Tail estimates for homogenization theorems in random media

DOI:10.1051/ps:2007036 www.esaim-ps.org

TAIL ESTIMATES FOR HOMOGENIZATION THEOREMS IN RANDOM MEDIA

Daniel Boivin

Abstract. Consider a random environment inZ^dgiven by i.i.d. conductances. In this work, we obtain tail estimates for the fluctuations about the mean for the following characteristics of the environment:

the effective conductance between opposite faces of a cube, the diffusion matrices of periodized envi- ronments and the spectral gap of the random walk in a finite cube.

Mathematics Subject Classification. 60K37, 35B27, 82B44.

Received April 13, 2007. Revised October 17, 2007.

1. Introduction

Consider the d-dimensional cubic lattice Z^d, d≥ 2, as a random electrical network. The conductances of the edges eof Z^d are given by a sequence of i.i.d. positive random variables (a(e, ω);eis an edge ofZ^d) on a probability space (Ω,F, IP) interpreted as the space of the environments.

The effective conductance of the electrical network between two opposite faces of the cube [0, N+ 1]^d∩Z^d, N >2 is the strength of the electrical flow through one of the faces needed to maintain a unit potential difference between the two opposite faces. It will be denoted byCN. See [13] Section 13.2 for a survey of some results and open questions about the asymptotic properties of the effective conductance.

We say that the conductances are uniformly elliptic if there is a constantκ≥1, called the ellipticity constant, such thatIP −a.s., for all edgeseofZ^d,

κ⁻¹≤a(e, ω)≤κ.

Using the basic laws of electric networks (see for example [24] Chap. 8), one can calculate that if all the edges of Q_N have unit conductance then the effective conductance between opposite faces isN^d−1/(N+ 1). Therefore by the variational representation of the effective conductance (12), or by the monotonicity law [24] Chapter 8, if the conductances are uniformly elliptic then

κ⁻¹≤(N+ 1)N^1−dCN ≤κ. (1)

Keywords and phrases.Periodic approximation, random environments, fluctuations, effective diffusion matrix, effective conduc- tance, non-uniform ellipticity.

1 Laboratoire de Math´ematiques UMR 6205, Universit´e de Bretagne Occidentale, 6, avenue Le Gorgeu, CS 93837, 29238 Brest Cedex 3, France;boivin@univ-brest.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2009

Furthermore, Jikovet al.[16] Chapters 8 and 9 showed by homogenization methods that for some differential operators with uniformly elliptic and stationary coefficients and for some percolation models,

A_N := (N+ 1)²N^−dCN converges a.s. asN→ ∞.

See also [22],[4] and [2] Section 10.

A first result here is an estimate of the fluctuations ofA_N around its mean when the conductances are i.i.d.

and uniformly elliptic. The convenience of working with A_N rather that CN is that we readily see for which dimensions the estimates are interesting. In particular, for i.i.d. and uniformly elliptic conductances, we see that althoughA_N converges for all dimensions, our estimates are interesting only for dimensions d≥3.

Denote byL^d the set of edges of the cubic latticeZ^d.

Theorem 1. Let (a(e);e∈ L^d),d≥3, be a sequence of i.i.d. uniformly elliptic conductances with ellipticity constant κ. SetC₀= 64κ³.

Then for allt≥0and N≥1,

IP(|AN −IEA_N| ≥tN^(2−d)/2)≤4 exp

−t/

κC₀

and

VVar(A_N)≤κC₀N^2−d

where the expectation and the variance with respect toIP are denoted byIE andVVarrespectively.

In Theorem3, tail estimates are obtained for environments given by the non uniformly elliptic conductances considered in Fontes and Mathieu [11]. They complement the lower bounds on the variance obtained by Wehr [28] for some distributions of the conductances.

Let us now use the random conductances to construct a reversible Markov chain (X_n;n≥0) on Z^d. In the environmentω, the probability transition between two neighboursx, y∈Z^d, which will be denoted byx∼y, is given by

p(x, y, ω) =a(x, y, ω)/a(x, ω), x∼y (2)

where a(x, y, ω) is the conductance of the edge joining x and y and a(x, ω) =

y∼x

a(x, y, ω). Note that the

Markov chain is reversible with a stationary measure given bya(x, ω).

In [26] Section 1, Sidoravicius and Sznitman showed that if the conductances are stationary and uniformly elliptic then a functional central limit theorem holds in almost all environment. Weaker versions of the invariance principle already appeared in [19–21]. LetD₀ be the diffusion matrix of the limiting Brownian motion. When the conductances are i.i.d., D₀ is of the formσ²I whereσ²>0 andIis the identity matrix.

A survey of various approximations and bounds forD₀can be found in [15] Chapters 5–7 and [16] Chapters 6–7 for this model and for related ones.

