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Deviation inequalities via coupling for stochastic processes

and random fields

Citation for published version (APA):

Chazottes, J. R., Collet, P., Külske, C., & Redig, F. H. J. (2005). Deviation inequalities via coupling for stochastic processes and random fields. (Report Eurandom; Vol. 2005019). Eurandom.

Document status and date: Published: 01/01/2005

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arXiv:math.PR/0503483 v1 23 Mar 2005

Deviation inequalities via coupling

for stochastic processes and random fields

J.-R. Chazottes

Centre de Physique Th´eorique, CNRS UMR 7644 F-91128 Palaiseau Cedex, France

jeanrene@cpht.polytechnique.fr P. Collet

Centre de Physique Th´eorique, CNRS UMR 7644 F-91128 Palaiseau Cedex, France

collet@cpht.polytechnique.fr C. K¨ulske

Department of Mathematics and Computing Sciences University of Groningen, Blauwborgje 3

9747 AC Groningen, The Netherlands kuelske@math.rug.nl

F. Redig

Faculteit Wiskunde en Informatica and Eurandom Technische Universiteit Eindhoven, Postbus 513

5600 MB Eindhoven, The Netherlands f.h.j.redig@tue.nl

Abstract

We present a new and simple approach to deviation inequalities for non-product measures, i.e., for dependent random variables. Our method is based on coupling. We illustrate our abstract results with chains with complete connections and Gibbsian random fields, both at high and low temperature.

Keywords and phrases: exponential deviation inequality, moment inequality, coupling matrix, Gibbsian random fields, chains with complete connections.

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1

Introduction

By now, deviation and concentration inequalities for product measures have be-come a standard and powerful tool in many areas of probability and statistics, such as density estimation [5], geometric probability [20], etc. A recent mono-graph about this area is [11] where the reader can find much more information. Deviation inequalities for dependent, strongly mixing random variables were obtained for instance in [18, 17]. Later, in the context of dynamical systems Collet et al. [3] obtained an exponential deviation inequality using spectral analysis of the transfer operator. In [10], C. K¨ulske obtained an exponential deviation inequality in the context of Gibbs random fields in the Dobrushin uniqueness regime. Therein the main input is Theorem 8.20 in [7] which al-lows to estimate uniformly the terms appearing in the martingale difference decomposition in terms of the Dobrushin matrix. Besides exponential devia-tion inequalities, moment inequalities have been obtained in, e.g., [2, 4, 5, 17]. In the dependent case, we also mention that K. Marton [13, 14, 15] obtained concentration inequalities based on “distance-divergence” inequalities and cou-pling (with a different approach than ours). In particular, she obtains in [14] results for a class of Gibbs random fields under a strong mixing condition close to Dobrushin-Shlosman condition. Let us notice that this method of “distance-divergence” inequalities inherently implies exponential deviation inequalities for Lipschitz functions (wrt to Hamming distance for instance).

In the present paper, we obtain abstract deviation inequalities using a cou-pling approach. We prove an upper bound for the probability of deviation from the mean for a general function of n variables, taking values in a finite alphabet, in term of a “coupling matrix” D. This matrix expresses how “well” one can couple in the far “future” if the “past” is given. If the coupling matrix can be uniformly controlled in the realization then an exponential deviation inequality follows. If the coupling matrix cannot be controlled uniformly in the realization then typically upper bounds for the moments are derived.

As a first application of our abstract inequalities, we obtain an exponential deviation inequality for Gibbsian random fields in a “high” temperature regime complementary to the Dobrushin uniqueness regime, and for chains with com-plete connections with a summable continuity rate. A second application is in the context of the low-temperature Ising model where we obtain upper bounds for the moments of a general local function. This is a typical situation where the coupling matrix cannot be controlled uniformly in the realization. Our de-viation inequalities are new and yield various non-trivial applications which will the subject of a forthcoming paper.

The paper is organized as follows. In Section 2, we state and prove our abstract inequalities, first in the context of random processes, and next in the context of random fields. Section 3 deals with the examples of high-temperature Gibbs measures, chains with complete connections, and finally of the low tem-perature Ising model.

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2

Main results

Let A be a finite set. Let g : An→ R be a function of n-variables. An element σ of the set AN is an infinite sequence drawn from A, i.e., σ = (σ

1, σ2, . . . , σi, . . .)

where σi∈ A. With a slight abuse of notation, we also consider g as a function

on AN which does not depend on σ

k, for all k > n. The variation of g at site i

is defined as

δig := sup

σj=σj′,∀j6=i

|g(σ) − g(σ′)| .

A deviation inequality is an estimate for the probability of deviation of the function g from its expectation, i.e., an estimate for

P{g − Eg ≥ t} (1)

for all n ≥ 1 and all t > 0, within a certain class of probability measures P. For example, an exponential deviation inequality is obtained by estimating the expectation

E h

eλ(g−Eg)i

for any λ ∈ R, and using the exponential Chebychev inequality.

