.73)-(5.4.75) together, we obtain the desired estimate. In the case where |M| = 2, we can improve this bound in the following way ; we observe that

|∇_{i}(E_{Θ∪Λ}f)^{\frac{1}{2}} | = |∇_{i}(E_{Θ}F)^{\frac{1}{2}} | = \frac{1}{2}|(E_{Θ}F_{|ω_{i}=+1})^{\frac{1}{2}} − (E_{Θ}F_{|ω_{i}=−1})^{\frac{1}{2}} | (5.4.76)

≤ \frac{1}{2} (E_{Θ}(F|ω_{i}=+1^{\frac{1}{2}} − F|ω_{i}=−1^{\frac{1}{2}})^{2})^{\frac{1}{2}} = (E_{Θ}|∇_{i}F^{\frac{1}{2}} |^{2})^{\frac{1}{2}}

with |∇_{i}F^{\frac{1}{2}} |^{2} = ν_{i}|∇_{i}F^{\frac{1}{2}} |^{2}. In the continuous case, the same estimate is true since

|∇_{i}(E_{Θ∪Λ}f)^{\frac{1}{2}} | = \frac{1}{2} \frac{|E_{Θ}∇_{i}F |}{(E_{Θ}F)^{\frac{1}{2}}} = \frac{|E_{Θ}F^{\frac{1}{2}} ∇_{i}F^{\frac{1}{2}} |}{(E_{Θ}F)^{\frac{1}{2}}} ≤ (E_{Θ}|∇_{i}F^{\frac{1}{2}} |^{2})^{\frac{1}{2}}

thanks to Cauchy-Schwarz’s inequality.

⋄

By m_{X,Y} we shall denote the best constant in the spectral gap inequality satisfied by the measure E_{X}|_{Σ_{(X\Y )^{c}}}. Let (E_{X}^{0_{i}}, X ⊂⊂ ZZ^{d}) be the Gibbs measures constructed with the potential {Φ_{X}δ_{{i∉X}}, X ⊂⊂ ZZ^{d}}, that is the Gibbs measures where the interaction with the spin at i has been removed. We shall denote

ξ_{i,Λ} ≡ \frac{dE_{Λ}^{0_{i}}|_{Σ_{Λ}}}{dE_{Λ}|_{Σ_{Λ}}} .

Finally, for any X ⊂ Λ and i ∈ Λ^{c}, we introduce a constant η_{iX}^{(Λ)}, given in the discrete setting by

η_{iX}^{(Λ)} ≡ Var_{Λ∩X} (E_{Λ\X}(ξ_{i,Λ}))  (5.4.77)

and in the continuous setting, if the potential U_{Λ} is νZZ_{d}-almost surely differentiable, by

η_{iX}^{(Λ)} ≡ Var_{Λ∩X} (E_{Λ\X}(|∇_{i}U_{Λ}|)) .  (5.4.78)

By definition, η_{iX}^{(Λ)} will be zero for any other couple (i, X).

Lemma 5.11 Let Λ ⊂⊂ ZZ^{d} and F be a measurable function. Let i ∈ Λ^{c}.

• In the continuous setting, assuming that F is almost surely differentiable, we have

|∇_{i}(E_{Λ}F)^{\frac{1}{2}} | ≤ (E_{Λ}|∇_{i}F^{\frac{1}{2}} |^{2})^{\frac{1}{2}} + ∑_{j∈Λ_{F} ∩Λ} ϵ_{ij}^{(Λ)} (E_{i}E_{Λ}|∇_{j}F^{\frac{1}{2}} |^{2})^{\frac{1}{2}} (5.4.79)

with

ϵ_{ij}^{(Λ)} = (\frac{2}{m_{Λ,j}} )^{\frac{1}{2}} η_{ij}^{(Λ)}.  (5.4.80)

• In the discrete setting, the following estimate is satisfied

(ν_{i}|∇_{i}(E_{Λ}F)^{\frac{1}{2}} |^{2})^{\frac{1}{2}} ≤ e^{\frac{1}{2} η_{i,Λ_{F} ∩Λ}^{(Λ)}} (E_{Λ}ν_{i}|∇_{i}F^{\frac{1}{2}} |^{2})^{\frac{1}{2}}  (5.4.81)

+2 (\frac{|M|}{m_{Λ,Λ_{F}}} )^{\frac{1}{2}} e^{||Φ||}η_{i,Λ_{F} ∩Λ}^{(Λ)} ∑_{j∈Λ∩Λ_{F}} (E_{i}E_{Λ}ν_{j}|∇_{j}F^{\frac{1}{2}} |^{2})^{\frac{1}{2}} .

Proof : Here we shall only focus on the discrete setting, the proof in the continuous being slightly easier and following almost the same lines. Writing

(E_{Λ}F)^{\frac{1}{2}} = (E_{Λ}^{0_{i}}F)^{\frac{1}{2}} + ((E_{Λ}F)^{\frac{1}{2}} − (E_{Λ}^{0_{i}}F)^{\frac{1}{2}} )  (5.4.82)

and applying the same arguments as in (5.4.75), we obtain

(ν_{i}|∇_{i}(E_{Λ}F)^{\frac{1}{2}} |^{2})^{\frac{1}{2}} ≤ (\frac{1}{2}ν_{i} ⊗ ν̃_{i} ((E_{Λ}^{0_{i}}F)^{\frac{1}{2}} − (E_{Λ}^{0_{i}} F̃)^{\frac{1}{2}} )^{2})^{\frac{1}{2}} (5.4.83)

+ (\frac{1}{2}ν_{i} ⊗ ν̃_{i} ((E_{Λ}F)^{\frac{1}{2}} − (E_{Λ}^{0_{i}}F)^{\frac{1}{2}} )^{2})^{\frac{1}{2}}

+ (\frac{1}{2}ν_{i} ⊗ ν̃_{i} ((Ẽ_{Λ}F̃)^{\frac{1}{2}} − (E_{Λ}^{0_{i}} F̃)^{\frac{1}{2}} )^{2})^{\frac{1}{2}} .

We can estimate the first term in the right hand side of (5.4.83) by using Minkowski’ s inequality so that

(\frac{1}{2}ν_{i} ⊗ ν̃_{i} ((E_{Λ}^{0_{i}}F)^{\frac{1}{2}} − (E_{Λ}^{0_{i}} F̃)^{\frac{1}{2}} )^{2})^{\frac{1}{2}} ≤ (\frac{1}{2}ν_{i} ⊗ ν̃_{i} E_{Λ}^{0_{i}} (F^{\frac{1}{2}} − F̃^{\frac{1}{2}} )^{2})^{\frac{1}{2}}
= (E_{Λ}^{0_{i}}ν_{i}|∇_{i}F^{\frac{1}{2}} |^{2})^{\frac{1}{2}} .  (5.4.84)

Moreover, by definition of ξ_{i,Λ}, we notice that for any non negative function G, we have

E_{Λ}^{0_{i}}G = E_{Λ}G + E_{Λ} (ξ_{i,Λ} − 1) G
= E_{Λ}G + E_{Λ} (E_{Λ\Λ_{G}}ξ_{i,Λ} − 1) G  (5.4.85)
≤ (1 + Var_{Λ∩Λ_{G}} (E_{Λ\Λ_{G}}(ξ_{i,Λ}))) E_{Λ}G = (1 + η_{i,Λ_{G}}^{(Λ)})E_{Λ}G.

Using this remark with G = ν_{i}|∇_{i}F^{\frac{1}{2}} |^{2}, we deduce from (5.4.84) and (5.4.85) that

(\frac{1}{2}ν_{i} ⊗ ν̃_{i} ((E_{Λ}^{0_{i}}F)^{\frac{1}{2}} − (E_{Λ}^{0_{i}} F̃)^{\frac{1}{2}} )^{2})^{\frac{1}{2}} ≤ [1 + η_{i,Λ_{F}}^{(Λ)} ]^{\frac{1}{2}} (E_{Λ}ν_{i}|∇_{i}F^{\frac{1}{2}} |^{2})^{\frac{1}{2}}

(5.4.86)

which provides the desired estimate of the first term in the right hand side of (5.4.83). The other terms only depend on the difference (E_{Λ}F)^{\frac{1}{2}} −(E_{Λ}^{0_{i}}F)^{\frac{1}{2}} that we are going to study. To this end, let us note that

|(E_{Λ}^{0_{i}}F)^{\frac{1}{2}} − (E_{Λ}F)^{\frac{1}{2}} | = \frac{|E_{Λ}(ξ_{i,Λ} − 1)F |}{(E_{Λ}^{0_{i}}F)^{\frac{1}{2}} + (E_{Λ}F)^{\frac{1}{2}}} .  (5.4.87)

The above numerator can be bounded above by noticing that

|E_{Λ}(ξ_{i,Λ} − 1)F | = |\frac{1}{2}E_{Λ} ⊗ Ê_{Λ} ((E_{Λ\Λ_{F}} ξ_{i,Λ} − Ê_{Λ\Λ_{F}} ξ̂_{i,Λ})(F − F̂)) | (5.4.88)
≤ \frac{1}{2}η_{i,Λ_{F}}^{(Λ)} E_{Λ} ⊗ Ê_{Λ}|F − F̂|,

with Ê_{Λ} is a copy of E_{Λ}. Moreover, further computations give

\frac{1}{2}E_{Λ} ⊗ Ê_{Λ}|F − F̂| ≤ (E_{Λ}F)^{\frac{1}{2}} (\frac{1}{2}E_{Λ} ⊗ Ê_{Λ}(F^{\frac{1}{2}} − F̂^{\frac{1}{2}} )^{2})^{\frac{1}{2}}  (5.4.89)

≤ (\frac{1}{m_{Λ,Λ_{F}}} )^{\frac{1}{2}} (E_{Λ}F)^{\frac{1}{2}} (E_{Λ}|∇_{Λ∩Λ_{F}} F^{\frac{1}{2}} |^{2})^{\frac{1}{2}} .

Inequalities (5.4.87) - (5.4.89) give the following control

|(E_{Λ}^{0_{i}}F)^{\frac{1}{2}} − (E_{Λ}F)^{\frac{1}{2}} | ≤ (\frac{1}{m_{Λ,Λ_{F}}} )^{\frac{1}{2}} η_{i,Λ_{F}}^{(Λ)} (E_{Λ}|∇_{Λ∩Λ_{F}} F^{\frac{1}{2}} |^{2})^{\frac{1}{2}} . (5.4.90)

Thanks to (5.4.90), we obtain the following bound for the second term in the right hand side of (5.4.83)

(\frac{1}{2}ν_{i} ⊗ ν̃_{i} ((E_{Λ}F)^{\frac{1}{2}} − (E_{Λ}^{0_{i}}F)^{\frac{1}{2}} )^{2})^{\frac{1}{2}} ≤ \frac{1}{mΛ,Λ_{F}^{\frac{1}{2}}} η_{i,Λ_{F}}^{(Λ)} (ν_{i}E_{Λ}|∇_{Λ∩Λ_{F}} F^{\frac{1}{2}} |^{2})^{\frac{1}{2}}

≤ \frac{e^{||Φ||}}{mΛ,Λ_{F}^{\frac{1}{2}}} η_{i,Λ_{F}}^{(Λ)} (E_{i}E_{Λ}|∇_{Λ∩Λ_{F}} F^{\frac{1}{2}} |^{2})^{\frac{1}{2}}  (5.4.91)

≤ \frac{e^{||Φ||}|M|^{\frac{1}{2}}}{mΛ,Λ_{F}^{\frac{1}{2}}} η_{i,Λ_{F}}^{(Λ)} ∑_{j∈Λ∩Λ_{F}} (E_{i}E_{Λ}ν_{j}|∇_{j}F^{\frac{1}{2}} |^{2})^{\frac{1}{2}} .

Similarly, we estimate the third term in the right hand side of (5.4.83) and, with (5.4.83), (5.4.86) and (5.4.91), conclude that

(ν_{i}|∇_{i}(E_{Λ}F)^{\frac{1}{2}} |^{2})^{\frac{1}{2}} ≤ [1 + η_{i,Λ_{F}}^{(Λ)} ]^{\frac{1}{2}} (E_{Λ}ν_{i}|∇_{i}F^{\frac{1}{2}} |^{2})^{\frac{1}{2}}  (5.4.92)

+2 (\frac{|M|}{m_{Λ,Λ_{F}}} )^{\frac{1}{2}} e^{||Φ||}η_{i,Λ_{F}}^{(Λ)} ∑_{j∈Λ∩Λ_{F}} (E_{i}E_{Λ}ν_{j}|∇_{j}F^{\frac{1}{2}} |^{2})^{\frac{1}{2}}

that is the desired result in the discrete setting.

⋄

For further use, we present here the following lemma

Lemma 5.12 For any Λ ⊆ Z^{d}

m_{Λ,Y} ≥ m_{0} e^{−4||Φ||·|Y |}  (5.4.93)

with m_{0} the spectral gap for the product measure νZZ_{d}.

The proof is a direct consequence of property 2.6. In fact, in finite range interaction models, this result can be improved as seen in property 2.7. The main interest of lemma 5.12 is that the estimate of m_{Λ,Y} there is independent of Λ.

We can now establish sweeping out relations.

Theorem 5.13 (Sweeping out relations) For any finite subset Λ of ZZ^{d} which can be represented as the union of a finite numbers of cubes obtained by translation of the same cube Y_{0}, there exist constants α_{ij}^{(Λ)} ∈ (0, ∞), so that for any i ∈ Λ^{c}, dist(i, Λ) ≤ R, we have

(ν_{i}|∇_{i}(E_{Λ}f)^{\frac{1}{2}} |^{2})^{\frac{1}{2}} ≤ α_{ii}^{(Λ)} (E_{Λ}ν_{i}|∇_{i}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}} + ∑_{j∈Λ∪\{i\}} α_{ij}^{(Λ)} (E_{i}E_{Λ}ν_{j}|∇_{j}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}}

(5.4.94)

If we additionnaly assume that the strong mixing condition (SMC)is satisfied, then

α_{ij}^{(Λ)} ≤ De^{−ε|i−j|}  (5.4.95)

for constants D, ε ∈ (0, ∞) independent of i ∈ Λ^{c}, d(i, Λ) ≤ R and j ∈ Λ ∪ {i}, but eventually depending on the size of Y_{0}.

We notice that the dependence of D and ε on the size of Y_{0} is not essential since the sets Λ can be as large as one wishes, for a given size of Y_{0}. Proof : The idea of the proof is to apply inductively lemma 5.11 and the uniform bound of lemma 5.12.

Let us assume that Λ is defined as the union of a family of disjoint cubes {Y_{k} ≡ Y_{0} + L_{0}y_{k}}, y_{k} ∈ Z^{d} with k = 1, …, N, for an integer number N, and a cube Y_{0} with diameter L_{0}. Let us remark that for any cube {Y_{k}, k ∈ {1, …, N}}, lemma 5.11 implies

(ν_{i}|∇_{i}(E_{Λ}f)^{\frac{1}{2}} |^{2})^{\frac{1}{2}} = (ν_{i}|∇_{i}(E_{Λ}(E_{Λ\Y_{k}}f))^{\frac{1}{2}} |^{2})^{\frac{1}{2}}  (5.4.96)
≤ e^{\frac{1}{2} η_{i,Y_{k}}^{(Λ)}} (E_{Λ}ν_{i}|∇_{i}(E_{Λ\Y_{k}}f)^{\frac{1}{2}} |^{2})^{\frac{1}{2}}

+Cη_{i,Y_{k}}^{(Λ)} ∑_{j∈Y_{k}} (E_{i}E_{Λ}ν_{j}|∇_{j}(E_{Λ\Y_{k}}f)^{\frac{1}{2}} |^{2})^{\frac{1}{2}}

with C ≡ 2(|M|/m_{Λ,Y_{k}})^{\frac{1}{2}} e^{||Φ||}. Using lemma 5.12, we know that

C ≤ A ≡ 2(|M|/m_{0})^{\frac{1}{2}} e^{||Φ||(1+2L_{0}^{d})}.