The approximation ofD0 by periodic environments is considered in [5,6,22]. Given an environment ω ∈Ω and an integerN >1, construct an environmentN-periodic onZ^d,d≥1, by setting

˙

a_N(x, y, ω) =a( ˙x,y, ω),˙ x∼y where ˙x,y˙ ∈[[0, N]]^d, ˙x∼y˙ and ˙x≡x, ˙y≡y modN coordinatewise.

Then consider the reversible random walk onZ^d with transition probabilities given by

˙

p_N(x, y, ω) = ˙a_N(x, y, ω)/a˙_N(x, ω), x∼y where ˙a_N(x, ω) =

y∼xa˙_N(x, y, ω).These induce a probability ˙P_z,N,ω on the paths starting atz∈Z^d. ForIP almost all environments, under ˙P_0,N,ω,n^−1/2X_[·n]converges in law to a Brownian motion. Denote by D˙N(ω) its covariance matrix and denote its entries by ˙D_N^ij, 1≤i, j≤d.

In view of applications to the theory of massless gradient fields onZ^d, where periodized states can be used to define the slope-independent surface tension [12], Caputo and Ioffe [6] considered periodic approximations of the homogenized diffusion matrix of a symmetric random walk onZ^d with i.i.d. and uniformly elliptic jump rates. They proved that there is a.s. convergence at a rate faster than CN^−ν for some constantsC andν >0 that depend on the dimension and the ellipticity constant.

Bourgeat and Piatnitski [5], considered elliptic differential operators with stationary and uniformly elliptic coefficients. They showed the a.s. convergence of the approximations of the effective diffusion matrix by cubic samples with periodic, Dirichlet or Neumann boundary conditions. Furthermore, using results from Yurinsky [29], they showed that under a uniform mixing condition the rate of convergence is faster thanCN^−ν for some constantsC andν >0.

In [22], Owhadi proved the a.s. convergence of the periodic approximations of the diffusion matricesD_N to the homogenized diffusion matrix D₀ for stationary and uniformly elliptic jump rates and for elliptic operator with stationary and uniformly elliptic coefficients. As noted in [22], the a.s. convergence could also be obtained from the principle of periodic localization [16], p. 155 and the a.s. G-convergence properties of the elliptic differential operators.

Our main results are the following estimates of the rate of convergence and of the fluctuations ofD_N around its mean.

Theorem 2. Let (a(e);e∈ L^d) be a sequence of i.i.d. uniformly elliptic conductances with ellipticity constant κ. Then there is a constantC,0< C <∞, which depends only ondandκsuch that, for all t >0 andN ≥1, maxi,j IP(|D˙^ij_N−IED˙_N^ij| ≥tN^−δ)≤4 exp (−Ct), (3) maxi,j VVar( ˙D_N^ij)≤8C⁻²N^−2δ (4)

where 2δ = max{α, d−4 +α} and α > 0 is the regularity exponent which appears in the Harnack inequal- ity (30).

In particular, ford≥5, it provides a lower bound for the exponentν of the estimate [6] (1.1) which does not depend on the ellipticity constant. More precisely, there isν > ^d₄−1 such that for all sufficiently largeN,

maxi,j IP(|D˙^ij_N −IED˙^ij_N| ≥N^−ν)≤exp (−N^ν).

Indeed, in (3), sett=N^δ/2 with 2δ=d−4 +αand takeν=d/4−1 +α/8< δ/2.

The proof uses the method of bounded martingale differences developed by Kesten in [18] for first passage percolation models. This method also applies to other models where there is homogenization and when some regularity results are available. See also [26] from (2.17). For both questions considered above, the quantities involved are similar to first-passage times in that they can be expressed as solutions of a variational problem.

It is this aspect that will be exploited.

This method also applies to the spectral gap of a random walk in cubes inZ^d,d≥3, with Dirichlet boundary conditions. In this situation, the regularity estimates are provided by the De Georgii-Nash-Moser theory.

The paper is organized as follows. In Section 2, the martingale estimates of [18] are given in the form that they will be used here. Theorem1 and its extension to non uniformly elliptic conductances are proved in Section3. The regularity estimate used for the effective conductance is simply a maximum principle. For periodic approximations of the diffusion matrices, more prerequisites are needed. They are gathered in Sections 4.1 and 4.2. Section4 ends with the proof of Theorem2. Tail estimates for the Dirichlet eigenvalues are given in the last section.

2. Martingale estimates and notations

When the conductances are assumed to be uniformly elliptic, a stronger version of Kesten’s martingale inequality can be used. The same proof applies with some simplifications. In particular, [18] Step (iii) is not needed. However the full generality of Kesten’s martingale inequalities will be used when we consider conductances that are bounded above but not necessarily uniformly elliptic. They are given in the second version.

