spectral gap and logarithmic sobolev inequality for ... - Numdam

Ann. I. H. Poincaré – PR 38, 5 (2002) 739–777  2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S0246-0203(02)01108-1/FLA

SPECTRAL GAP AND LOGARITHMIC SOBOLEV INEQUALITY FOR UNBOUNDED CONSERVATIVE SPIN SYSTEMS TROU SPECTRAL ET INÉGALITÉS DE SOBOLEV LOGARITHMIQUES POUR DES SYSTÉMES DE SPINS CONSERVATIFS ET NON BORNÉS

C. LANDIM a,b , G. PANIZO a , H.T. YAU c a IMPA, Estrada Dona Castorina 110, CEP 22460 Rio de Janeiro, Brazil b CNRS UMR 6085, Université de Rouen, 76128 Mont Saint Aignan, France c Courant Institute, New York University, 251 Mercer street, New York, NY 10 012, USA

Received 28 September 2000, revised 6 July 2001

A BSTRACT. – We consider reversible, conservative Ginzburg–Landau processes, whose potential are bounded perturbations of the Gaussian potential, evolving on a d-dimensional cube of length L. Following the martingale approach introduced in (S.L. Lu, H.T. Yau, Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics, Comm. Math. Phys. 156 (1993) 433–499), we prove in all dimensions that the spectral gap of the generator and the logarithmic Sobolev constant are of order L−2 .  2002 Éditions scientifiques et médicales Elsevier SAS Keywords: Interacting particle systems; Spectral gap; Logarithmic Sobolev inequality R ÉSUMÉ. – Nous considérons des processus de Ginzburg–Landau réversibles, dont le potentiel est une perturbation bornée du potential Gaussien, évoluent sur un cube d-dimensionel de largeur L. Suivant la méthode martingale introduite dans (S.L. Lu, H.T. Yau, Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics, Comm. Math. Phys. 156 (1993) 433–499), nous démontrons qu’en toute dimension le trou spectral et la constante de Sobolev logarithmique sont d’ordre L−2 .  2002 Éditions scientifiques et médicales Elsevier SAS

E-mail addresses: [email protected] (C. Landim), [email protected] (G. Panizo), [email protected] (H.T. Yau).

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1. Introduction In recent years some progress has been made in the investigation of convergence to equilibrium of reversible conservative interacting particle systems [1,2,9,8,11,4,5]. In finite volume the techniques used to obtain the rate of convergence to equilibrium rely mostly on the estimation of the spectral gap of the generator. In general, one shows that the generator of the particle system restricted to a cube of length N has a gap of order N −2 in any dimension. This estimate together with standard spectral arguments permits to prove that the particle system restricted to a cube of size N decays to equilibrium in the variance sense at the exponential rate exp{−ct/N 2 }: for any function f in L2 , Pt f − Eπ [f ]22
exp{−ct/N 2 }f − Eπ [f ]22 , where {Pt , t  0} stands for the semi-group of the process, π for the invariant measure, Eπ [f ] for the expectation of f with respect to π and  · 2 for the L2 norm with respect to π . In infinite volume, since the spectrum of the generator of a conservative system has no gap at the origin, instead of exponential convergence to equilibrium, one expects a polynomial convergence. In this context, the main difficulty is to use the local information on the gap of the spectrum of the generator restricted to a finite cube to deduce the global behavior of the system in infinite volume. On the other hand, the relation between the logarithmic Sobolev inequality and the hypercontractivity has long been established. The hypercontractivity in turn permits to prove upper and lower Gaussian estimates of the transition probability of a reversible Markov process (cf. [6,11]). In this article we present a sharp estimate of the spectral gap and of the logarithmic Sobolev constant for the Ginzburg–Landau process whose potential is a bounded perturbation of the Gaussian potential. The precise assumptions are given in Section 2. We follow here the martingale approach introduced in [14]. The main ideas are essentially the same but there are several technical difficulties coming from the unboundedness of the spins. The main ingredients are a local central limit theorem, uniform over the parameter and from which follows the equivalence of ensembles, and some sharp large deviations estimates. The article is divided as follows. In Section 2 we state the main results and introduce the notation. In Section 3 we prove the spectral gap and in Section 4 the logarithmic Sobolev inequality. In Section 5 we prove a uniform local central limit theorem and deduce some results regarding the equivalence of ensembles. In Section 6 we obtain some large deviations estimates which play a central role in the proof of the logarithmic Sobolev inequality. 2. Notation and results For L  1, denote by L the cube {1, . . . , L}d . Configurations of the state space R are denoted by the Greek letters η, ξ , so that ηx indicates the value of the spin at x ∈ L for the configuration η. The configuration η undergoes a diffusion on R L L

C. LANDIM ET AL. / Ann. I. H. Poincaré – PR 38 (2002) 739–777

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whose infinitesimal generator L L is given by L L =

1 2




(∂ηx − ∂ηy )2 −

x,y∈ L |x−y|=1

1 2






V  (ηy ) − V  (ηx ) (∂ηy − ∂ηx ).

x,y∈ L |x−y|=1

V : R → R represents the potential V (a) = (1/2)a 2 + F (a), where F : R → R is a smooth bounded function such that F  ∞ < ∞, 

e−V (x) dx = 1.

We assumed the convex part of the potential to be Gaussian for simplicity. All proofs of the results presented in Section 5 on the uniform local central limit theorems rely strongly on this hypothesis. We believe, however, that the approach presented here extend to the case where we have a bounded perturbation of a convex potential. In this respect, it was recently observed by Caputo [3] that when the potential is a purely convex function, the L2 behavior of the inverse of the spectral gap and of the logarithmic Sobolev constant can be easily obtained by techniques introduced for models with convex interactions (see [13] and references therein). Denote by Z : R → R the partition function Z(λ) =

∞

eλa−V (a) da,

(2.1)

−∞

by R : R → R the density function ∂λ log Z(λ), which is smooth and strictly increasing, and by  the inverse of R so that 1 α= Z((α))

∞

a e(α)a−V (a) da −∞

for each α in R. For λ in R, denote by ν¯ λ L the product measure on R L defined by ν¯ λ L (dη) =



1 ληx −V (ηx ) e dηx Z(λ) x∈ L

L and let να L = ν¯ (α) . Notice that Eνα [ηx ] = α for all α in R, x in L . Most of the times omit the superscript L . For each M in R, denote by ν L ,M the canonical measure on L with total spin equal to M:

ν L ,M ( · ) =

να L

 
   · ηx = M . x∈ L

Expectation with respect to ν L ,M is denoted by E L ,M .

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An elementary computation shows that the product measures {ν¯ λ , λ ∈ R} are reversible for the Markov process with generator L L . The Dirichlet form D L associated to L L is given by D L (µ, f ) =




1 2

(T x,y f )2



µ.

x,y∈ L |x−y|=1

In this formula and below, for a probability measure µ, ·µ stands for expectation with respect to µ. Furthermore, for x, y in Zd , T x,y represents the operator that acts on smooth functions f as ∂f ∂f − ∂ηx ∂ηy

T x,y f =

and µ stands for the invariant measures να , ν L ,M . For a positive integer L and M in R, denote by W (L, M) the inverse of the spectral gap of the generator L L with respect to the measure ν L ,M : W (L, M) = sup f

f ; f ν L ,M D L (ν L ,M , f )

.

In this formula the supremum is carried over all smooth functions f in L2 (ν L ,M ) and f ; f µ stands for the variance of f with respect to µ. We also denote this variance by the symbol Var(µ, f ). Let W (L) = sup W (L, M). M∈R

T HEOREM 2.1. – There exists a finite constant C0 depending only on F such that W (L)
C0 L2 for all L  2. A lower bound of the same order is easy to derive. Fix a smooth function d H : [0, 1] → R such that H (u) du = 0 and let fH (η) = x∈ L H (x/L)ηx . An elementary computation shows that fH ; fH ν L ,M =




2

H (x/L)

x

+




η2e1 ; ηe1 ν L ,M



H (x/L)2 ηe1 ; ηe1 ν L ,M − η2e1 ; ηe1 ν L ,M ,

x

D L (ν L ,M , fH ) = (1/2)




2

H (y/L) − H (x/L) .

|x−y|=1

In this formula {ej , 1
j
d} stands for the canonical basis of Rd . By Corollary 5.3, as L ↑ ∞, M/Ld → α, fH ; fH ν L ,M /L2 D L (ν L ,M , fH ) converges to

C. LANDIM ET AL. / Ann. I. H. Poincaré – PR 38 (2002) 739–777

ηe1 ; ηe1 να



743



H (u)2 du/ (∇H )(u)2 du. This proves that lim inf L−2 W (L) > 0. L→∞

For L  2, a probability measure ν on R L and a function f such that f 2 ν = 1, denote by S L (ν, f ) the entropy of f 2 dν with respect to ν: S L (ν, f ) =



f 2 log f 2 dν

and by θ(L, M) the inverse of the logarithmic Sobolev constant of the Ginzburg–Landau process on the cube L with respect to the measure ν L ,M : θ(L, M) = sup f

S L (ν L ,M , f ) . D L (ν L ,M , f )

In this formula, the supremum is carried over all smooth functions f in L2 (ν L ,M ) such that f 2 ν L ,M = 1. Let θ(L) = sup θ(L, M). M∈R 

T HEOREM 2.2. – Assume that F ∞ < ∞. There exists a finite constant C depending only on F such that θ(L)
CL2 for all L  2. We follow here the martingale method developed by Lu and Yau [14] to prove the spectral gap and a bound on the logarithmic Sobolev constant for a conservative interacting particle system. This approach relies on two a-priori estimates. First, a local central limit theorem for i.i.d. random variables with marginals equal to the marginals of the product measure ν¯ λ , uniform over the parameter λ in R. Second, a spectral gap or a logarithmic Sobolev inequality, uniform over the density, for a Glauber dynamics on one site which is reversible with respect to the one-site marginal of the canonical invariant measure. 3. Spectral gap To fix ideas, we prove Theorem 2.1 in dimension 1. The reader can find in Section A.3.3 of [10] the arguments needed to extend the proof to higher dimensions. To detach the main ideas, we divide the proof in four steps. The proof goes by induction on L. We start with L = 2. In this section all constants are allowed to depend on F ∞ , F  ∞ . In the case they depend on some other parameter, the dependence is stated explicitly. Step 1. One-site spectral gap. Consider a smooth function f : R 2 → R. We want to estimate f ; f ν 2 ,M in terms of the Dirichlet form of f . Since for the measure ν 2 ,M the total spin is fixed to be equal to M, let g(a) = f (M −a, a) and notice that f ; f ν 2 ,M is equal to g; gν 2 ,M .

