Antoine Gloria

WHEN ARE INCREMENT-STATIONARY RANDOM POINT SETS STATIONARY?

Keywords: random geometry, random point sets, thermodynamic limit, stochastic homogenization. 2010 Mathematics Subject Classification: 60D05, 39A70, 60H25. x, y∈ℓ∩D, x

In a recent work, Blanc, Le Bris, and Lions defined a notion of incrementstationarity for random point sets, which allowed them to prove the existence of a thermodynamic limit for two-body potential energies on such point sets (under the additional assumption of ergodicity), and to introduce a variant of stochastic homogenization for increment-stationary coefficients. Whereas stationary random point sets are increment-stationary, it is not clear a priori under which conditions increment-stationary random point sets are stationary. In the present contribution, we give a characterization of the equivalence of both notions of stationarity based on elementary PDE theory in the probability space. This allows us to give conditions on the decay of a covariance function associated with the random point set, which ensure that increment-stationary random point sets are stationary random point sets up to a random translation with bounded second moment in dimensions d > 2. In dimensions d = 1 and d = 2, we show that such sufficient conditions cannot exist.

Introduction and main results

1.1. Increment-stationarity. Let d be the dimension, and denote by L(R d ) the set of locally finite simple point sets of R d . In what follows we consider random variables on some probability space (Ω, F, P) taking values in L(R d ). We call such random variables random point sets. In [START_REF] Blanc | The energy of some microscopic stochastic lattices[END_REF], Blanc, Le Bris and Lions addressed the issue of defining the thermodynamic limit of the energy of random sets ℓ of particles (seen as simple random point sets). Typical energies to be considered are given by two-body potentials V : R d \ {0} → R, so that the energy in some bounded region D ⊂ R d writes

E(ℓ, D) := 1 2
and a measure-preserving group action {θ k } k∈Z d such that ℓ satisfies for almost all ω ∈ Ω and all k ∈ Z d ,

ℓ(θ k ω) = ℓ(ω) + Y k (ω) (1.2)
that is, ℓ(θ k ω) is the translation of ℓ(ω) by the vector Y k (ω). Assume in addition that

Y 0 ≡ 0, (1.3)
Y j (θ k ω) -Y i (θ k ω) = Y j+k (ω) -Y i+k (ω) (1.4)
for all i, j, k ∈ Z and almost every ω ∈ Ω (properties which directly follow from (1.2) if Y k is uniquely defined by (1.2)). Let us check that such a definition ensures that the limit in (1.1) is deterministic if it exists. For almost all ω, Y k (ω) is finite for all k ∈ Z d . Hence, for all D ⊂ R d , the points of ℓ(θ k ω)∩D are the translations of the points of ℓ(ω)∩(D -Y k (ω)).

Since the two-body potential is non-negative and translation-invariant, this yields

E(ℓ(θ k ω), D) = E(ℓ(ω), D -Y k (ω)) ≤ E(ℓ(ω), B |Y k (ω)| (D)),
where for all t > 0, B t (D) := {x ∈ R d , d(x, ∂D) < t}. Likewise,

E(ℓ(ω), D) = E(ℓ(θ k ω), D + Y k (ω)) ≤ E(ℓ(θ k ω), B |Y k (ω)| (D)).
Hence, since for all t ≥ 0, lim D↑R d

|Bt(D)| |D|

= 1, these two estimates yield for all k ∈ Z d , lim sup

D↑R d 1 |D| E(ℓ(ω), D) = lim sup D↑R d 1 |D| E(ℓ(θ k ω), D),
and the thermodynamic limit is invariant by the group action, and therefore constant if the group action is ergodic. We say that ℓ is stationary if it is increment-stationary and if Y k is given for all k ∈ Z d by

Y k := T k (1.5)
for some deterministic d × d matrix T .

In [START_REF] Blanc | The energy of some microscopic stochastic lattices[END_REF], the authors prove that if the positions of an infinite set of nuclei are given by a stationary random point set (satisfying in addition uniform hard-core and non-empty space properties), then the thermodynamic limit of the associated electronic cloud exists in the sense that the notions of averaged energy and cloud density are well-defined, in the case of Thomas-Fermi models. This was later extended by Blanc and Lewin [START_REF] Blanc | Existence of the thermodynamic limit for disordered Coulomb quantum systems[END_REF], and Cancès, Lahbabi and Lewin [START_REF] Cancès | Mean-field models for disordered crystals[END_REF], to quantum models with Coulomb forces and to Hartree-Fock and Kohn-Sham type models, respectively. In terms of point sets, their proofs essentially rely on the stationarity of two-body interactions and an ergodic theorem, so that, using the argument above, these proofs should extend to the more general case of incrementstationary random point sets.

