Exponential concentration for first passage percolation through modified Poincaré inequalities

www.imstat.org/aihp 2008, Vol. 44, No. 3, 544–573

DOI: 10.1214/07-AIHP124

© Association des Publications de l’Institut Henri Poincaré, 2008

Exponential concentration for first passage percolation through modified Poincaré inequalities ¹

Michel Benaïm and Raphaël Rossignol

Institut de Mathématiques, Université de Neuchâtel, 11 rue Emile Argand, 2000 Neuchâtel, Switzerland.

E-mail: [email protected]; [email protected] Received 3 October 2006; revised 20 March 2007; accepted 10 April 2007

Abstract. We provide a new exponential concentration inequality for first passage percolation valid for a wide class of edge times distributions. This improves and extends a result by Benjamini, Kalai and Schramm (Ann. Probab.31(2003)) which gave a variance bound for Bernoulli edge times. Our approach is based on some functional inequalities extending the work of Rossignol (Ann. Probab.35(2006)), Falik and Samorodnitsky (Combin. Probab. Comput.16(2007)).

Résumé. On obtient une nouvelle inégalité de concentration exponentielle pour la percolation de premier passage, valable pour une large classe de distributions des temps d’arêtes. Ceci améliore et étend un résultat de Benjamini, Kalai et Schramm (Ann.

Probab.31(2003)) qui donnait une borne sur la variance pour des temps d’arêtes suivant une loi de Bernoulli. Notre approche se fonde sur des inégalités fonctionnelles étendant les travaux de Rossignol (Ann. Probab.35 (2006)), Falik et Samorodnitsky (Combin. Probab. Comput.16(2007)).

AMS 2000 subject classifications:Primary 60E15; secondary 60K35

Keywords:Modified Poincaré inequality; Concentration inequality; Hypercontractivity; First passage percolation

1. Introduction

First passage percolation was introduced by Hammersley and Welsh [10] to model the flow of a fluid in a randomly porous material (see [12] for a recent account on the subject). We will consider the following model of first passage percolation inZ^d, whered≥2 is an integer. LetE=E(Z^d)denote the set of edges inZ^d. The passage time of the fluid through the edgeeis denoted byx_e and is supposed to be nonnegative. Randomness of the porosity is given by a product probability measure onR^E₊. Thus,R^E₊is equipped with the measureμ=ν^⊗E, whereνis a probability measure onR+according to which each passage time is distributed, independently from the others. Ifu, vare two vertices ofZ^d, the notationα: {u, v}means thatαis a path with end pointsuandv. Whenx∈R^E₊,d_x(u, v)denotes the first passage time, or equivalently the distance fromutovin the metric induced byx,

d_x(u, v)= inf

α:{u,v}

e∈α

x_e.

1We acknowledge financial support from the Swiss National Science Foundation Grants 200021-1036251/1 and 200020-112316/1.

The study ofd_x(0, nu)whennis an integer which goes to infinity is of central importance. Kingman’s subadditive ergodic theorem implies the existence, for each fixedu, of a “time constant”t (u)such that:

dx(0, nu) n

ν−→-^a.s.

n→+∞t (u).

It is known (see [15], pages 127 and 129) that ifν({0})is strictly smaller than the critical probability for Bernoulli bond percolation onZ^d, thent (u)is positive for everyudistinct from the origin. Under such an assumption, one can say that the random variabled_x(0, nu)is located aroundnt (u), which is of order O(|nu|), where we denote by| · |the L¹-norm of vertices inZ^d. In this paper, we are interested in the fluctuations of this quantity. Precisely, we define, for any vertexv,

∀x∈R^E₊, f_v(x)=d_x(0, v).

It is widely believed that the fluctuations off_v are of order|v|^1/3when d=2. Apart from some predictions made by physicists, this faith relies on recent results for related growth models [2,13,14]. Until recently, the best results rigourously obtained for the fluctuations offv were some moderate deviation estimates of order O(|v|^1/2)(see [16, 24]). In 1993, Kesten [16] proved that

Var_μ(f_v)=O

|v| ,

providedνadmits a finite second-order moment. If, furthermore,νadmits a finite moment of exponential order, there exist two constantsC₁andC₂such that for anyt≤ |v|,

νfv−E(fv)> t

|v|

≤C1e^−C2t. (1)

Later, Talagrand improved the right-hand side of the above inequality to exp(−C₂t²). In 2003, Benjamini, Kalai and Schramm [5] proved that for Bernoulli edge times bounded away from 0, the variance off_vis of order O(|v|/log|v|), and therefore, the fluctuations are of order O(|v|^1/2/(log|v|)^1/2).