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The following result will be of much help. Fix L  2 and M in R. Denote by ν 1 L ,M the marginal distribution of ηL with respect to ν L ,M . The Glauber dynamics has a positive spectral gap which is uniform with respect to M: L EMMA 3.1. – There is a finite constant C0 depending only on F ∞ such that 

Var(ν 1 L ,M , f )


C0 Eν 1

L ,M

∂f ∂ηL

2 

for every L  2, every M in R and every smooth function f : R → R in L2 (ν 1 L ,M ). Remark 3.2. – In the case of grand canonical measures, this result is true under the more general hypothesis of strict convexity at infinity of the potential (cf. [13] and references therein). In case of canonical measures the main problem is to obtain a good approximation of the one-site marginal in terms of the one-site marginal of grand canonical measures. Before proving this result, we conclude the first step. Applying this result to the function g defined above, we obtain that its variance is bounded by C0 Eν 1 [(∂g/∂η2 )2 ]. 2 ,M Since ∂g/∂η2 = (∂f/∂η2 − ∂f/∂η1), we have that f ; f ν 2 ,M = g; gν 2 ,M = g; gν 1 


C0 Eν 1

2 ,M

∂g ∂η2



= C0 Eν 2 ,M

2 

2 ,M



= C0 Eν 2 ,M

∂f ∂f − ∂η2 ∂η1

2 

∂g ∂η2

2 

.

This shows that W (2)
C0 , proving Theorem 2.1 in the case L = 2. We conclude this step with the Proof of Lemma 3.1. – We first prove the lemma for the grand canonical measure. Fix λ in R and denote by ν¯ λ1 the one-site marginal of the product measure ν¯ λ L . Fix xλ in R, that will be specified later, and f in L2 (ν¯ λ1 ). The variance of f is bounded above by 



2

f (x) − f (xλ ) e−Vλ (x) dx,

R

where Vλ (x) = −λx + log Z(λ) + V (x). By Schwarz inequality, the previous expression is less than or equal to ∞

 

2 −Vλ (x)

dx [f (x)] e

∞ Vλ (x)

e

xλ

 −Vλ (y)

dy (y − xλ )e

x

+

xλ

−∞

 

2 −Vλ (x)

dx[f (x)] e

x Vλ (x)

e

−∞

 −Vλ (y)

dy (xλ − y)e

.

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745

It remains to show that the expressions inside braces are uniformly bounded in x and λ for an appropriate choice of xλ . Both expressions are handled in the same way and we consider, to fix ideas, the first one where we need to estimate  (1/2)(x−λ)2+F (x)

∞

sup e x
xλ

 −(1/2)(y−λ)2−F (y)

dy (y − xλ )e

.

x

Choose xλ = λ and change variables to reduce the previous expression to  (x 2 /2)+Fλ (x)

−(y 2 /2)−Fλ (y)

dy ye

sup e x
0



∞

,

x

where Fλ (a) = F (a + λ). In the case where F = 0, this expression is bounded above by some universal constant C0 . Since F is bounded, this expression is less than or equal to C0 exp{2F ∞ } uniformly over λ. This concludes the proof of the lemma in the case of grand canonical measures. We turn now to the case of canonical measures. We need to introduce some notation. For λ in R, let {Xjλ , j  1} be a sequence of i.i.d. random variables with density Z(λ)−1 exp{λx − V (x)}. For a positive integer L, denote by fλ,L the density of (σ (λ)2L)−1/2 1j L {Xjλ − γ1 (λ)}, where γk (λ) is the kth truncated moment of X1λ and σ (λ)2 is its variance: γ1 (λ) = E[X1λ ], γk (λ) = E[(X1λ − γ1 (λ))k ]. We prove in Section 5 an Edgeworth expansion for fλ,L uniform over the parameter λ. 1 (dx) in terms of the density fλ,L . Choose λ so We may write the measure νL,M √ 1 (dx) = L/(L − 1)gλ (x)fλ,L−1 ([γ1 − that γ1 (λ) = M/L: λ = (M/L). Then, νL,M √ x]/σ L − 1) fλ,L (0)−1 dx, where gλ stands for the density Z(λ)−1 exp{λx − V (x)}. Hereafter, we will omit the dependence of γj and σ on λ. 1 (dx) with respect to the Lebesgue Denote the Radon–Nikodym derivative of νL,M 2 1 measure by R(x) = RL,M (x). Fix a function f in L (ν L ,M ) and xλ in R to be specified later. Following the proof for the grand canonical measure, we bound the variance of f by 



2

f (x) − f (xλ ) R(x) dx.

R

We now repeat the arguments presented in the case of the grand canonical measures. After few steps, we reduce the proof of the lemma to the proof that 

sup R(x) x
xλ

−1

∞ x



dy(y − xλ )R(y)

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is bounded, uniformly in M. Choose xλ = λ, change variables and recall the notation introduced above to rewrite the previous expression as ∞

sup x
0

x

 √ gλ (y + λ) fλ,L−1 ([γ1 − y − λ]/σ L − 1) √ dy y . gλ (x + λ) fλ,L−1 ([γ1 − x − λ]/σ L − 1)

By the explicit formula for the density gλ and since F is bounded, this expression is less than or equal to  2F ∞

e

x 2 /2

∞

dy ye

sup e x
0

−y 2 /2

x

 √ fλ,L−1([γ1 − y − λ]/σ L − 1) √ . fλ,L−1([γ1 − x − λ]/σ L − 1)

We need now to estimate the ratio of the densities inside the integral. For a positive integer L, denote by gλ,L (x) the density of 1j L Xjλ . An elementary induction argument shows that gλ,L (x) = Z(λ)−L exp{λx}g0,L (x) so that gλ,L (x)/gµ,L (x) = (Z(µ)/Z(λ))L exp{(λ − µ)x} for any parameter µ. Choose µ so that γ1 (λ) − γ1 (µ) = x/(L − 1) and notice that µ
λ because x  0 and γ1 is an increasing function. The previous identity gives that √ fλ,L−1([γ1 (λ) − y − λ]/σ (λ) L − 1) √ fλ,L−1([γ1 (λ) − x − λ]/σ (λ) L − 1) √ fµ,L−1 ([γ1 (λ) − λ + x − y]/σ (µ) L − 1) (λ−µ)(x−y) √ e . = fµ,L−1 ([γ1 (λ) − λ]/σ (µ) L − 1) The exponential is bounded by 1 because µ
λ and x
y. To conclude the proof of the lemma it is therefore enough to show that the previous ration is bounded. In the proof of Lemma 5.1 we show that |γ1 (λ) − λ| is bounded, uniformly in λ, by a constant C1 which depends only on F ∞ and that σ (µ) is bounded above and below by a finite positive constant for all λ in R and x in R+ . In particular, by Theorem 5.2, there exists L0 such that for L  L0 , the ratio on the right hand side of the previous formula is bounded by a constant that depends only on F ∞ . On the other hand, for 2
L
L0 , by Lemma 5.6 and explicit computations to express fµ,L in terms of f˜µ,L , this ratio is bounded by exp{CL} for some constant C depending only on F ∞ . This concludes the proof of the lemma. ✷ Step 2. Decomposition of the variance. We will obtain now a recursive equation for W (L). Assume that we already estimated W (K) for 2
K
L − 1. Let us write the identity





f − E L ,M [f ] = f − E L ,M [f | ηL ] + E L ,M [f | ηL ] − E L ,M [f ] . Through this decomposition we may express the variance of f as 

E L ,M (f − E L ,M [f ])2 









= E L ,M (f − E L ,M [f | ηL ])2 + E L ,M (E L ,M [f | ηL ] − E L ,M [f ])2 . (3.1)

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The first term on the right-hand side is easily analyzed through the induction assumption and a simple computation on the Dirichlet form. We write 







E L ,M (f − E L ,M [f | ηL ])2 = E L ,M E L ,M (f − E L ,M [f | ηL ])2 | ηL 







= E L ,M E L−1 ,M−ηL (fηL − E L−1 ,M−ηL [fηL ])2 . Here we used the fact that E L ,M [· | ηL ] = E L−1 ,M−ηL [·]. In this formula and below fηL stands for the real function defined on R L−1 whose value at (ξ1 , . . . , ξL−1 ) is given by fηL (ξ1 , . . . , ξL−1 ) = f (ξ1 , . . . , ξL−1 , ηL ). By the induction assumption this last expectation is bounded above by 



W (L − 1)E L ,M D L−1 (ν L−1 ,M−ηL , fηL )
W (L − 1)D L (ν L ,M , f ). In conclusion, we proved that 



E L ,M (f − E L ,M [f | ηL ])2
W (L − 1)D L (ν L ,M , f ).