The aim of this contribution is to investigate in which respect increment-stationarity is more general than stationarity, and identify under which conditions on the sequence Y k one can conclude that an increment-stationary random point set is stationary. In particular, both the probability space and the group action are fixed, and we are indeed investigating the rigidity of increment-stationary random point sets. As we shall see, the validity of such a rigidity result depends on the dimension. •] the associated expectation, variance, and covariance, respectively. We denote by L 2 0 (Ω, R d ) the space of random vectors with bounded second moments and vanishing expectations. We let {e l } l∈{1,...,d} denote the canonical basis of R d .

Let ℓ be a stationary point set in R d and Y ∈ L 2 0 (Ω, R d ) be some non-identically zero random vector. Then ℓ + Y is not stationary but is clearly increment-stationary. We shall say that a random point set ℓ is stationary up to translation if there exists some Ỹ ∈ L 2 0 (Ω, R d ) such that ℓ + Ỹ is a stationary random point set. The following theorem gives a characterization of the equivalence between increment-stationarity and stationarity up to translation. This result is directly inspired by the treatment of the corrector equation in stochastic homogenization by Papanicolaou and Varadhan in [START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients[END_REF] (see [START_REF] Künnemann | The diffusion limit for reversible jump processes on Z d with ergodic random bond conductivities[END_REF] for the case of discrete elliptic equations). It relies on the differential calculus in the probability space generated by the group action {θ k } θ∈Z d . Proposition 1. Let ℓ : Ω → L(R d ) be an increment-stationary random point set for the group action

{θ k } k∈Z d . Let {Y k } k∈Z d ∈ L 2 (Ω, R d ) be a sequence satisfying (1.2)-(1.4).
For all µ > 0 and i ∈ {1, . . . , d}, consider the unique weak solution φ µ,i ∈ L 2 (Ω) of the equation: For all ψ ∈ L 2 (Ω),

µφ µ,i ψ + Dψ • Dφ µ,i = Dψ • ζ i , (1.6)
where

ζ i := (Z e 1 • e i , . . . , Z e d • e i ) ∈ L 2 (Ω, R d ), Z e l := Y e l -Y e l for all l ∈ {1, . . . , d}, and
D := (D 1 , . . . , D d ) is the differential operator from L 2 (Ω) → L 2 (Ω, R d ) defined by D l ψ(ω) = ψ(θ e l ω) -ψ(ω). If {φ µ,i } i∈{1,.
..,d} is bounded in L 2 (Ω) uniformly wrt µ > 0, then ℓ is stationary up to translation. Conversely, if ℓ is stationary up to translation and if ℓ is non-degenerate (in the sense for almost all ω, if X is such that ℓ(ω) + X = ℓ(ω) then X = 0), then {φ µ,i } i∈{1,...,d} is bounded in L 2 (Ω) uniformly wrt µ > 0.

Remark 1.1. The condition that ℓ be non-degenerate ensures that the sequence Y k is unique and rules out periodicity. In this case, (1.3) and (1.4) follow from (1.2). If the point set is degenerate, the sequence Y k is not uniquely defined due to some translationinvariance.

It remains to identify sufficient conditions on {Y e l } l∈{1,...,d} for the boundedness of the functions φ µ,i . These conditions are written in terms of the decay of a covariance function as follows.

Hypothesis 1 (Decay of order α > 0). The random point set ℓ : Ω → L(R d ) is incrementstationary for the group action {θ k } k∈Z d , and the associated random vectors Y e 1 , . . . , Y e d display the following covariance decays: There exists α > 0 such that for all k ∈ Z d and l, l ′ , n ∈ {1, . . . , d}

cov Y e l • θ k • e n ; Y e l ′ • e n 1 1 + |k| α ,
where means ≤ up to a universal multiplicative constant.

As the following result shows, there are two types of behavior, depending on the dimension d (d = 2 is critical).

Theorem 1. We have: 3

• For d = 1 and d = 2: There exists an increment-stationary random point set ℓ : Ω → L(R d ) satisfying Hypothesis 1 with a finite range of dependence and which is not stationary up to translation. • For all d > 2: If ℓ : Ω → L(R d ) is increment-stationary and satisfies Hypothesis 1 for some α > 2, then ℓ is stationary up to translation.

Let us comment on this result. On the one hand, there is a rather natural similarity with the behavior of the corrector in stochastic homogenization. On the other hand, there is also some connection with the Palm-Khinchin theory for point processes on the real line.