In order to bound the first term on the right hand side of (5.4.96), we apply inductively this relation. It will soon appear that the optimal sequence of cubes {Y_{k}, k ∈ {1, …, N}} one can choose is to order them in decreasing order of their distance to i. We hence let (k_{j}^{i}, 1 ≤ j ≤ N) be a lexicographical order on {1, …, N} such that j → d(i, Y_{k_{j}^{i}}) is decreasing and let Λ_{i}^{l} = ⋃_{m=l}^{N} Y_{k_{m}^{i}}.

To simplify the notations, we set η_{i,Y_{k{i \atop l}}}^{(Λ_{i}^{l})} = η_{i,l} and Y_{k_{l}^{i}} = Y_{i,l}, i, l ∈ \{1, …, N\}.

Using repeatedly (5.4.96), we obtain

(ν_{i}|∇_{i}(E_{X}f)^{\frac{1}{2}} |^{2})^{\frac{1}{2}} ≤ e^{\frac{1}{2} ∑_{l=1}^{N} η_{i,l}} (E_{X}ν_{i}|∇_{i}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}}  (5.4.97)

+ ∑_{l=1}^{N} A · η_{i,l} ∑_{j∈Y_{i,l}} (E_{i}E_{Λ}ν_{j}|∇_{j}(E_{Λ_{i}^{l}}f)^{\frac{1}{2}} |^{2})^{\frac{1}{2}} .

To bound the second term in the above inequality,

• we apply, for j ∈ Y_{i,l} so that d(j, Λ_{i}^{l}) > R, lemma 5.10 (with Θ = Λ_{i}^{l} and Λ = ∅).

• We notice that for the j ∈ Y_{i,l} so that d(j, Λ_{i}^{l}) ≤ R, we can write a bound for Λ_{i}^{l} similar to that obtained in (5.4.97),

(ν_{j}|∇_{j}(E_{Λ_{i}^{l}}f)^{\frac{1}{2}} |^{2})^{\frac{1}{2}} ≤ e^{\frac{1}{2} ∑_{l_{2}=1}^{N−l} η_{j,l_{2}}^{(2)}} (E_{X}ν_{j}|∇_{j}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}}  (5.4.98)

+ ∑_{l_{2}=1}^{N−l} A · η_{j,l_{2}}^{(2)} ∑_{j_{2}∈Y_{j,l_{2}}} (E_{i}E_{Λ}ν_{j_{2}}|∇_{j}(E_{Λ_{j}^{l,2}}f)^{\frac{1}{2}} |^{2})^{\frac{1}{2}}

where this time (k_{l_{2}}^{j}, 1 ≤ l_{2} ≤ N − l) is a lexicographical order on (k_{m}^{i}, m ∈ {l, …, N}) so that l_{2} → d(j, Y_{k_{l_{2}}^{j}}) is decreasing, Λ_{j}^{m,2} = ⋃_{l_{2}=1}^{N−l} Y_{k_{l_{2}}^{j}} and we set

Y_{j,l_{2}} = Y_{k_{l_{2}}^{j}}, η_{j,Y_{j,l_{2}}}^{(Λ_{j}^{l_{2},2})} = η_{j,l_{2}}^{(2)}.

Applying these two arguments inductively, we finally arrive at

(ν_{i}|∇_{i}(E_{X}f)^{\frac{1}{2}} |^{2})^{\frac{1}{2}} ≤ α_{ii}^{(Λ)} (E_{X}ν_{i}|∇_{i}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}} + ∑_{j∈Λ} α_{ij}^{(Λ)} (E_{i}E_{X}ν_{j}|∇_{j}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}}

(5.4.99)

with

α_{ii}^{(Λ)} ≤ e^{\frac{1}{2} ∑_{l=1}^{N} η_{l}^{i}}  (5.4.100)

and

α_{ij}^{(Λ)} ≤ ∑_{1≤n≤N} ∑_{Y_{1}^{i},Y_{2}^{i},...,Y_{n}^{i}} A^{n}η_{i,Y_{1}^{i}}^{(Λ)}η_{Y_{1}^{i},Y_{2}^{i}}^{(Λ_{2})} ⋯ η_{Y_{n−1}^{i},Y_{n}^{i}}^{(Λ_{n})} max{b, max_{i∈Λ} α_{ii}^{(Λ)}} (5.4.101)

where we have denoted η_{X,Y}^{(Λ)} = max_{i∈X} η_{i,Y}^{(Λ)} and where the sum holds over all the random walks (Y_{1}^{i}, Y_{2}^{i}, …, Y_{n}^{i}) on the cubes in Λ such that d(i, Y_{1}^{i}) = max_{1≤k≤N} d(i, Y_{k}) and

d(Y_{k}^{i}, Y_{k−1}^{i}) = max_{Y ∈Λ\∪_{p=1}^{k−1}Y_{p}^{i}} d(Y_{k−1}, Y)  (5.4.102)

and j ∈ Y_{n}^{i}. Further, Λ_{k} = Λ \ ∪_{p=1}^{k−1}Y_{p}^{i}.

Now, assuming that condition (SMC) is satisfied, we see that by definition of the coefficients η_{i,X}^{(Λ)} there exist two constants C, ε ∈ (0, ∞) such that for any Λ ⊂ ZZ^{d}, any X ⊂ ZZ^{d} and i ∈ ZZ^{d},

η_{i,X}^{(Λ)} ≤ Ce^{−εd(i,X)}

and hence

η_{X,Y}^{(Λ)} ≤ Ce^{−εd(Y,X)}.

In particular, if d(Y_{k}^{i}, Y_{k−1}^{i}) ≥ D_{0} ≡ (2/ϵ) (||Φ||(1 + 2L_{0}^{d}) + \frac{1}{2} log 2(|M|/m_{0})),

Aη_{Y_{k−1}^{i},Y_{k}^{i}}^{(Λ_{k})} ≤ Ce^{−(ε/2)d(Y_{k}^{i},Y_{k−1}^{i})}.  (5.4.103)

We then conclude that

α_{ij}^{(Λ)} ≤ C_{0}e^{−C_{1}|i−j|}  (5.4.104)

for constants C_{0}, C_{1} ∈ (0, ∞) depending only on L_{0}. The proof is complete. ⋄

5.4.4 Comments on the strong mixing assumptions

To complete this section, we comment here on the mixing conditions we assumed to prove the logarithmic Sobolev inequality . We shall see that they are optimal with respect to our strategy in the sense that when Π satisfies condition (Ciii), some strong mixing property will be satisfied. We shall also show how to check that (SDC)(and hence (SMC)) is satisfied in high temperature models.

Optimality of strong mixing assumption

Let us consider the symmetric transfer matrix

T ≡ Π^{∗}Π

with Π as in section 5.4.2. Its adjoint Π^{∗} is given by a similar formula but with the conditional expectations coming in reverse order. It is clear that by using the same argument as in the last paragraph 5.4.3, and assuming (SMC), one finds the following analogous of property (5.4.46) for T

|∇ (Tf)^{\frac{1}{2}} |^{2} ≤ λ^{2}|∇f^{\frac{1}{2}} |^{2}  (5.4.105)

with a constant λ ∈ (0, 1). We shall see that this bound implies itself the strong mixing assumption. For further use, we recall we have the following estimate of the variance of T^{N}f, for N ∈ IN,

μ(T^{N}f; T^{N}f) = \frac{1}{2}μ ⊗ μ̃ (T^{N}f − T̃^{N}f)^{2}
≤ 2|| (T^{N}f)^{\frac{1}{2}} ||_{∞} · ∑_{i} ||∇_{i} (T^{N}f)^{\frac{1}{2}} ||_{∞}. (5.4.106)

If f is localized in Λ(f), then by construction of T, the function T^{N}f is localized in the set Λ_{N} ≡ {i : d(i, Λ(f)) ≤ 4d(L + R)N} satisfying |Λ_{N}| ≤ CN^{d} with a constant C ≤ (4d(L + R) + D(f))^{d} if D(f) is the diameter of Λ(f). We obtain in particular that

∑_{i} ||∇_{i} (T^{N}f)^{\frac{1}{2}} ||_{∞} ≤ CN^{d} (|| |∇ (T^{N}f)^{\frac{1}{2}} |^{2} ||_{∞})^{\frac{1}{2}} .  (5.4.107)

Thus, applying (5.4.105) with (5.4.106)-(5.4.107), we arrive at the following bound

μ(T^{N}f; T^{N}f) ≤ 2||f^{\frac{1}{2}} ||_{∞} · CN^{d}λ^{N} (|| |∇f^{\frac{1}{2}} |^{2} ||_{∞})^{\frac{1}{2}} .  (5.4.108)

Since T is self-adjoint with non negative eigenvalues, its spectral radius (see Bourbaki, [9], p. 15) is given by

sup_{f∈C_{1}} lim_{N→∞} (μ(T^{N}f; T^{N}f))^{\frac{1}{2N}} ≤ λ^{\frac{1}{2}}  (5.4.109)

and so

μ(T^{N}f; T^{N}f) ≤ λ^{N}μ(f; f).  (5.4.110)

Let us now assume that we are given two functions f and g localized in two disjoint sets Λ(f) and Λ(g). Denoting N ≡ [d(Λ(f), Λ(g))/C], and assuming N even (up to replace it by N + 1) we see that

Λ(T^{\frac{N}{2}} f) ∩ Λ(T^{\frac{N}{2}} g) = ∅

so that T^{\frac{N}{2}} (fg) = T^{\frac{N}{2}} (f)T^{\frac{N}{2}} (g). In particular, as μ is invariant for T,

|μ(f; g)| = |μ(T^{\frac{N}{2}} f; T^{\frac{N}{2}} g)| ≤ (μ(T^{\frac{N}{2}} f; T^{\frac{N}{2}} f))^{\frac{1}{2}} (μ(T^{\frac{N}{2}} g; T^{\frac{N}{2}} g))^{\frac{1}{2}} (5.4.111)
≤ λ^{N/2} (μ(f; f))^{\frac{1}{2}} (μ(g; g))^{\frac{1}{2}}

Since (μ(f; f))^{\frac{1}{2}} ≤ C|||f||| for a finite constant C, we conclude that we also have

|μ(f; g)| ≤ C^{2}e^{−M[dist(Λ(f),Λ(g))−1]}|||f||| · |||g|||  (5.4.112)

with

M ≥ −\frac{1}{2} log λ

Similar arguments can be developed in any finite volume (by modifying T accordingly) so that the strong mixing condition is indeed equivalent with condition (5.4.105).

Proof of (SMC)in the high temperature models

Here, we consider local Gibbs measures (E_{Λ}^{ω}, Λ ⊂⊂ ZZ^{d}, ω ∈ Ω) defined by a finite range potential Φ. We shall assume that ||Φ|| is sufficiently small (corresponding to the high temperature situations) and shall show that the strong decay of correlation property (or equivalently the strong mixing property) is satisfied. We follow here the papers [60] and [31].

On Λ ⊂⊂ ZZ^{d}, let us consider a lexicographical order (j_{i}, i = 1, …, |Λ|) and set

V_{Λ}^{i} = ∑_{∗} Φ_{X}

where the sum is taken over all the finite subsets X with non empty intersection with Λ, containing j_{i} but not (j_{k}, k < i). With such a definition,

U_{Λ}(x) = ∑_{i=1}^{|Λ|} V_{Λ}^{i}(x).

Up to replace U_{Λ} by U_{Λ} − ∫ U_{Λ}dν_{Λ}, we may always assume that the energy U_{Λ} is centered with respect to the product measure ν_{Λ}. Then, Jensen’s inequality implies that the partition function Z_{Λ}^{ω} is bounded below by 1.

Let f and g be two bounded measurable functions localized respectively in Λ(f) and Λ(g). We are going to prove that, if ||Φ|| is sufficiently small, there exists a constant ϵ > 0 so that

E_{Λ}^{ω}(f; g) ≤ e^{−ϵd(Λ(f),Λ(g))}|||f|||.|||g|||.  (5.4.113)

To this end, let us first notice that

E_{Λ}^{ω}(f; g) = \frac{1}{2(Z_{Λ}^{ω})^{2}} ∫ (f − f̃)(g − g̃)e^{−U_{Λ}−_{U_{Λ}}}dν_{Λ}(x)dν_{Λ}(x̃).  (5.4.114)

Denoting z_{Λ}^{i} = e^{−V_{Λ}^{i}−_{V_{Λ}^{i}}} − 1, we can write

e^{−U_{Λ}−_{U_{Λ}}} = ∏_{i=1}^{|Λ|}(z_{Λ}^{i} + 1) = ∑_{i} ∏_{k∈i} z_{Λ}^{k}

where the sum i goes over all the subsets of {1, …, |Λ|}. Observe that the (z_{Λ}^{i}, i ∈ {1, …, N}) are localized into sets of radius bounded by the range of the interaction R. Hence, introducing this decomposition into the right hand side of (5.4.114), we see that only the i’ such that the (j_{k}, k ∈ i) make a path of points in Λ at distance less or equal to R joining Λ(f) and Λ(g) will contribute. Denote W_{Λ(f),Λ(g)} the set of these paths. We then have, since Z_{Λ}^{ω} ≥ 1,

|E_{Λ}^{ω}(f; g)| ≤ \frac{1}{2} ∑_{i∈W_{Λ(f),Λ(g)}} ∫ |f − f̃||g − g̃| ∏_{k∈i} |z_{Λ}^{k}|dν_{Λ}(x)dν_{Λ}(x̃) (5.4.115)

According to (5.4.66), we have |f −f̃| ≤ a|||f|||. Moreover, the following uniform bound

|z_{Λ}^{k}| ≤ δ ≡ 2||Φ||e^{2||Φ||}

holds. Further, we observe that, in any dimension d, we can find a finite constant C_{d,R} depending on the number of neighbours of a point in ZZ^{d} and of the range R so that

|W_{Λ(f),Λ(g)}| ≤ C_{d,R}^{d(Λ(f),Λ(g))}.

Hence, we conclude according to (5.4.115) that

|E_{Λ}^{ω}(f; g)| ≤ a^{2}C_{d,R}^{d(Λ(f),Λ(g))}δ^{\frac{1}{R} d(Λ(f),Λ(g))}|||f|||.|||g|||

which gives the announced statement if δ^{\frac{1}{R}} C_{d,R} < 1, that is if ||Φ|| is sufficiently small.

Logarithmic Sobolev inequalities and cellular automata

In this part, we introduce and study cellular automata. This approach to obtain log-Sobolev inequalities for measures possibly non related to a given potential was introduced by one of the authors (as an extension of the result for finite convolutions contained in [103]). It was later studied by G. Gielis ([42]) who used an idea based on disagreement of percolation to cover an extended high temperature domain. Here we shall use cellular automata to establish logarithmic Sobolev inequality for dynamical systems with possibly infinite range of interaction. The transition probability of a parallel cellular automata is described by a product probability ; if C is a countable set, we consider the transition probability on Ω = M^{C} given by

Pf(ω) = ∫ f(ω̃) ⊗_{i∈C} p_{i}^{ω}(dω̃_{i})  (6.0.1)

where (p_{i}^{ω}, i ∈ C) are probability measures on M which are absolutely continuous with respect to the uniform measure

p_{i}^{ω}(dω̃_{i}) = ρ_{i}^{ω}(ω̃_{i})ν_{i}(dω̃_{i}).