However, there are natural examples where the exponential deviation in-equality does not hold (see the example of the low-temperature Ising model below). In that case we are interested in bounding moments of the form

E(g − Eg)2p to control the probability (1).

In this section, we use a combination of the classical martingale decompo-sition of g − Eg and optimal coupling to perform a further telescoping which is adequate for the dependent case. This will lead us to a “coupling matrix” depending on the realization σ ∈ AN. This matrix quantifies how “good” future

symbols can be coupled if past symbols are given according to σ. Typically, we have in mind applications to Gibbsian random fields. In that framework, the elements of the coupling matrix can be controlled uniformly in σ in the “high-temperature regime”. This uniform control leads naturally to an exponential deviation inequality. At low temperature we can only control the coupling ma-trix for “good” configurations, but not uniformly. Therefore the exponential deviation fails and instead we will obtain Rosenthal-type inequalities for the moments of g − Eg; see e.g. [17] for the case of sums of random variables. De-vroye inequality [4] is an example of such an inequality for the second moment (in the i.i.d. case).

2.1 The coupling matrix D

We now present our method. For i = 1, 2, . . . , n, let Fi be the sigma-field

generated by the random variables σ1, . . . , σi, and F0 be the trivial sigma-field

{∅, Ω}. We write g(σ1, . . . , σn) − Eg(σ1, . . . , σn) = n X i=1 Vi(σ) (2)

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where

Vi(σ) := E[g|Fi](σ) − E[g|Fi−1](σ) =

Z

P(dηi+1· · · dηn|σ1, . . . , σi) g(σ1, . . . , σi, ηi+1, . . . , ηn)

− Z

P(dηi· · · dηn|σ1, . . . , σi−1) g(σ1, . . . , σi−1, ηi, ηi+1, . . . , ηn) =

Z P(dηi+1· · · dηn|σ1, . . . , σi) g(σ1, . . . , σi, ηi+1, . . . , ηn) − Z P(dηi|σ1. . . , σi−1) Z

P(dηi+1· · · dηn|σ1. . . , σi−1ηi) g(σ1, . . . , σi−1, ηi, ηi+1, . . . , ηn) ≤

max

α∈A

Z

P(dηi+1· · · dηn|σ1, . . . , σi= α) g(σ1, . . . , σi−1, α, ηi+1, . . . , ηn)

− min

β∈A

Z

P(dηi+1· · · dηn|σ1, . . . , σi= β) g(σ1, . . . , σi−1, β, ηi+1, . . . , ηn) .

=: Yi(σ) − Xi(σ) . (3)

Denote by Pσ

i,α,β = P

σ<i

i,α,β the optimal coupling of the conditional

distribu-tions P(dη≥i+1|σ1· · · σi = α) and P(dη≥i+1|σ1· · · σi = β), and introduce the

(infinite) upper-triangular matrix D = Dσ defined for i, j ∈ N by Dii:= 1

Di,i+j = Dσi,i+j := max

α,β∈AP

σ i,α,β

n

σi+j(1) 6= σi+j(2)o. (4) Notice that if the σi’s are mutually independent, then D is the identity matrix

because the conditional distributions P(dη≥i+1|σ1· · · σi = α) and

P(dη≥i+1|σ1· · · σi = β) are equal. Hence we have a perfect coupling in this case.

We proceed with the following simple telescoping identity:

g(σ1, . . . , σi−1, α, σi+1(1), . . . , σ(1)n ) − g(σ1, . . . , σi−1, β, σ(2)i+1, . . . , σ(2)n ) =

[g(σ1, . . . , σi−1, α, σ(1)i+1, . . . , σ(1)n ) − g(σ1, . . . , σi−1, β, σ(1)i+1, . . . , σn(1))]+

[g(σ1, . . . , σi−1, β, σ(1)i+1, . . . , σn(1)) − g(σ1, . . . , σi−1, β, σ(2)i+1, σ (1)

i+2, . . . , σ(1)n )]+

[g(σ1, . . . , σi−1, β, σ(2)i+1, σ (1)

i+2, . . . , σ(1)n )−g(σ1, . . . , σi−1, β, σ(2)i+1, σ (2)

i+2, σ

(1)

i+3, . . . , σ(1)n )]

+ · · · +

[g(σ1, . . . , σi−1, β, σi+1(2), σi+2(2), . . . , σn−1(2) , σ(1)n ) − g(σ1, . . . , σi−1, β, σ(2)i+1, . . . , σ(2)n )]

=:

n−i

X

j=0

∇12i,i+jg .

We have the following implications:

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σi+j(1) 6= σ(2)i+j ⇒ ∇12i,i+jg ≤ δi+jg .

Therefore using (3) and (4) we arrive at the inequalities Vi≤ Yi− Xi≤

n−i

X

j=0

Di,i+jδi+jg = (Dδg)i (5)

where δg denotes the column vector with coordinates δjg, for j = 1, . . . , n, and

0 for j > n.