(3.2)

The second term in (3.1) is nothing more than the variance of E L ,M [f | ηL ], a function of one variable. Lemma 3.1 provides an estimate for this expression: 





E L ,M (E L ,M [f | ηL ] − E L ,M [f ])
C0 E L ,M 2

2 

∂ E ,M [f | ηL ] ∂ηL L

(3.3)

for some constant C0 depending only on F ∞ . Step 3. Bounds on Glauber dynamics, small values of L. We now estimate the right hand side of (3.3), which is the Glauber Dirichlet form of E L ,M [f | ηL ], in terms of the Kawasaki Dirichlet form of f . A straightforward computation gives that 


1 L−1 ∂f ∂f ∂ E L ,M [f | ηL ] = E L ,M − ∂ηL L − 1 x=1 ∂ηL ∂ηx 

+ E L ,M

    ηL  




 1 L−1 f; V  (ηx )  ηL . L − 1 x=1

(3.4)

In this formula E[g; h | F] stands for the conditional covariance of g and h: E[g; h | F] = E[gh | F] − E[g | F]E[h | F]. We examine these two terms separately. The first expression on the right hand side of (3.4) is easily estimated. Recall the definition of the operator T x,y f . Since T L,x f = xyL−1 T y+1,y f , by Schwarz inequality, we have that 

E L ,M




E L ,M


1 L−1 T L,x f L − 1 x=1

 2    ηL 

L−1

1 L−1 (L − x) E L ,M [(T y,y+1 f )2 ] L − 1 x=1 y=x


L

L−1
x=1

E L ,M [(T x,x+1f )2 ] = LD L (ν L ,M , f ).

(3.5)

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The second term in (3.4) is also easy to handle for small values of L. Since V (ϕ) = 2 (1/2)ϕ + F (ϕ) and since 1xL−1 ηx is fixed for the measure E L ,M [· | ηL ], the square of the second term on the right hand side of (3.4) is equal to 

E L ,M




 1 L−1 f; F  (ηx )  ηL L − 1 x=1

2



= E L−1 ,M−ηL


1 L−1 f ηL ; F  (ηx ) L − 1 x=1

2




E L−1 ,M−ηL [fηL ; fηL ] E L−1 ,M−ηL


1 L−1 F (ηx ) L − 1 x=1

2 

.

In this formula, F stands for F  − F  ν L−1 ,M−ηL . The second term is bounded by 4F  2∞ . On the other hand, by the induction assumption, the first term is bounded by W (L − 1)D L−1 (ν L−1 ,M−ηL , fηL ). Hence, taking expectation with respect to ν L ,M , we obtain that 

E L ,M



E L ,M




 1 L−1 f; V  (ηx )  ηL L − 1 x=1

2 


C0 W (L − 1)D L (ν L ,M , f )

for some constant C0 depending on F only. Without much effort and using the local central limit theorem, one can obtain an estimate of type C0 W (L − 1)L−1 D L (ν L ,M , f ) for the left hand side of the last expression. However, for small values of L this improvement is irrelevant. From this estimate and (3.5) we get that the left hand side of (3.3), which is the second term of (3.1), is bounded above by C0 {L + W (L − 1)}D L (ν L ,M , f ). Putting together this estimate with (3.2), we obtain that



f ; f ν L ,M
[1 + C0 ]W (L − 1) + C0 L D L (ν L ,M , f ) or, taking a supremum over smooth functions f , that W (L)
C1 W (L − 1) + C0 L.

(3.6)

This inequality permits to iterate the estimate W (2)
C obtained in Step 1 to derive estimates of W (L) for small values of L. We now consider large values of L. Step 4. Bounds on Glauber dynamics, large values of L. Here again we want to estimate the second term of (3.1). Applying Lemma 3.1, we bound this expression by the right hand side of (3.3). The first term of (3.4) is handled as before, giving (3.5). The

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second one requires a deeper analysis. Its square is equal to 




 1 L−1 f; F  (ηx )  ηL L − 1 x=1

E L ,M

2



= E L−1 ,M−ηL

2


1 L−1 f; F  (ηx ) . L − 1 x=1

(3.7)

√ Here and below we omit the subscript ηL of f . Fix 1
K
L and divide the interval {1, . . . , L − 1} into 4 = (L − 1)/K adjacent intervals of length K or K + 1, where a represents the integer part of a. Denote by Ij the j th interval, by Mj the total spin on Ij : Mj = x∈Ij ηx and by EIj ,Mj the expectation with respect to the canonical measure νIj ,Mj . The right hand side of the previous formula is bounded above by 

2E L−1 ,M−ηL

4
1
f; {F  (ηx ) − EIj ,Mj [F  ]} L − 1 j =1 x∈Ij



+ 2E L−1 ,M−ηL

2

2

4 1
f; |Ij |EIj ,Mj [F  ] . L − 1 j =1

(3.8)

Taking conditional expectation with respect to Mj , we rewrite the first term as 





4
1
E L−1 ,M−ηL EIj ,Mj f ; F  (ηx ) 2 L − 1 j =1 x∈Ij




2

   4

24  E Var(ν , f )Var ν , F (η ) . ,M−ηL Ij ,Mj Ij ,Mj x (L − 1)2 j =1 L−1 x∈I j

By the induction assumption, Var(νIj ,Mj , f ) is bounded above by W (|Ij |)DIj (νIj ,Mj , f ). On the other hand, by Corollary 5.4, the variance of |Ij |−1 x∈Ij F  (ηx ) with respect to νIj ,Mj is bounded above by C0 |Ij |−1 F  2∞ uniformly over Mj , where C0 is a finite constant depending only on F ∞ . The previous expression is thus less than or equal to 4 C1 4
W (|Ij |)|Ij |E L−1 ,M−ηL [DIj (νIj ,Mj , f )] L2 j =1




4 C2
W (|Ij |)E L−1 ,M−ηL [DIj (νIj ,Mj , f )]. L j =1

Since W (K + 1)
CW (K), which follows from (3.6) and from the bound W (K)  CK 2 , and since the previous sum is bounded by the global Dirichlet form D L−1 (ν L−1 ,M−ηL , f ), we proved that the first term of (3.8) is bounded above by C3 W (K) D L−1 (ν L−1 ,M−ηL , f ). L

(3.9)

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C. LANDIM ET AL. / Ann. I. H. Poincaré – PR 38 (2002) 739–777

We turn now to the second term of (3.8). It is equal to 



2 E L−1 ,M−ηL

4 1
f; |Ij |(EIj ,Mj [F  ] − a − b[mj − m]) L − 1 j =1

2

,

where mj = Mj /|Ij |, m = (M − ηL )/(L − 1) and a, b are constants to be chosen later. We were allowed to add the terms a, b[mj − m] in the covariance because a, b 4j =1 |Ij |[mj − m] are constants. Let G(mj ) = EIj ,Mj [F  ] − a − b[mj − m]. By Schwarz inequality, the previous expression is bounded above by 

2E L−1 ,M−ηL [f ; f ] E L−1 ,M−ηL

4 1
|Ij |G(mj ) L − 1 j =1

2 

.

We claim that 

E L−1 ,M−ηL

4 1
|Ij |G(mj ) L − 1 j =1

2 




C0 KL

(3.10)

for some finite constant C0 . Indeed, developing the square, we write this expectation as 4
1 |Ij |2 E L−1 ,M−ηL [G(mj )2 ] 2 (L − 1) j =1

+


1 |Ij ||Ii |E L−1 ,M−ηL [G(mi )G(mj )]. 2 (L − 1) i=j

(3.11)

Recall that m = (M − ηL )/(L − 1),

mj = Mj /|Ij |.

By Corollary 5.3, E L−1 ,M−ηL [G(mj ) ] is bounded above by 2

Eνm [G(mj )2 ] +

C0 |Ij |  Eνm [G(mj )4 ]. L

(3.12)

Let A(α) = Eνα [F  (η1 )] and set a = A(m), b = A (m). With this choice, G(mj ) = EIj ,Mj [F  (ηx )] − A(mj ) + A(mj ) − A(m) − A (m)[mj − m]. By Corollary 5.3, |EIj ,Mj [F  (ηx )] − A(mj )| is less than or equal to CF  ∞ /|Ij |. On the other hand, A(mj )−A(m)−A (m)[mj −m] is bounded in absolute value by (1/2)A ∞ [mj −m]2 . In particular, 4 4
C0 4 A 2∞
1 2 2 |I | E [G(m ) ]
+ |Ij |2 Eνm [(mj − m)4 ] j νm j (L − 1)2 j =1 L2 2L2 j =1

(3.13)

for some constant C0 depending only on F . By Lemma 5.1, since νm is a product measure, the expectation on the right hand side of the previous inequality is bounded

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751

above by C|Ij |−2 . In Lemma 3.3 below we prove that A ∞ is bounded by a constant. The last expression is thus less than or equal to C0 4/L2
C/KL. The same arguments give that 4 
C0 C0 3 |I | Eνm [G(mj )4 ]
2 . j (L − 1)3 j =1 L

Therefore, the first line of (3.11) is bounded above by C0 /KL. We proceed in the same way to bound the second term of (3.11). Fix i = j . By Corollary 5.3, E L−1 ,M−ηL [G(mi )G(mj )] is bounded above by Eνm [G(mi )G(mj )] +

C0 K  Eνm [G(mi )2 G(mj )2 ]. L

Notice that the first term vanishes because νm is a product measure and 







  
 F (ηx )  ηy = Mj = Eνm [F  (ηx )] = A(m),



Eνm EIj ,Mj [F (ηx )] = Eνm Eνm



y∈Ij

Eνm [mj ] = m. On the other hand, since νm is a product measure, Eνm [G(mi )2 G(mj )2 ] = Eνm [G(mi )2 ]Eνm [G(mj )2 ]. Hence, 4

1 C0 K 1 Eνm [G(mi )2 ] |I ||I |E [G(m )G(m )]
|Ii | j i L−1 ,M−ηL i j 2 (L − 1) i=j (L − 1) i=1 L



because j |Ij | = L − 1. The right hand side of the previous formula is exactly the 1 /KL. This estimate together first term in (3.13) that we showed to be bounded by C with the bounds obtained in (3.13) and in the paragraph that follows (3.13) prove (3.10). Therefore, the second term of (3.8) is bounded above by C0 (KL)−1 E L−1 ,M−ηL [f ; f ]. This bound together with (3.9) gives that (3.8), and therefore (3.7), is less than or equal to C C3 W (K) D L−1 (ν L−1 ,M−ηL , f ) + E ,M−ηL [f ; f ]. L KL L−1 Since (3.7) is just the square of the second term of (3.4), taking expectation with respect to ν L ,M in (3.7) and recalling (3.5), we have that (3.3) is bounded above by 