As before, we shall concentrate on the case where M is a finite set, the continuous setting being easier to analyze. The operator P given by (6.0.1) can be considered as the transition matrix of a Markov chain.

Let (P^{n,ω}, n ∈ IN^{∗}) be the family of transition probability measures defined by induction as follows P^{n+1,ω}f = P(P^{n,.}f)(ω), P^{0,ω}f = f(ω). We shall show that, under some conditions specified later, the following logarithmic Sobolev inequality is satified with a constant c ∈ (0, ∞) independent of n ∈ IN^{∗} and ω ∈ Ω

P^{n,ω} (f log f) − P^{n,ω}f log P^{n,ω}f ≤ c P^{n,ω}|∇f^{\frac{1}{2}} |^{2}.  (6.0.2)

We shall see that under proper assumptions P^{n,ω} converges as n goes to infinity towards a probability measure μ on Ω, the limit being independent of the configuration ω ∈ Ω. This limit law μ will also satisfy a logarithmic Sobolev inequality . The limiting probability measure has a priori no link with the Gibbs measures introduced in the previous chapter. Also, we point out that the transition matrix P is in general not symmetric in L^{2}(μ).

We begin with the following central proposition

Proposition 6.1 Suppose that

A ≡ sup_{i} sup_{ω,_{ω}} ∏_{k} sup_{ω_{\{i\}^{c}} =ω_{\{i\}^{c}}} ∥\frac{ρ_{k}^{ω}}{ρ_{ω} ; k} ∥_{∞} < ∞  (6.0.3)

and

sup_{i} max [∑_{j∈C} ξ_{ij} , ∑_{j∈C} ξ_{ji}] < ∞  (6.0.4)

with the notation

ξ_{ij} ≡ sup_{ω_{\{i\}^{c}} =ω_{\{i\}^{c}}} ∥\frac{ρ_{j}^{ω}}{ρ_{ω} ; j} − 1∥_{∞} .  (6.0.5)

Then, there exists a constant λ ∈ (0, ∞) such that for any non negative function f, we have

|∇(Pf)^{\frac{1}{2}} |^{2} ≤ λP|∇f^{\frac{1}{2}} |^{2}.  (6.0.6)

\underline{Remark} 6.2: The constant λ can be chosen for example as follows

λ ≡ b^{2}A^{4}c_{0} sup_{i} max [∑_{j∈C} ξ_{ij} , ∑_{j∈C} ξ_{ji}]^{2}  (6.0.7)

with b ≤ |M|^{\frac{1}{2}} if |M| ≥ 3 and b = 1 if M has cardinality 2 or is a Riemannian manifold. The constant c_{0} can be chosen equal to sup_{j,ω} ∥ρ_{j}^{ω}∥_{∞} · ∥ (ρ_{j}^{ω})^{−1} ∥_{∞}. Proof of Proposition 6.1. For any i ∈ C and any non negative function f, we can show as in (5.4.72) that

|∇_{i}(Pf)^{\frac{1}{2}} |^{2} ≤ b^{2} ν_{i}|∇_{i}(Pf)^{\frac{1}{2}} |^{2} = b^{2} ν_{i} ((Pf)^{\frac{1}{2}} ; (Pf)^{\frac{1}{2}} )  (6.0.8)

The right hand side of this last inequality can be rewritten as

ν_{i} ((Pf)^{\frac{1}{2}} ; (Pf)^{\frac{1}{2}} ) = \frac{1}{2} ∫ ν_{i}(dω_{i})⊗ν_{i}(dω̃_{i}) (\frac{Pf(ω) − Pf(ω̃)}{(Pf)^{\frac{1}{2}} (ω) + (Pf)^{\frac{1}{2}} (ω̃)})^{2} (6.0.9)

with the notation ω̃ ≡ ω_{{i}^{c}} ∘ ω̃_{i}. Using a lexicographic order {j_{l} ∈ C}_{l∈N}, we obtain

Pf(ω) − Pf(ω̃) = ∑_{l∈N} P_{l} (p_{j_{l}}^{ω}f − p_{ω} ; j_{l}f)  (6.0.10)

where P_{l} ≡ P_{l,i,ω,_{ω}} ≡ ⊗_{s<l} p_{j_{s}}^{ω} ⊗_{s>l} p_{ω} ; j_{s}. With the notation (6.0.5), we have

|p_{j_{l}}^{ω}f − p_{ω} ; j_{l}f| = |p_{j_{l}}^{ω}(\frac{ρ_{ω} ; j_{l}}{ρ_{j_{l}}^{ω}}; f)| ≤ 2^{\frac{1}{2}} ∥ξ_{ij_{l}}∥_{∞} (p_{j_{l}}^{ω}(f))^{\frac{1}{2}} (p_{j_{l}}^{ω}(f^{\frac{1}{2}} ; f^{\frac{1}{2}} ))^{\frac{1}{2}}
≤ 2^{\frac{1}{2}} c0^{\frac{1}{2}} ∥ξ_{ij_{l}}∥_{∞} (p_{j_{l}}^{ω}(f))^{\frac{1}{2}} (p_{j_{l}}^{ω}|∇_{j_{l}}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}}  (6.0.11)

with a constant c_{0} ≤ sup_{j,ω} ∥ρ_{j}^{ω}∥_{∞} · ∥ (ρ_{j}^{ω})^{−1} ∥_{∞}. We now deduce from (6.0.10) and (6.0.11) that

|Pf(ω) − Pf(ω̃)| ≤ ∑_{l∈N} 2^{\frac{1}{2}} c0^{\frac{1}{2}} ∥ξ_{ij_{l}}∥_{∞} (P_{l} ⊗ p_{j_{l}}^{ω}(f))^{\frac{1}{2}} (P_{l} ⊗ p_{j_{l}}^{ω}|∇_{j_{l}}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}} .

(6.0.12)

Now, if

A ≡ sup_{i} sup_{ω,_{ω}∈Ω} ∏_{k∈C} sup_{ω_{\{i\}^{c}} =ω_{\{i\}^{c}}} ∥\frac{ρ_{k}^{ω}}{ρ_{ω} ; k} ∥_{∞} ,

let us observe that for any l ∈ N, and all ω, ω̃ ∈ Ω, we have

A^{−1} · PF (ω) ≤ P_{l} ⊗ p_{j_{l}}^{ω}F (ω, ω̃) ≤ A · PF (ω)  (6.0.13)

so that we can estimate the right hand side of (6.0.9) by

(\frac{Pf(ω) − Pf(ω̃)}{(Pf)^{\frac{1}{2}} (ω) + (Pf)^{\frac{1}{2}} (ω̃)})^{2} ≤ (∑_{j∈C} [2^{\frac{1}{2}} Ac0^{\frac{1}{2}} ξ_{ij}] (P|∇_{j}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}} )^{2} (ω) (6.0.14)

≤ [2A^{2}c_{0} ∑_{j∈C} ξ_{ij}] ∑_{j∈C} ξ_{ij}P|∇_{j}f^{\frac{1}{2}} |^{2}(ω)

where - we recall - that ω and ω̃ only differ at the site i. Inserting this result in (6.0.9) and using (6.0.8), we obtain

|∇_{i}(Pf)^{\frac{1}{2}} |^{2} ≤ b^{2} [A^{2}c_{0} ∑_{j∈C} ξ_{ij}] ∑_{j∈C} ξ_{ij} ∫ ν_{i}(dω_{i})P|∇_{j}f^{\frac{1}{2}} |^{2}

≤ b^{2} [A^{4}c_{0} ∑_{j∈C} ξ_{ij}] ∑_{j∈C} ξ_{ij}P|∇_{j}f^{\frac{1}{2}} |^{2}  (6.0.15)

which gives, after summation over the i ∈ C,

|∇(Pf)^{\frac{1}{2}} |^{2} ≤ (b^{2}A^{4}c_{0} sup_{i} max [∑_{j∈C} ξ_{ij} , ∑_{j∈C} ξ_{ji}]^{2}) P|∇f^{\frac{1}{2}} |^{2}  (6.0.16)

which finishes the proof of the proposition. ⋄ Proposition 6.1 will be the key ingredient in the proof of the following theorem.

Theorem 6.3 Assume λ of (6.0.6) is strictly smaller than one. Then there exists c ∈ (0, ∞) so that

P^{n,ω} (f log f) − P^{n,ω}f log P^{n,ω}f ≤ c P^{n,ω}|∇f^{\frac{1}{2}} |^{2}  (6.0.17)

for any n ∈ IN and ω ∈ Ω, and every non negative differentiable function f. Consequently, the probability measure μ ≡ lim_{n→∞} P^{n,ω} satisfies the logarithmic Sobolev inequality with the same constant.

Proof. For a non negative differentiable function f, we set f_{n} ≡ Pf_{n−1} with the convention f_{0} ≡ f.

P^{ω}f_{n−1} log f_{n−1} − P^{ω}f_{n−1} log P^{ω}f_{n−1} ≤ c̃_{0} P^{ω}|∇fn−1^{\frac{1}{2}}|^{2}  (6.0.18)

By proposition 6.1, we find that

P^{ω}f_{n−1} log f_{n−1} − P^{ω}f_{n−1} log P^{ω}f_{n−1} ≤ 2c̃_{0}λ^{n−1} P^{ω}|∇f^{\frac{1}{2}} |^{2}.  (6.0.19)

Thus, we deduce, as in section 5.2 (see (5.2.5) and (5.2.6)), (6.0.17) from (6.0.19) with c = (c̃_{0}/1−λ) when λ < 1. To prove the second part of the theorem, let us remark that using (6.0.10), for any (ω, ω̃) ∈ Ω^{2} so that ω_{{i}^{c}} = ω_{{i}^{c}}, we have

|P^{ω}f − P_{ω}f| = |∑_{l} P_{l} (p_{j_{l}} ((\frac{ρ_{j}^{ω}}{ρ_{ω} ; j} − 1)∇_{j_{l}}f))|  (6.0.20)

and therefore

|P^{ω}f − P_{ω}f| ≤ A ∑_{j∈C} ξ_{ij}P|∇_{j}f|.  (6.0.21)

Now, for any n, m ∈ IN, n > m, we see that

|P^{n,ω}f − P^{m,ω}f| ≤ sup_{ω,ω̄} |P^{m}f(ω) − P^{m}f(ω̄)|.  (6.0.22)

It is not hard to see that

|P^{m}f(ω) − P^{m}f(ω̄)| ≤ ∑_{i∈C} ||∇_{i}P^{m}f||_{∞} ≡ |||P^{m}f|||.  (6.0.23)

Since (6.0.21) implies that for any i, ||∇_{i}Pf||_{∞} ≤ A ∑_{j} ξ_{ij}||∇_{j}f||_{∞}, we deduce that

|||P^{m}f||| ≤ (A sup_{i} ∑_{j} ξ_{ij})^{m} |||f|||  (6.0.24)

which, under our hypotheses, converges towards zero as m → ∞. Hence, we have shown that (P^{n}f)_{n∈IN} is a Cauchy sequence, uniformly with respect to the boundary conditions ω ∈ Ω. This is sufficient to guarantee, as we saw in part 5.4, that (P^{n})_{n∈IN} converges towards a unique probability measure μ.

⋄

In the last part of this chapter, we discuss the case where the cellular automata is described by a potential. In other words, if for a potential Φ of possibly infinite range of interaction and a subset C of ZZ^{d}, we set U_{j} ≡ ∑_{X⊂⊂C:X∋j} Φ_{X}, then we define the transition matrix P_{Φ} by (6.0.1) with ρ_{j} = e^{−U_{j}}/ν_{j}e^{−U_{j}}.

To study the ergodic properties of P_{Φ}, we shall try to find natural conditions under which the assumptions of the last proposition are satisfied. With the same notation as above, let us first note that for any ω_{{i}^{c}} = ω_{{i}^{c}}, we have

∥\frac{ρ_{k}^{ω}}{ρ_{ω} ; k} ∥_{∞} ≤ exp (2 ∑_{X⊂⊂C:X∋i,k} V ar_{X}(Φ_{X}))  (6.0.25)

In this case, we see that

A ≤ exp (2 sup_{i} ∑_{X⊂⊂C:X∋i} |X| · V ar_{X}(Φ_{X})) ≤ e^{4∥Φ∥_{B_{2}}}  (6.0.26)

and that

ξ_{ij} ≤ 2 ∑_{X⊂⊂C:X∋i,k} V ar_{X}(Φ_{X}) exp (2 ∑_{X⊂⊂C:X∋i,k} V ar_{X}(Φ_{X})) . (6.0.27)

Here, we used the notation

||Φ||_{B_{2}} ≡ sup_{i∈Z^{d}} ∑_{X⊂⊂Z^{d}: X∋i} |X| · ||Φ_{X}||_{∞}.  (6.0.28)

We shall assume in the sequel that Φ ≡ {Φ_{X}}_{X⊂⊂Z^{d}} belongs to the Banach space B_{2} of potential with finite || · ||_{B_{2}} norm. Then,

sup_{i} max [∑_{j∈C} ξ_{ij} , ∑_{j∈C} ξ_{ji}] < 2e^{2∥Φ∥_{B_{2}}} ∥Φ∥_{B_{2}}  (6.0.29)

Using the previous proposition, we can hence write down the following result.

Theorem 6.4 Assume that the cellular automata P_{Φ} is described by a potential Φ ∈ B_{2}. If ||Φ||_{B_{2}} < β_{0} ∈ (0, ∞), for a sufficiently small β_{0} ∈ (0, ∞), then there exists a unique invariant measure μ for the semi-group P ≡ P_{Φ} satisfying for some constant ε > 0 and for any differentiable function f,

||P^{n}f − μf||_{∞} ≤ e^{−nε}|||f|||.  (6.0.30)

Moreover, there exists a finite constant c ∈ (0, ∞) such that

μf log f/μf ≤ 2cμ|∇f^{\frac{1}{2}} |^{2}  (6.0.31)

for any non negative function for which the above right hand side is finite.

\underline{Remark} 6.5: The reader interested in the ergodicity questions of the cellular automata may like to consult for example [67], [?] and [43] and the references therein. It is interesting to note that the question whether the limiting probability measure of a cellular automata is a Gibbs measure or not has not been addressed in the general setting of a potential Φ ∈ B_{2}.

Logarithmic Sobolev inequalities for spin systems with long range interaction, Martingale expansion.

We now come back to a spin system described by a potential Φ ≡ {Φ_{X}}_{X⊂⊂Z^{d}} as introduced in chapter 5. However, we remove the assumption of finite range of the interaction to extend the previous results to Φ ∈ B_{2}, that is satisfying

||Φ||_{B_{2}} ≡ sup_{i∈Z^{d}} ∑_{X⊂⊂Z^{d}: X∋i} |X| · ||Φ_{X}||_{∞} < ∞.

We shall prove that when the uniqueness condition of Dobrushin is fulfilled, the unique Gibbs measure in infinite volume satisfies a logarithmic Sobolev inequality. To this end, we shall first recall a few facts from the uniqueness theory of Dobrushin ([24], [38], [88], [41]). Then, to prove the logarithmic Sobolev inequality, we follow an approach based on martingale expansion introduced in [64]. To simplify the notations, we restrict ourselves to the case M = {−1, +1}.