2.2 Uniform decay of D: exponential deviation inequality Let Di,j∞ := supσ∈ANDσi,j. We assume that

kD∞k22:= sup

x∈ℓ2(N),kxk2=1

kD∞xk22 < ∞ . (6) We then have the following theorem.

THEOREM 1. If (6) is valid, then for all n ∈ N and all functions g : An→ R,

we have the inequality

P{g − Eg ≥ t} ≤ e−

2t2

kD∞k22 kδgk22 . (7)

PROOF. We have the following lemma which appears in [5].

LEMMA 1. Suppose F is a sigma-field and Z1, Z2, V are random variables such

that

1. Z1≤ V ≤ Z2

2. E(V |F) = 0

3. Z1 and Z2 are F-measurable

Then, for all λ ∈ R, we have

E(eλV|F) ≤ eλ2(Z2−Z1)2/8. (8)

We apply this lemma with V = Vi, F = Fi−1, Z1 = Xi− E[g|Fi−1], Z2 =

Yi− E[g|Fi−1]. Remember the inequality

Yi− Xi ≤ (Dδg)i. We obtain E(eλVi|F i−1) ≤ eλ 2(Dδg)2 i/8. (9)

Therefore, by successive conditioning, and the exponential Chebychev inequal-ity, P{g − Eg ≥ t} ≤ e−λtEeλPni=1Vi  ≤ e−λtE  E(eλVn|F n−1)eλ Pn−1 i=1 Vi  ≤ · · · ≤ e−λteλ2kD∞δgk22/8≤ e−λteλ2kD∞k22 kδgk22/8. (10)

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Now choose the optimal λ = 4t/(kD∞k2

2 kδgk22) to obtain the result.

From (7) we deduce

P{|g − Eg| ≥ t} ≤ 2e−

2t2 kD∞k22kδgk2

2 .

2.3 Non-uniform decay of D: moment inequalities

If the dependence on σ of the elements of the coupling matrix cannot be con-trolled uniformly, then in many cases we can still control the moments of the coupling matrix. To this aim, we introduce the (non-random, i.e., not depend-ing on σ) matrices

D(p)i,j := E(Di,jp )1/p (11) for all p ∈ N.

A typical example of non-uniformity which we will encounter, for instance in the low-temperature Ising model, is an estimate of the following form:

i,i+j ≤ 1I{ℓi(σ) ≥ j} + ψi,i+j (12)

where ψi,i+j does not depend on σ, and where ℓi are unbounded functions of

σ with a distribution independent of i. The idea is that the matrix elements

Di,i+j “start to decay” when j ≥ ℓi(σ). The “good” configurations σ are those

for which ℓi(σ) is “small”.

In the particular case when (12) holds, in principle one still can have an exponential deviation inequality provided one is able to bound

E 

eλPni=1ℓ2i

 .

However, in the examples given below, the tail of the ℓi will be exponential or

stretched exponential. Henceforth, we cannot deduce an exponential deviation inequality from these estimates.

We now prove an inequality for the variance of g which is a generalization of an inequality derived in [4] in the i.i.d. case.

THEOREM 2. For all n ∈ N, all functions g : An→ R we have the inequality

E(g − Eg)2 ≤ kD(2)k2

2 kδgk22. (13)

PROOF. We start again from the decomposition (2). Recall the fact that

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Using (5) and Cauchy-Schwarz inequality we obtain E(g − Eg)2 = E n X i=1 Vi2 ≤ E n X i (Dδg)2i ! = n X i=1 n X k=1 n X l=1 E(Di,kDi,l) δkgδlg ≤ n X i=1 n X k=1 n X l=1 E Di,k2 12 E Di,l2 12 δkgδlg = kD(2)δgk22 ≤ kD(2)k22 kδgk22.

REMARK 1. In the i.i.d. case, the coupling matrix D is the identity matrix.

Hence inequality (13) reduces to

E(g − Eg)2 ≤ kδgk2 2

which the analogue of Devroye inequality [4].

In case (12) holds, we have the following proposition.

PROPOSITION 1. Assume that there exists ǫ > 0 such that E(ℓ2+ǫ

0 ) < ∞, and

assume moreover that kψk2 < ∞. Then kD(2)k2 < ∞.

PROOF. Let Pi,i+j := E(1I{ℓi ≥ j})

1

2 = P(ℓ0 ≥ j) 1

2, where we used that the

distribution of ℓi is independent of i by assumption. It suffices to prove that

kP k2< ∞. Since

kP k22 ≤ kP k1kP k∞

it suffices to prove that kP k1, kP k∞< ∞. We have

kP k∞ = sup i X j j1+ǫ2 P(ℓi ≥ j) 1 2j− 1+ǫ 2 (14) ≤   X j j1+ǫP(ℓ0 ≥ j)   1 2   X j j−(1+ǫ)   1 2 = Cǫ E(ℓ2+ǫ0 )

where Cǫ> 0. We have for the other norm:

kP k1 = sup j X i |Pi,j| = sup j X i≤j |Pi,j| (15) = sup j X i≤j P(ℓi≥ j − i) 1 2 = sup j X i≤j P(ℓ0≥ j − i) 1 2 ≤ CǫE(ℓ2+ǫ0 ) .