C L+



W (K) C E ,M [f ; f ]. D L (ν L ,M , f ) + L KL L

Choose K large enough for ε = C/K to be strictly smaller than 2. Adding this term to (3.2), in view of the decomposition (3.1), we deduce that E L ,M



2



ε (f − E L ,M [f ])
1 − L

−1 



C3 W (L − 1) + C3 L + D L (ν L ,M , f ). L

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Taking supremum over smooth functions f : R L → R in L2 (ν L ,M ), we obtain that 

ε W (L)
1 − L

−1 



C3 W (L − 1) + C3 L + . L

It is not difficult to deduce from this recursive relation the existence of a constant C4 such that W (L)
C4 L2 for all L  2. This concludes the proof of Theorem 2.1. ✷ We conclude this section proving a result needed earlier in the proof. L EMMA 3.3. – There exists a constant C0 , depending only on F ∞ , such that sup |A (α)|
C0 . α∈R

Proof. – We claim that A(α) = (α) − α. Indeed, since A(α) = Eνα [F  (η1 )], we have that A(α) = (α) − α +

1 Z((α))



{−(α) + a + F  (a)}e(α)a−V (a) da.

An integration by parts shows that the integral vanishes proving that A(α) = (α) − α. It follows from this identity that A (α) =  (α). On the other hand, since  = R −1 ,  (α) = −

R  ((α)) . [R  ((α))]3

Recall that {γk , k  2} stands for the truncated moments of the variables X1λ . We obtain from the definition of R that R  ((α)) = γ2 ((α)), R  ((α)) = γ3 ((α)). Therefore, A (α) = −γ3 ((α))/γ2((α))3 and the statement follows from Lemma 5.1. ✷ 4. Logarithmic Sobolev inequality We prove in this section Theorem 2.2. The approach is similar to the one presented in last section for the spectral gap. We will derive a recursive formula for θ(L) in terms of θ(L − 1) and L in four steps. As before, all constants are allowed to depend on F ∞ , F  ∞ and F  ∞ . Step 1. One-site logarithmic Sobolev inequality. We start our proof with the case L = 2. Let f : R 2 → R be a smooth function such that f 2 ν 2 ,M = 1. Let g(η2 ) = f (M −η2 , η2 ). Since the total spin is fixed to be M, we have that g 2 ν 2 ,M = f 2 ν 2 ,M = 1 and that S 2 (ν 2 ,M , g) = S 2 (ν 2 ,M , f ). The next lemma permits to estimate the entropy of S 2 (ν 2 ,M , g) in terms of the Glauber Dirichlet form of g. This result is in fact a logarithmic Sobolev inequality for the Glauber dynamics obtained when restricting the Kawasaki exchange dynamics to one site. Recall that ν 1 L ,M represents the one-site marginal of ν L ,M .

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L EMMA 4.1. – There exists a finite constant C0 depending only on F ∞ such that 

 2

H (ηL ) log H (ηL )

2

dν 1 L ,M (ηL )


C0 Eν 1

L ,M

∂H ∂ηL

2 

(4.1)

for every L  2, every M in R and every smooth function H : R → R in L2 (ν 1 L ,M ) such = 1. that H 2 ν 1 L ,M

Same comments presented in Remark 3.2 apply here. We conclude the first step before proving the lemma. From the previous statement applied to L = 2 and H = g we have that 

S 2 (ν 2 ,M , f ) = S 2 (ν 2 ,M , g)
C0 Eν 1

2 ,M



= C0 Eν 2 ,M

∂g ∂η2

2 

∂g ∂η2

2 



= C0 Eν 2 ,M

∂f ∂f − ∂η2 ∂η1

2 

because ∂g/∂η2 = ∂f/∂η2 − ∂f/∂η1 . This proves that θ(2)
C0 . Proof of Lemma 4.1. – We first prove the lemma in the case of grand canonical measures. Recall that we denote by ν¯ λ1 the one-site marginal of the measure ν¯ λ L . We want to show that there exists a constant C0 , independent of λ, such that  2

H (a) log H (a)

2

ν¯ λ1 (da)


C0



[H  (a)]2 ν¯ λ1 (da)

(4.2)

for all smooth functions H : R → R such that H 2 ν¯ 1 = 1. Since the potential V is a λ bounded perturbation of the Gaussian potential, by Corollary 6.2.45 in [7], the previous inequality holds with a constant C0 that might depend on λ. All the matter here is to show that we may find a finite constant independent of λ. Recall the definition of the potential Vλ introduced in the proof of Lemma 3.1. A change of variable permits to rewrite the left hand side of (4.2) as 

1 2 Hλ (a)2 log Hλ (a)2 e−Fλ (a) √ e−(a /2) da, 2π

  and Z(λ) is a normalizing where Hλ (a) = H (a + λ), Fλ (a) = F (a + λ) + log Z(λ)  constant. It is easy to check that  exp{±Fλ }∞
exp 2F ∞ . In particular, by Corollary 6.2.45 in [7], the previous expression is bounded above by 4F ∞

2e



1 [Hλ (a)]2 e−Fλ (a) √

2π

−(a 2 /2)

e

4F ∞

da = 2e



[H  (a)]2 ν¯ λ1 (da).

This proves the lemma in the case of grand canonical measures with C0 = 2 exp{4F ∞ }. For canonical measures, we just need to use the local central limit theorem for large values of L and explicit computations for small values of L. We start with the case of = 1 and recall the large values of L. Fix a smooth function H : R → R with H 2 ν 1 L+1 ,M

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notation introduced in the proof of Lemma 3.1. The left hand side of (4.1) can be written as √ √  L+1 fλ,L ([γ1 − a]/σ L) 2 2 √ da, H (a) log H (a) gλ (a) fλ,L+1 (0) L where gλ stands for the density Z(λ)−1 exp{λx − V (x)} and λ = (M/[L + 1]). We base our proof on two facts. First, that if a function W is strictly convex then the measure µW (dx) = Z −1 exp{−W (x)} dx associated to the potential W satisfies a logarithmic Sobolev inequality. Secondly, if µ(dx) satisfies a logarithmic Sobolev inequality, and f is a density with respect to µ, which is bounded below and above (0 < C1
f
C1−1 ), then f dµ satisfies a logarithmic Sobolev inequality. The proof of these two well known sentences can be found, for instance, in [13]. In view of these statements, we just need to show that the above density is equivalent to the density of a measure associated to a convex potential. Here and below two functions g, f are said to be equivalent if there exists a finite, strictly positive constant C0 depending only on V (and not on M, λ or L) such that C0 g
f
C0−1 g. We shall rely on the local central limit theorem to show the equivalence of the above density with some density associated to a convex potential. By Theorem 5.2, for L large enough fλ,L+1 (0) is bounded above and below by a constant. We may therefore ignore the denominator in the previous integral. Recall from the previous section that we denote by gλ,L the density of the random variable λ 1j L Xj . An elementary computation, already mentioned in the proof of Lemma 3.1, gives that 

gλ,L (a) = e

(λ−µ)a

Z(µ) Z(λ)

L

gµ,L (a)

for all λ, µ in R. In particular, writing fλ,L in terms of gλ,L , we get that 

fλ,L

γ1 (λ) − x √ σ (λ) L







 σ (λ) Z(µ) L = exp (λ − µ)(γ1 (λ) − x) + L(λ − µ)γ1 (λ) σ (µ) Z(λ)   γ1 (λ) − x √ γ1 (λ) − γ1 (µ) √ + L × fµ,L . σ (µ) σ (µ) L

We will now choose µ for the variable on the right hand side to vanish. In this case, we will be able to apply the local central limit theorem to claim that fµ,L (0) is bounded above and below by positive finite constants. Set 

µ(x) = 





1 x 1+ γ1 (λ) − . L L

(4.3)

With this choice, γ1 (µ) = (1 + L−1 )γ1 (λ) − (x/L) so that the right hand side of the previous formula becomes 

σ (λ) Z(µ) σ (µ) Z(λ)

L

eL(λ−µ)γ1(µ) fµ,L (0).

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By Lemma 5.1, σ (·) is a function bounded below and above by strictly positive finite constants. By Theorem 5.2, fµ,L (0) is bounded below and above by strictly positive finite constants. It follows from this observation and from the previous estimate on fλ,L+1(0) that the density of the measure ν 1 L+1 ,M with respect to the Lebesgue measure, denoted by RL+1,M (x), is equivalent in the sense defined above to the function



exp − (1/2)(x − λ)2 + L log Z(λ) − log Z(µ) − (λ − µ)γ1 (µ)



because, by (5.4), gλ (x) is equivalent to exp{−(1/2)(x − λ)2 }. In this formula µ = µ(x) is defined by (4.3). It remains to show that the function inside braces, denoted by 0. Here, Fj (ηx ) = F  (ηx ) − EIj ,Mj [F  ]. By definition of θ(|Ij |), the second term is bounded above by θ(|Ij |)β −1 DIj (νIj ,Mj , gj ). On the other hand, by Lemma 6.1, the first one is bounded above by C0 βK for some finite constant C0 . Minimizing over β and summing over j , since θ(K + 1)
Cθ(K), we get that (4.11) is less than or equal to 4   C0 θ(K) C0 θ(K)
D L (ν L ,M , g). E L ,M EIj ,Mj [g 2 ] DIj (νIj ,Mj , gj )
L L j =1

This concludes the proof of the lemma.