7.0.5 Long range interaction systems

Given a family of probability kernels {E_{k} ≡ E_{k,Φ}}_{k∈Z^{d}} on a probability space Ω equipped with a σ-algebra Σ, the interaction matrix C_{kl}, k, l ∈ Z^{d} of Dobrushin is defined by

C_{kl} ≡ sup_{ω,_{ω}∈Ω:ω_{{l}^{c}} =ω_{{l}^{c}}} sup_{A} |E_{k}^{ω}(A) − E_{ω} ; k(A)|  (7.0.1)

where the supremum is taken over all Σ-measurable sets A.

Hereafter, we shall assume that the following condition is satisfied Dobrushin’s uniqueness condition

sup_{k} ∑_{l∈Z^{d}} C_{kl} < 1  (7.0.2)

Let us recall the very nice result due to H. Föllmer [38]

Theorem 7.1 Assume Dobrushin’s uniqueness condition holds. Then, for any Λ ⊂ Z^{d} and ω ∈ Ω,we have

|E_{Λ}^{ω}(f ; g)| ≤ ∑_{k,l∈Λ} Var_{k}(f)D_{kl}Var_{l}(g)  (7.0.3)

with

D_{kl} ≡ ∑_{n=0}^{∞} C_{kl}^{n}.  (7.0.4)

\underline{Remark} 7.2: One can see as in section 5.4.4 that Dobrushin uniqueness condition is satisfied when for a sufficiently small positive real number β_{0}

||Φ||_{B_{2}} < β_{0}

but that this condition is not necessary ( see [88], [41] for examples of possibly large potentials and for some other types of potentials see [74] , [35] ). Theorem 7.1 shows that the local Gibbs measures satisfy a property of decay of correlations, the crucial step to obtain a logarithmic Sobolev inequality in the previous parts. In order to use it to prove a logarithmic Sobolev inequality, we shall first prove an auxiliary lemma which shall be essential to get sweeping out relations. To this end, let us denote as in section 5.4.3,

ξ_{i,Λ}(ω) ≡ \frac{dE_{Λ}^{ω^{(i)}}}{dE_{Λ}^{ω}} = \frac{e^{−\{U_{Λ}(ω^{(i)})−U_{Λ}(ω)\}}}{[Z_{Λ}^{ω^{(i)}}/Z_{Λ}^{ω}]}  (7.0.5)

with ω_{Z^{d}\i}^{(i)} = ω_{Z^{d}\i} and ω_{i}^{(i)} = −ω_{i}, then

Lemma 7.3 If Dobrushin uniqueness condition holds, for any Λ ⊂⊂ Z^{d} and all i ∈ Λ^{c}, j ∈ Λ, the quantity

η_{ij} ≡ sup_{Λ⊂⊂Z^{d}} Var_{j}(E_{Λ\j}ξ_{i,Λ})  (7.0.6)

satisfies

sup_{i} max ( ∑_{j∈Z^{d}} η_{ij} , ∑_{j∈Z^{d}} η_{ji}) < ∞.  (7.0.7)

Proof : To estimate the quantities Var_{j}(E_{Λ\j}ξ_{i,Λ}), let us remark that

|E_{Λ\j}^{ω}ξ_{i,Λ} − E_{Λ\j}^{ω^{(j)}}ξ_{i,Λ}| ≤ |E_{Λ\j}^{ω} (ξ_{i,Λ}; ξ_{j,Λ\j}) | + Var_{j}(ξ_{i,Λ}).  (7.0.8)

Using theorem 7.1, we obtain

|E_{Λ\j}^{ω} (ξ_{i,Λ}; ξ_{j,Λ\j}) | ≤ ∑_{k,l∈Λ\j} Var_{k}(ξ_{i,Λ})D_{kl}Var_{l}(ξ_{j,Λ\j}).  (7.0.9)

Consequently, to bound (7.0.9) as well as the second term in the right hand side of (7.0.8), we have to bound Var_{k}(ξ_{i,Λ}). To this end, let us go back to the explicit definition (7.0.5), to obtain

Var_{k}(ξ_{i,Λ}) ≤ ∥Z_{Λ}^{ω}/Z_{Λ}^{ω^{(i)}} ∥_{∞} Var_{k}(e^{−{U_{Λ}(ω^{(i)})−U_{Λ}(ω)}}) ≤ C̃_{ik}  (7.0.10)

where we have set

C̃_{ik} ≡ 2e^{2||Φ||} ∑_{X∋i,k} Var_{X}(Φ_{X}).

To use this bound, we note first that C̃_{ik} is summable in i or k. In fact, changing the order of summation according to Fubini’s theorem for non negative variables and since Var_{X}(Φ_{X}) ≤ 2||Φ_{X}||_{∞}, we get

\frac{1}{2}e^{−2||Φ||} ∑_{k∈Z^{d}} C̃_{ik} = ∑_{k∈Z^{d}} ∑_{X∋i,k} Var_{X}(Φ_{X}) ≤ 2 ∑_{X∋i} |X| · ||Φ_{X}||_{∞} ≤ 2||Φ||_{B_{2}}

(7.0.11)

with the right hand side finite with our hypothesis. We can thus bound (7.0.9) as follows

|E_{Λ\j}^{ω} (ξ_{i,Λ}; ξ_{j,Λ\j}) | ≤ ∑_{k,l∈Λ\j} C̃_{ik}D_{kl}C̃_{jl}.  (7.0.12)

Since under the Dobrushin uniqueness condition, the family D_{kl} is summable with respect to k, l according to theorem 7.1, we deduce from (7.0.10) and (7.0.11) that the right hand side of (7.0.12) is also summable over i and j. Combining this remark with the bounds (7.0.8) and (7.0.9) completes the proof. ⋄ Lemma 7.3 is the key to the following theorem.

Theorem 7.4 Assume that the Dobrushin uniqueness condition is satisfied for a system with potential Φ ∈ B_{2}. Then, there exists a constant c ∈ (0, ∞) such that for any Λ ⊂ Z^{d} and ω ∈ Ω, we have

E_{Λ}^{ω}flog (\frac{f}{E_{Λ}^{ω}f}) ≤ c E_{Λ}^{ω}|∇_{Λ}f^{\frac{1}{2}} |^{2}  (7.0.13)

for any non negative function f such that the right hand side is well defined.

E. Laroche [59] proved a similar result for exponentially fast decreasing interactions. (Also, note that the method used in [70] proves similar results for exponentially decaying interactions.) Here we present a proof based on lemma 7.3 and a martingale expansion of relative entropy, (introduced in [64]). Proof : Let us consider a lexicographical order {i_{k}, k = 1, .., N = |Λ|} of Λ and set Λ_{1} = {i_{1}}, Λ_{n+1} = Λ_{n} ∪ {i_{n+1}}. Let E_{n} = E_{Λ_{n}}, and, for a non negative measurable function f, set f_{n} = E_{n}f, f_{0} = f. We then deduce from lemma 7.5 applied with cubes reduced to single point sets that

E_{N}f log \frac{f}{E_{N}f} ≤ c_{0} ∑_{n=1}^{N} E_{N}|∇_{i_{n}}fn−1^{\frac{1}{2}}|^{2}  (7.0.14)

with c_{0} the smallest constant in the log-Sobolev inequalities satisfied by all the local Gibbs measures restricted to a single spin σ algebra related to points {i_{k}, 1 ≤ k ≤ N}, uniformly with respect to boundary conditions. Applying lemma 7.3 and proceeding as in section 5.4.3, it is not difficult to see that there exist non negative coefficients α_{ij}^{(Λ)} so that for any n ∈ {1, .., N},

|∇_{i_{n+1}} (E_{n}f)^{\frac{1}{2}} | ≤ ∑_{j∈Λ_{n}∪\{i_{n+1}\}} α_{i_{n+1}j}^{(Λ_{n})} (E|∇_{j}f^{\frac{1}{2}} |^{2})^{\frac{1}{2}}  (7.0.15)

with

γ_{ij} ≡ sup_{Λ⊂⊂Z^{d}} α_{ij}^{(Λ)}  (7.0.16)

satisfying

γ ≡ sup_{i} max ( ∑_{j∈Z^{d}} γ_{ij}, ∑_{j∈Z^{d}} γ_{ji}) < ∞.  (7.0.17)

It is then easy to deduce from (7.0.14) that

E_{N}f log \frac{f}{E_{N}f} ≤ c_{0} · γ^{2}E_{N}|∇_{Λ_{N}} f^{\frac{1}{2}} |^{2}.  (7.0.18)

Indeed, inequality (7.0.15) gives

|∇_{i} (E_{n}f)^{\frac{1}{2}} |^{2} ≤ ( ∑_{j∈Λ_{n}∪\{i\}} α_{ij}^{(Λ_{n})}) ∑_{j∈Λ_{n}∪\{i\}} α_{ij}^{(Λ_{n})}E|∇_{j}f^{\frac{1}{2}} |^{2} (7.0.19)

≤ γ · ∑_{j∈Λ_{n}} γ_{i_{n}j}E_{n}|∇_{j}f^{\frac{1}{2}} |^{2}

so that

∑_{n=1}^{N} E_{N}|∇_{Y_{n}}fn−1^{\frac{1}{2}}|^{2} ≤ γ ∑_{n=1}^{N} · ∑_{j∈Λ_{n}∪\{i\}} γ_{ij}E_{N}|∇_{j}f^{\frac{1}{2}} |^{2}  (7.0.20)
≤ γ^{2} ∑_{j∈Λ_{N}} E_{N}|∇_{j}f^{\frac{1}{2}} |^{2} = γ^{2}E_{N}|∇_{Λ_{N}} f^{\frac{1}{2}} |^{2}.

(7.0.15) thus implies the desired estimate. ⋄

7.0.6 Martingale expansions

In this section we present a different useful way of organizing the proofs using a martingale expansion of relative entropy(introduced first in [64]; see also [70]). We consider spin variables with values in a finite set or a smooth connected Riemaniann manifold M. Let L_{0} ∈ 2IN and Y_{k} ≡ Y_{0} + L_{0} · y_{k} be the translation of the cube Y_{0} = [−L_{0}/2, L_{0}/2]^{d} centered at the origin by the vector L_{0}y_{k}, y_{k} ∈ ZZ^{d}. The vectors (y_{k}, k ∈ IN) are ordered according to a lexicographic order compatible with the distance d(·, ·) on ZZ^{d}. We define a sequence of finite subsets of ZZ^{d} by Λ_{1} ≡ Y_{1} and Λ_{n+1} ≡ Λ_{n}∪Y_{n+1}. We denote in short E_{n} ≡ E_{Λ_{n}}. For a continuously differentiable function f, we set f_{0} ≡ f and f_{n+1} ≡ E_{n+1}f_{n} = E_{n+1}f. We then have the following

Lemma 7.5 There exists a constant c_{0} ∈ (0, ∞) depending only on the size of Y_{0} such that for any N ∈ IN and any continuously differentiable non negative function f, we have

E_{N}f log \frac{f}{E_{N}f} ≤ c_{0} ∑_{1}^{N} E_{N}|∇_{Y_{n}}fn−1^{\frac{1}{2}}|^{2}.  (7.0.21)

Proof : Let us first notice that

E_{n} (f_{n−1} log \frac{f_{n−1}}{E_{n}f_{n−1}} ) = E_{n,Y_{n}} (f_{n−1} log \frac{f_{n−1}}{E_{n,Y_{n}}f_{n−1}} )  (7.0.22)

with E_{n,Y_{n}} the restriction of E_{n} to the σ-algebra Σ_{Y_{n}}. According to lemma 7.6 below, at any point ω ∈ Ω, the measure E_{n,Y_{n}}^{ω} satisfies a logarithmic Sobolev inequality with a constant c_{0} ∈ (0, ∞) independent of ω ∈ Ω and n ∈ IN. Consequently, we get

E_{n} (f_{n−1} log \frac{f_{n−1}}{E_{n}f_{n−1}} ) ≤ c_{0} E_{n}|∇_{Y_{n}}fn−1^{\frac{1}{2}}|^{2}  (7.0.23)

Using this inequality with (7.0.22), we obtain lemma 7.5. It thus remains to prove the

Lemma 7.6 Let c_{Y,X} be the constant in the logarithmic Sobolev inequality for the restriction E_{X}|Σ_{(X\Y)^{c}} of E_{X} to the σ-algebra Σ_{(X\Y)^{c}}. Then, for any continuously differentiable non negative function f localized in a set Y ⊂ X, we have

E_{X}f log \frac{f}{E_{X}f} ≤ c_{Y,X} E_{X}|∇_{Y} f^{\frac{1}{2}} |^{2}  (7.0.24)

with

0 < c_{Y,X} < c̃_{0} · e^{4||Φ||·|Y |}  (7.0.25)

if c_{0} is the constant in the logarithmic Sobolev inequality for the probability measure ν.

86 7. LSI for spin systems in long range interaction This lemma is a direct consequence of property 4.6. Indeed, the probability measure E_{X}^{ω} is absolutely continuous with respect to the probability measure Ẽ_{X,Y}^{ω} ≡ E_{X}^{ω}(e^{U_{Y}} ·)/E_{X}^{ω}(e^{U_{Y}}) = δ_{ω_{X^{c}}} ⊗ν_{Y} ⊗μ_{X\Y}^{ω} if ν_{Y} is the product probability measure on (x_{i}, i ∈ Y). The corresponding density ρ_{Y,X}^{ω}(ω̃_{Y}) ≡ \frac{dE_{X}^{ω}}{d_{E_{X,Y}^{ω}}} satisfies

e^{−2||Φ||·|Y |} ≤ ρ_{Y,X}^{ω}(ω̃_{Y}) ≤ e^{+2||Φ||·|Y |}  (7.0.26)

so that property 4.6 provides the desired estimate.

⋄

Markov semi-group in infinite volume, ergodic properties

In this chapter, we study Markov semi-groups acting on functions of infinitely many variables of M^{ZZ^{d}}. We first construct them as limits of semi-groups described in section 5.1 with localized potentials. This construction is important to insure that such semi-groups are Feller continuous, but also to be able to approximate them by Markov semi-group in finite volume (see the exponential approximation property of theorem 8.2) which are easier to study. Such a construction can be found in the literature in [59] (for the continuous setting) and [62], [94] (for the discrete setting); see also the references given there. We then study the uniform ergodicity of these infinite volume semi-groups when the corresponding Gibbs measure satisfies a logarithmic Sobolev inequality. We show that they converge uniformly towards this Gibbs measure with an exponential rate. This result is actually part of the so-called equivalence theorem which states equivalence between such a uniform convergence of the semi-groups with a logarithmic Sobolev property of the Gibbs measure, but also with other properties of the Gibbs measure such as spectral gap inequality. We discuss this theorem in the last section of this chapter.

8.1 Construction of Markov semi-groups in infinite volume

In this section, we present a construction of Markov semi-groups in infinite volume when M is a finite set (the continuous setting being left as an exercise to the reader).

Let Ω = M^{ZZ^{d}} be the configuration set and {E_{X}^{ω}}_{X⊂⊂ZZ^{d},ω∈Ω} be a local specification constructed with the product measure μ_{0} = ν^{⊗ZZ^{d}} and a interaction potential Φ with finite range. We shall use the notations of the chapter 5 (see section 5.1).

We can easily define, for X ⊂⊂ ZZ^{d}, finite volume semi-groups by their generators

L_{Λ} = ∑_{j:X+j⊂Λ} L_{X+j}

with

L_{X+j}f = E_{X+j}f − f.

We denote by P_{t}^{(Λ)} the corresponding semi-group.

Exercise 8.1 Show that for any subset Λ the operator L_{Λ} formally given above is well defined on the space of functions f for which the following semi-norm is finite

|||f||| ≡ ∑_{j∈ZZ^{d}} ||∇_{j}f||_{u}

were ∇_{j}f ≡ f − ν_{j}f.