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Notice that this proposition can also be proved using Young’s inequality since P is a convolution operator.

We now turn to higher moment estimates. We have the following theorem.

THEOREM 3. There exists a constant C > 0 such that for all n ∈ N, all

functions g : An→ R, for any p ∈ N, we have

E(g − Eg)2p ≤ Cp p2p kD(2p)k2p

2 kδgk

2p

2 .

PROOF. We start from (2) and get

E(g − Eg)2p =X

i1

· · ·X

i2p

E Vi1· · · Vi2p .

This sum can be estimated by applying the martingale version of the Marcinkiewicz-Zygmund inequality [19, Theorem 3.3.6] since E[Vi|Fj] = 0 for all i > j. This

gives

E(g − Eg)2p ≤ Cp p2p E(X

i

Vi2)p

where the constant Cpp2p can be deduced from the proof of Theorem 3.3.6 in [19].

We now estimate the rhs by using (5): E(X i Vi2)p (16) = X i1 · · ·X ip EVi21· · · Vi2p (17) ≤ X i1 · · ·X ip Eh(Dδg)2i1· · · (Dδg)2ipi = X i1···ip X j1···jp X k1···kp E p Y r=1 Dir,jrDir,kr ! p Y r=1 δjrg δkrg ! ≤ X i1···ip X j1···jp X k1···kp p Y r=1  D(2p)ir,jrD(2p)ir,krδjrg δkrg  = kD(2p)δgkp2p≤ kD(2p)k2p2 kδgk2p2 where in the fourth step we used the inequality

E(f1· · · f2p) ≤ 2p

Y

i=1

(E(fi2p))2p1

which follows from H¨older inequality.

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PROPOSITION 2. Let p ∈ N. Assume that there exists ǫ > 0 such that

E(ℓ2p+ǫ0 ) < ∞, and that there exist a constant c > 0 and 0 < α ≤ 1 such that ψi,i+j < e−cj

α

for all i ∈ N. Then kD(p)k2 < ∞.

PROOF. The proof follows the line of the proof of Proposition 1. Now let

Pi,i+j = P(ℓi ≥ j)

1

2p = P(ℓ

0 ≥ j)

1

2p. It suffices to show that kP k

2 < ∞. In

turn, it is sufficient to prove that kP k∞ < ∞ and kP k1 < ∞. Let ǫ′ > 0 to be

fixed later on. We have kP k∞ = X j j2p−12p (1+ǫ′) P(ℓ 0≥ j) 1 2p j− 2p−1 2p (1+ǫ′) (18) ≤   X j j(2p−1)(1+ǫ′) P(ℓ0≥ j)   1 2p  X j j−1−ǫ′   2p−1 2p = Cǫ,p′ E(ℓ2p+ǫ0 )2p1

where in the last step we have chosen ǫ′ = ǫ/(2p − 1) and where Cǫ,p′ > 0. Using Theorem 3, Proposition 2 and Chebychev inequality, we immediately obtain the following deviation inequality:

P{|g − Eg| > t} ≤ Kpkδgk 2p 2

t2p

for all t > 0, where Kp := 2(p(2p − 1))pkD(2p)k2p2 .

REMARK2. The assumption on ψ in the proposition is far from being optimal.

However, it will be satisfied in all examples below.

2.4 Random fields

We now present the extension of our previous results to random fields. This requires mainly notational changes. We work with lattice spin systems. The configuration space is Ω = {−, +}Zd, endowed with the product topology. We could of course take any finite set A instead of {−, +}. For Λ ⊂ Zdand σ, η ∈ Ω we denote σΛηΛc the configuration coinciding with σ (resp. η) on Λ (resp. Λc).

For σ ∈ Ω and x ∈ Zd, σxdenotes the configuration obtained from σ by flipping the spin at x. A local function g : Ω → R is such that there exists a finite subset Λ ⊂ Zdsuch that for all σ, η, ω, g(σΛωΛc) = g(σΛηΛc).

We denote δxg = supσ|g(σx) − g(σ)| the variation of g at x. δg denotes the

map Zd→ R : x 7→ δxg.

We introduce the spiraling enumeration Γ : Zd→ N illustrated in the figure for the case d = 2.

We will use the abbreviation (≤ x) = {y ∈ Zd : Γ(y) ≤ Γ(x)} and similarly we introduce the abbreviations (< x). By definition F≤xdenotes the sigma-field

generated by σ(y), y ≤ x and F<0 denotes the trivial sigma-field.

For any local function g : Ω → R, we have the analog decomposition as in (2):

g − E(g) = X

x∈Zd

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1 2 4 5 3 6 7 8 9 10 where Vx:= E [g|F≤x] − E[g|F<x] .