Martingale estimates I.

Let (M_n;n≥0) be a martingale with respect to the filtration(Fn;n≥0).

LetΔ_k=M_k−M_k−1,k≥1. If there are positive random variablesU_k (not necessarilyFk-measurable) such that for some constantB₀<∞

IE(Δ²_k|Fk−1)≤IE(U_k|Fk−1)for allk≥1 and ∞

1

U_k≤B₀, (5)

thenM_n→M_∞ inL² and a.s., andIE|M_∞−M₀|²≤B₀.

Moreover, if there is a constant B₁<∞such that for all k≥1,

|Δk| ≤B₁, (6)

then for all t >0,

IP(|M∞−M₀|> t)≤4 exp

− t 4√ B

whereB = max{B₀, eB²₁}. Martingale estimates II.

Let (M_n;n≥0) be a martingale with respect to the filtration(Fn;n≥0).

Let Δ_k =M_k−M_k−1,k≥1. If there is a constantB₁<∞ such that for all k≥1,

|Δk| ≤B₁, (7)

if for some random variables U_k ≥0 (not necessarilyFk-measurable)

IE(Δ²_k|Fk−1)≤IE(U_k|Fk−1)for allk≥1, (8) and if there are constants0< C₁, C₂<∞ands₀≥e²B₁² such that,

IP

k

U_k> s

≤C₁exp(−C₂s²), for alls≥s₀, (9)

then there are universal constantsc₁ andc₂ which do not depend on B₁,s₀,C₁,C₂ nor on the distribution of (M_k)and(U_k) such that for alls >0,

IP(|M∞−M₀|> s)≤c₁

1 +C₁+ C₁ s²₀C₂

exp −c2 s

s^1/2₀ + (s/(s₀C₂))^1/3

·

We end this section with some notations that will be used throughout this article. OnR^d,d≥1, the₁-distance, the Euclidean distance and the_∞-distance will respectively be denoted by| · |₁,| · |and| · |_∞. Foru:Z^d→R^d, letu∞= sup_x∈Zd|u(x)|∞.

η∈R^d will be considered as a column vector and its transpose will be denoted byη so thatη²= tr(ηη).

For m < n ∈ N, [[m, n]] = [m, n]∩N. With this notation, for an integer N ≥1, Q_N := [[1, N]]^d, Q_N :=

[[0, N+ 1]]^d and its boundary is ∂Q_N :=Q_N \Q_N.

Foru:Q_N →Randx∈Q_N, letP u(x) :=E_x(u(X₁)) andHu(x) :=u(x)−P u(x).

In an environmentω, the conductance of an edgeewith endpointsx∼y is denoted bya(e, ω) ora(x, y, ω).

Similarly, for a function vwhich is defined forxandy, the endpoints ofe, letv(e, ω) :=|v(x, ω)−v(y, ω)|.

Because we consider conductances that are independent, identically distributed and bounded by κ, we will assume that IP is the product measure on Ω =]0, κ]^Ld and that a(e,·) are the coordinate functions. The expectation with respect toIP will sometimes be denoted by·instead ofIE.

Let{ek;k≥1}be a fixed ordering ofL^dand letFk,k≥1, be theσ-algebra generated by{a(ej,·); 1≤j≤k}.

Then for an integrable random variableh: Ω→R,

IE(h|Fk) =IE_σh([ω, σ]_k)

where [ω, σ]_k ∈ Ω agrees with ω for the firstk coordinates and with σ for all the other coordinates andIE_σ denotes the integration with respect todIP(σ).

3. Effective conductance

In this section, we obtain tail estimates for the effective conductance of an increasing sequence of cubes under mixed boundary conditions and an upper bound on the variances.

Consider boundary conditions which can be interpreted as maintaining a fixed potential difference between two opposite faces ofQ_N = [[1, N]]^dwhile the other faces are insulated.

To describe more precisely the boundary conditions, denote the first coordinate of x∈ Z^d byx(1). Then denote by∂⁻Q_N and∂⁺Q_N the following two opposite faces ofQ_N,

∂⁻Q_N ={x∈∂Q_N;x(1) = 0} and ∂⁺Q_N ={x∈∂Q_N;x(1) =N+ 1}

and by∂^insQ_N the set of vertices on the insulated faces,

∂^insQ_N =∂Q_N \(∂⁻Q_N ∪∂⁺Q_N).

LetVN be the set of real-valued functions onQ_N such that

u(x) = 0 ifx∈∂⁻Q_N, u(x) =N+ 1 ifx∈∂⁺Q_N

and u(x) =u(y) ifx∈∂^insQ_N, y∈Q_N andx∼y.