It is natural to ask whether the work of Benjamini, Kalai and Schramm [5] can be extended to other distributions, notably continuous distributions which are not bounded away from zero. This has been done in a preliminary version of the present paper [4] by extending the tools of [5], namely a modified Poincaré inequality due to Talagrand. It is also natural, and even more desirable, to try to improve the result of Benjamini, Kalai and Schramm [5] into an exponential inequality in the spirit of (1), with√

|v|/log|v|instead of√

|v|. In this article, we show that some different modified Poincaré inequalities arising from the context of “threshold phenomena” for Boolean functions (see [9,21]) may be used successfully instead of Talagrand-type inequalities from [4,23]. This is the main result of this paper stated in Theorem 5.4. Whereas we focused on the percolation setting, the argument is fairly general and we present also an abstract exponential concentration result, Theorem 4.2, which is very likely to have applications outside the setting of percolation.

This article is organized as follows. In Section 2, we extend the modified Poincaré inequalities of Falik and Samorodnitsky [9] to non-Bernoulli and countable settings where logarithmic Sobolev inequalities are available, no- tably to a countable product of Gaussian measures. Section 3 is devoted to the obtention of similar inequalities for other continuous measures by a simple mean of change of variable. In Section 4, we show how to deduce new general exponential concentration bounds from the modified Poincaré inequalities of Section 2. This allows us to obtain in Section 5 an exponential version of the bound of Benjamini et al. in some continuous and discrete settings.

Notation. Given a probability space(X,X, μ)and a real valued measurable functionf defined onXwe let fp,μ=

|f|^pdμ

1/p

∈ [0,∞],

andL^p(μ)denote the set off such thatfp,μ<∞.Themeanoff∈L¹(μ)is denoted Eμ(f )=

fdμ,

thevarianceoff ∈L²(μ)is Var_μ(f )=f−Eμ(f )²

2,μ,

and theentropyof any positive measurable functionf is Ent_μ(f )=

flogfdμ−

fdμlog

fdμ.

When the choice ofμis unambiguous we may writefp (respectivelyL^p,E(f ),^Var(f )and^Ent(f ))forfp,μ

(respectivelyL^p(μ),Eμ(f ),Var_μ(f )andEnt_μ(f )).

2. Logarithmic Sobolev and modified Poincaré inequalities onR^N

The relevance of a “modified Poincaré inequality” due to Talagrand [23] in the context of first passage percolation was shown by Benjamini, Kalai and Schramm [5]. Let us explain this point a little bit more. A classical Poincaré inequality has the following form:

Var_μ(f )≤CEμ(f ),

whereCis a constant, andEμ(f )is an “energy” off, that is, usually, the mean againstμof the square of some kind of gradient. There is a good theory for this in the context of Markov semi-groups (see [1,3], for instance). By “modified Poincaré inequality,” we mean a functional inequality which improves upon the classical Poincaré inequality for a certain class of functionsf. This is usually achieved through a hypercontrativity property (see [4] and [19]).

In this section, we will show how to build a modified Poincaré inequality on a product of probability spaces each of which satisfies a Sobolev logarithmic inequality. This approach was initiated independently by Rossignol [21], Falik and Samorodnitsky [9] in the Bernoulli setting.

We shall need some notation for tensorisation. Suppose that we are given a countable collection of probability spaces(Xi,Xi, μi)i∈I. Ifibelongs toI, andx⁻ⁱis an element of

j∈I,j=iXj, then for everyxi inXi, we denote by (x⁻ⁱ, xi)the elementxof

j∈IXj. For every functionf from

i∈IXitoR, everyj∈I, and everyx^−jin

i=jXi, we denote byf_x−j the function fromXj toRobtained fromf by keepingx^−jfixed:

∀xj∈Xj, f_x−j(xj)=f (x).

Now, suppose that we are given a collection(Ai)_i∈I of linear subspaces,Ai ⊂L²(Xi, μ_i), containing the constant functions. Then, we introduce

A^I=

f ∈L²

i∈I

Xi,

i∈I

μ_i s.t.∀j ∈I, f_x−j∈Ajfor

i=j

μ_i-a.e.x^−j

.

An operatorRj fromAj toL²(Xj, μj)is naturally extended onA^I, “acting only on coordinatej”:

∀f ∈A^I,∀x∈

i∈I

Xi, R_j(f )(x):=R_j(f_x−j)(x_j).