The goal of this section is to prove the following result

Theorem 8.2 For any local function f, the following limit in the uniform norm exists

P_{t}f := lim_{Λ↑ZZ^{d}} P_{t}^{(Λ)}f

and defines a Feller-continuous Markov semi-group on C(Ω).

Moreover, we have the following exponential approximation property : for any A ∈ IR^{+}, there exists a constant B ∈ [1, ∞) depending only on A such that if Λ ⊂ ZZ^{d} is sufficiently large, contains Λ(f) and dist(Λ(f), Λ^{c}) ≥ Bt, then

||P_{t}f − P_{t}^{(Λ)}f||_{∞} ≤ e^{−At}|||f|||.  (8.1.1)

Proof :

We will show that for any couple (Λ_{1}, Λ_{2}) ∈ (ZZ^{d})^{2}, Λ_{1} ⊂ Λ_{2} so that Λ_{2}\Λ_{1} is a cube of a given size, the semi-groups P^{1} := P^{(Λ_{1})} and P^{2} := P^{(Λ_{2})} satisfy

||P_{t}^{1}f − P_{t}^{2}f||_{∞} ≤ e^{−At−AN}|||f|||  (8.1.2)

if

dist(Λ(f), Λ_{2}\Λ_{1}) ≥ Bt

for a finite constant B depending only on A for

N = [\frac{dist(Λ(f), Λ_{2}\Λ_{1})}{R + diam(X)}]

with the square bracket denoting the entire part.

To prove this estimate, note that

P_{t}^{2}f − P_{t}^{1}f = ∫_{0}^{t} ds\frac{d}{ds}P_{t−s}^{1}P_{s}^{2}f = ∫_{0}^{t} dsP_{t−s}^{1}(L_{2} − L_{1})P_{s}^{2}f (8.1.3)

where we have denoted in short L_{i} := L_{Λ_{i}} for i = 1, 2.

From (8.1.3), we immediately deduce that

||P_{t}^{2}f − P_{t}^{1}f||_{∞} ≤ ∫_{0}^{t} ds||(L_{2} − L_{1})P_{s}^{2}f||_{∞}
≤ C_{0} ∫_{0}^{t} ds ∑_{j∈Λ_{2}\Λ_{1}} ||∇_{j}P_{s}^{2}f||_{∞}  (8.1.4)

with a finite constant C_{0} depending only on the size of Λ_{2}\Λ_{1} and the interaction Φ.

We are hence naturally interested in the quantities ∇_{j}P̃_{s}f for a finite volume semi-group P̃. We present below a simple study of these quantities.

Let us first remark that

∇_{j}P̃_{t}f − P̃_{t}∇_{j}f = ∫_{0}^{t} ds\frac{d}{ds}P̃_{t−s}∇_{j}P̃_{s}f
= ∫_{0}^{t} dsP̃_{t−s}[∇_{j}, L̃]P̃_{s}f  (8.1.5)

with [∇_{j}, L̃] = ∇_{j}L̃ − L̃∇_{j}. We also point out that, due to the local structure of the generator, we have

[∇_{j}, L̃] = ∑_{k:d(k,j)≤R+diamX}[∇_{j}, L_{X+k}].  (8.1.6)

Moreover, one can easily see that for any smooth function F, we have

||[∇_{j}, L_{X+k}]F ||_{∞} ≤ ∑_{l∈{j}∪X+k} α_{jl}||∇_{l}F ||_{∞}  (8.1.7)

for uniformly bounded constants α_{jl}. We deduce from (8.1.5), (8.1.6) and (8.1.7) that

||∇_{j}P̃_{t}f||_{∞} ≤ ||∇_{j}f||_{∞} + ∫_{0}^{t} ds ∑_{l} D_{jl}||∇_{l}P̃_{s}f||_{∞}  (8.1.8)

with a matrix D with uniformly bounded coefficients such that

D_{jl} = 0  if  d(j, l) > R + diam(X).

Since ∇_{j}f = 0 if j does not belong to Λ(f), we can use (8.1.8) inductively to conclude that

||∇_{j}P̃_{t}f||_{∞} ≤ ∑_{n=N_{j}}^{∞} \frac{t^{n}}{n!} ∑_{l} D_{jl}^{(n)}||∇_{l}f||_{∞}  (8.1.9)

where

N_{j} = [\frac{dist(j, Λ(f))}{R + diam(X)}]

and D_{jl}^{(n)} is the jl^{th} entry of the matrix D^{n} the n-th power of the matrix D. We can estimate these coefficients by noting that if C is a finite constant satisfying

D_{jl} ≤ \frac{C}{[2(R + diam(X)) + 1]^{d}}

for any (jl), we have

D_{jl}^{(n)} ≤ C^{n}.

As a consequence,

||∇_{j}P̃_{t}f||_{∞} ≤ ∑_{n=N_{j}}^{∞} \frac{t^{n}}{n!} C^{n}|||f||| ≤ \frac{(tC)^{N_{j}}}{N_{j}!} e^{tC}|||f|||.  (8.1.10)

Observing that for any n ∈ IN^{∗},

n! > e^{n log n−2n}

and choosing B ∈ (0, ∞) such that

2 − log B + log C + \frac{C}{B} ≤ −2A

we conclude that for

N = inf_{j} N_{j} = [\frac{dist(Λ_{2}\Λ_{1}, Λ(f))}{R + diam(X)}] ≥ Bt

we have

||∇_{j}P̃_{t}f||_{∞} ≤ e^{−At−AN}|||f|||.  (8.1.11)

Using (8.1.11) with P̃ = P^{2} in (8.1.4), we deduce that for any sequence {P^{(Λ_{i})}, i ∈ IN} of Markov semi-groups such that the Λ_{i}’s are a Van Hove sequence (i.e are constructed by addition of the translation of a given cube), {P^{(Λ_{i})}f, i ∈ IN} is Cauchy for any local function f. Hence, the limit

P_{t}f = lim_{Λ↑ZZ^{d}} P_{t}^{(Λ)}f

exists for any local function f. Furthermore, if Λ = Λ_{N}, we have

||P_{t}f − P_{t}^{(Λ)}f||_{∞} ≤ ∑_{n=N}^{∞} ||P_{t}^{(Λ_{n+1})}f − P_{t}^{(Λ_{n})}f||_{∞}

≤ tC_{0} ∑_{n=N}^{∞} ∑_{j∈Λ_{n}\Λ_{n−1}} ||∇_{j}P_{t}^{(Λ_{n})}f||_{∞}

≤ tC_{0}e^{−At} ∑_{n=N}^{∞} ∑_{j∈Λ_{n}\Λ_{n−1}} e^{−AN_{n}}|||f|||

≤ te^{−At} (C_{0}|Λ_{0}| ∑_{L=N}^{∞} L^{d−1}e^{−AL}) |||f||| (8.1.12)

with the notation

N_{n} = [\frac{dist(Λ_{n}\Λ_{n−1}, Λ(f))}{R + diam(X)}] .

This last estimate completes the proof of the exponential approximation. As a consequence, since in finite volume P^{(Λ)} is strongly continuous, the same is true for P. Since the local functions are dense in C(Ω), P extends naturally to a Feller-continuous Markov semi-group with generator

L = ∑_{j∈ZZ^{d}} L_{X+j}.

Exercise 8.3 Generalize theorem 8.2 to the continuous case, i.e. to the case where M is a smooth connected Riemannian manifold and L_{Λ} has the form

L_{Λ} = ∑_{i∈ZZ^{d}}(Δ_{i} + ∇_{i}H_{Λ}.∇_{i}).

where H_{Λ} ≡ ∑_{X⊂Λ} Φ_{X}. Hint : Follow the arguments of the proof of theorem 8.2.

Exercise 8.4 In the case where the potential Φ is not of finite range, but satisfies

||Φ||_{B_{2}} ≡ sup_{i∈ZZ^{d}} ∑_{X∋i} |X| · ||Φ_{X}||_{u} < ∞

show that the following exponential approximation property holds ; for any A > 0 and any δ > 0, there exists a finite constant C so that

||P_{t}f − P_{t}^{(Λ)}f||_{∞} ≤ e^{−At}|||f|||

provided that

d(Λ, Λ(f)^{c}) ≥ Ce^{δt}.

8.2 Uniform ergodicity of Markov semi-groups in infinite volume

In this section, we summarize the main links between the log-Sobolev inequality and the uniform ergodicity of Markov semi-groups in infinite volume, (see [55],[56] and [92]-[95]).

Again, we are given a local specification {E_{X}^{ω}}_{X⊂⊂ZZ^{d},ω∈Ω} defined by a uni-

form product measure μ_{0} = ν^{⊗ZZ^{d}} and a finite range interaction potential Φ. Then, the following relations are satisfied

Theorem 8.5 Assume that we can define a Gibbs measure μ in infinite volume and that it satisfies the logarithmic Sobolev inequality with a coefficient c.

In finite volume, assume that there exists a finite constant C_{1} such that for any non negative function f, any Λ ⊂⊂ ZZ^{d},

||P_{1}^{(Λ)}f||_{∞} ≤ e^{C_{1}|Λ|}E_{Λ}[f].  (8.2.13)

Moreover, assume that the finite volume exponential approximation is satisfied, that is that for any local function f, any A ∈ IR^{+}

||P_{t}f − P_{t}^{(Λ)}f||_{∞} ≤ e^{−At}|||f|||  (8.2.14)

provided that for sufficiently large Λ,

dist(Λ(f), Λ^{c}) ≥ Bt

for a finite constant B depending only on A.

Then, for any θ ∈ (0, 1), there exist a finite constant C(θ, Λ(f)) and a positive real number m ∈ [1/c, ∞[ such that for any t ≥ 0,

||P_{t}f − μf||_{∞} ≤ C(θ, Λ(f))e^{−θmt}|||f|||.

\underline{Remark} 8.6: If \{E_{X}, X ⊂⊂ ZZ^{d}\} satisfy a classical Nash inequality with constants growing at most exponentially with the volume, (8.2.13) is satisfied according to theorem 3.3. Proof : Let us first note that the following decomposition holds

|P_{t}f − μf| ≤ |P_{t}^{(Λ)}f − μf| + |P_{t}^{(Λ)}f − P_{t}f|.  (8.2.15)

The second term will be estimated by the exponential approximation property. For the first term on the right hand side of (8.2.15), we note that, by Hölder’s inequality, for any q ≥ 1,

|P_{t}^{(Λ)}f − μf| = |P_{1}^{(Λ)}(P_{t−1}^{(Λ)}f − μf)|
≤ |P_{1}^{(Λ)}|P_{t−1}^{(Λ)}f − μf|^{q}|^{\frac{1}{q}}
≤ e^{\frac{C_{1}|Λ|}{q}} (E_{Λ}|P_{t−1}^{(Λ)}f − μf|^{q})^{\frac{1}{q}}  (8.2.16)

where we used hypothesis (8.2.13). Moreover, by definition of P_{t}^{(Λ)}, for any time t, P_{t}^{(Λ)}f − μf is Σ_{Λ^{R}} measurable if f is when Λ^{R} = {i, d(i, Λ) ≤ R}. Hence, (using the assumption that the potential Φ is bounded and of finite range), we can find a finite constant C_{2} so that

E_{Λ}|P_{t−1}^{(Λ)}f − μf|^{q} ≤ e^{C_{2}|∂Λ|}μ(P_{t−1}^{(Λ)}f − μf)^{q}.

Noting as well that

(μ|P_{t−1}^{(Λ)}f − μf|^{q})^{\frac{1}{q}} ≤ ||P_{t−1}^{(Λ)}f − P_{t−1}f||_{∞} + (μ|P_{t−1}f − μf|^{q})^{\frac{1}{q}}  (8.2.17)

we deduce from (8.2.16) that

||P_{t}^{(Λ)}f − μf||_{∞} ≤ e^{\frac{C_{1}|Λ|+C_{2}|∂Λ|}{q}} \{(μ|P_{t−1}f − μf|^{q})^{\frac{1}{q}} + ||P_{t−1}^{(Λ)}f − P_{t−1}f||_{∞}\}
+||P_{t}^{(Λ)}f − P_{t}f||_{∞}.  (8.2.18)

By the exponential approximation property, we conclude that if f is a local function and Λ is chosen sufficiently large, so that Λ(f) ⊂ Λ and its diameter is of order t ≥ 1,

||P_{t}^{(Λ)}f − μf||_{∞} ≤ e^{\frac{C′_{1}(t+1)t^{d−1}}{q}} (μ(P_{t−1}f − μf)^{q})^{\frac{1}{q}} + e^{\frac{C′_{1}(t+1)t^{d−1}}{q}} e^{−At}|||f|||

for a finite constant C′_{1}. If moreover μ satisfies a logarithmic Sobolev inequality with a coefficient c < ∞, let us recall that theorem 4.1 implies that for any θ ∈ (0, 1) such that q ≤ q(\frac{t−1}{θt}, 2, c) = 1 + e^{\frac{2(1−θ)t−2}{c}},

(μ|P_{t−1}f − μf|^{q})^{\frac{1}{q}} ≤ (μ(P_{θt}f − μf)^{2})^{\frac{1}{2}} ≤ e^{−mθt}||f − μf||_{2}

where m is the spectral gap of L which is bounded below by c^{−1} (see theorem

4.9). With such a choice of q, the factor e^{\frac{C′_{1}(t+1)t^{d−1}}{q}} is uniformly bounded in t and we obtain the desired estimate.

⋄

Exercise 8.7 Extend the theorem to the case where Φ ∈ B_{2}.

8.3 Equivalence Theorem

We consider a local specification {E_{Λ}^{ω}}_{Λ⊂⊂ZZ^{d},ω∈Ω} defined by a product measure

μ_{0} = ν^{⊗ZZ^{d}} and an interaction potential Φ = (ϕ_{X})_{X⊂⊂ZZ^{d}} with finite range R. When needed, we shall indicate more explicitly the potential defining the local specification by the following notation {E_{Λ,Φ}^{ω}}_{Λ⊂⊂ZZ^{d},ω∈Ω}.

For Λ ⊂⊂ ZZ^{d}, we shall denote P_{t}^{(Λ,ω)} = e^{tL_{Λ,ω}} the Markov semi-group generated by

L_{Λ,ω} = δ_{ω_{Λ^{c}}} (∑_{i∈Λ} L_{i})

that is the semi-group acting on the variables {σ_{i}, i ∈ Λ} and with fixed boundary conditions. There, we used the notation

(δ_{ω_{Λ^{c}}} L_{i})f(ω̃) = E_{{i}}f(ω_{Λ∩{i}^{c}} ∘ ω_{Λ^{c}∩{i}^{c}}) − f(ω̃_{Λ} ∘ ω_{Λ^{c}})

in the discrete case and

(δ_{ω_{Λ^{c}}} L_{i})f(ω̃) = (Δ_{i} − ∇_{i}U_{i}(ω̃_{Λ} ∘ ω_{Λ^{c}}).∇_{i})f(ω̃_{Λ} ∘ ω_{Λ^{c}})

in the continuous setting.

Now, we present the result from [92] relating mixing conditions and ergodicity.

Theorem 8.8 The following conditions are equivalent (i) \underline{Strong mixing conditions} There exists M ∈ (0, ∞) such that for any Λ ⊂⊂ ZZ^{d}, any ω ∈ Ω, and any

local functions f and g localized in Λ,

|E_{Λ}^{ω}(f, g)| ≤ e^{−Md(Λ(f),Λ(g))}|||f|||.|||g|||.