Assume for the moment that the conductances (a(e);e∈ L^d) are non-random and satisfy

0< a(e)≤κ, for all edgese. (10)

For two functions u, v:Q_N →R, the Dirichlet form, denoted by EN, is defined by EN(u, v) =

x,y

a(x, y)(u(x)−u(y))(v(x)−v(y)) (11)

where the sum is over all ordered pairs{x, y}such thatx∈Q_N andy∈Q_N.

For allN ≥1, there is a uniquev_N ∈ VN, called the equilibrium potential, such that Hv_N = 0 on Q_N.

The equilibrium potential is also a solution of a variational problem : it is the unique element ofVN such that EN(v_N, v_N) = inf

EN(u, u);u∈ VN

.

This provides a variational representation of the effective conductance between two opposite sides of the cube Q_N since

CN = (N+ 1)⁻²EN(v_N, v_N). (12) Finally by the maximum principle, ifu:Q_N →RverifiesHu= 0 onQ_N then

max

Q_N

u= max

∂QN

u.

Reintroduce a random media by assuming that (a(e);e∈ L^d) is a sequence of i.i.d. random variables satisty- ing (10) a.s.

We will writeEN(v_N, v_N)_ωto indicate that bothEN andv_N are random and are calculated with the conduc- tancesa(e, ω) given by the environmentω. Then

A_N(ω) = (N+ 1)²N^−dCN(ω) =N^−dEN(v_N, v_N)_ω.

Since the maximum principle does not require uniform ellipticity, it is possible to obtain good estimates under weaker conditions on the conductances with these boundary conditions. Theorem 1 given in the introduction gives tail estimates for uniformly elliptic conductances while Theorem3given below applies when they are not.

This model was considered by Wehr [28]. He showed that for some lawsIP, which include the exponential and the one-sided normal distributions, and assuming that IE(A_N) is bounded below by a positive constant, then

lim inf

N N^dVVar(A_N)>0.

See also the recent paper of Benjamini and Rossignol [1] Section 5.

If the conductances are uniformly elliptic and stationary, thenA_N convergesIP a.s. and inL¹(IP), asN → ∞.

This was done in [2] Section 10 by adapting the homogenization methods of [16] Chapter 7.

For conductances that are not necessarily uniformly elliptic, we have the following estimates. Additional properties of non-uniformly elliptic reversible random walks can be found in [11].

Theorem 3. Let (a(e);e ∈ L^d) be a sequence of i.i.d. conductances on Z^d such that for some constant 1≤κ <∞,0< a(e)≤κfor all edgese. LetC₀= 64κ³.

Then, for d≥5,VVar(A_N)≤128dκ²N^4−d and for allt >0 andN≥1, IP(|AN −IEA_N| ≥tN^(4−d)/2)≤4 exp

−t/(8κ√ 2d)

. (13)

If moreover, for some constants D₀<∞andγ,0< γ <2,

IP(a⁻¹(e)≥s)≤D₀s^1−2/γ, for alls≥1, (14) then, if d≥4 or if d= 3and1/2≤γ <1,

VVarA_N ≤16C₀(D₀+ 1)N^2−d+γ and for all 0< t <

C0(D0+1)⁵ 8

_1/2

N^{(d−6+5γ)/2},

IP(|A_N −IEA_N|> t)≤11c₁exp − c₂ 2

2C₀(D₀+ 1)N^{(d−2−γ)/2}t

. (15)

wherec₁ andc₂ are the constants that appear in the martingale estimates II. In particular, they do not depend on κ,dor N.

Ford= 3 and0< γ≤1/2,VVarA_N ≤16C₀(D₀+ 1)N^−1/2.

Lemma 1. Letω andσbe two environments such thata(e, ω) =a(e, σ)for all edges eexcept maybe fore=e_k. Then for allN ≥1,

|AN(ω)−A_N(σ)| ≤κN^−d(v_N²(e_k, ω) +v_N²(e_k, σ)).

Proof. Sincev_N ∈ VN is the solution of a variational problem,

A_N(ω)−A_N(σ) =N^−d(EN(v_N, v_N)_ω− EN(v_N, v_N)_σ)

≤N^−d

e

(a(e, ω)−a(e, σ))v_N²(e, σ)≤κN^−dv_N²(e_k, σ).

Proof of Theorem 1. With the notations of Section 2, let Δ_k = IE(A_N | Fk)−IE(A_N | F_k−1), M₀ = 0 and M_n=_n

1Δ_k,n≥1.