Proposition 2.1. Let(Xi,Xi, μi)i∈I,be a sequence of probability spaces.Let(Ai)i∈I be a collection of sets such that for everyi,Aiis a linear subspace ofL²(Xi, μi)which contains the constant functions.Suppose that for everyi inI,μ_i satisfies a logarithmic Sobolev inequality of the following form:

∀f ∈Ai, Ent_μi f²

≤Eμ_i

Ri(f )² ,

whereR_i is a linear operator fromAi toL²(μ_i)with value zero on any constant function.Furthermore,suppose that the following commutation property holds:

∀i,∀f ∈A^I,

fd

j=i

μ_j∈Ai and R_i

fd

j=i

μ_j =

R_i(f )d

j=i

μ_j.

Then,μ^I=

i∈Iμi satisfies the following modified Poincaré inequality:

∀f ∈A^N, Var_μI(f )log ^VarμI(f )

i∈IΔ_if²_μI,1

≤

i∈I

Eμ^I

Ri(f )² ,

whereΔ_i is the following operator onL²(μ^I):

∀f ∈L² μ^I

, Δ_if =f−

fdμi.

Proof. To shorten the notations, we shall writeμinstead ofμ^I.

First, suppose thatI is finite,I = {1, . . . , n}. The tensorisation property of the entropy (see [1] or [17], Proposi- tion 5.6, page 98, for instance) states that for every positive measurable functiong,

Ent_μ(g)≤ n

i=1

Eμ

Ent_μi(g) .

Thus, the logarithmic Sobolev inequalities for eachμi imply that:

∀g∈Aⁿ, Ent_μ g²

≤ n

i=1

Eμ

Ri(g)²

. (2)

Now, letf be a function inAⁿ. Following Rossignol [21], Falik and Samorodnitsky [9], we writef −Eμ(f )as a sum of martingale increments, and apply the logarithmic Sobolev inequality (2) to each increment:

n

j=1 Ent_μ

V_j²

≤ n

j=1

n

i=1

Eμ

R_i(V_j)²

, (3)

where

f −Eμ(f )= n

j=1

V_j, and

V_j=

fdμ1⊗ · · · ⊗dμj−1−

fdμ1⊗ · · · ⊗dμj=

Δ_jfdμ1⊗ · · · ⊗dμj−1.

The following inequality, which is a clever application of Jensen’s inequality, is shown in [9] and is cleaner than the corresponding one in [21]:

n

j=1 Ent_μ

V_j²

≥^Varμ(f )log ^Varμ(f ) _n

j=1V_j²_μ,1. Jensen’s inequality implies that:

n

j=1 Ent_μ

V_j²

≥^Varμ(f )log ^Varμ(f ) _n

j=1Δjf²_μ,1. (4)

On the other hand, for everyi, the termEμ(R_i(g)²)in (2) is called an “energy” forg, and we claim that the sum of the energies of the increments off equals the energy off:

∀i∈ {1, . . . , n}, n

j=1

Eμ

R_i(V_j)²

=Eμ

R_i(f )²

. (5)

Indeed, sinceR_i is linear, using the commutation hypothesis, and the fact thatR_i(f )is zero on any functionf which is constant on coordinatei, we get:

∀i < j, R_i(V_j)=0, Ri(Vi)=

Ri(f )dμ1⊗ · · · ⊗dμi−1, and

∀i > j, R_i(V_j)=

R_i(f )dμ1⊗ · · · ⊗dμj−1−

R_i(f )dμ1⊗ · · · ⊗dμj. Therefore,

n

j=1

Eμ

R_i(V_j)²

=

i−1

j=1

Eμ

R_i(V_j)² +Eμ

R_i(V_i)²

=Eμ

R_i(f )² .

Now, claim (5) is proved and the result follows from (3), (4) and (5), at least whenI is finite.

Now, suppose thatI is strictly countable, let us sayI=N, and letFn be theσ-algebra generated by the firstn coordinate functions inR^N. Letf ∈A^Nandf_n=E(f|Fn)be the conditional expectation off with respect toFn. Then, the commutation property tells us thatf_nbelongs toA^N, andR_i(f_n)=E(R_i(f )|Fn). Therefore, we can apply the first part of Proposition 2.1, the one that we just proved:

Var_μn(fn)log ^Varμn(f_n) _n

i=1Δ_if_n²_μn,1

≤ n i=1

Eμⁿ

Ri(fn)² .

This may be written as:

Var_μN(f_n)log ^VarμN(f_n) _n

i=1Δ_if_n²_μn,1

≤ n

i=1

E_μN E

R_i(f )|Fn

2 .