We start with the connection to stochastic homogenization. In this case the gradient of the corrector is stationary (which corresponds to the increment-stationarity of the point set), and we investigate whether the corrector can be stationary itself. In dimension d = 1, there cannot exist stationary correctors in L 2 (Ω) (for this would contradict the central limit theorem, see discussion [7, p. 790]), dimension d = 2 is critical and stationary correctors do not exist either, whereas in dimensions d > 2, stationary correctors exist under some assumptions on the statistics (a spectral gap estimate, see [START_REF] Gloria | An optimal variance estimate in stochastic homogenization of discrete elliptic equations[END_REF]). Equation (1.7) can indeed be seen as the corrector equation in the regime of vanishing ellipticity contrast (the variable-coefficients elliptic operator is replaced by a constant-coefficients elliptic operator). In particular, in dimensions d = 1 and d = 2, the corrector in the regime vanishing ellipticity contrast for independent and identically distributed random conductivities provides with an example of increment-stationary point set which is not stationary up to translation, see Step 2 in the proof of Theorem 1 for details.

Let us now turn to the connection to the Palm-Khinchin theory. Let d = 1, and consider a random point set ℓ on the real line. Recall (see for instance [START_REF] Daley | An Introduction to the theory of point processes. Volumes I and II[END_REF]Chapter 3]) that the random point set ℓ is characterized by the sequence of (measurable) random variables {X i } i∈Z , where X i < X i+1 and X 0 is the closest point to the origin on the negative axis. The associated sequence of intervals is denoted by {τ i } i∈Z d and defined by τ i = X i+1 -X i . The random point set ℓ is said to be interval-stationary if the sequence {τ i } i∈Z is stationary in the following sense: For all m ∈ N, i 1 , . . . , i m ∈ Z, the distribution of (τ i 1 +k , . . . , τ im+k ) does not depend on k ∈ Z. In turn, this implies (and is indeed equivalent to, up to changing the probability space, see for instance such a construction in [10, Section 16.1]) the existence of a discrete group action {θ z } z∈Z which preserves the probability measure and is such that for all i, k ∈ Z and almost every ω ∈ Ω,

τ i (θ k ω) = τ i+k (ω).
In terms of the random point set ℓ, this turns into: For all i, j, k ∈ Z and almost every ω ∈ Ω,

X i+k (ω) -X j+k (ω) = X i (θ k ω) -X j (θ k ω). The latter implies that the random point set ℓ is increment-stationary with Y k (ω) = X 0 (θ k ω) -X 0 (ω).
The Palm-Khinchin theory (see for instance [START_REF] Daley | An Introduction to the theory of point processes. Volumes I and II[END_REF]Theorem 13.3.I]) establishes a one-to-one relation between interval-stationary and stationary random point sets, the proof of which shows that to pass from one to the other one has to consider some random translation which is not in L 2 (Ω). The argument behind this is best illustrated by the "waiting time paradox" (see e. g. [6, p6]), which shows that the Poisson point process, which is obviously stationary, is not interval-stationary (the probability that 0 belongs to a large interval is larger than the one it belongs to a small interval, so that the length of the interval around the origin cannot be exponentially distributed: this interval 4

has to be sent to infinity to turn the Poisson process into an interval-stationary process).

The general incompatibility between stationary and interval-stationary point sets gives another interpretation of the example of Theorem 1 for d = 1. Define ℓ = {X k } k∈Z by X 0 = 0, and for all k ∈ N by

X k = X k-1 + τ k-1 and X -k = X -(k-1) -τ -k
, where {τ k } k∈Z is an iid sequence of non-negative random variables with bounded second moment. This point set is interval-stationary, has finite range of dependence but is not stationary up to translation according to Theorem 1. We believe there could the following dichotomy in dimension 1: an increment-stationary point set that satisfies Hypothesis 1 for some α > 2 is either stationary up to translation or interval-stationary up to translation.

Before we turn to the proofs of these results, let us focus on a specific class of examples of increment-stationary random point sets given by the image of Z d by an "incrementstationary stochastic diffeomorphism". This case is of interest for disordered crystals and their thermodynamic limits, and was a motivation for [START_REF] Blanc | Stochastic homogenization and random lattices[END_REF]. It will also allow us to emphasize that, although Hypothesis 1 can be interpreted as a condition on the decay of the correlation between "stationary increments" (whenever this notion is well-defined), the distance is the one given by the group action (that is, k ∈ Z d associated with θ k ). In particular Hypothesis 1 is not a condition on the decay of the correlation between the "stationary increments" wrt to the Euclidean distance.