To check that (M_n;n≥0) is a martingale that verifies conditions (5) and (6), we have by Lemma1,

|Δk| ≤ IE_σ|AN([ω, σ]_k)−A_N([ω, σ]_k−1)|

≤ κN^−dIE_σv_N²(e_k,[ω, σ]_k−1) +κN^−dIE_σv²_N(e_k,[ω, σ]_k)

≤ 8κN^2−d

since by the maximum principle, 0≤v_N(x)≤N+ 1, for allx∈Q_N. Then,

Δ²_k ≤ IE_σ(|AN([ω, σ]_k)−A_N([ω, σ]_k−1)|²)

≤ 2κ²N^−2dIE_σv_N⁴(e_k,[ω, σ]_k−1) + 2κ²N^−2dIE_σv⁴_N(e_k,[ω, σ]_k) and by the maximum principle,

≤ 8κ²N^2−2dIE_σv_N²(e_k,[ω, σ]_k−1) + 8κ²N^2−2dIE_σv_N²(e_k,[ω, σ]_k).

Fork≥1, let

U_k(ω) = 16κ²N^2−2dv²_N(e_k, ω). (16)

We see thatIE(Δ²_k | F_k−1)≤IE(U_k | F_k−1).

By (1) and (12),EN(v_N, v_N)_ω≤2κN^d. Then by uniform ellipticity,

k

U_k = 16κ²N^2−2d

k

v²_N(e_k, ω)

≤ 16κ³N^2−2dEN(v_N, v_N)_ω<16κ³(2κ)N^2−2d+d.

Hence condition (5) holds with B₀ = 64κ⁴N^2−d and condition (6) holds with B₁ = 4κN^2−d. Therefore, A_N −IEA_N =

∞ 1

Δ_k a.s. and inL² and ford≥3, by the martingale estimates I, we have that for allN ≥1 andt≥0,

IP(|AN −IEA_N| ≥t)≤4 exp

− t 4√

B

whereB = max{B₀, eB²₁}= 64κ⁴N^2−d. Accordingly, VVar(A_N)≤64κ⁴N^2−d.

Proof of Theorem 3. To obtain (13), the preceding proof can be used up to (16) where uniform ellipticity is first needed.

LetU_k(ω) = 16κ²N^2−2dv_N²(e_k, ω),k≥1.

Then simply boundv_N²(e_k, ω) by 4N² to obtain that

k

U_k ≤128dκ²N^4−d.

Hence the martingale estimates I hold with B₀= 128dκ²N^4−d, B₁= 4κN^2−dandB = max{B₀, eB₁²}=B₀. These estimates can be improved if we assume that (14) holds for some 0< γ <2. Starting from (16), we have that for allN≥1,

k

v²_N(e_k, ω) ≤ N^γ

k

a(e_k, ω)v_N²(e_k, ω) + 4N²{k;a(e_k, ω)≤N^−γ}

≤ 2κN^d+γ+ 4N²{k;a(e_k, ω)≤N^−γ}.

Therefore,

k

U_k ≤32κ³N^2−d+γ+ 64κ²N^4−2d{k;a(e_k, ω)≤N^−γ}.

LetC₀= 64κ³. Then fort >1,N ≥1 andγ >0,

IP

k

U_k > C₀tN^2−d+γ

≤ IP

{k;a(e_k, ω)≤N^−γ}>(t−1)N^d−2+γ

≤ 2 exp

−N^d

4 ((t−1)N^γ−2−p_N)²

by Bernstein’s inequality withp_N =IP(a⁻¹(e)> N^γ). By (14), we have that for all N ≥1, p_N ≤D₀N^γ−2. Therefore, if t >2(1 +D₀) then (t−1−D₀)²> t²/4 and

IP

k

U_k> C₀tN^2−d+γ

≤ 2 exp

−N^d

4 (t−1−D₀)²N^2γ−4

≤ 2 exp

−t²

16N^d−4+2γ

.

Equivalently, for allEs≥2C₀(D₀+ 1)N^2−d+γ, IP(

k

U_k> s)≤2 exp

− s²

16C₀²N^3d−8

.

We see that the conditions for the martingale estimates II hold with B₁= 4κN^2−d, C₁= 2, C₂= 1

16C₀²N^3d−8 ands₀= 2C₀(D₀+ 1)N^2−d+γ sinceC₀≥8e²κ².

In particular, we have the tail estimates (15).

Moreover,

VVarA_N ≤ IE

k

U_k= _∞

0 IP

k

U_k > s

ds

≤ s₀+ _∞

s0

C₁e^−C2s2ds.

Hence, if 2−d+γ≥6−2d−γ,

VVarA_N ≤s₀+ C₁

s₀C₂ ≤16C₀(D₀+ 1)N^2−d+γ while ford= 3 andγ−1≤ −1/2,

VVarA_N ≤s₀+ C₁

√C₂ ≤16C₀(D₀+ 1)N^−1/2.