Obviously,Δ_if_n=E(Δ_i(f )|Fn). Therefore, Jensen’s inequality implies:

Var_μN(f_n)log ^VarμN(fn) n

i=1Δ_if²_μn,1

≤ n

i=1

Eμ^N

R_i(f )2 .

Of course,fn converges tof inL²(μ^N), and we may letntend to infinity in the last inequality to get the desired

result.

Remark 1. Actually,a logarithmic Sobolev inequality associated to a probability measureμi which is reversible with respect to an operatorLmay always be written in the form of Proposition2.1.Indeed,such an inequality may be written as:

∀f ∈Ai, Ent_μi f²

≤cEμ_i(−fLf ),

wherecis a positive constant.SinceLis a self-adjoint operator inL²(μ_i),it admits a spectral representation(see [27],page313).It is easy to show that its eigenvalues are nonnegative(see,for instance[3],page7).The spectral decomposition of−Lmay,therefore,be written as:

−L= _∞

0

λdE(λ),

and a suitable candidate forR_i may be deduced from it:

R_i= _∞

0

√cλdE(λ). (6)

Nevertheless,in the applications which follow,it is essential that the operatorR_i is nice enough to allow the quantity _n

i=1R_i(f )²to be easily controlled,and the one given in(6)may not be appropriate for this.Another candidate, which we shall see to be the right one for certain continuous probability measures,is the square root of the “carré du champ” operatorΓ^1/2(f, f ),where:

Γ (f, g)=1

2(Lf g−fLg−gLf ).

But in the discrete case,this is not the most natural choice.Therefore,we prefer not to try to generalize any longer, and rather give some examples.

2.1. Examples

Not surprisingly, we start to illustrate Proposition 2.1 with the Bernoulli and Gaussian cases. Our choice to present them “mixed” might look a little weird at first sight, but this will prove to be useful in the percolation context (see Section 5).

Example 1 (The Bernoulli and Gaussian cases). We let

β_p=(1−p)δ₀+pδ₁

be the Bernoulli measure with parameterpon{0,1}.Ifpbelongs to]0,1[,β_p(see,for instance[22],Theorem2.2.8, page336,or[1])satisfies the following logarithmic Sobolev inequality:for any functionf from{0,1}toR,

Ent_βp f²

≤c_LS(p)Eβp

(Δf )² , where

c_LS(p)=logp−log(1−p) p−(1−p) and

Δf =f−

fdβp.

IfSis a countable set,for anysinS,letXs be a copy of{0,1},As be the set of functions fromXs toR,andR_s be the operatorΔacting onAs.We denote also a product measureλ^S_pon{0,1}^S: λ^S_p=β_p^⊗S.

Now we introduce the Gaussian setting.Let

γ (dy)= 1

√2πe^−y2/2dy

denote the standard Gaussian measure on R.A mapf is said to be weakly differentiableprovided there exists a locally integrable function denotedf(x)such that

f(x)g(x)dx= −

g(x)f (x)dy

for every smooth functiong:R→Rwith compact support.Theweighted SobolevspaceH₁²(γ )is defined to be the space of weakly differentiable functionsf onRsuch that

f²_H2

1 = f²2+f²

2<∞.

It is well known thatγ (see, for instance[18],Theorem5.1, page92)satisfies the following logarithmic Sobolev inequality:for any functionf inH₁²(γ ),

Ent_γ f²

≤2Eγ

f(x)2 .

For anyiinN,letXi be a copy ofR,Ai a copy ofH₁²(γ )andR_i the derivation operator onAi.We letγ^N=γ^⊗N denote the standard Gaussian measure onR^N.

The setA^S∪N thus defined is the so-called weighted Sobolev spaceH₁²(λ^S_p⊗γ^N),which contains the functions f∈L²(λ^S_p⊗γ^N)verifying the following condition.For alli∈N,there exists a functionhiinL²(λ^S_p⊗γ^N)such that

−

Rg(yi)f (x, y)dyi=

Rg(yi)hi(x, y)dyi, λ^S_p⊗γ^Na.s.

for every smooth functiong:R→Rhaving compact support.The functionh_iis called the partial derivative off with respect toyi,and is denoted by_∂y^∂f

i.

Thus,we deduce from Proposition2.1the following result.

Corollary 2.2. For anyp∈ ]0,1[,and anyf ∈H₁²(λ^S_p⊗γ^N), Var(f )log ^Var(f )

s∈SΔ_sf²₁+

i∈NΔ_if²₁≤cLS(p)

s∈S

E

(Δsf )² +2

i∈N

E ∂f

∂x_i

2

.