The analog of the coupling matrix is the following matrix indexed by lattice sites x, y ∈ Zd

Dx,y(σ) := ˆPσx,+,−{X1(y) 6= X2(y)} (20)

where ˆPσx,+,− denotes the optimal coupling between the conditional measures P(·|σ<x,+x) and P(·|σ<x,−x).

We first consider the case of uniform decay of D. In that case, the expo-nential deviation inequality of Theorem 1 holds with the norm of ℓ2(Zd), i.e.,

kδgk2

2 =

P

x∈Zd(δxg)2.

THEOREM 1’Assume that

D∞x,y:= sup

σ

Dx,y(σ) (21)

is a bounded operator in ℓ2(Zd). Then for all local functions g we have the

following inequality

P{g − Eg ≥ t} ≤ e−

2t2 kD∞k22 kδgk2

2 . (22)

In the non-uniform case, the moment inequalities of Theorems 2 and 3 extend immediately as follows. The analog of (12) is

Dx,y(σ) ≤ 1I{ ℓx(σ) ≥ |y − x|} + ψ(|y − x|) (23)

where ψ(n) decays at least as a stretched exponential, i.e., there exist C, c > 0 and 0 < α ≤ 1, such that ψ(n) ≤ Ce−cnα for all n ≥ 1. We assume that the distribution of ℓx is independent of x. We extend the matrix D defined in (11)

by putting

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for x, y ∈ Zd.

THEOREM 2’For all local functions g, for any p ∈ N, we have

E(g − Eg)2p ≤ (p(2p − 1))pkD(2p)k2p2 kδgk2p2 .

The analog of Propositions 1 and 2 is the following:

PROPOSITION 2’ Let p ∈ N. Assume (23) is satisfied and that there exists

ǫ > 0 such that E(ℓ2dp+ǫ0 ) < ∞. Then kD(2p)k

2 < ∞.

REMARK 3. It is immediate to extend the previous inequalities to integrable

functions g belonging to the closure of the set of local functions with the norm |||g||| := kδgk2. Notice that Theorem 1’ implies that such functions are Lp(P) for

any p ∈ N.

2.5 Existence of the coupling by bounding the variation

We continue with random fields and state a proposition which says that if we have an estimate of the form

Vx≤ (Dδg)x

for some matrix D, then there exists a coupling with coupling matrix ˆD such that its matrix elements decay at least as fast as the matrix elements of D. We formulate the proposition more abstractly:

PROPOSITION 3. Suppose that P and Q are probability measures on Ω and

g : Ω → R such that we have the estimate |EP[g] − EQ[g]| ≤

X

x∈Zd

ρ(x)δxg (24)

for some “weights” ρ : Zd→ R+. Suppose ϕ : Zd→ R+ is such that

X

x∈Zd

ρ(x)ϕ(x) < ∞ .

Then there exists a coupling ˆµ of P and Q such that X x∈Zd ˆ µ {X1(x) 6= X2(x)} ϕ(x) ≤ X x∈Zd ϕ(x)ρ(x) < ∞ .

PROOF. Let Bn:= [−n, n]d∩ Zd. Define the “cost” function

Cnϕ(σ, σ′) := X

x∈Bn

|σx− σ′x| ϕ(x) .

Denote by Pn, resp. Qn, the joint distribution of {σx, x ∈ Bn} under P, resp.

Q. Consider the class of functions GCϕ

n := {g| g ∈ FBn, |g(σ) − g(σ

)| ≤ X

x∈Zd

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It is obvious from the definition that g ∈ GCϕ

n, if, and only if, g is FBn

-measurable and

(δxg)(σ) ≤ ϕ(x) ∀x ∈ Bn, ∀σ ∈ Ω .

Therefore, if (24) holds, then for all g ∈ GCϕ n, |EP[g] − EQ[g]| ≤ X x∈Zd ρ(x)δxg ≤ X x∈Zd ρ(x)ϕ(x) .

Hence, by the Kantorovich-Rubinstein duality theorem [16], there exists a cou-pling ˆµn of Pnand Qn such that

ˆ µn Cnϕ(σ, σ′) = ˆµn X x∈Bn ϕ(x)1I{X1(x) 6= X2(x)} ! ≤ X x∈Zd ϕ(x)ρ(x) .

By compactness (in the weak topology), there exists a subsequence along which ˆ

µn converges weakly to some probability measure ˆµ. For any k ≤ n, we have

ˆ µn   X x∈Bk ϕ(x)1I{X1(x) 6= X2(x)}  ≤ ˆ µn X x∈Bn ϕ(x)1I{X1(x) 6= X2(x)} ! ≤ X x∈Zd ϕ(x)ρ(x) . Therefore, taking the limit n → ∞ along the above subsequence yields

ˆ µ   X x∈Bk ϕ(x)1I{X1(x) 6= X2(x)}  ≤ X x∈Zd ϕ(x)ρ(x) .

We now take the limit k → ∞ and use monotonicity to conclude that

ˆ µ   X x∈Zd ϕ(x)1I{X1(x) 6= X2(x)}  ≤ X x∈Zd ϕ(x)ρ(x) .