4. The periodic approximation

The first step of the proof will be to express the covariance matrices ˙DN in terms of periodic corrector fields.

The second step will be to obtain regularity estimates for the corrector fields. The proof will then be completed with the martingale estimates.

The expectation with respect to ˙P_0,N,ω will be denoted by ˙E_z,N,ω. We will also use the Laplacian ˙H_N,ω which is defined for a functionu:Z^d→Rby ˙H_N,ωu(x) =u(x)−E˙_x,N,ωu(X₁).

4.1. Periodic corrector fields

A periodic corrector field for the random walk (X_n;n≥0) in the periodic environment is a function ˙χ_N : Z^d×Ω→R^d with the property thatIP-a.s.

X_n+ ˙χ_N(X_n), n≥0, is a martingale with respect to ˙P_0,N,ω.

Therefore, IP-a.s., ˙χ_N must verify the equations ˙E_x,N(X₁ + ˙χ_N(X₁)) = x+ ˙χ_N(x) for all x ∈ Z^d, or equivalently,

H˙_Nχ˙_N(x) = ˙d_N(x), for allx∈Z^d, (17) where ˙d_N(x) = ˙E_x,N(X₁)−xis the drift of the walk. Note that each coordinate of ˙d_N isN-periodic.

The vector space ofN-periodic functions onZ^d can be identified with

H˙N ={u:Q_N →R; u(x) =u(y)∀x, y∈Q_N such that x≡y modN}.

Letu∈H˙N. By consideringuas anN-periodic function onZ^d, ˙H_Nu(x) is defined for allx∈Z^d. Furthermore, since ˙H_Nuis N-periodic onZ^d, its restriction to Q_N belongs to ˙HN. This procedure defines a bounded linear operator ˙H_N : ˙HN →H˙N.

For two functionsu, v:Q_N →R, define the norm, the scalar product, and the Dirichlet form respectively by u^p_p,N˙ =

x∈QN

|u(x)|^pa˙_N(x), 1≤p <∞, (u, v)N˙ =

x∈QN

u(x)v(x) ˙a_N(x) and

E˙N(u, v) =

x,y

˙

a_N(x, y)(u(x)−u(y))(v(x)−v(y))

where the sum is over all ordered pairs{x, y}such that x∈Q_N andy∈[[0, N]]^d. This expression makes sense for all functionsu, v:Z^d→R. But if bothu, vareN-periodic, then the Green-Gauss formula holds

E˙_N(u, v) = (u,H˙_Nv)_N˙. (18)

Let ˙H⁰_N(ω) ={u∈H˙N ; (u,1)_N˙ = 0}.Note that ˙H⁰_N depends on the environment but ˙HN does not.

All the properties of the solutions of a Poisson equation that will be needed are gathered in the following proposition.

Proposition 1. Let (a(e);e ∈ L^d),d ≥1, be a stationary sequence of uniformly elliptic conductances. Then IP-a.s.,

(1) ˙H_N : ˙H⁰_N →H˙⁰_N is a bounded invertible linear operator.

(2) Forf :Q_N →R,H˙_Nu=f possesses a unique solution u∈H˙N if and only iff ∈H˙⁰_N. (3) Let f ∈H˙⁰_N.

(a) The infimuminf

H˙⁰_N[ ˙EN(u, u)−2(u, f)]is attained by the solutionu∈H˙⁰_N of the equation H˙_Nu=f. (b) The infimumγ := inf

ME˙N(u, u)where M={u∈H˙⁰_N ;E (f, u)_N˙ = 1} is attained by the solution u∈H˙⁰_N of the equation H˙_Nu=γf.

(4) If f ∈H˙⁰_N then the unique solution u∈H˙⁰_N of H˙_Nu=f is u=

_∞

0

e^−tH˙Nfdt, x∈Q_N.

(5) There is a constant C =C(d, κ) <∞such that for all N ≥1 and f ∈H˙⁰_N,u∈ H˙⁰_N, the solution of H˙_Nu=f, verifies the regularity estimates,

u_∞≤CN²f_2,N˙ and u_∞≤CN²(logN)f_∞. Proof. For allu∈H˙⁰_N, ˙H_Nu∈H˙⁰_N by the Green-Gauss formula (18).

H˙_N : ˙H⁰_N →H˙⁰_N is invertible since if ˙H_Nu= 0 then uis constant by the maximum principle.

The variational principle3bholds for the Poisson equation on a smooth compact Riemannian manifold with f ∈ C^∞, see [14] Proposition 2.6 due to Druet. The same arguments can be used.

Suppose thatf is not identically 0. ThenMis a closed convex set which is not empty sincef⁻²_2,Nf ∈ M.