Example 2 (The gamma case (associated to the Laguerre generator)). We let

ν_a,b(dy)= b^a

Γ (a)y^a−1e^−by1y>0dy

denote the gamma probability measure with parametersa and b.This measure is the invariant distribution of the Laguerre semi-group,with generator:

La,bf (x)=bxf(bx)−(a−bx)f(bx).

Whena≥1/2andb=1,it can be easily seen that this generator satisfies theCD(ρ,∞)curvature inequality:

Γ₂(f )≥1 2Γ (f ).

This implies thatνa,bsatisfies the following logarithmic Sobolev inequality (see Definition3.1, page28and Theo- rem3.2,page29in[3],see also[1]).For any weakly differentiable functionf ∈L²(ν_a,b),ifa≥1/2,

Ent_νa,b f²

≤ 4 bEνa,b

√xf(x)2 .

Therefore,we deduce the following result from Proposition2.1.

Corollary 2.3. Suppose thata≥1/2,b >0,and letR⁺_∗^Nbe equipped with the product measureν_a,b^N .For any weakly differentiable functionf inL²(ν_a,b^N ),define

∇if (x)= ∂f

∂xi

(x)√ x_i. Suppose that,

∀i∈N, ∇if∈L² ν_a,b^N

. Then,

Var(f )log ^Var(f )

i∈NΔif²₁≤ 4 b

i∈N

E (∇if )²

.

Remark that whena∈ ]0,1/2[,ν_a,bstill satisfies a logarithmic Sobolev inequality with a positive constantC_a,b instead of 4/b,but the precise value of Ca,b is not known;see [20].This gives the analogue of Corollary2.3for a∈ ]0,1/2[,withC_a,binstead of4/b.

Example 3 (The uniform case). We let

λ(dy)=1_0≤y≤1dy

denote the uniform probability measure on[0,1].It is known thatλsatisfies the following logarithmic Sobolev inequal- ity(it is a direct consequence of the logarithmic Sobolev inequality on the circle[8]).For any weakly differentiable functionf inL²(λ),

Ent_λ f²

≤ 2 π²Eλ

f(x)2 .

Corollary 2.4. Let[0,1]^Nbe equipped with the product measureλ^N.Suppose that,

∀i∈N, ∂f

∂xi ∈L² λ^N

.

For any weakly differentiable functionf inL²(λ^N),

Var(f )log ^Var(f )

i∈NΔ_if²₁≤ 2 π²

i∈N

E ∂f

∂x_i

2

.

We shall see in Section3thatλsatisfies another logarithmic Sobolev inequality with an energy whose form “looks like” the energy appearing in the gamma case.

3. Extension from the Gaussian case to other measures

As usual, we can deduce from Corollary 2.2 other inequalities by mean of change of variables. To make this precise, letΩ be a measurable space andΨ: R^N→Ω a measurable isomorphism (meaning thatΨ is one to one withΨ and Ψ⁻¹measurables). LetΨ^∗γ^Ndenote the image ofγ^NbyΨ. That is,Ψ^∗γ^N(A)=γ^N(Ψ⁻¹(A)). Forg:S×Ω→R such thatg◦(I d, Ψ )∈H₁²(λ⊗γ^N), one obviously has

Var_{λ⊗Ψ∗γN}(g)=^Varλ⊗γ^N(g◦Ψ )

and

∂_i,Ψgp,Ψ^∗γ^N= ∂g◦Ψ

∂y_i

p,γ^N

,

where∂_i,Ψgis defined as

∂_i,Ψg(x, ω)=∂(x◦Ψ )

∂y_i

q, Ψ⁻¹(ω) .

Hence inequality in Corollary 2.2 forf=g◦(I d, Ψ )transfers to the same inequality forgprovided _∂y^∂f

i is replaced by∂i,Ψg.

Example 4. Letk≥2 be an integer,S^k−1⊂R^k the unit k−1 dimensional sphere,Ω =(R⁺_∗ ×S^k−1)^N,and let E=(R^k_∗)^N.A typical point inΩ will be written as(ρ, θ )=(ρⁱ, θⁱ)and a typical point inE as y =(y^j).Now consider the change of variablesΨ:E→Ω given byΨ (y)=(Ψ (y)^j)with

Ψ^j(y)=y^j2, y^j y^j .

The image of(γ^k)^N=γ^NbyΨ is the product measureγ˜^Nwhereγ˜is the probability measure onR⁺_∗ ×S^k−1defined by

˜

γ (dtdv)= 1

r_ke^{−t /2}t^k/2−11t >0dtdv.