(ii)\underline{Complete analyticity condition :}

For any n ∈ IN and any potentials {ψ_{k}, k = 1, .., n} with finite range, the map

(z_{1}, .., z_{n}) → E_{Λ,Φ+∑_{k=1}^{n} z_{k}ψ_{k}}^{ω}(f)

is analytic in a neighborhood ∩{|z_{k}| ≤ ϵ} of the origin for some ϵ independent of Λ ⊂⊂ ZZ^{d}, ω ∈ Ω and of the local function f.

(iii)\underline{Uniform spectral gap inequality :}

There exists a constant m ∈ (0, ∞) such that for any Λ ⊂⊂ ZZ^{d}, and all ω ∈ Ω,

m E_{Λ}^{ω}(f − E_{Λ}^{ω}f)^{2} ≤ E_{Λ}^{ω}(|∇f|^{2})

for any function f for which the right hand side of the above inequality is well defined and finite.

(iv)\underline{Uniform logarithmic Sobolev inequality :}

There exists a constant c ∈ (0, ∞) such that for any Λ ⊂⊂ ZZ^{d}, and all ω ∈ Ω,

E_{Λ}^{ω}(f log \frac{f}{E_{Λ}^{ω}f}) ≤ cE_{Λ}^{ω}(|∇f|^{2})

for any non negative function f for which the right hand side of the above inequality is well defined and finite.

Remarks :

1) The equivalence between (i) and (ii) is due to Dobrushin and Shlosman [19] who proved the equivalence between some 14 conditions encountered in statistical mechanics. The proof of the equivalence with (iii) and (iv) was given first in [92].

2) Similar assumptions remain equivalent if one replaces the sequence of finite sets by a Van Hove sequence. In this setting, the equivalence between (i) and (ii) is due to E. Olivieri and P. Picco [72], the others to S.Lu and H.T. Yau [64] as well as F.Martinelli and E. Olivieri [70]. These results allow to extend the domain of validity of logarithmic Sobolev inequalities to a larger class of potentials.

3) The implication (iv) gives (iii) was already seen and is a direct consequence of the definition of logarithmic Sobolev inequality . The converse implication is in general rather unusual. It is intimately related to the assumption that the Spectral Gap inequalities are assumed to be obtained uniformly with respect to the finite sets Λ and the boundary conditions.

4) The implication (i) gives (iv) can be proved as discussed in details in the previous chapters.

5) The fact that (iii) implies (i) is due to the exponential approximation property. Indeed, we have

E_{Λ}^{ω}(f, g) = E_{Λ}^{ω}(fg) − E_{Λ}^{ω}(f)E_{Λ}^{ω}(g)
= E_{Λ}^{ω}(P_{t}^{(Λ,ω)}(fg)) − E_{Λ}^{ω}(f)E_{Λ}^{ω}(g)

Using the exponential approximation property, we get

|P_{t}^{(Λ,ω)}fg − P_{t}^{(Λ,ω)}fP_{t}^{(Λ,ω)}g| ≤ e^{−At}|||f|||.|||g|||

as long as

d(Λ(f), Λ(g)) ≈ Bt

for some B = B(A) ∈ (0, ∞). As a consequence,

|E_{Λ}^{ω}(f, g)| ≤ |E_{Λ}^{ω}(P_{t}^{(Λ,ω)}f, P_{t}^{(Λ,ω)}g)| + e^{−At}|||f|||.|||g|||
≤ (E_{Λ}^{ω}(P_{t}^{(Λ,ω)}f − E_{Λ}^{ω}f)^{2}E_{Λ}^{ω}(P_{t}^{(Λ,ω)}g − E_{Λ}^{ω}g)^{2})^{\frac{1}{2}} + e^{−At}|||f|||.|||g|||

Using the spectral gap inequality, we obtain (i).

Disordered systems ; uniform ergodicity in the high temperature regime

The simplest example of disordered systems we consider is described by the following formal Hamiltonian

H(σ) = ∑_{|i−j|=1} J_{ij}σ_{i} · σ_{j}

with the spins σ_{i} in {−1, +1} for the Ising type models, and for continuous models such as the rotator, the spins take on values in a smooth manifold such as an N-dimensional sphere. (For example if N = 2, one has the representation

σ_{i}.σ_{j} = cos(ϕ_{i} − ϕ_{j})

with ϕ_{i} ∈ [0, 2π].)

In this chapter, the J_{ij} will be taken at random. It is assumed that the J_{ij} are independent and identically distributed.

One can easily imagine that in such systems on the infinite lattice ZZ^{d}, large regions will have strong couplings. Thus the interaction in such regions will be of low temperature type. However, if the inverse of the temperature β, (used to scale the Hamiltonian in the formal expresion for the Gibbs measure), is sufficiently small, ”most” of the spins in the system will effectivelly interact weakly. The system is therefore mostly of the type studied previously, but exhibits with probability one (with respect to the realization of the couplings) large regions of strong interaction. As a consequence of this general picture, the Glauber dynamics should still converge at high temperature towards the unique Gibbs measure but its convergence will be slowed down due to these regions with strong couplings.

To illustrate these ideas, we shall first describe the proof given in [104] of the absence of spectral gap at any temperature for models in which the J_{ij} can be as large as one wishes with positive probability.

Secondly, we shall bound almost surely the growth of constant in logarithmic Sobolev inequality for local Gibbs measures in dimension 2 as a function of the volume and deduce a stretched exponential decay of the dynamics towards equilibrium (that is a decay with rate going to zero as the exponential of −t^{θ} for some θ ∈ (0, 1)).

9.1 Absence of spectral gap for disordered ferromagnetic Ising model

Here, we restrict ourselves to the Ising model

σ_{i} ∈ {−1, +1}, Ω = {−1, +1}^{ZZ^{d}}.

We shall also assume d ≥ 2, We consider the Hamiltonian given, for any Λ ⊂⊂ ZZ^{d}, by

H_{Λ}^{ω}(σ) ≡ H_{Λ}^{ω}(J, σ) ≡ − ∑_{|i−j|=1,i,j∈Λ} J_{ij}σ_{i}σ_{j} + ∑_{|i−j|=1,i∈Λ,j∈Λ^{c}} J_{ij}σ_{i}ω_{j}.

where the couplings (J) = (J_{ij}, (i, j) ∈ ZZ^{d}) are independent identically distributed real valued random variables on a probability space (JJ, B_{J}, IP). We denote by IE the expectation under IP. We shall assume that IE[|J_{ij}|] is finite. We can associate to H_{Λ}^{ω} the local Gibbs measure E_{Λ,J}^{ω} on {−1, +1}^{Λ} given by

E_{Λ,J}^{ω}(dσ_{Λ}) = \frac{1}{Z_{Λ,J}^{ω}} e^{−βH_{Λ}^{ω}(σ)}dμ_{0}^{⊗|Λ|}(σ_{Λ})

with

dμ_{0}(σ) = \frac{1}{2}(δ_{σ=−1} + δ_{σ=+1}).

In case when the couplings (J) can take with positive probability values as small as one wishes, that is (IP(|J_{ij}| ≤ ϵ) > 0 for any ϵ > 0), and IEJ_{ij} = 0, for any β > 0 one can define with IP-probability one a unique Gibbs measure μ_{J} in infinite volume as a limit of the local Gibbs measures E_{Λ,J}^{ω} (see [39], [26], [31]). Indeed, with IP-probability one, the couplings J_{ij}) will be as small as needed in a set C_{L}^{l} = {i ∈ ZZ^{d}, |i_{1}| + |i_{2}| + .. + |i_{d}| ∈ [L, L + l]} with l as large as one wishes provided L is large enough (as a consequence of the Borel-Cantelli lemma). Thus, the expectation under E_{Λ,J}^{ω}, C_{L}^{l} ⊂ Λ, of functions localized in the sphere S_{L} = {i ∈ Z^{d}, |i_{1}| + |i_{2}| + .. + |i_{d}| ≤ L} will weakly depend on the boundary conditions ω (with a correction going to zero exponentially with l growing to infinity). Consequently, with IP-probability one, one can construct a Gibbs measure in infinite volume and it is unique. [This phenomenon is even more sharp when one deals with couplings (J) which can be null with positive probability; as soon as one can draw a closed loop surrounding the origin of null couplings, the expectation inside this loop is independent of the spins outside it and therefore does not depend on the choice of the local Gibbs measure E_{Λ}^{ω} as soon as Λ contains this loop.]

Despite this decay of correlations and uniqueness of Gibbs measures, we will show that the generators of the corresponding Glauber dynamics have no spectral gap. To this end, let us consider the Glauber dynamics generated by

L_{J}f = ∑_{i∈ZZ^{d}}(E_{{i}}f − f).

The construction of the associated infinite volume semi-group P^{J} can be done in a similar way as in nonrandom case considered before, (see [49]). We shall see that the following result is true.

Theorem 9.1 For any β > 0, if

IP(J_{ij} ≥ J) > 0  et IP(|J_{ij}| ≤ a) > 0  (9.1.1)

for a sufficiently large J > 0 and a constant a > 0 sufficiently small (depending on β), with IP-probability one,

inf_{f⊥1} \frac{μ_{J} (−L_{J}(f)f)}{μ_{J} (f − μ_{J}f)^{2}} = 0  (9.1.2)

e.g. the generator L_{J} has no spectral gap IP-almost surely.

Proof : Let us consider the configurations of the couplings (J) such that, if we denote Λ_{L} = [−L, L]^{d}, (J) is in the set

P_{k+Λ_{L}}^{J,a} = {∀(i, j) ⊂ k + Λ_{L} J_{ij} ≥ J and ∀i ∈ Λ_{L}, j ∈ ∂Λ_{L}, |J_{ij}| ≤ a}

with k ∈ ZZ^{d}. Here,

k + Λ_{L} = {(i, j) = k + (i′, j′); (i′, j′) ∈ Λ_{L}}.

We claim that, for any L ∈ IN, with IP-probability one, (J) belongs to P_{k+Λ_{L}}^{J,a} for some point k of the lattice. Indeed, IP(P_{k+Λ_{L}}^{J,a}) = IP(P_{Λ_{L}}^{J,a}) is strictly positive according to our hypotheses for any L ∈ IN and k ∈ ZZ^{d}. Consequently, Borel-Cantelli’s lemma implies that for any L ∈ IN,

IP( ⋃_{k∈ZZ^{d}} P_{k+Λ_{L}}^{J,a}) = 1.

We can assume without loss of generality that k is the origin and thus assume that (J) ∈ P_{Λ_{L}}^{J,a}.

For such a configuration of the couplings, we shall compare the restriction of the Gibbs measure μ_{J} to Σ_{Λ_{L}} with the measure E_{Λ_{L},J}^{0} with Dirichlet boundary conditions (e.g. ω = 0^{∂Λ_{L}} at the boundary of Λ_{L}).

9.2 Upper bound for LS constant and uniform ergodicity, d=2 99

In fact, for any function f localized in Λ_{L}, we have

\frac{μ_{J} (∑_{i}(∂_{i}f)^{2})}{μ_{J} (f − μ_{J}f)^{2}} ≤ e^{6aβ|∂Λ|} \frac{E_{Λ_{L},J}^{0} (∑_{i}(∂_{i}f)^{2})}{E_{Λ_{L},J}^{0} (f − μ_{Λ,J}^{0}f)^{2}} .  (9.1.3)

Moreover, in Λ_{L}, if βJ is large enough, we can use the results of Thomas [98]. We then find a constant α > 0 such that

inf_{f⊥1} \frac{E_{Λ_{L},J}^{0} (∑_{i}(∂_{i}f)^{2})}{E_{Λ_{L},J}^{0} (f − E_{Λ_{L},J}^{0}f)^{2}} ≤ C_{1}(J)e^{−Jβα|∂Λ_{L}|}.  (9.1.4)

(9.1.3) and (9.1.4) imply that

inf_{f⊥1} \frac{μ_{J} (−L_{J}(f)f)}{μ_{J} (f − μ_{J}f)^{2}} ≤ e^{(6a−Jα)β|∂Λ_{L}|}.  (9.1.5)

Since L can be taken as large as desired, we conclude that when 6a < Jα,

inf_{f⊥1} \frac{μ_{J} (−L_{J}(f)f)}{μ_{J} (f − μ_{J}f)^{2}} = 0  IPa.s.  (9.1.6)

⋄

Absence of spectral gap was also proved for the rotator model in [49] where the spins take on values in a unit circle and the interaction is described by the potential

Φ_{X} = {J_{ij} cos(φ_{i} − φ_{j}) if X = {ij} 0  otherwise  (9.1.7)

with the (J_{ij}) taking again arbitrarily large and small values with positive probability. Thomas’s estimates need then to be replaced, in order to estimate the spectral gap of the generator in the low temperature regions, by the use of Ginibre’s inequalities [33].

9.2 Upper bound for the constant of logarithmic Sobolev inequality in finite volume and uniform ergodicity, d=2

In this section, we prove an upper bound for the constant in logarithmic Sobolev inequality in finite volume in dimension 2 and show that it implies the uniform ergodicity for the dynamics of the corresponding infinite system. We describe the strategy developed in [49]. It relies on the controls obtained in [93],[94] and [95] and described in chapter 5. Under some additional but physically quite general assumptions, sharper results were obtained later by Cesi, Maes and Martinelli [11] giving optimal controls in any dimension. These results show that if the tail of the random variables (J_{ij}, (i, j) ∈ ZZ^{d}) decreases faster than exponentially, the log-Sobolev constant c(Λ) for the local Gibbs measure localized in Λ decreases very slowly and more precisely, when Λ = [−L, L]^{d},

c(Λ) ≤ exp cβ(log log L)^{d−1}(log L)^{\frac{d−1}{d}} IP.a.s.

However, when the tail of the random variables (J_{ij}, (i, j) ∈ ZZ^{d}) is exponential, c(Λ) decreases only polynomially with the volume. In this setting, the estimate obtained in [49] is on the right scale.

We restrict ourselves here to the discrete setting, the generalization to the continuous setting being straightforward and given in sufficient detail in [49]. We shall denote by Ω = M^{ZZ^{2}} the configuration space with a finite set M. (JJ, B_{J}, IP) will denote the probability space on which the external random coupling live. The Hamiltonian of the system is given by a potential Φ ≡ (Φ_{X})_{X∈ZZ^{d}} of realvalued measurable functions on JJ × Ω such that

- For any X ⊂ ZZ^{d}, the function Φ_{X}(J, ·) is continuous and Σ_{X} measurable.

- For any i ∈ ZZ^{d}, we have

∑_{{X∈F \atop X∋i}} ||Φ_{X}(J, ·)||_{∞} < ∞

-The family {Φ_{X}(·, ω), X ∈ ZZ^{d}} of random variables is mutually independent. Moreover, the random variables Φ_{X+j}(·, T_{j}ω) et Φ_{X}(·, ω) are identically distributed.

Furthermore, we shall assume that the interaction is of finite range, that is there exists a finite positive real number R such that Φ_{X} ≡ 0 when diam(X) > R, IP-almost surely.

The Hamiltonian of the system in finite volume Λ is then given by

H_{Λ}^{ω}(σ) = ∑_{X∩Λ≠∅} Φ_{X}(J, σ_{Λ} ∘ ω_{Λ^{c}})

where σ_{Λ} ∘ ω_{Λ^{c}} is the configuration described by σ inside the cube Λ and by ω outside Λ.

\underline{Remark} 9.2: In the previous section, we considered the particular case (but most commonly studied) where

Φ_{X}(J, σ) = J_{ij}σ_{i}σ_{j}

if X = {i, j} when |i − j| = 1 and Φ_{X} ≡ 0 otherwise.