Therefore the infimum, γ, is attained for some u₀ ∈ M and γ > 0 since u₀ is not constant. Then using a theorem by Lagrange, there are two constantsα, β∈Rsuch that for allx∈Q_N,

2

y∼x

(u₀(x)−u₀(y)) ˙a_N(x, y)−αf(x) ˙a_N(x)−βa˙_N(x) = 0

and 2 ˙H_Nu₀(x)−αf(x)−β= 0.Therefore, for allϕ∈H˙N,

2 ˙EN(u₀, ϕ) =α(f, ϕ)_N˙ +β(1, ϕ)_N˙.

Forϕ= 1, one finds thatβ = 0 while for ϕ=u₀, one finds thatα= 2γ. Hence (Hu₀, ϕ)_N˙ =γ(f, ϕ)_N˙ for all ϕ∈H˙N.

The variational principle 3acan be proven similarly.

To prove the last two properties, the estimate of the speed of convergence to equilibrium of a Markov chain on a finite state space given in terms of the spectral gap is needed. A more detailed survey is given in [25]

Section 2.1

Let ˙K_N(t, x, y), t ≥ 0 and x, y ∈ Q_N, be the heat kernel of e^−tH˙N. Then for all t ≥ 0 and x, y ∈ Q_N, K˙_N(t, x, y)≥0 andE

y

K˙_N(t, x, y) = 1. In particular, for allf ∈H˙N andt≥0,

e^−tH˙Nf_∞≤ f_∞. (19) Denote the volume of the torusQ_N, the invariant probability for the random walk onQ_N and the smallest non zero eigenvalue of ˙H_N on ˙HN respectively by

˙

a_N(Q_N) =

x∈QN

˙

a_N(x), π˙_N(x) = ˙a_N(x)/a˙_N(Q_N) and λ˙_N.

With these notations, we have the following very useful inequality. A proof can be found in [25] p. 328 for instance. For allt >0 andx, y∈Q_N,

|K˙_N(t, x, y)−π˙_N(y)| ≤κexp

−tλ˙_N

. (20)

Therefore, for allt >0 andf ∈H˙_N⁰, e^−tH˙Nf∞= sup

x

y

( ˙K_N(t, x, y)−π˙_N(y))f(y)

≤κexp(−tλ˙_N)f_1,N˙. (21) By the Riesz-Thorin interpolation theorem, from (19) and (21), we obtain that

e^−tH˙Nf∞≤√

κexp(−tλ˙_N/2)f_2,N˙. (22) This will be completed by the following lower bound on ˙λ_N. There exists a constantC₁ >0 which depends only on the dimension and on the ellipticity constantκsuch thatIP a.s. and for allN ≥1,

N²λ˙_N > C₁. (23)

This follows from the Courant-Fischer min-max principle [25], p. 319 by comparison with the eigenvalues of the simple symmetric random walk which corresponds to the case where the conductance of every edge is 1.

For Neumann boundary conditions the expressions are not as explicit but for Dirichlet and periodic boundary conditions onQ_N, the eigenvalues can be calculated explicitely much as in [27]. We find that for eachξ∈[[0, N[[^d, there is an eigenvalue for the periodic boundary conditions onQ_N,λ_ξ(Q_N), that verifies

Nlim→∞N²λ_ξ(Q_N) =π² d

|z|=1

(ξ·z)² as N → ∞.

The representation formula given in 4 follows from the spectral estimates (20) and (23). See [23] for another recent application of4.

The first regularity result follows from the representation formula (4) and (22): for f ∈H˙_N⁰, u_∞≤

_∞

0 e^−sH˙Nf_∞ds≤ _∞

0

√κe^−sλ˙N/2f_2,N˙ds≤CN²f_2,N˙.

The second one follows from (19) and (21) : forf ∈H˙⁰_N andt >0, u∞≤

_t

0 e^−sH˙Nf∞ds+ _∞

t

e^−sH˙Nf∞ds≤tf∞+e^−tλ˙N

λ˙_N f_1,N˙.

Usef_1,N˙ ≤2dκN^df∞and lett= d

C₁N²logN.

For a functionu:Q_N →R^d, define ˙H_NuinQ_N by applying it coordinatewise.

Letg:Z^d→Z^d be the function defined byg(x) =x.

Corollary 1. Let (a(e);e∈ L^d),d≥1, be a stationary sequence of uniformly elliptic conductances. Then for all N ≥1, there is a unique functionχ˙_N :Q_N ×Ω→R^d such that IP-a.s., in each coordinate, it is in H˙⁰_N(ω) and

H˙_Nχ˙_N =−H˙_Ng, onQ_N. (24)

Moreover, there is a constantC=C(d, κ)such that

χ˙_N∞≤CN²logN. (25) Proof. Note that for each coordinate of−H˙_Ng belongs to ˙H⁰_N :

Indeed ˙H_Ng isN-periodic and

x∈QN

H˙_Ng(x) ˙a_N(x) =

x

y∼x

˙

a_N(x) ˙p_N(x, y)(g(x)−g(y))

=

x

y∼x

˙

a_N(x, y)(x−y) = 0.