(4.12)

✷

We turn now to the second term of (4.10). Recall that m, mj stand for M/L, Mj /|Ij |, respectively. Let G(mj ) = EIj ,Mj [F  ] − A(m) − A (m)[mj − m], where A(m) = Eνm [F  ]. Since we may add constants in a covariance, the expectation in the

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second term of (4.10) is equal to 

E L ,M



4 1
g ; |Ij |G(mj ) . L j =1 2

(4.13)

To estimate this covariance we need to consider two cases. Let β0 , be the constant given by Lemma 6.5 and fix 0 < δ < 2. By Lemma 6.5, there exists K0 for which the left hand side of (6.10) is bounded by δβ for all β
β0 and all L  2K  2K0 . L EMMA 4.5. – Fix L  2K  2K0 , M in R and a smooth function g in L2 (ν L ,M ) such that g 2 ν L ,M = 1. Assume that θ(L)L−1 D L (ν L ,M , g) < δβ02 . Then, 



E L ,M

4 1
g ; |Ij |G(mj ) L j =1

2

2




δθ(L) D L (ν L ,M , g). L

Proof. – Fix a density g 2 satisfying the assumptions. By the entropy inequality, the expectation in the statement of the lemma is bounded by 



4
1 log E L ,M exp β |Ij |G(mj ) βL j =1



+

1 S (ν ,M , g) βL L L

(4.14)

for every β > 0. By Lemma 6.5 and our choice of K, L, the first term is bounded above by δβ for all β < β0 . The second one, by definition of θ , is bounded above by (θ(L)/βL)D L (ν L ,M , g). Therefore, (4.13) is less than or equal to δβ +

θ(L) D L (ν L ,M , g). βL

The value of β that minimizes this expression is β12 =

θ(L) D L (ν L ,M , g). δL

By hypothesis, β1 < β0 and we may therefore minimize in β < β0 to obtain that the square of (4.13) is bounded above by δθ(L) D L (ν L ,M , g), L which concludes the proof of the lemma.

✷

L EMMA 4.6. – Fix L  2K  2K0 , M in R and a smooth function g in L2 (ν L ,M ) such that g 2 ν L ,M = 1. Assume that θ(L)L−1 D L (ν L ,M , g)  δβ02 . Then, there exists a finite constant C such that 



E L ,M

4 1
g ; |Ij |G(mj ) L j =1 2

2





δθ(L)
CL + D L (ν L ,M , g). L

(4.15)

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Proof. – The covariance E L ,M [g 2 ; 1j 4 |Ij |G(mj )] is equal to the covariance of g 2 and 1j 4 |Ij |HK (mj ), where HK (mj ) = E Ij ,Mj [F  ]. Since g 2 is a density with respect to ν L ,M , by Schwarz inequality, the left hand side of (4.15) is bounded above by 2



4   1
|Ij |E L ,M g 2 (HK (mj ) − HK (m)) 2 L j =1



2

4 1
+2 |Ij |E L ,M [HK (mj ) − HK (m)] L j =1

.

(4.16)

By Lemma 6.6, HK is uniformly Lipschitz. In particular, since g 2 is a density, by Schwarz inequality the first term is bounded above by 4   CK C
|Ij |E L ,M g 2 [mj − m]2
L j =1 4L






E L ,M [g 2 mj − mi ]2



1i=j 4

for some finite constant C because m is just the average of the densities mi . By Lemma 4.7 below, each expectation is bounded by 



C1 (K) + C2 (K) DIi (νIi ,Mi , g) + DIj (νIj ,Mj , g) + E L ,M

∂g ∂g − ∂ηyi ∂ηxj

2 

,

where C2 (K) is a finite constant and C1 (K) is a constant that can be made as small as one wishes by letting K ↑ ∞. Here we are assuming that the cubes Ij are ordered, that i < j and that yi is the rightmost site in Ii and xj is the leftmost site in Ij . An elementary computation shows that the expectation in the previous formula is bounded above by LD L (ν L ,M , g). Therefore, the first term in (4.16) is less than or equal to C1 (K) + C2 (K)LD L (ν L ,M , g). The second term in (4.16) is easy to estimate. Since HK is uniformly Lipschitz, by Schwarz inequality, this term is bounded by 4 CK
E ,M [(mj − m)2 ] L j =1 L

for some finite constant C. By Corollary 5.5, this term is bounded above by CK −1 . In conclusion, we proved that (4.16) is bounded above by C1 (K) + C2 (K)LD L (ν L ,M , g), where C1 (K) is a constant that can be made as small as one wishes by letting K ↑ ∞. In particular, choosing K large enough for C1 (K)
δ 2 β02 , by assumption, the previous

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term is less than or equal to 



δθ(L) + CL D L (ν L ,M , g) L

for some finite constant C depending on δ. This concludes the proof of the lemma.

✷

Proposition 4.3 follows from the decomposition (4.10) and Lemmas 4.4–4.6. We conclude this section with a technical result needed above. Consider the cube 2, the average spin over the first and second half: m1 = 2K . Denote by mi , i = 1, K −1 1xK ηx , m2 = K −1 K 0 and that log(1 + x)
x to estimate the first term by     1 4βE 2K ,M (m1 − m)2 + 16β 2 E 2K ,M (m1 − m)4 exp{4β(m1 − m)2 } β

because m1 − m2 = 2(m1 − m). By Corollary 5.5, we may replace the expectation with respect to canonical measures by expectation with respect to grand canonical measures, paying the price of a finite constant. Since the grand canonical measures are product measures, by Schwarz inequality, the previous expression is bounded above by  1/2  1/2 C + CβEνm (m1 − m)8 Eνm exp{8β(m1 − m)2 } . K

Since exp{ax 2 } is a convex function for a > 0 and since νm is a product measure, this sum is less than or equal to  1/2 Cβ C + 2 Eνm exp{8β(η1 − m)2 } . K K

For β small enough, the previous expectation is bounded, uniformly in m. This proves the lemma. ✷

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5. Local central limit theorem We prove in this section some estimates that follow from the local central limit theorem and play a central role in the proof of the spectral gap and the logarithmic Sobolev inequality. For λ in R, denote by Pλ the probability measure on the product space RN that makes the coordinates {Xk , k  1} independent random variables with marginal density Z(λ)−1 exp{λx − V (x)}. Denote by Eλ expectation with respect to Pλ . Recall that γ1 (λ), σ (λ)2 , {γk (λ), k  3} stand for the expectation, the variance and the kth truncated moment of the coordinate variables under the distribution Pλ : 

γ1 (λ) = Eλ [X1 ],





σ (λ)2 = Eλ (X1 − γ1 (λ))2 ,



γk (λ) = Eλ (X1 − γ1 (λ))k .

For N  1, denote by fλ,N the density of the random variable (σ (λ)2 N)−1/2 − γ1 (λ)).



1j N (Xj

L EMMA 5.1. – Assume that F ∞ < ∞. Then, there exist finite constants {Cj , j  1}, depending only on j and F ∞ , such that 0 < C1−1 < σ (λ)2 < C1 ,

0 < Cj−1 < γ2j (λ) < Cj

for all λ in R. Proof. – We first claim that Z(λ) exp{−λ2 /2} is bounded above an below by finite positive constants. Indeed, by definition, Z(λ) = eλ

2 /2



da e−(1/2)(a−λ) −F (a) = eλ 2

2 /2



da e−(1/2)a

2−F

λ (a)

,

+ λ). Since F is absolutely bounded, this expression is bounded where Fλ (a) = F (a √ below and above by 2π exp{λ2 /2} exp{±F ∞ }, proving the claim. We now claim that |γ1 (λ) − λ| is bounded by F ∞ exp{2F ∞ }. Indeed, by definition, the difference γ1 (λ) − λ is equal to 1 Z(λ)



(x − λ)eλx−(x

2 /2)−F (x)

dx.

R

Changing variables, we may rewrite this integral as 

xe−(x

2 /2)−F



λ (x)

dx

R

e−(x

2 /2)−F

λ (x)

dx.

R



Since dx x exp{−(1/2)x 2 } vanishes, by Schwarz inequality, the absolute value of this expression is bounded above by F ∞

e



1  √  2π

which proves the claim.

−(1/2)x 2

xe R



−Fλ (x)

e

  − 1 dx 
F ∞ e2F ∞ , 

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We now prove a lower bound for σ (λ)2 . The same ideas permit to derive an upper bound for σ (λ)2 or upper and lower bounds for the truncated moments {γ2j (λ), j  2}. A change of variables and the estimate on Z(λ) exp{−λ2 /2} gives that  2 2 −2F ∞ 1 √ da [a + λ − γ1 (λ)]2 e−a /2 σ (λ)  e 2π  1 2 −2F ∞ √ e inf da [a + β]2 e−a /2  C1 > 0, β, |β|λ−γ1 (λ)∞ 2π where C1 depends only on F ∞ . This concludes the proof of the lemma. ✷ It follows from this lemma that

   γj (λ)  j  
C sup σ (λ)j 

(5.1)

λ∈R

for all j  3, which is the estimate needed in order to prove the uniform local central limit theorem. T HEOREM 5.2. – Assume that F ∞ < ∞. There exists N0  1 and a finite constant C depending only on F ∞ such that      C fλ,N (x) − √1 e−x 2 /2 1 − γ3 (λ)x 
  N 3 1/2 6σ (λ) N 2π

for all N  N0 , x in R and λ in R. For a fixed parameter λ this is just the usual statement of the local central limit theorem for i.i.d. random variables with finite fourth moments. The important point here is the uniformity over the parameter λ. This uniformity can be obtained in virtue of (5.1) and the estimates presented in the Lemma 5.1. The local central limit theorem gives asymptotic expansions of the expectation of a function with respect to a canonical measure. This is the content of the next result. C OROLLARY 5.3. – Fix 4  1 and fix a function G : R4 → R. There exist N0  1 and a finite constant C depending only on F ∞ such that for all N  N0 and all M in R C4 G∞ if G is bounded and |E N ,M [G] − Eνm [G]|
| N | C4  Eνm [G; G]. |E N ,M [G] − Eνm [G]|
| N | In these formulas, m = M/| N |. The proof is elementary (cf. Corollary A2.1.4 in [10]). Of course, by changing the value of the constant C, the first inequality remains valid for all values of N  4. C OROLLARY 5.4. – Let G : R → R be a smooth bounded function and let GL,M = G − Eν L ,M [G]. There exists a finite constant C0 , depending only on F ∞ such that 

E L ,M for all L  1 and M in R.