1.3. Increment-stationary stochastic diffeomorphisms. In [START_REF] Blanc | Stochastic homogenization and random lattices[END_REF], Blanc, Le Bris and Lions introduced a variant of stochastic homogenization of linear elliptic equations where the diffusion coefficients are random but not necessarily stationary. These diffusion coefficients are obtained using a stochastic diffeomorphism Φ : R d × Ω → R d which is increment-stationary in the following sense: For all k ∈ Z d , all x ∈ R d , and almost every ω ∈ Ω, ∇Φ(x + k, ω) = ∇Φ(x, θ k ω). Such random fields Φ allow one to define a specific class of increment-stationary random point sets. Set ℓ := Φ(Z d ). Since Φ is a diffeomorphim, ℓ is a simple point process almost surely. The interest of such a definition is the natural labeling of the points by Z d . Define

X i = Φ(i) for all i ∈ Z d so that ℓ = ∪ i∈Z d {X i }.
The increment-stationarity of Φ then implies that for all i, j, k ∈ Z d and almost every ω ∈ Ω we have

X i (θ k ω) -X j (θ k ω) = X i+k (ω) -X j+k (ω),
from which we deduce that ℓ is increment-stationary. Such increment-stationary point sets are very specific in the sense that they satisfy the so-called hard-core and non-empty space properties (that is, positive minimal distance between any point x ∈ ℓ and ℓ \ {x}, and existence of R < ∞ such that any ball of radius R contains at least a point of ℓ almost surely) and that they inherit the invariance group of Z d (in a statistical way), but not more. In particular such point sets cannot be statistically isotropic (see [START_REF] Alicandro | Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity[END_REF]Theorem 10]), as opposed e. g. to the random parking point set (defined in [START_REF] Penrose | Random parking, sequential adsorption, and the jamming limit[END_REF], the properties of which are listed in [8, Proposition 2.1]). Anisotropy of the point set is the necessary drawback of the natural labeling of such point sets. The labeling of ℓ = Φ(Z d ) has the advantage to allow one to define the notion of increments in the form of the quantities X i -X j . As mentioned above, these increments are stationary, and one may consider the associated covariances, that is, cov [(Φ(k + e l ) -Φ(k)) • e n ; (Φ(e l ) -Φ(0)) • e n ]. As we shall show, conditions on the decay of such quantities may ensure the stationarity of the random field Φ, but not the stationarity of the point process ℓ = Φ(Z d ). More precisely, we 5

have the following counterpart of Proposition 1 and Theorem 1 for increment-stationary random fields.

∂Φ(z + k, ω) = ∂Φ(z, θ k ω).
Assume that ∂Φ(0) ∈ L 2 (Ω, R d×d ), and for all µ > 0 and i ∈ {1, . . . , d} consider the unique weak solution φ µ,i ∈ L 2 (Ω) of the equation: For all ψ ∈ L 2 (Ω),

µφ µ,i ψ + Dψ • Dφ µ,i = Dψ • ∂(Φ • e i )(0) . (1.7)
Then, Φ is stationary up to translation, that is, there exists a unique random vector

X ∈ L 2 0 (Ω, R d ) such that (z, ω) → Φ(z, ω) + X(ω)
is stationary in the sense that for all k, z ∈ Z d and almost every ω ∈ Ω,

Φ(z + k, ω) + X(ω) = Φ(z, θ k ω) + X(θ k ω),
if and only if the family {φ µ,i } i∈{1,...,d} is bounded in L 2 (Ω) uniformly wrt µ > 0.

The conditions corresponding to Hypothesis 1 are now Hypothesis 2 (Decay of order α > 0). The random field Φ : Z d × Ω → R d is incrementstationary for the ergodic group action {θ k } k∈Z d , and there exists α > 0 such that for all k ∈ Z d and l, l ′ , n ∈ {1, . . . , d}

cov [∂ l Φ(k) • e n ; ∂ l ′ Φ(0) • e n ] 1 1 + |k| α ,
where means ≤ up to a universal multiplicative constant.

Theorem 2. We have:

• For d = 1 and d = 2: There exists an increment-stationary random field Φ : for some α > 2, then it is stationary up to translation.