We associate to Φ the local specification (E_{Λ,J}^{ω}, Λ ⊂⊂ ZZ^{d}) as before. We consider the dynamics generated by

Lf = ∑_{i}{E_{i+X}f − f}

for a finite set X of ZZ^{d}. The construction, for IP almost all J, of the semi-group P^{J} in infinite volume can be achieved as in chapter 8.1. We then obtain the following exponential approximation property.

Theorem 9.3 Assume that Φ ≡ {Φ_{X}}_{X∈F} is a finite range potential such that

sup_{X} sup_{i,j∈ZZ^{d}} IE[||∇_{j}ϕ_{X}||_{∞}^{K}] < ∞

for a finite real number K > d. Then, the limit

P_{t}^{J}f ≡ lim_{Λ↑ZZ^{d}} P_{t}^{J,Λ}f

exists IP-almost surely. Moreover, for any cube Λ, any A ∈ IR^{+} and any local function f, we have

||P_{t}^{J}f − P_{t}^{J,Λ}f||_{∞} ≤ e^{−At}D(J)|||f|||  (9.2.8)

provided that d(f, Λ^{c}) ≥ L̄ for a constant L̄ ≡ L̄(Λ(f), R, J) almost surely finite and d(f, Λ^{c})^{1−δ} ≥ Ctfor a finite constant C ∈ IR^{+} depending on J, Λ(f), A and for some δ ∈ (0, 1).

9.2.1 Bound on the log-Sobolev constant

The basic idea to control the log-Sobolev constant is to show that a decay of correlation property is satisfied by the local Gibbs measures E_{Λ,J}^{ω} but that it will depend on the size |Λ| of the finite volumes under consideration. More precisely, we prove the following

Property 9.4 Let Λ_{L} = [−L, L]^{d}.

Assume that

(H1) For any ξ ∈ IR, we have

IE exp{ξ||Φ̃^{(j)}(J, ·)||_{1}} < ∞  (9.2.9)

with

||Φ̃^{(j)}(J, ·)||_{1} ≡ ∑_{{X∈F \atop X∋j}} ||Φ_{X}(J, ·)||_{∞}.

(H2) There exists J_{0} > 0 sufficiently small so that

p_{1} ≡ sup_{X} IP { ||Φ_{X}||_{∞} > J_{0}}  (9.2.10)

belongs to (0, p_{c}^{b}(2, R)) with p_{c}^{b}(2, R) a universal critical percolation exponent.

If (H1) and (H2) are satisfied, there exist a finite constant r_{0} > 0 and a constant M ∈ (0, ∞) such that for almost all J, L sufficiently large, for any Λ ⊂ Λ_{L}, and any ω ∈ M^{|Λ_{L}|},

E_{Λ_{,}J}^{ω}(f, g) ≤ e^{−Md(Λ(f),Λ(g))}|||f|||.|||g|||

for any functions (f, g) localized in Λ_{L} such that

d(Λ(f), Λ(g)) ≥ r_{0} log L.

Moreover, r_{0} goes to zero when p_{1} goes to zero.

Using the result of exercise 5.7, one deduces from this property the following almost-sure bound on the log-Sobolev constants.

Theorem 9.5 Let Λ_{L} = [−L, L]^{d} for L ∈ IN. If (H1) and (H2) are satisfied, there exists a finite constant c_{0} > 0 such that IP almost surely, for L sufficiently large (L ≥ L(J) with L(J) being IP-a.s. finite)

c(E_{Λ_{L},J}^{ω}) ≤ exp{c_{0}(log L)^{d−1}}.

Moreover, c_{0} goes to zero when p_{1} does.

To complete our proof, we shall prove property 9.4. The proof is in fact a slight generalization of the computation presented in section 5.4.4. To simplify the notation, we assume R = 1, the generalization being given in details in [49]. Let us notice, following [31], that

E_{Λ,J}^{ω}(f, g) ≤ e^{4 sup_{j∈Λ} ∑_{X:j∈X} ||ϕ_{X}||_{∞}} ∑_{γ∈W (f,g)} ∏_{{i,j}∈γ} z_{i,j} (9.2.11)

with W (f, g) the set of paths in Λ connecting Λ(f) and Λ(g) and

z_{i,j} = \{ \begin{matrix}e^{4J_{0}} − 1 if |ϕ_{ij}| ≤ J_{0} \\ 1 otherwise. \end{matrix}

Under (H1), we know by Chebychev’s inequality that for any ϵ > 0, there exists a finite constant C_{ϵ} such that

P ( sup_{j∈Λ_{n}} ∑_{X:j∈X} ||ϕ_{X}||_{∞} ≥ \frac{ϵ}{4} log n) ≤ C_{ϵ}n^{−2}.

Hence, for any ϵ > 0, any sufficiently large n

sup_{j∈Λ_{n}} ∑_{X:j∈X} ||ϕ_{X}||_{∞} ≤ \frac{ϵ}{4} log n  (9.2.12)

almost surely, by the Borel-Cantelli lemma.

Moreover, according to Kesten [60] , if

λ_{j}(r) = inf_{γ∈W (f,g),d(Λ(f),Λ(g))≥r} \frac{card(\{i, j\} ∈ γ : ||ϕ_{ij}||_{∞} ≤ J_{0})}{card(\{i, j\} ∈ γ)}

and if

p_{1} = IE[||ϕ_{ij}||_{∞} ≥ J_{0}] < p_{c}^{b}(2, 1),

there exists r_{0}(p_{1}) > 0, r_{0}(p_{1}) → 0 when p_{1} → 0, such that for any η > 0,

IP[ ⋂_{n∈IN} ⋃_{n≥k} inf_{j∈Λ_{n}}{λ_{j}(r_{0} log n) ≥ η}] = 1.

With (9.2.11) and (9.2.12), we conclude that almost surely, for any n and all Λ ⊂ Λ_{n}, we have

E_{Λ,J}^{ω}(f, g) ≤ const.(e^{4J_{0}} − 1)^{ηd(Λ(f),Λ(g))}|||f|||.|||g|||

provided that d(Λ(f), Λ(g)) ≥ r_{0} log n.

⋄

9.2.2 Ergodicity in infinite volume

We shall deduce from theorem 9.5 a stretched exponential decay towards equilibrium of the Markov semi-group in infinite volume. We follow the steps of chapter 8.2. The theorem states as follows.

Theorem 9.6 Assume that conditions (H 1) and (H 2) are satisfied with p_{1} > 0 sufficiently small. Then, there exists θ ∈ (0, 1) such that for every local function f, with IP-probability one, we have

||P_{t}^{J}f − μ_{J}(f)||_{∞} ≤ C(Λ(f), J)e^{−t^{θ}}|||f|||  (9.2.13)

for any t ≥ T (Λ(f), J) and with almost surely finite positive random variables C(Λ(f), J) and T (Λ(f), J).

Proof : Let a local function f be given. Assume n_{0} large enough so that Λ(f) ⊂ Λ_{n_{0}} ≡ [−n_{0}, +n_{0}]^{2}. Let JJ_{0} ⊂ JJ be a measurable set with IP-probability one such that the conclusions of property 9.4 and theorem 9.3 are satisfied on JJ_{0}. Choosing L sufficiently large, we know that for such configurations of the disorder, the conclusions of theorem 9.5 are satisfied. We thus have that

|P_{t}^{J}f(ω) − μ_{J}f|
≤ |P_{t}^{J,Λ_{n}}f(ω) − E_{J,Λ_{n}}^{0}f| + ||P_{t}^{J}f − P_{t}^{J,Λ_{n}}f||_{∞} + |μ_{J}f − E_{J,Λ_{n}}^{0}f|
≤ |P_{t}^{J,Λ_{n}}f(ω) − E_{J,Λ_{n}}^{0}f| + e^{−At}|||f||| + e^{−M(n−n_{0})}|||f|||  (9.2.14)

if d(Λ(f), Λ_{n}^{c})^{1−η} ≥ Bt and E_{J,Λ_{n}}^{0} is the Gibbs measure in finite volume with free boundary conditions. We saw above that, with the result of theorem 9.5, we have for any δ ∈ (0, 1) and q = 1 + e^{\frac{2δt}{cn}},

|P_{t}^{J,Λ_{n}}f(ω) − E_{J,Λ_{n}}^{0}f| ≤ e^{2||H_{J,Λn} ||_{∞}/q} (E_{J,Λ_{n}}^{0}|P_{t}^{J,Λ_{n}}f(·) − E_{J,Λ_{n}}^{0}f|^{q})^{\frac{1}{q}}
≤ e^{2||H_{J,Λn} ||_{∞}/q}e^{\frac{(δ−1)t}{cn}} .

(9.2.15)

We thus obtain that for n sufficiently large such that d(Λ(f), Λ_{n}^{c})^{1−η} ≥ Bt, or in other words for n of order t^{\frac{1}{1−η}},

|P_{t}^{J}f(ω) − μ_{J}f| ≤ e^{2||H_{J,Λn} ||_{∞}/q}e^{\frac{(δ−1)t}{cn}} + e^{−At}|||f||| + e^{−M(n−n_{0})}|||f|||.

(9.2.16)

Clearly, this bound can only help us if \frac{t}{c_{n}} (that is \frac{n^{1−η}}{c_{n}}) goes to infinity with n. In particular, if this ratio goes to infinity faster than logarithmically with n, we almost surely have

lim_{n→∞} ||H_{J,Λ_{n}}||_{∞}/q = 0.

When d = 2, we have

c_{n} ≈ n^{ξ}

with ξ going to zero as p_{1} goes to zero. Consequently, when β is sufficiently small, choosing n^{1−η} ≈ Bt, we obtain

|P_{t}^{J}f(ω) − μ_{J}f| ≤ e^{−c_{0}t^{α}}|||f|||

(9.2.17)

with a constant c_{0} > 0 and

α = \frac{1 − η − ξ}{1 − η} .

Since we took t = n^{1−η}/B′ with n ≥ N(J) and an almost surely finite random variable N(J), this result is obtained for t ≥ T (J), with an almost surely finite random time T (J). ⋄

Low temperature regime : L^{2} ergodicity in infinite volume

At low temperature, it frequently happens that there are several coexisting phases. (For classical references to this phenomenon, we refer to [90], [41] and [84].) For example, let us consider the ferromagnetic Ising model given by the interaction Hamiltonian

H_{Λ}^{ω}(σ) = − ∑_{|i−j|=1,i,j∈Λ} σ_{i}σ_{j} − ∑_{|i−j|=1,j∈Λ^{c}} σ_{i}ω_{j}

and

E_{Λ}^{ω}(f) = \frac{1}{Z_{Λ}^{ω}} ∫ f(σ)e^{−βH_{Λ}^{ω}(σ)}dν_{Λ}

with ν_{Λ} the product measure on the spins (σ_{i}, i ∈ Λ) with marginal law ν = (1/2)(δ_{+1} + δ_{−1}). One knows that for sufficiently large β’s the following two limits exist and are different,

lim_{Λ↑ZZ^{d}} E_{Λ}^{+} = μ^{+},  lim_{Λ↑ZZ^{d}} E_{Λ}^{−} = μ^{−}

where + (resp. −) means that the spins at the boundary of the set Λ take all the value +1 (resp. −1). In such a setting, one cannot hope anymore the infinite volume semi-group to be uniformly ergodic. However, one can look for L^{2} ergodicity. For instance, it is expected that for ferromagnetic Ising model in dimension 2, there exists a positive constant c such that for any local function f

μ^{ϵ}(P_{t}f − μ^{ϵ}f)^{2} ≤ e^{−c\sqrt{t}}|||f|||

with ϵ = + or − (see Fisher and Huse [37]).

106 10.Low temperature regime : L^{2} ergodicity in infinite volume

F. Martinelli [68] has shown that for any local function f and any M ∈ IR^{+}, there exists a finite constant c(Λ(f), M) such that for all times t ≥ 1,

μ^{+}(P_{t}f − μ^{+}f)^{2} ≤ c(Λ(f), M)t^{−M}|||f|||.

His proof relies on the spectral gap estimate

m^{+}(Λ_{L}) = inf_{f} \frac{E_{Λ_{L}}^{+}(f(−L_{Λ})f)}{E_{Λ_{L}}^{+}(f − E_{Λ_{L}}^{+}f)^{2}} ≥ e^{−o(L)L}  (10.0.1)

if Λ_{L} = [−L, L]^{2} and lim_{L→∞} o(L) = 0. If m^{0}(Λ_{L}) denotes the spectral gap constant with free boundary conditions, he also proved that

lim_{L↑∞} \frac{1}{2βL} log m^{0}(Λ_{L}) = −τ_{β}  (10.0.2)

where τ_{β} > 0 is the surface tension.

For related results on this subject, one can read [98] and [53].

We shall discuss hereafter the proof of (10.0.1) and (10.0.2), as well as that of the L^{2}(μ^{+})- ergodicity of the semi-group in infinite volume.

10.1 Spectral gap estimate

10.1.1 Strategy

The strategy we propose below slightly differs from that originally proposed by F. Martinelli relying on a clever use of auxiliary bloc dynamics. Here, we may as well consider other auxiliary dynamics. However, both approaches are more or less equivalent and require the same kind of basic estimates.

We shall denote by μ a probability measure and will present a general scheme to establish a spectral gap inequality for μ with respect to a Dirichlet form E.

The basic idea is to decouple the difficulties by introducing auxiliary operators. In fact, we see that if π is an operator in L^{2}(μ), for any function f ∈ L^{2}(μ), the triangle inequality yields

||f − μf||_{2} ≤ ||f − πf||_{2} + ||πf − μf||_{2}.  (10.1.3)

Let us now assume that π was chosen so that there exists δ > 0 such that for every bounded measurable function f,

||πf − μf||_{∞} ≤ (1 − δ)||f − μf||_{∞}.  (10.1.4)

Then, if π is self-adjoint in L^{2}(μ), we have

||πf − μf||_{2} ≤ (1 − δ)||f − μf||_{2}  (10.1.5)

since (see [71] or [9])

sup_{f⊥1} \frac{||πf − μf||_{2}}{||f − μf||_{2}} = lim_{n→∞} (sup_{f⊥1} \frac{||π^{n}f − μf||_{∞}}{||f − μf||_{∞}} )^{\frac{1}{n}} .

Hence, we deduce from (10.1.3) and (10.1.5) that

||f − μf||_{2} ≤ \frac{1}{δ}||f − πf||_{2}.  (10.1.6)

To obtain the desired result for the Dirichlet form under consideration, we need the following additional property

μ(f − πf)^{2} ≤ CE(f, f)  (10.1.7)

for a finite constant C. Then, we conclude that

μ(f − μf)^{2} ≤ \frac{C}{δ} E(f, f).  (10.1.8)

We recall that F. Martinelli [69], (following an idea of Holley), applied the above strategy with the specific choice of

μ = E_{Λ}^{ω}, π = \frac{1}{n} ∑_{i=1}^{n} E_{Λ_{i}}^{ω}

for subsets Λ_{i} of Λ with possibly non empty intersection. Such the π generates the so-called bloc spin flip dynamics.

The next exercise shows how it can be used to control the spectral gap in the high temperature regime.

Exercise 10.1 [ See [69](Thm 4.5) ]

Let Λ_{L} = [−L, L]^{2} ⊂ ZZ^{d}. We assume in this exercise that the interaction is of finite range and that the local Gibbs measure E_{Λ}^{ω} on Σ_{Λ} satisfies the mixing condition

Var_{Y} E_{Λ}^{ω}f ≤ Ce^{−Md(Y,X∩Λ_{f})}||f||_{∞}

for finite constants (C, M) > 0 (see section 5.4.3 for the notations ). As a consequence, there exists a unique Gibbs measure μ in infinite volume for the local specification (E_{Λ}^{ω}, Λ ⊂ ZZ^{d}).