We shall illustrate below this proposition with the example of Gibbs random fields at high-temperature under the Dobrushin uniqueness condition.

3

Examples

3.1 High-temperature Gibbs measures

For the sake of convenience, we briefly recall a few facts about Gibbs measures. We refer to [7] for details.

A finite range potential (with range R) is a family of functions U (A, σ) indexed by finite subsets A of Zd such that the value of U (A, σ) depends only

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on σA and such that U (A, σ) = 0 if diam(A) > R. If R = 1 then the potential

is nearest-neighbor.

The associated finite volume Hamiltonian with boundary condition η is then given by

HΛη(σ) = X

A∩Λ6=∅

U (A, σΛηΛc) .

The specification is then defined as γΛ(σ|η) =

e−HΛη(σ)

ZΛη ·

We then say that P is Gibbs measure with potential U if γΛ(σ|·) is a version of

the conditional probability P(σΛ|FΛc).

Before we state our result, we need some notions from [6]. What we mean by “high temperature” will be an estimate on the variation of single-site conditional probabilities, which will imply a uniform estimate for disagreement percolation. Let px:= 2 sup σ,σ′ P(σx = +|σZd\x) − P(σ ′ x = +|σ′Zd\x) .

From [6, Theorem 7.1] it follows that there exists a coupling Pσx,+,− of the conditional distributions P(·|σ<x, +x) and P(·|σ<x, −x) such that under this

coupling

1. For x > y, the event X1(y) 6= X2(y) coincides with the event that there

exists a path γ ⊂ Zd\ (< x) from x to y such that, for all z ∈ γ, X1(z) 6= X2(z). We denote this event by “x=y”.

2. The distribution of 1I{X1(y) 6= X2(y)} for y ∈ Zd\ (≤ x) under Pσx,+,− is

dominated by the product measure Y

y∈Zd\(≤x)

νpy.

Let pc = pc(d) be the critical percolation threshold for site-percolation on Zd.

It then follows from statements 1 & 2 above that, if

sup{py : y ∈ Zd} < pc (25)

then we have the uniform estimate

x,+,−{X1(y) 6= X2(y)} ≤

Y

y∈Zd\(≤x)

νpy(x=y) ≤ e

−c|x−y|. (26)

Then we can apply Theorem 1’ to obtain

THEOREM 4. Let U be a nearest-neighbor potential such that (25) holds. Then

for the coupling matrix (20) we have the uniform estimate Dx,y(σ) ≤ e−C|x−y|

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for some C > 0. Hence we have the following exponential deviation inequality: for any local function g and for all t > 0

P{g − Eg ≥ t} ≤ e−

2t2 kD∞k22 kδgk22 .

REMARK 4. Theorem 4 can easily be extended to any finite range potential.

Theorem 4 was obtained in [10] in the Dobrushin uniqueness regime using a different approach. The high-temperature condition which we use here is sometimes less restrictive than Dobrushin uniqueness condition, but sometimes it is more restrictive. However, Dobrushin uniqueness condition is not limited to finite range potentials. We now apply Proposition 3 to show that in the Dobrushin uniqueness regime, there does exist a coupling of P(·|σ<x,+x) and

P(·|σ<x,−x) such that the elements of the associated coupling matrix decay at

least as fast as the elements of the Dobrushin matrix. The Dobrushin uniqueness condition is based on the matrix

Cx,y := 2 sup σ,σ′ Zd\y=σ′Zd\y P(σx= +|σZd\x) − P(σx= +|σ ′ Zd\x) . This condition is defined by requiring that

sup

x∈Zd

X

y∈Zd

Cx,y < 1

and the Dobrushin matrix is then defined as ∆x,y:=

X

n≥0

Cx,yn .

We now have the following proposition:

PROPOSITION 4. Assume that the Dobrushin uniqueness condition holds. For

any ϕ : Zd→ R+ such that for any x ∈ Zd,

X

y∈Zd

ϕ(y)∆y,x< ∞

then there exists a coupling ˆPσx,+,− of P(·|σ<x,+x) and P(·|σ<x,−x) such that

X

y∈Zd

ϕ(y) ˆPσ<x

x,+,−{X1(y) 6= X2(y)} < ∞ .

PROOF. From [10, Lemma 1], we have the estimate

Z P(dη|σ<x,+x)g(η) − Z P(dη|σ<x,−x)g(η) ≤ X y∈Zd

(1Ix,y+ ∆y,x)δyg

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We can apply Proposition 3 to conclude the proof.

As an example we mention that if the potential is finite-range and translation-invariant and satisfies the Dobrushin uniqueness condition, we have for large enough |x − y|

∆y,x ≤ e−c|x−y|

and hence there exists a coupling ˆPσ<x

x,+,− such that

ˆ Pσ<x

x,+,−{X1(y) 6= X2(y)} ≤ e−c

|x−y|

for all c′ < c and large enough |x − y|.