Hererk=_∞

0 e^{−t /2}t^k/2−1dt,anddvstands for the uniform probability measure onS^k−1.Forg:{0,1}^S×Ω→R withg◦(I d, Ψ )∈H₁²(λ⊗γ^N),let

∂g

∂ρ^j(x, ρ, θ ),∇θ^jg(x, ρ, θ ) ∈R×T_θiS^k−1⊂R×R^k

denote the partial gradient ofgwith respect to the variable(ρⁱ, θⁱ),whereT_θiS^k−1⊂R^kstands for the tangent space ofS^k−1atθ_i.It is not hard to verify that for alli∈Nandj∈ {1, . . . , k},

∂_i,j,Ψg(x, ρ, θ )=2∂g

∂ρⁱ(x, ρ, θ )

ρⁱθ_jⁱ+ 1 ρⁱ

∇θⁱg(x, ρ, θ )

j. (7)

As a consequence,we may recover in this way Corollary2.3when the parameteraequalsk/2−1,withkan integer. Indeed,this follows from(7)applied to the map(x, ρ, θ )→g(_2α^ρ ).Concentrating on the angular part instead of the radial one,we obtain the following modified Poincaré inequality on the sphere.

Corollary 3.1 (Uniform distribution onSⁿ). Letdv_n denote the normalized Riemannian probability measure on Sⁿ⊂Rⁿ⁺¹.Forg∈H₂¹(dvn)andi=1, . . . , n+1let∇ig(θ )denote theith component of∇g(θ )inRⁿ⁺¹(we see T_θSⁿas the vector space ofRⁿ⁺¹consisting of vector that are orthogonal toθ).Then,forn≥2,

Var(g)log ^Var(g) _N

i=1Δ_ig²₁ ≤ 1 n−1

i∈N

E (∇ig)²

. (8)

Proof. follows from (7) applied to the map(x, ρ, θ )→g(θ ). Details are left to the reader.

Example 5. If one wants to get a result similar to Corollary2.2withγ replaced by another probability measureν, one may of course perform the usual change of variables through inverse of repartition function.In the sequel,we denote by

g(x)= 1

√2πe^−x2/2 (9)

the density of the normalized Gaussian distribution,and by

G(x)= x

−∞g(u)du (10)

its repartition function.For any function φ fromR toR,we shall note φ˜ the function from R^N toR^N such that (φ(x))˜ _j=φ (x_j).

Corollary 3.2 (Unidimensional change of variables). Let ν be a probability onR⁺ absolutely continuous with respect to the Lebesgue measure,with densityhand repartition function

H (t )= _t

0

h(u)du.

Let{0,1}^S×Rⁿbe equipped with the probability measureλ^S_p⊗ν^⊗N.Then,for every functionf on{0,1}^S×Rⁿsuch thatf ◦(I d,H⁻¹◦G)∈H₁²(λ^S_p⊗γ^N),

Var(f )log ^Var(f )

s∈SΔ_sf²₁+

i∈NΔ_if²₁ ≤c_LS(p)

s∈S

E

(Δ_sf )² +2

i∈N

E (∇if )²

,

where for every integeri,

∇if (x, y)=ψ (yi)∂f

∂y_i(x, y),

andψis defined onI= {t≥0s.t.h(t ) >0}:

∀t∈I, ψ (t )=g◦G⁻¹(H (t )) h(t ) .

Proof. It is a straightforward consequence of Corollary 2.2, applied tof◦(I d,H⁻¹◦G).

4. A general exponential concentration inequality

In this section, we show how one can deduce from Proposition 2.1 an exponential concentration inequality for a function F of independent variables. We shall prove in Section 5, in the context of first passage percolation, that this new general concentration inequality may in certain cases improve on the ones due to Talagrand [24–26], or Boucheron et al. [7]. The reason why we can get stronger results is that Proposition 2.1 is generally stronger than a simple Poincaré inequality, and it is well known (see [17], Corollary 3.2, page 49 and Theorem 3.3, page 50) that a Poincaré inequality for a measureμimplies an exponential concentration inequality for any Lipschitz function of a random variable with distributionμ. This can be achieved through applying the Poincaré inequality to exp(θf ), and then performing some recurrence. This last step is essentially contained in the following simple version, adapted to our case, of Corollary 3.2, page 49 in [17].

Lemma 4.1. Let f be a measurable real function on a probability space (X,X, μ), and K a positive constant.