L 1
GL,M (ηx ) L x=1

2 


C0

G2∞ L

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765

Proof. – The variance is equal to 



  1 1 E L ,M (GL,M (η1 ))2 + 1 − E L ,M [GL,M (η1 )GL,M (η2 )]. L L

The first expression is bounded by 4G2∞ L−1 for all L  1 and M ∈ R. The second one, by definition of GL,M is equal to 



 1 1− E L ,M [G(η1 )G(η2 )] − E L ,M [G(ηL )]2 . L

By Corollary 5.3, since να is a product measure, the first term of the previous expression is equal to Eα [G(ηL )]2 ± CL−1 G∞ , where C is a finite constant depending only on F ∞ . By the same result, the second term is equal to Eα [G(ηL )]2 ± CL−1 G2∞ , which concludes the proof. ✷ For 1
K < L, denote by ν KL ,M the marginal on R K of the canonical measure ν L ,K . An elementary computation shows that ν KL ,M is absolutely continuous with respect to the Lebesgue measure and that its Radon–Nikodym derivative RK,L,M (xK ) is given by L1/2 (L − K)−1/2 gλK (xK )fλ,L−K

  √

σ L−K

−1




[γ1 − xi ] fλ,L (0)−1 dxK ,

1iK



where xK = (x1 , . . . , xK ), gλK stands for the density Z(λ)−K exp{ 1iK λxi − V (xi )} K (xK )/gλK (xK ) is bounded and λ = (M/L). The next result shows that the ratio RL,M above, uniformly over λ, provided K/L is bounded away from 1. [4] has obtained the same result in the case of lattice gases under strong mixing assumptions. C OROLLARY 5.5. – There exists a finite constant C0 depending only on F ∞ , such that RK,L,M (xK )
C0 gλK (xK ) for all L/2  K  1 and xK in R K . In this formula, λ = (M/L). In particular, if K
L/2, for any local function H : R K → R, E L ,M [H (η1 , . . . , ηK )]
C0 EνM/| L | [ |H | ].

(5.2)

Proof. – In view of the explicit formula for the density RK,L,M and since K
L/2, we only have to show that √ fλ,L−K ({σ L − K}−1 1iK [γ1 − xi ])
C0 . (5.3) fλ,L (0) We prove separately that the numerator is bounded and that the denominator is bounded below by a strictly positive constant. Consider, for instance the denominator. For L large enough, the lower bound follows from Theorem 5.2. For L small, it follows by

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inspection. The same argument applies to the numerator with L − K in place of L. This proves the corollary since (5.2) follows at once form (5.3). ✷ Theorem 5.2 and its corollaries permit to estimate expectation with respect to a canonical measure µ L ,M , provided L is large. The next result provides an estimate for small values of L. The important point in this result is once again the uniformity over the parameter λ. Denote by f˜λ,N the density of the random variable N −1/2 1j N {Xj − λ} under the measure Pλ . Note that we are not renormalizing by σ (λ) and that we are subtracting λ instead of γ1 (λ). L EMMA 5.6. – There exists a positive and finite constant C1 , depending only on F ∞ , such that 1 1 2 2 C1−N √ e−x /2
f˜λ,N (x)
C1N √ e−x /2 2π 2π for every λ in R. Proof. – The proof is elementary. We present the upper bound. For N  1, let gλ,N be the density with respect to the Lebesgue measure of the random variable 1j N Xj under Pλ . By the estimate on Z(λ) exp{λ2 /2} obtained in the proof of Lemma 5.1 and by the explicit formula for gλ,N , we have that gλ,N (x) is bounded by 2 C1N eλx−(λ N/2)

1 (2π )N/2





dx1 . . . dxN−1 RN−1



N−1

1 N−1 1 exp − xi2 − x− xi 2 i=1 2 i=1

2 

for some constant C1 depending only on F ∞ . Since the integral with the renormalization factor in front is equal to (2π )−1/2 exp{−x 2 /2N}, the previous expression is equal to C1N (2π )−1/2 exp{−(x − λN)2 /2N}. To conclude the proof of the lemma, it remains to express f˜λ,N in terms of gλ,N (x). ✷ The same argument shows that gλ (x) = Z(λ)−1 exp{λx − V (x)} is bounded above and below by a Gaussian density. More precisely, there exists a finite, strictly positive constant C0 depending only on F ∞ , such that 1 1 2 2 C0 √ e−(x−λ) /2
gλ (x)
C0−1 √ e−(x−λ) /2 2π 2π

(5.4)

for every λ in R. L EMMA 5.7. – There exists β0 > 0 and a finite constant C0 such that 



Eνα exp{β0 | L |{m L − α}2 }
C0 for every α in R and L  1. In this formula, m = | L |−1

x∈ L

ηx .

Proof. – For small values of L this statement is a straightforward consequence of the previous lemma, the fact that γ1 (λ) − λ is absolutely bounded, proved in Lemma 5.2, and the fact that the statement holds for Gaussian distributions.

C. LANDIM ET AL. / Ann. I. H. Poincaré – PR 38 (2002) 739–777

767

For large values of L, with the notation introduced in the beginning of this section, the expectation can be written as 

eβ0 σ (λ)

2x 2

fλ,L (x) dx

R

for some appropriate choice of λ. Notice that the local central limit theorem, stated in Theorem 5.1, gives a good bound only for small values of x. The idea is therefore to replace in the previous formula λ by a variable µ which makes x a typical value. By (4.3) or a direct computation, √     σλ Zµ L (λ−µ)[xσλ√L+Lγ1 (λ)] xσλ L(γ1 (λ) − γ1 (µ)) e fµ,L + . fλ,L (x) = σµ Zλ σµ σµ Choose µ for the expression inside fµ,L to be small (in order to be able to use the local central limit estimate): √ xσλ L = L[γ1 (µ) − γ1 (λ)]. With this choice, since by Theorem 5.1 C1−1
fµ (0)
C1 for some universal constant C1 , and since by Lemma 5.2 σλ is bounded, √

 fλ,L (x) ∼ exp L log{Zµ /Zλ } + (λ − µ)[xσλ L + Lγ1 (λ)] , where ∼ means that the left hand side is bounded above and below by the right hand side multiplied by finite positive constants. The expression inside the exponential vanishes at x = 0. It is also not difficult to show that it is strictly concave in x (cf. computation right after (4.3)). In particular, fλ,L (x) ∼ e−C2 x

2

for some finite constant C2 and we are back to the Gaussian case. 6. Large deviations estimates Fix a differentiable function R : R → R with bounded derivative: R  ∞ < ∞. Let = R(a) − Rνα in the case of grand canonical measures and let Rα (a) = R(a) − Rν L ,M in the case of canonical measures. Notice that Rα (ηx ) − Rαg (ηx ) = Eνα [R(ηx )] − E L ,M [R(ηx )]. It follows from Corollary 5.3 that Rαg (a)

   Eν [R(ηx )] − E ,M [R(ηx )]
CR ∞ α L

| L |

(6.1)

for some finite constant C depending only on F ∞ because Eνα [R; R]
Eνα [{R(η1 ) − R(α)}2]
R  2∞ σ ((α))2 . We claim that there exists a finite constant C0 depending only on F ∞ for which |Rαg (a)|
C0 R  ∞ (1 + |a − α|),

|Rα (a)|
C0 R  ∞ (1 + |a − α|)

(6.2)

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C. LANDIM ET AL. / Ann. I. H. Poincaré – PR 38 (2002) 739–777

for all a, α in R (in the canonical case for all L  1, M in R). Consider first the grand canonical case. Notice that |Rαg (a)|
Eνα [|R(a) − R(η1 )|]
R  ∞ Eνα [|a − η1 |]




R  ∞ |a − α| + Eνα [(η1 − α)2 ]1/2 . By Lemma 5.1 the second term inside braces in the last expression is bounded above by some finite constant C1 that depends on F ∞ only. This proves the claim in the grand canonical case. The same arguments apply to the canonical case provide we show that E L ,M [(η1 − α)2 ] is uniformly bounded. But this is part of the content of Corollary 5.5. L EMMA 6.1. – Fix a differentiable function R : R → R with bounded derivative and L  2. There exists a constant C, depending only on F ∞ , such that 1 log β| L |





exp β






Rα (ηx ) dν L ,M
CR  2∞ β

(6.3)

x∈ L

for all β > 0 and all M in R. Here Rα = R − Rν L ,M . Proof. – We first prove this result for the grand canonical measure in place of the canonical measure. In this case we replace Rα by Rαg and we only need to show that 1 log β



exp{βRαg (η1 )} dνα
CR  2∞ β

(6.4)

for all β > 0 because να is a product measure. We consider first the case of β small. By the spectral gap for the Glauber dynamics (Lemma 3.1), there exists a universal constant C0 such that f 2 να − f 2να
C0 (∂η1 f )2 να −1/2

for all smooth functions f in L2 (να1 ). Let C1 = C0 R  2∞ and assume that β < C1 Applying this inequality to the function f = exp{(β/2) Rαg }, we obtain that 



g

βRα

Eνα e




Eνα e

g

(β/2)Rα

2

 2

β 2

+ C0



g

R  2∞ Eνα eβRα

.



so that 

g



Eνα eβRα


 (β/2)R g 2 1 α E e ν α 1 − C0 R  2∞ (β/2)2  2



g


e(1/2)C0R ∞ β Eνα e(β/2)Rα 2

2

because (1 − x)−1
e2x for 0
x < 1/2. Iterating this estimate n − 1 times we obtain that 

g

βRα

Eνα e






exp C1 β

2

n
j =1



−j

2



Eνα e(β/2

n )R g α

2n

.