Unhappily, we are not able to construct explicitly such a coupling. 3.2 Chains with complete connections

Here we deal with a class of chains with complete connections and use a cou-pling estimate proved in [9]. Let A be a finite set (the alphabet). A chain with complete connections (σj)j∈Z, σj ∈ A, distributed according to P has the

property that the sequence defined as εn:= sup k∈Z sup σ,ξ∈AZ Pk(σk|σ k−1 −∞) − Pk(σk|σk−nk−1ξk−n−1−∞ )

converges to 0 as n tend to ∞. This sequence is called the continuity rate of the chain. We further assume that the continuity rate is summable, i.e.

X

n

εn< ∞ . (27)

We also assume the following non-nullness condition to hold: inf

k∈Zσ∈AinfZPk(σk|σ

k−1

−∞) =: ϑ > 0 . (28)

For this class of chain with complete connections we have the following exponential deviation inequality.

THEOREM 5. Assume that (σj)j∈Z is a chain with complete connections such

that (27) and (28) hold. Then there exists a constant C > 0 such that for all n ∈ N, all functions g : An→ R and all t > 0, we have the estimate

P{g − Eg ≥ t} ≤ e−

Ct2 kδgk22 .

PROOF. The theorem will be proved if the assumption of Theorem 1, i.e. (6),

is satisfied by our class of chains with complete connections. But the proof of the main theorem in [9] contains an estimate which immediately implies that

sup

i∈N

D∞i,i+j ≤ (1 − ϑ)j

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3.3 The low-temperature Ising model

It is clear that for the Ising model in the phase coexistence region, no ex-ponential deviation inequalities can hold. Indeed, this would contradict the surface-order large deviations for the magnetization in that regime (see e.g. [8] and reference therein). Nevertheless, we shall show that we can control the moments of all local functions.

We consider the low-temperature plus phase of the Ising model on Zd, d ≥ 2. This is a probability measure P+β on lattice spin configurations σ ∈ Ω, defined as the weak limit as Λ ↑ Zd of the following finite volume measures:

P+Λ,β(σΛ) = exp  β X <xy>∈Λ σxσy+ β X <xy>,x∈∂Λ, y /∈Λ σx   . ZΛ,β+ (29)

where β ∈ R+, and ZΛ,β+ is the partition function. In (29) < xy > denotes nearest neighbor bonds and ∂Λ the inner boundary, i.e. the set of those x ∈ Λ having at least one neighbor y /∈ Λ. The existence of the limit Λ ↑ Zdof P+

Λ,β is

by a standard and well-known monotonicity argument, see e.g. [7]. For any η ∈ Ω , Λ ⊂ Zd we denote by Pη

Λ,β the corresponding finite volume

measure with boundary condition η:

Λ,β(σΛ) = exp  β X <xy>∈Λ σxσy+ β X x∈Λ, y /∈Λ σxηx   . ZΛ,βη .

Later on we will have to choose β large enough.

We can now formulate our result on moments of arbitrary local functions. We shall show that we can apply Theorem 2’ and Proposition 2’.

THEOREM 6. Let P = P+

β be the plus phase of the low-temperature Ising model

defined above. There exists β0 > βc, such that for all β > β0, for al p ∈ N,

there exists a constant Cp ∈ (0, ∞) such that for all local functions g, we have

E (g − Eg)2p ≤ Cp kδgk2p2 .

Consequently, for all t > 0, we have the deviation inequalities P[|g − Eg| > t] ≤ Cp

kδgk2p2 t2p .

PROOF. The theorem follows from Theorem 2’ and Proposition 2’ if we obtain

the bound (23) with good decay properties for the tail of the distribution of ℓx.

This is the content of the following proposition.

PROPOSITION 5. Let P = P+

β be the plus phase of the low-temperature Ising

model. There exists β0 > βc such that for all β > β0, the inequality (23) holds

together with the estimate

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for all n ∈ N and

P{ℓ0 ≥ n} ≤ C′e−c

nα

for some c, c′, C, C′ > 0 and 0 < α ≤ 1.

PROOF. We shall make a coupling of the conditional measures P(·|σ<x,+

x)

and P(·|σ<x,−x). This coupling already appeared in [1] (see also [6]). Both

conditional measures are a distribution of a random field ωy, y /∈ (≤ x).

We start with the first site y1 > x according to the order induced by Γ.

We generate X1(y1) and X2(y1) as a realization of the optimal coupling

be-tween P(σy1 = ·|σ<x,+x) and P(σy1 = ·|σ<x,−x). Given that we have

gener-ated X1(y), X2(y), . . . , X1(yn), X2(yn) for y = y1, . . . , yn, we generate X1(yn+1),

X2(yn+1) for the smallest yn+1 > yn as a realization of the optimal coupling

between

P(σyn+1= ·|X1(y1) · · · X1(yn)σ<x,+x) and P(σyn+1 = ·|X2(y1) · · · X2(yn)σ<x,−x) .