Show that if for any L ∈ IN, we set

m_{L} = inf_{max_{1≤i≤d}(k′_{i}−k_{i})≤L} inf_{i∈ZZ^{d}} inf_{f} \frac{E_{i+Λ_{K},J}^{ω}(∑_{i}(∂_{i}f)^{2})}{E_{i+Λ_{K},J}^{ω}(f, f)}

with Λ_{K} = [k_{1}, k′_{1}] × ⋯ × [k_{d}, k′_{d}], there exist two finite constants c, c′ such that

m(2L) ≥ (1 − \frac{c}{\sqrt{L}})m(L + c′\sqrt{L} log L).  (10.1.9)

Show that for ρ ∈ (1, 2), we can find L_{0} ∈ IN so that m(L_{0}) ≥ m > 0 and ρL ≥ 2^{−1}(L + c′\sqrt{L} log L) for L ≥ L_{0}, to conclude that there exists a positive constant m′ such that for all n ∈ IN, m(L_{0}ρ^{n}) ≥ m′ > 0. Conclude.

Hint: Proceed by induction; Consider a set R = k +∏_{1≤i≤d}[0, l_{i}] with l_{i} ≤ L for 1 ≤ i ≤ d, with, to simplify the notations k = 0 and l_{1} ≥ l_{2} ≥ .. ≥ l_{d}, l_{1} ≥ \sqrt{L}. Let, for k, n ∈ IN, kn ≤ l_{1}/2, Λ_{1} = [0, l_{1}/2 + nk] × [0, l_{2}] × ⋯ × [0, l_{d}] and Λ_{2} = [l_{1}/2 + (n − 1)k, l_{1}] × [0, l_{2}] × ⋯ × [0, l_{d}]. Notice that, under our hypotheses and since d(Λ(E_{Λ_{2}}f), ∂Λ_{1}) = d(Λ(E_{Λ_{1}}f), ∂Λ_{2}) = k, for i = 0, 1,

||E_{Λ_{i+1}}E_{Λ_{2−i}}f − E_{R}f||_{∞} ≤ e^{−Mk}||f − E_{R}f||_{∞}.

Let π = \frac{1}{2}(E_{Λ_{1}} + E_{Λ_{2}}). Show that for any integer number p ≥ 2

||π^{p}f − E_{R}f||_{∞} ≤ \frac{1}{2^{p}} (1 + e^{−Mk})^{p}||f − E_{R}f||_{∞}

Deduce that since

(E_{R}(πf − f)^{2})^{\frac{1}{2}} ≤ \frac{1}{2}E_{R} (E_{Λ_{1}}(f − E_{Λ_{1}}f)^{2})^{\frac{1}{2}} + \frac{1}{2}E_{R} (E_{Λ_{2}}(f − E_{Λ_{2}}f)^{2})^{\frac{1}{2}} ,

we have

(E_{R}(f − E_{R}f)^{2})^{\frac{1}{2}} ≤ (1 − e^{−Mk})^{−1} max\{m(Λ_{1})^{−1}, m(Λ_{2})^{−1}\}

×(E_{R}( ∑_{i∈R}(∂_{i}f)^{2} + ∑_{i∈Λ_{1}∩Λ_{2}}(∂_{i}f)^{2}))^{\frac{1}{2}} .

Summing over n ∈ \{1, …, n_{0}\}, bound (E_{R}(f − E_{R}f)^{2})^{\frac{1}{2}} in terms of m(Λ_{i}^{n})^{−1}, i = 1, 2 and proceed by induction to bound similarly max{m(Λ_{1})^{−1}, m(Λ_{2})^{−1}} to arrive at

m(L) ≥ (1 − e^{−Mk})^{d}(1 + n_{0}^{−1})^{−d}m(2^{−1}L + n_{0}k).

Choose wisely M and n_{0} to conclude.

10.1.2 Spectral gap estimate

In the high temperature regime and in dimension 2, we choose, (compare [69]),

μ = E_{Λ}^{+}, π = E_{Λ_{1}} ⋯ E_{Λ_{n−1}}E_{Λ_{n}}E_{Λ_{n−1}}..E_{Λ_{1}}

with, for i ∈ {1, …, n = 2[2L/[ϵL]]} and ϵ > 0

Λ_{i} = [[ϵL](i − 1)/2 − L; [ϵL](i − 1)/2 + [ϵL] − L] × [0, L] .

Following the strategy stated below, we need to bound ||f − πf||_{2} in terms of the Dirichlet form and πf − μf uniformly. For the first term, the triangular inequality and local Gibbs measures property imply that

||f − πf||_{2} ≤ 2 ∑_{i=1}^{n} (E_{Λ}^{+} (E_{Λ_{1}} ⋯ E_{Λ_{i−[i/n]n}} − E_{Λ_{1}} ⋯ E_{Λ_{i+1−[i/n]n}})f)^{2})^{\frac{1}{2}}

≤ 2 ∑_{i=1}^{n} (E_{Λ}^{+} (f − E_{Λ_{i}}f)^{2})^{\frac{1}{2}} .  (10.1.10)

Moreover, property 2.7 shows that there exists a constant c(β) < ∞ such that

E_{Λ_{i}} (f − E_{Λ_{i}}f)^{2} ≤ e^{c(β)ϵL}E_{Λ_{i}} ( ∑_{j∈Λ_{i}}(∂_{j}f)^{2}) .

Consequently, we obtain

||f − πf||_{2} ≤ 2e^{\frac{1}{2} c(β)ϵL} ∑_{i=1}^{n} E_{Λ}^{+}[|∇_{Λ_{i}}f|^{2}]^{\frac{1}{2}}
≤ 4[ϵL]ne^{c(β)ϵL} (E_{Λ}^{+}[|∇_{Λ}f|^{2}])^{\frac{1}{2}} .  (10.1.11)

Moreover, F. Martinelli proved that for the ferromagnetic Ising model (see [69], p. 68) the following bound is true

Property 10.2 For any ϵ > 0, L sufficiently large,

||πf − E_{Λ}^{+}f||_{∞} ≤ e^{−ϵL}||f − E_{Λ}^{+}f||_{∞}.

Consequently, for this model, we deduce that for any ϵ > 0

E_{Λ}^{+}(f − E_{Λ}^{+}f)^{2} ≤ \frac{8L}{ϵ(1 − e^{−mϵL})}e^{c(β)ϵL}E_{Λ}^{+} (∑_{j∈Λ}(∂_{j}f)^{2}) (10.1.12)

which gives the desired estimate on the spectral gap.

Property 10.2 relies on the precise knowledge of the fluctuations of the interface between the phases of +1 spins and −1 spins in dimension 2 (see [27]) which allows to see that the restriction of E_{[0,L]×[0,l]}^{(+,+,+,ω)} to the σ-algebra generated by (σ_{i}, i ∈ [0, L] × [0, l − M\sqrt{L}]) depends very weakly on the boundary conditions ω. Here, (+, +, +, ω) denotes the boundary conditions where all the spins on the sides at north, south and west are equal to +1 whereas they take the configuration ω on the east side. This observation allows roughly speaking to see that E_{Λ_{1}}E_{Λ_{2}}f ≃ E_{Λ_{1}∪Λ_{2}} for overlapping sets Λ_{1} = [k_{1}, k_{1} + L] × [k′_{1}, k′_{1} + L], Λ_{2} = [k_{2}, k_{2} + L] × [k′_{2}, k′_{2} + L] with intersection of the form Λ_{1} ∩ Λ_{2} = [k, k + M\sqrt{L}] ∩ [l, l + L], and by induction that πf ≃ μf.

In case of free boundary conditions, F. Martinelli [69] proved that Property 10.3 For any ϵ > 0,

||πf − E_{Λ}^{0}f||_{∞} ≤ (1 − e^{−βτ_{β}(1+14ϵ)(2L+1)})||f − E_{Λ}^{+}f||_{∞}

which in turn implies that

E_{Λ}^{0}(f − E_{Λ}^{0}f)^{2} ≤ \frac{8L}{ϵ}e^{c(β)ϵL+βτ_{β}(1+14ϵ)(2L+1)}E_{Λ}^{+} (∑_{j∈Λ}(∂_{j}f)^{2}) (10.1.13)

and gives a lower bound on the spectral gap. F. Martinelli has also shown that the bound (10.1.13) is on the right scale by bounding above the spectral gap by choosing, in its definition as an infimum on test functions, the test function

f_{Λ}(σ) = 1I_{∑_{i∈Λ} σ_{i}>0} − 1I_{∑_{i∈Λ} σ_{i}<0}.

10.2 L^{2} ergodicity in infinite volume

We obtained in the previous section spectral gap lower bounds in finite volume decreasing with the volume faster than |Λ|^{− \frac{1}{d}}. Following the remarks we did when we studied the ergodicity for disordered systems, the method employed there is in this case useless. In fact, one does not know in general how to deduce ergodicity in infinite volume from such estimates. This is due to the lack of balance between the finite volume approximation of the semi-group, which says that we can approximate P_{t}f by P_{t}^{Λ_{n}}f for n of order t, and the spectral gap estimate which provides an exponential decay to equilibrium of P_{t}^{Λ_{n}}f with the speed m_{n}t, which is going to zero when n is of order t.

Fortunately, in ferromagnetic models, the finite volume approximation of the semi-group can be avoided. We briefly describe the arguments used in this case restricting ourselves to the discrete spins and considering a generator

Lf = ∑_{i}(E_{{i}}f − f) = ∑_{i} c_{i}(σ)∂_{i}f

with

c_{i}(σ) = \frac{1}{1 + e^{2βσ_{i} ∑_{|i−j|=1} σ_{j}}}

and

∂_{i}f = f(σ^{(i)}) − f(σ).

Then, it is not hard to see that for any configuration (σ, η) such that σ_{i} ≤ η_{i} for any i and any j ∈ ZZ^{d},

c_{j}(σ) ≤ c_{j}(η) if σ_{j} = η_{j} = −1

and

c_{j}(σ) ≥ c_{j}(η) if σ_{j} = η_{j} = +1.

10.2 L^{2} ergodicity in infinite volume 111

Thus, the more spins with state +1 will be contained in a given configuration, the higher the probability that the Glauber dynamics will turn the spins into the state +1. This heuristic can be characterized by the construction of a coupling (see [86]) ρ_{t}^{η,η′}(dσ, σ′) of P_{t}(η) and P_{t}(η′)( that is a probability measure on the product space Ω^{2} with marginals P_{t}(η) and P_{t}(η′)) such that, with ρ_{t}^{η,η′} probability one, if for any j ∈ ZZ^{d}, η_{j} ≤ η′_{j}, then

σ_{t}(j) ≤ σ′_{t}(j) ∀j ∈ ZZ^{d}, ρ_{t}^{η,η′} − a.s..

Also, one can construct a coupling r_{t}^{η,+,Λ} of P_{t}(η) with the semi-groupP_{t}^{Λ,+}(+1)⊗ ∏_{j∈Λ^{c}} δ_{σ_{j}=+1} such that

σ_{t}(j) ≤ σ′_{t}(j) ∀j ∈ ZZ^{d}, ρ_{t}^{η,+,Λ} − a.s.

Similarly, one can couple P_{t}(η), P_{t}(η′) and P_{t}^{Λ,+}(+1) ⊗ ∏_{j∈Λ^{c}} δ_{σ_{j}=+1}, the process representing the last marginal then dominating the two others. In particular, for any t ≥ 0,

P_{t}(σ_{0} = +1)(η) ≤ P_{t}^{Λ,+}(σ_{0} = +1)(+1).  (10.2.14)

This property is the key point to overcome the finite volume approximation of Markov semi-groups.

Note that this kind of property also exists for continuous models when, roughly speaking, the potential is convex (see [50]).

We shall use it to show that

Property 10.4 For any function f, any M ∈ IR^{+}, and all t ≥ 1,

μ^{+}(P_{t}f − μ^{+}f)^{2} ≤ t^{−M}|||f|||.

Proof : Let us first notice that, on the discrete set {−1, +1}^{ZZ^{d}}, the local functions can be decomposed into sums of monomial functions of the type

f(σ) = ∏_{i∈X} σ_{i}

for finite sets X ⊂ ZZ^{d}.

For such a function, we have

μ^{+}(P_{t}f − μ^{+}f)^{2} = \frac{1}{2}μ^{+} ⊗ μ^{+}(P_{t}f(η) − P_{t}f(η′))^{2}
= \frac{1}{2}μ^{+} ⊗ μ^{+}(ρ_{t}^{η,η′}(∏_{i∈X} σ_{i}(t) − ∏_{i∈X} σ′_{i}(t))^{2}
≤ |X| ∑_{i∈X} μ^{+} ⊗ μ^{+}(ρ_{t}^{η,η′}|σ_{i}(t) − σ′_{i}(t)|)^{2}
= 4|X| ∑_{i∈X} μ^{+} ⊗ μ^{+}(ρ_{t}^{η,η′}[σ_{i}(t) ≠ σ′_{i}(t)])
≤ 4|X|^{2}μ^{+} ⊗ μ^{+}(ρ_{t}^{η,η′}[σ_{0}(t) ≠ σ′_{0}(t)])

(10.2.15) with ρ_{t}^{η,η′} a coupling between P_{t}(η) and P_{t}(η′). Here we used the invariance by translation of μ^{+}. Therefore we only need to prove the estimate for f(σ) = σ_{0}. According to the last remark, we have for any finite cube Λ,

\frac{1}{2}μ^{+} ⊗ μ^{+}(ρ_{t}^{η,η′}[σ_{0}(t) ≠ σ′_{0}(t)])

≤ μ^{+} ⊗ μ^{+}(r_{t}^{η,+,Λ}[σ_{0}(t) ≠ σ′_{0}(t)])
= μ^{+} ⊗ μ^{+}(P_{t}^{Λ,+}(σ_{0} = +1)(+) − P_{t}(σ_{0} = +1)(η))
= P_{t}^{Λ,+}σ_{0}(η) − μ^{+}P_{t}σ_{0}
= P_{t}^{Λ,+}σ_{0}(+) − μ^{+}σ_{0}
≤ e^{2β|Λ|} (E_{Λ}^{+}(P_{t}^{Λ,+}(σ_{0}) − E_{Λ}^{+}σ_{0})^{2})^{\frac{1}{2}} + E_{Λ}^{+}σ_{0} − μ^{+}σ_{0}
≤ e^{2β|Λ|−m(Λ)t} + E_{Λ}^{+}σ_{0} − μ^{+}σ_{0}  (10.2.16)

where we used the definition of the spectral gap constant and introduced by ”force” the Gibbs measure E_{Λ}^{+}. Finally, let us recall that (see [27]), if Λ = [−L, L]^{2},

|E_{Λ}^{+}σ_{0} − μ^{+}σ_{0}| ≤ e^{−L}.

Thus, by the estimate on the spectral gap inequality obtained in the last section,

μ^{+}(P_{t}f − μ^{+}f)^{2} ≤ 4|X|^{2} (e^{2βL^{2}−e^{−o(L)L}t} + e^{−L}) . (10.2.17)

Optimizing on L, we take L = [M log t] for M ∈ IR^{+} showing that we can find a finite constant C(M) so that

μ^{+}(P_{t}f − μ^{+}f)^{2} ≤ C(M)|X|^{2}t^{−M}.  (10.2.18)

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