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769

The exponential is obviously bounded by exp{C1 β 2 }. On the other hand, we claim that 

g



lim n log Eνα e(1/n)Rα = 0,

(6.5)

n→∞

showing that the left hand side of (6.4) is bounded above by C1 β = C0 βR  2∞ provided −1/2 β < C1 . To prove (6.5), just notice that exp{(1/n)Rαg } is bounded above by 1 + (1/n)Rαg + (1/n2 )(Rαg )2 exp{(1/n)|Rαg |}. Since log(1 + x)
x and since Rαg has mean zero with respect to να , we obtain that g    1 n log Eνα e(1/n)Rα
Eνα (Rαg )2 exp{(1/n)|Rαg |} . n

By (6.2), the right hand side is bounded above by   C Eνα {1 + (η1 − α)2 } exp{C|η1 − α|/n} n

for some finite constant C depending only on F ∞ , R  ∞ . The expectation is bounded −1/2 for all n  1 because να has Gaussian tails. This proves (6.4) for β < C1 . −1/2 We now turn to the case of large β, which is simpler. Assume that β  C1 . It follows from (6.2) that the left hand side of (6.4) is bounded above by 





C2 R  ∞ + β −1 log Eνα eβR ∞ C2 |η1 −α| .

(6.6)

Since e|x|
ex + e−x , we need only to estimate Eνα [exp{βR  ∞ C2 (η1 − α)}] for β and −β. Recall the definition of the partition function Z given in Eq. (2.1). The logarithm of the previous expectation is equal to log Z((α) + βR  ∞ C2 ) − log Z((α)) − βR  ∞ C2 α. An elementary computation gives that (log Z) ((α)) = α so that the previous difference can be written as 











log Z (α) + βR  ∞ C2 − log Z (α) − (log Z) (α) βR  ∞ C2 . By Taylor’s expansion, this difference is bounded by (1/2)(βR  ∞ C2 )2 (log Z) (λ) for some λ between (α) and (α) + βR  ∞ C2 . Since (log Z) (λ) = σ 2 (λ) and since, by Lemma 5.1, σ 2 (λ) is bounded uniformly in λ, we have that 



log Eνα exp{βR  ∞ C2 (η1 − α)}
CR  2∞ β 2 for some constant depending only on F ∞ . Since log{a + b}
log 2 + max{log a, log b}, (6.6) is bounded above by C2 R  ∞ +

log 2 + C3 R  2∞ β, β −1/2

which is obviously bounded above by C4 R  2∞ β because β  C1 the proof of the lemma in the case of the grand canonical measure.

. This concludes

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We now turn to the canonical measure. We need to consider two cases. Assume first that βR  ∞
| L |−1 . By Schwarz inequality, the left hand side of (6.3) is bounded above by 1 log β| L |










exp 2β

Rα (ηx ) dν L ,M .

1xL/2

The difference is that we are now summing only over one half of the cube and that we had to pay a factor 2 in the exponential to do it. Since ex
1+x +x 2 e|x| , since log(1+x)
x and since Rα has mean zero, the previous expression is bounded above by 4β | L |

 


2

Rα (ηx )

 



1xL/2

 




exp 2β 

Rα (ηx ) dν L ,M .

(6.7)

1xL/2

Since e|x|
ex + e−x , we may remove the absolute value in the exponential, provide we estimate the expression for Rα , as well as for −Rα . Moreover, by Corollary 5.5, we may replace the canonical measure by the grand canonical one paying the price of a finite constant and turning Rα into a non-mean-zero function. At this point, we need to estimate C0 β | L |

 


2

Rα (ηx )



exp 2β

1xL/2






Rα (ηx ) dνα ,

1xL/2

with α = M/| L |. Since να is a product measure, expanding the square, we get that the previous integral is less than or equal to 





C0 βEνα Rα (η1 )2 e2βRα (η1 ) Eνα e2βRα (η1 ) 



+ C0 β| L | Eνα Rα (η1 )e2βRα (η1 )

2

(L/2)−1 

Eνα e2βRα (η1 )

(L/2)−2

.

(6.8)

There are three different types of terms in the previous formula and we estimate them separately. We first examine the exponentials. By (6.1), 

Eνα e2βRα (η1 )

(L/2)





g


eCβR ∞ Eνα e2βRα (η1 )

(L/2)

.

On the range considered βR  ∞
1, so that the exponential term is less than some finite constant C. On the other hand, since Rαg has mean zero with respect to να , since ex
1 + x + x 2 e|x| , since by (6.2) |Rαg (a)|
C0 R  ∞ [1 + |a − α| ] and since β 2 R  2∞
1, 

g



Eνα e2βRα (η1 )
1 + C0 β 2 R  2∞ . Here we took advantage of the fact that there exists some finite constant C2 depending only on F ∞ such that 



Eνα {1 + |η1 − α|2 }e2|η1 −α|
C2 for all α in R because να has uniform Gaussian tails. Since 1 + x
ex , 



Eνα e2βRα (η1 )

because β 2 R  2∞
L−1 .

L


exp{C0 β 2 R  2∞ L}
C2

C. LANDIM ET AL. / Ann. I. H. Poincaré – PR 38 (2002) 739–777

771

We now turn to the remaining expectations in (6.8). By (6.1), 





g

Eνα Rα (η1 )2 e2βRα (η1 )
CEνα Rα (η1 )2 e2βRα (η1 )



because βR  ∞
1. The same estimate (6.1) gives that the previous expression is less than or equal to g g     CR  2∞ Eνα e2βRα (η1 ) + CEνα Rαg (η1 )2 e2βRα (η1 ) . 2 | L |

By (6.2), |Rαg (η1 )|
C0 R  ∞ (1 + |η1 − α|). The previous sum is thus bounded by 



CR  2∞ + CR  2∞ Eνα {1 + |η1 − α|}2 e2C0 |η1 −α| . This expression is less than CR  2∞ because να has uniform exponential tails. It remains to estimate 



| L | Eνα Rα (η1 )e2βRα (η1 )

2

.

As before, we may replace Rα by Rαg in the exponential. After this replacement, applying (6.1), we bound the previous expression by 



C| L | Eνα Rαg (η1 )e2βRα (η1 )

2

+C

R  2∞   2βRαg (η1 ) 2 Eνα e . | L |

The second term is seen to be less than or equal to CR  2∞ /| L |, while the first, since aeb
a + |ab|e|b| and since Rαg has mean zero, is bounded by 



g

C| L |β 2 Eνα Rαg (η1 )2 e2β|Rα (η1 )| 

2




C| L |β 2 R  4∞ Eνα {1 + |η1 − α|2 }e2C0 |η1 −α|

2

.

This expression is bounded by CR  2∞ because να has uniform exponential tails and because β 2 R  2∞
| L |−1 . This proves the lemma in the case of small β. We now turn to the case of large β. Assume that β 2 R  2∞ > | L |−1 . We first replace Rα by Rαg . By (6.1), the left hand side of (6.3) is bounded above by 1 log β| L |





exp β






Rαg (ηx ) dν L ,M +

x∈ L

C0 R  ∞ . | L |

Since | L |−2
| L |−1 < β 2 R  2∞ , | L |−1
βR  ∞ . In particular, the second term is less than or equal to C0 βR  2∞ . It remains to estimate the first term. By Schwarz inequality, this expression is bounded above by 1 log β| L |





exp 2β


1xL/2



Rαg (ηx ) dν L ,M .

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C. LANDIM ET AL. / Ann. I. H. Poincaré – PR 38 (2002) 739–777

By Corollary 5.5, this expression is less than or equal to 1 log C + log β| L | β| L |





exp 2β






Rαg (ηx ) dνα ,

1xL/2

−1 where α = M/| L |. Since β 2 > C1 R  −2 ∞ | L | , the first term is bounded by  2 CβR ∞ . It remains to consider the second one which is equal to

1 log 2β



exp{2βRαg (η1 )} dνα

(6.9)

because να is a product measure. This expression is just (6.4) and we proved in the first part of the lemma that it is bounded by CβR  2∞ . This concludes the proof. ✷ The same proof gives the following estimate that we state for further use. L EMMA 6.2. – Fix a differentiable function R : R → R with bounded derivative: R  ∞ < ∞ and L  2. There exists a constant C, depending only on F ∞ , such that  1 log exp{βRα (η1 )} dν L ,M
CR  2∞ β β for all β > 0 and all M in R. Lemma 6.1 provides an estimate, uniform over the charge M, on the expectation of −1 | L | x∈ L Rα (ηx ) with respect to some measure f dν L ,M in terms of the entropy of this measure. C OROLLARY 6.3. – Fix L  2, M in R, a differentiable function R : R → R with bounded derivative and a density f with respect to ν L ,M . There exists a constant C0 , depending only on F ∞ , such that  



1
Rα (ηx ) f dν L ,M | L | x∈ L

2


C0

   R  2∞ S L ν L ,M , f . | L |

Proof. – By the entropy inequality, the integral on the left hand side of the statement of the lemma is bounded above by √ 
  S L (ν L ,M , f ) 1 log exp β Rα (ηx ) dν L ,M + β| L | β| L | x∈ L for all β > 0. By Lemma 6.1, the first term is bounded above by C0 R  2∞ β for some finite constant depending only on F ∞ . Minimizing in β we conclude the proof of the lemma. ✷ Lemma 6.2 provides a similar estimate in the case of a one-site function: C OROLLARY 6.4. – Fix L  2, M in R, a differentiable function R : R → R with bounded derivative and a density f with respect to ν 1 L ,M . There exists a constant C0 ,

C. LANDIM ET AL. / Ann. I. H. Poincaré – PR 38 (2002) 739–777

773

depending only on F ∞ , such that 

2

Rα (η1 ) f (η1 ) dν 1 L ,M




C0 R  2∞ S{0} ν 1 L ,M ,

 

f .