Then use property 5of proposition1 with the functionf =−H˙_Ng. The regularity estimate (25) follows since

f∞≤1.

The next step is to express the covariance matrix of the walk in a periodic environment in terms of ˙χ_N. By (17),M_n=X_n+ ˙χ_N(X_n),n≥0 is a martingale with uniformly bounded increments: Z_n=M_n−M_n−1.

Leth(x) = ˙E_x,N(Z₁Z₁). Sinceh∈H˙N, by the ergodic theorem for a Markov chain onQ_N, ˙P_0,N a.s., 1

n n

1

E˙_0,N(Z_jZ_j |X_j−1) = 1 n

n 1

h(X_j−1)→

QN

˙

π_N(x)h(x) asn→ ∞.

Then by the martingale central limit theorem (see [10], (7.4) Chap. 7), 1

√nM_n converges to a Gaussian law.

Hence √1

nX_n also converges to a Gaussian with the same covariance matrix which is given by D˙N =

QN

˙

π_N(x)h(x) (26)

= ˙a_N(Q_N)⁻¹

x∼y

˙

a_N(x, y)( ˙v_N(y)−v˙_N(x))( ˙v_N(y)−v˙_N(x)).

where ˙v_N(x) =x+ ˙χ_N(x).

For uniformly elliptic, stationary and ergodic conductances, the effective diffusion matrix of the jump process in a periodic environment converges to the homogenized effective diffusion matrix D0 (see [5] Th. 1 and [22]

Th. 4.1). And since the jump process and the random walk onZ^dhave the same diffusion matrix, we have that D˙N → D0IP-a.s. asN→ ∞.

To write ˙DN in terms of the Dirichlet form on ˙HN, we will extend the definition of ˙EN toR^d-valued functions so that the expression of ˙DN given in (26) becomes

D˙N = ˙a_N(Q_N)⁻¹E˙N( ˙v_N,v˙_N) where ˙v_N =g+ ˙χ_N.

For two functions u, v:Q_N →R^d, define (u, v)_N˙ =

x∈QN

u(x)v(x)a˙_N(x), and E˙N(u, v) =

x,y

˙

a_N(x, y)(u(x)−u(y))(v(x)−v(y))

where the sum is over all ordered pairs{x, y}such thatx∈Q_N andy∈[[0, N]]^d.

Coordinatewise, the periodic corrector fields are the solutions of variational problems. Indeed, from the variational formula3a of proposition1, we have thatIP a.-s. and for allN ≥1,

inf{tr ˙EN(g+u, g+u);u∈( ˙H⁰_N)^d} = tr ˙EN(g, g) + inf{tr ˙EN(u, u)−2 tr(f, u);u∈( ˙H⁰_N)^d}

= tr ˙EN( ˙v_N,v˙_N) whereg(x) =xandf =−H˙_Ng as in Corollary1.

In particular, since

tr ˙EN( ˙χ_N,χ˙_N)≤2 tr ˙EN( ˙v_N,v˙_N) + 2 tr ˙EN(g, g)≤4 tr ˙EN(g, g), there is a constantC=C(d, κ)<∞such thatIP-a.s. and for allN≥1,

tr ˙EN( ˙χ_N,χ˙_N)≤CN^d. (27)

The second variational principle,3bof Proposition1, could be used to obtain a lower bound on tr ˙EN(χ_N, χ_N).

4.2. Further regularity results

In the following proposition, we improve the estimate given in (25) for dimensions 2≤d≤4.

Proposition 2. Let (a(e);e ∈ L^d),d ≥2, be a stationary sequence of uniformly elliptic conductances. Then there is a constantC=C(d, κ)<∞, such that for all N≥1

χ˙_N∞≤

CN^d/2 for d≥3

CN(logN)^1/2 for d= 2. (28)

Proof. Forη∈R^d, letz₀andz₁ be two vertices ofZ^d such that η·χ˙_N(z₀) = min

x∈Z^dη·χ˙_N(x) and η·χ˙_N(z₁) = max

x∈Z^dη·χ˙_N(x).

Since ˙χ_N isN-periodic, we can assume that|z₀−z₁|_∞≥N.

Letz₀=x₀, x₁, . . . , x_n−1, x_n=z₁ be a path fromz₀to z₁such that 1≤n≤dN.