By the Markov property of P we have the following: if there exists a contour separating y from x such that for all sites z belonging to that contour we have X1(z) = X2(z), then X1(y) = X2(y). The complement of this event (of having

such a contour) is contained in the event that there exists a path of disagreement from x to y, i.e., a path γ ⊂ Zd\ (< x) such that for all z ∈ γ, X1(z) 6= X2(z).

Denote that event by Exy. Clearly its probability is bounded from above by the

probability of the same event in the product coupling. In turn the event Exy is

contained in the event Exy+ that there exists a path γ from x to y in Zd\ (< x) such that for all z ∈ γ, (X1(z), X2(z)) 6= (+, +). In [12] the probability of that

event in the product coupling is precisely estimated from above by

Ce−c|x−y|+ 1I{ℓx(σ) ≥ |x − y|} (30)

for some C, c > 0, where ℓx(σ) is an unbounded function of σ with tail estimate

P(ℓx(σ) ≥ n) = P(ℓ0(σ) ≥ n) ≤ C′e−c

nα

for some C′, c′ > 0 and 0 < α < 1. For the reader’s convenience, we briefly comment on these estimates. The ideas is that the conditional measure P(·|ξ≤x)

resembles the original unconditioned plus phase (in Zd \ (≤ x)) provided ξ contains “enough” pluses. “Containing enough pluses” is exactly quantified by the random variable ℓx(ξ): (ℓx(ξ) ≤ n) is the event that for all self-avoiding

path γ of length at least n, the magnetization along γ, mγ(ξ) := 1 |γ| X z∈γ ξz

is close “enough to one”. If this is the case then under the conditional measure we still have a Peierls’ estimate, which produces the exponential term in (30). We refer to [12] for more details.

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References

[1] J. van den Berg, C. Maes, Disagreement percolation in the study of Markov fields, Ann. Probab. 22, 749–763 (1994).

[2] J.-R. Chazottes, P. Collet, B. Schmitt, Devroye inequality for a class of non-uniformly hyperbolic dynamical systems, preprint, 2005.

[3] P. Collet, S. Mart´ınez, B. Schmitt, Exponential inequalities for dynamical measures of expanding maps of the interval, Probab. Theor. Rel. Fields 123, 301–322 (2002).

[4] L. Devroye, Exponential inequalities in nonparametric estimation, Nato ASI series C, Math. Phys. Sci 335, p. 31-44, Kluwer Academic Publishers (1991).

[5] L. Devroye and G. Lugosi, Combinatorial methods in density estimation, Springer Series in Statistics, Springer, New York (2001).

[6] H.O. Georgii, O. H¨aggstr¨om and C. Maes, The Random Geometry of Equi-librium Phases in Phase transitions and critical phenomena, Vol. 18, Eds. C. Domb and J.L. Lebowitz, 1–142, Academic Press London (2001) [7] H.-O. Georgii. Gibbs Measures and Phase Transitions. Walter de Gruyter

& Co., Berlin, 1988.

[8] D. Ioffe, Exact large deviation bounds up to Tc for the Ising model in two

dimensions, Probab. Theory Related Fields 102 (1995), no. 3, 313–330. [9] M. Iosifescu, A coupling method in the theory of dependence with complete

connections according to Doeblin, Rev. Roumaine Math. Pures Appl. 37 (1992), no.1, 59–66.

[10] C. K¨ulske, Concentration inequalities for functions of Gibbs fields with ap-plications to diffraction and random Gibbs measures, Comm. Math. Phys. 239, 29–51 (2003).

[11] M. Ledoux, The concentration of measure phenomenon, Mathematical Sur-veys and Monographs 89, American Mathematical Society, Providence R.I., 2001.

[12] C. Maes, F. Redig, S. Shlosman and A. Van Moffaert, Percolation, Path large deviations and weak Gibbsianity, Comm. Math. Phys. 209, 517–545 (2000).

[13] K. Marton, Measure concentration for a class of random processes, Probab. Theory Related Fields 110 (1998), no. 3, 427–439.

[14] K. Marton, Measure concentration and strong mixing, Studia Sci. Math. Hungar. 40 (2003), no. 1-2, 95–113.

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[15] K. Marton, Measure concentration for Euclidean distance in the case of dependent random variables, Ann. Probab. 32 (2004), no. 3B, 2526–2544. [16] S.T. Rachev, Probability metrics and the stability of stochastic models.

Wi-ley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley and Sons, Ltd., Chicheter, 1991.

[17] E. Rio, Th´eorie asymptotique des processus al´eatoires faiblement d´ependants, Math´ematiques & Applications 31, Springer, 2000.

[18] P.-M. Samson, Concentration of measure inequalities for Markov chains and Φ-mixing processes, Ann. Probab. 28 (2000), no. 1, 416–461.

[19] W. Stout, Almost sure convergence, Probability and Mathematical Statis-tics 24, Academic Press, New York-London, 1974.

[20] J. E. Yukich, Probability theory of classical Euclidean optimization prob-lems, Lecture Notes in Mathematics 1675, Springer-Verlag, Berlin, 1998.

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