Suppose that for any real numberθ < ¹

2√

K,the functionx→e^{θf (x)}is inL¹(μ),and:

Var e^{θf /2}

≤Kθ²E e^θf

. Then,

∀t≥0, μ

f−

fdμ > t√

K ≤4e^−t and

∀t≥0, μ

f−

fdμ <−t√

K ≤4e^−t.

Now, we can state our general concentration inequality. For any functionF on a product space(Xi,Xi, μ_i)_i∈I, we define the following quantities, which play an important role in Theorem 4.2 (the notation is that of Section 2).

Wi,+(x)= F x⁻ⁱ, yi

−F (x)

+dμi(yi), whereh₊=sup{h,0}.

W₊(x)=

i∈I

Wi,+.

Remark that similar quantities are involved in the work of Boucheron et al. [6,7].

Theorem 4.2. Let (Xi,Xi, μ_i)_i∈I,(Ai)_i∈I and(R_i)_i∈I be as in Proposition2.1,and satisfying all the hypotheses therein.LetF be a function inL²(

i∈IXi).Define r=sup

i∈I

E W_i,²₊

, s=

E W₊²

.

Define,for every real numberK > ers:

l(K)= K

log(K/(rslog(K/(rs)))).

Suppose that there exists a real numberK > erssuch that,for everyθsuch that|θ| ≤₂^√1_l(K), e^(θ/2)F belongs toA^I, and:

i∈I

E Ri

e^(θ/2)F

≤Kθ²E e^{θ F}

. (11)

Then,denotingμ=

i∈Iμi,for everyt >0:

μ

F−E(F )≥t l(K)

≤4e^−t, μ

F−E(F )≤ −t l(K)

≤4e^−t.

Proof. For any functionf inL¹(

i∈IXi), anyi∈I,x∈

i∈IXi andy_i∈Xi, Δ_if1= f (x)−f

x⁻ⁱ, y_i

dμi(y_i) dμ(x)

≤ f (x)−f

x⁻ⁱ, y_idμ(x)dμi(y_i)

=2

f (x)−f x⁻ⁱ, yi

+dμ(x)dμi(yi)

=2

f (x)−f

x⁻ⁱ, y_i

−dμ(x)dμi(y_i), whereh₋=sup{−h,0}. On the other hand,

e^{(θ/2)F (x−i,yi)}−e^{(θ/2)F (x)}

+=e^{(θ/2)F (x)}

e^{(θ/2)(F (x−i,yi)−F (x))}−1

+

≤e^{(θ/2)F (x)} θ

2 F

x⁻ⁱ, y_i

(x)−F (x)

+

= _|θ|

2e^{(θ/2)F (x)} F

x⁻ⁱ, yi

−F (x)

+ ifθ >0,

|θ|

2e^{(θ/2)F (x)} F

x⁻ⁱ, y_i

−F (x)

− ifθ <0.

Therefore,

e^{(θ/2)F (x−i,yi)}−e^{(θ/2)F (x)}

+≤ _|θ|

2e^{(θ/2)F (x)} F

x⁻ⁱ, y_i

−F (x)

+ ifθ >0,

|θ|

2e^{(θ/2)F (x−i,yi)}

F (x)−F x⁻ⁱ, yi

+ ifθ <0.

SinceF (x)andF (x⁻ⁱ, y_i)have the same distribution underμ⊗μ_i, we get, for any real numberθ, Δ_ie^{θ F /2}

1≤ |θ| F

x⁻ⁱ, y_i

−F (x)

+dμi(y_i)e^{(θ/2)F (x)}dμ(x).

And, using Cauchy–Schwarz inequality,

i∈I

Δie^{θ F /2}

1≤ |θ| E

W₊² E

e^{θ F} .

But we also have, again using Cauchy–Schwarz inequality, Δ_ie^{θ F /2}

1≤ |θ| E

W_i,²₊ E

e^{θ F} . Therefore,

i∈I

Δie^{θ F /22}

1≤θ²rsE e^{θ F}

. (12)

Inequality (12), the Poincaré inequality for e^{θ F} (Proposition 2.1) and hypothesis (11) imply that:

∀|θ| ≤ 1 2√

l(K), Var e^(θ/2)F

log^Var(e^(θ/2)F)

θ²rsE(e^{θ F})≤Kθ²E e^{θ F}

. (13)

Therefore, we are left in front of the following alternative:

• eitherVar(e^{θ F /2})≤θ²log(K/(rs))^K E(e^{θ F}),

• orVar(e^{θ F /2}) > θ²log(K/(rs))^K E(e^{θ F}). But in this case, plugging this minoration into the logarithm of inequality (13) leads to:

Var e^{θ F /2}

≤θ² K

log(K/(rslog(K/(rs))))E e^θf˜

. In any case, for any|θ| ≤₂^√1_l(K),

Var e^{θ F /2}

≤θ²l(K)E e^θf˜

.