The proof is the same as the one of Corollary 6.3. Fix K  1, L  K 2 and divide the interval {1, . . . , L} into 4 = L/K adjacent intervals of length K or K + 1, where a represents the integer part of a. Denote by Ij the j th interval, by Mj the total spin on Ij : Mj = x∈Ij ηx and by EIj ,Mj the expectation with respect to the canonical measure νIj ,Mj . Let m, mj stand for M/L, Mj /|Ij |, respectively and let G(mj ) = EIj ,Mj [F  ] − E L ,M [F  ] − A (m)[mj − m], where A(m) = Eνm [F  ]. L EMMA 6.5. – There exist β0 > 0 and a finite constant C0 depending only on F ∞ , F  ∞ such that 



4
1 log E L ,M exp β |Ij |G(mj ) βL j =1






C0 β K

(6.10)

for all β
β0 , all L  K 2 and all M in R. Proof. – We first prove the lemma in the grand canonical case with G replaced by the  given by: mean-zero function G  j ) = EI ,M [F  ] − Eν [F  ] − A (m)[mj − m]. G(m m j j

Fix a density m. To keep notation simple, assume that all cubes Ij have the same length K. Since νm is a product measure, the left hand side of (6.10) is equal to 1  1 )}]. log Eνm [exp{βK G(m βK  = 0, the previous Since ex
1 + x + x 2 e|x| , since log(1 + x)
x and since Eνm [G] expression is less than or equal to   β  1 )}2 exp{βK|G(m  1 )|} . Eνm {K G(m K

We claim that there exists β1 and a finite constant C0 such that 



 1 )}2 exp{βK|G(m  1 )|}
C0 Eνm {K G(m

(6.11)

 1) = for all m in R, all K  1 and all β
β1 . Indeed, let A(α) = Eνα [F  ]. Since G(m    {EI1 ,M1 [F ] − Eνm1 [F ]} + A(m1 ) − A(m) − A (m)[m1 − m], by Lemma 3.3 and  is bounded in absolute value by CK −1 + C(m1 − m)2 for some finite Corollary 5.3, G constant C. In particular, the left hand side of (6.11) is bounded above by 

CeCβ Eνm {1 + K 2 (m1 − m)4 } exp{CβK(m1 − m)2 } 




CEνm exp{C  βK(m1 − m)2 } .



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By Lemma 5.7, there exists β1 > 0 such that for β < β1 , the expectation is bounded uniformly in K and m. This proves claim (6.11) and that the left hand side of (6.10) is bounded by Cβ/K for β
β1 , which concludes the proof of the lemma in the grand canonical case. We now turn to the canonical measure. Notice first that  1 )|
|G(m1 ) − G(m

CF  ∞ L

(6.12)

for some finite constant C = C(F ∞ ). We now turn to the proof of (6.10). By Schwarz inequality, the left hand side of (6.10) is bounded by 




1 log E L ,M exp 2β |Ij |G(mj ) βL j =1 4/2



.

The difference is that we are now summing only over the first 4/2 cubes of length K so that we can use Corollary 5.5 to estimate the expectation with respect to canonical measure by the expectation with respect to grand canonical measure. Assume that K/L
β 2
β02 = β12 /4. By (6.12), the previous expression is bounded by 




1  j) log E L ,M exp 2β |Ij |G(m βL j =1 4/2



+

C L

for some finite constant C = C(F ∞ , F  ∞ ). In the range considered, L−1
β 2 /K
Cβ/K because β
β0 . The remainder term is thus bounded by Cβ/K for some finite constant C = C(F ∞ , F  ∞ ). On the other hand, by Corollary 5.5, the previous expression is bounded above by 




1 C0  j) + log Eνm exp 2β |Ij |G(m βL βL j =1 4/2



.

Since β 2  K/L, the first term is bounded by C0 β/K. On the other hand, by the first part of the proof, the second term is bounded by C0 β/K because 2β
β1 . This proves (6.10) provided K/L
β 2
β02 . Assume now that β 2
min{K/L, β02 }. In this case, since exp{x}
1 + x + 2 x exp{|x|}, since log(1 + x)
x and since the sum that appears in the exponential of (6.10) has mean zero with respect to the canonical measure, the left hand side in (6.10) is bounded above by 4β E L ,M L

 4/2
j =1

2

|Ij |G(mj )

 4/2  
   exp 2β  |Ij |G(mj ) .   

j =1

Since e|x|
ex + e−x , we may remove the absolute value in the exponential provide we estimate the previous expression with −β in place of β in the exponential. Consider the

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C. LANDIM ET AL. / Ann. I. H. Poincaré – PR 38 (2002) 739–777

case with β. By Corollary 5.5, the previous expression without the absolute value in the exponential is less than or equal to Cβ Eν L m

 4/2


2

|Ij |G(mj )



exp 2β

j =1

4/2




|Ij |G(mj )

.

j =1

Since νm is a product measure, expanding the square we obtain that this term is equal to    (4/2)−1 Cβ Eνm {KG(m1 )}2 e2βKG(m1 ) Eνm e2βKG(m1 ) K  2  (4/2)−2 CβL + Eνm KG(m1 )e2βKG(m1 ) Eνm e2βKG(m1 ) . 2 K We estimate separately each of the expectations appearing in this formula. We start examining the exponential terms. By (6.12), we have that 

Eνm e2βKG(m1 )

(4/2)



(m1 )
eCβ Eνm e2βK G

(6.13)

(4/2)

for some finite constant C = C(F ∞ , F  ∞ ). Since β
β0 , exp{Cβ}
C. Since  1 ) has mean zero with respect to νm , expanding the exponential up to the second G(m  1 )}] is bounded above by order, we get that Eνm [exp{2βK G(m 



(m1 )|  1 )}2 e2βK|G . 1 + 4β 2 Eνm {K G(m

Since β
β0 , by (6.11) the previous expression is less than or equal to 1 + Cβ 2
exp{Cβ 2 }. Therefore, 

(m1 ) Eνm e2βK G

4

2


eCβ 4
C1

because β 2
K/L = 4−1 . We now estimate Eνm [{KG(m1 )}2 exp{2βKG(m1 )}]. Here again we first replace  1 ). By (6.12), this expression is bounded above by G(m1 ) by G(m 





 1 )}2 exp{2βK G(m  1 )} + CEν exp{2βK G(m  1 )} CEνm {K G(m m



for some finite constant C = C(F ∞ , F  ∞ ) because β
β0 . We have already seen that the exponential term is bounded. On the other hand, by (6.11) the first expectation is bounded by a constant because β
β0
β1 /2. It remains to estimate  2 L Eνm KG(m1 )e2βKG(m1 ) . K  By (6.12) the previous Here again we start replacing G by the mean-zero function G. expression is less than or equal to   CL (m1 ) 2 CK (m1 ) 2  1 )e2βK G Eνm K G(m Eνm e2βK G + K L

for some finite constant C = C(F ∞ , F  ∞ ) because β
β0 . We have seen that the expectation of the exponential term is bounded. On the other hand, since a exp{a}


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 1 ) has mean zero with respect to νm , a + a 2 exp{|a|} and since G(m 







(m1 ) (m1 )|  1 )e2βK G  1 )}2 e2βK|G
2βEνm {K G(m . Eνm K G(m

Since β
β0 , by (6.11) the previous expression is bounded by Cβ. In view of the previous estimates, (6.13) is bounded above by Cβ Cβ 3 L Cβ +
K K2 K because β 2
K/L. This proves (6.10) in the case where β 2
min{K/L, β02 } and concludes the proof of the lemma. ✷ L EMMA 6.6. – Fix a bounded function H : R → R and L  2. The function HL : R → R defined by HL (m) = E L ,M [H (η1 )] is Lipschitz continuous on R and the Lipschitz constant does not depend on L. Proof. – An elementary computation shows that ∂M E L ,M [H (η1 )] = −E L ,M [η2 ; H (η1 )] = −E L ,M [H (η1 ){η2 − m}]. By Corollary 5.3, the absolute value of the previous expression is bounded above by C0 L−1 σ ((m)) for some finite constant C0 depending on H ∞ because νm is a product measure. Since HL = L∂M E L ,M [H (η1 )], it remains to recall the statement of Lemma 5.1 to conclude the proof of the lemma. ✷ Acknowledgements C.L. was partially supported by CNPq grant 300358/93-8, FAPERJ and PRONEX 41.96.0923.00. H.T. Yau is partially supported by US National Science Foundation grant 9703752. REFERENCES [1] L. Bertini, B. Zegarlinski, Coercive inequalities for Kawasaki dynamics: The product case, Markov Proc. Related Fields 5 (1999) 125–162. [2] L. Bertini, B. Zegarlinski, Coercive inequalities for Gibbs measures, J. Funct. Anal. 162 (1999) 257–289. [3] P. Caputo, A remark on spectral gap and logarithmic Sobolev inequalities for conservative spin systems, Preprint, 2001. [4] N. Cancrini, F. Martinelli, Comparison of finite volume canonical and grand canonical Gibbs measures under a mixing condition, Markov Proc. Related Fields 6 (2000) 23–72. [5] N. Cancrini, F. Martinelli, On the spectral gap of Kawasaki dynamics under a mixing condition revisited, J. Math. Phys. 41 (2000) 1391–1423. [6] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989. [7] J.D. Deuschel, D.W. Stroock, Large Deviations, Academic Press, Boston, 1989. [8] P.A. Ferrari, A. Galves, C. Landim, Rate of convergence to equilibrium of symmetric simple exclusion processes, Markov Proc. Related Fields 6 (2000) 73–88.

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