The result follows from Lemma 4.1.

Remark 2. It is well known,through Herbst’s argument(see,e.g., [18],Theorem5.3,page95),that condition(11) implies a subexponential concentration inequality of the form:

μF−E(F )≥2t√ K

≤2e^−t2.

In the applications to follow,K/(rs)is big,and thereforel(K)is small compared toK.Therefore,at the price of trading the sub-Gaussian behavior against a subexponential one,Theorem4.2shows that whenK/(rs)is big;the fluctuations ofF are lower than√

l(K),which is small compared to√ K.

Let us give a closer look at the case where for everyi,μ_iis the invariant measure of a diffusion process with carré du champΓ_i. Naturally associated with this diffusion process, a Sobolev logarithmic inequality forμ_i has the form (if it exists):

Ent_μi f²

≤c_iEμ_i

Γ_i(f, f ) ,

wherec_i is a positive constant. Furthermore, we have the following property:

Γ_i

Φ(f ), g

=Φ(f )Γ_i(f, g), which leads to:

R_i

e^(θ/2)F2

=c_iΓ_i

e^(θ/2)F,e^(θ/2)F

=c_iθ²

4 e^{θ F}Γ_i(F, F ).

Therefore, condition (11) becomes:

E

e^(θ/2)F

i∈I

c_iΓ_i(F, F ) ≤4KE e^{θ F}

.

The main work to satisfy condition (11) is to bound from below, and somewhat independently from e^(θ/2)F, the quantity

i∈IΓ_i(F, F ). In some particular cases, and notably percolation, this quantity is upperbounded byF itself.

And it is possible to show, following Boucheron et al. [7], that, at least for smallθ,E(Fe^{θ F})is upper bounded by a constant timesE(F )E(e^{θ F}). More generally, one can state the following result.

Corollary 4.3. Let(Xi,Xi, μ_i)_i∈I,(Ai)_i∈I and(R_i)_i∈I be as in Proposition2.1,and satisfying all the hypotheses therein.LetF be a function inL²(

i∈IXi).Define r=sup

i∈I

E W_i,²₊

, s=

E W₊²

.

Define,for every real numberK > ers:

l(K)= K

log(K/(rslog(K/(rs)))).

Suppose that there exists two constantsCandDsuch that,denoting:

ACD=4CE(F )+D

1+ 2 C , we have

(i) C≤√ l(A_CD),

(ii) A_CD4CE(F )+D(1+_C²)≥ers, (iii) for everyθsuch that|θ| ≤₂^√_l(A¹

CD), e^{θ F} belongs toA^I,and:

i∈I

E R_i

e^(θ/2)F

≤Cθ²E Fe^{θ F}

+Dθ²E e^{θ F}

. (14)

Then,denotingμ=

i∈Iμi,for everyt >0:

μ

F −E(F )≥t

l(ACD)

≤4e^−t, μ

F −E(F )≤ −t

l(A_CD)

≤4e^−t.

Proof. The only thing to prove is that condition (11) holds withK=4CE(F )+D(1+_C²). This will follow from condition (14) and a variation on the theme of Herbst’s argument due to Boucheron et al. [7]. Indeed, recall that using the tensorisation of entropy, the logarithmic Sobolev inequalities for eachμ_i imply that:

∀g∈A^I, Ent_μ g²

≤

i∈I

Eμ

Ri(g)² .

Let us apply this inequality tog=e^(θ/2)F, and use condition (14). For everyθsuch that|θ| ≤₂^√_l(4C¹_{E(F ))}, Ent_μ

e^{θ F}

≤Cθ²E Fe^{θ F}

. This may be written as:

θE Fe^{θ F}

−E e^{θ F}

logE e^{θ F}

≤Cθ²E Fe^{θ F}

+Dθ²E e^{θ F}

. (15)

First, suppose thatθis positive. The proof of Theorem 5 in [7] shows that, for everyθ <_C¹, logE

e^{θ F}

≤ θ

1−θ CE(F )+D θ² 1−θ C, and Eq. (15) implies that, for everyθ <_C¹,

E Fe^{θ F}

≤E(F )+Dθ (1−θ C)² E

e^{θ F} .

Ifθis negative, e^{θ F} is decreasing inF, and it follows from Chebyshev’s association inequality that (see, e.g., [11], page 43):

E Fe^{θ F}

≤E(F )E e^{θ F}

.