Z d × Ω → R d satisfying
As a direct corollary of Theorem 2, we have the following result on random Lipschitz fields with stationary gradients: Corollary 1. Let d > 2 and Φ : R d × Ω → R d be a random Lipschitz field such that its (continuum) gradient ∇Φ is stationary and uniformly bounded. If there exists α > 2 such that for all k ∈ Z d , x ∈ [0, 1) d and l, l ′ , n ∈ {1, . . . , d}

cov [∂ l Φ(x + k) • e n ; ∂ l ′ Φ(x) • e n ] 1 1 + |k| α , then there exists a [0, 1) d -periodic random field X ∈ W 1,∞ per ([0, 1) d , L 2 0 (Ω, R d )) such that (x, ω) → Φ(x, ω) + X(x, ω) is a stationary field: For all x ∈ R d , k ∈ Z d , and almost every ω ∈ Ω, Φ(x + k, ω) + X(x + k, ω) = Φ(x, θ k ω) + X(x, θ k ω). 6
To conclude, let us compare the two notions of stationarity for a random field Φ and for the associated random point set ℓ = Φ(Z d ). The following result shows that these notions are essentially incompatible. Proposition 3. Let Φ : Z d × Ω → R d be an increment-stationary (discrete) injective random field for the ergodic group action {θ k } k∈Z d , that is, such that its discrete gradient ∂Φ = (∂ 1 Φ, . . . , ∂ d Φ), with ∂ i Φ := Φ(• + e i ) -Φ(•), is stationary. If Φ and ℓ = Φ(Z d ) are stationary up to translation, then Φ is linear and ℓ is periodic up to a random translation. Indeed, we then have Φ(•, ω) : x → ∂Φ x + Φ(0, ω), and ℓ(ω) = ∂Φ Z d + Φ(0, ω) for almost every ω ∈ Ω. This proposition illustrates that even in the case when one may properly define the notion of "stationary increments", Hypothesis 1 cannot be turned into a condition on the decay of the correlation between the "stationary increments" wrt to the Euclidean distance (which corresponds to Hypotheses 2).

Proofs

The proofs of Proposition 2 and Theorem 2 are straightforward adaptations of the proofs of Proposition 1 and Theorem 1, and we only prove the latter.

2.1. Proof of Proposition 1. We split the proof into two steps.

Step 1. Proof that boundedness of {φ µ,i } i∈{1,...,d} implies stationarity up to translation. Let ℓ be an increment-stationary random point set. Since the action group is ergodic, it generates a Weyl decomposition of L 2 (Ω, R d ) into potential fields (that is, the closure in L 2 (Ω, R d ) of gradient fields) and solenoidal fields (that is, the vector fields that are orthogonal to potential fields for the L 2 (Ω, R d )-scalar product), see for instance [9, Lemma 7.3], the adaptation of which is straightforward in the case of a discrete group action. Taking ψ = φ µ,i in (1.7) yields

µ φ 2 µ,i + |Dφ µ,i | 2 = Dφ µ,i • ζ i , which
turns, by Cauchy-Schwarz' inequality and the assumption

ζ i ∈ L 2 (Ω, R d ), into the energy estimate µ φ 2 µ,i + |Dφ µ,i | 2 1.
By the Banach-Alaoglu theorem the sequence Dφ µ,i is weakly compact, and converges weakly in L 2 (Ω, R d ) to some potential field χ i ∈ L 2 (Ω, R d ) up to extraction. Passing to the limit along the subsequence in the defining equation (1.7) for φ µ,i and using the a priori estimate µ φ 2 µ,i 1 yield for all ψ ∈ L 2 (Ω),

Dψ • χ i = Dψ • ζ i .
The above identity for arbitrary ψ ∈ L 2 (Ω) shows that the L 2 (Ω)-projections of χ i and ζ i onto potential fields coincide. Since ζ i = ((Z e 1 -Z 0 ) • e i , . . . , (Z e d -Z 0 ) • e i ) = D(Z • e i ) and χ i (as limit of potential fields Dφ µ,i ) are potential fields themselves, this implies χ i = ζ i and yields the uniqueness of the limit. Assume in addition that the family φ µ,i is bounded in L 2 (Ω) uniformly wrt µ. Then, by weak compactness, there exists some φ i ∈ L 2 (Ω) such that, up to extraction, φ µ,i converges to φ i weakly in L 2 (Ω). Note that φ i = 0 since φ µ,i = 0 for all µ > 0. Combined with the argument above, this shows that Dφ i = ζ i , and this implies in turn the uniqueness of φ i and the convergence of the entire sequence. 7

Indeed, let ϕ ∈ L 2 (Ω) be such that ϕ = φ i = 0 and Dϕ

= ζ i . Then ϕ -φ i ∈ L 2 (Ω) is such that |D(ϕ -φ i )| 2 = 0.
This implies by ergodicity that ϕφ i is constant, and therefore ϕ = φ i by the mean-free condition.

We then define Ỹ : Ω → R d , ω → d i=1 φ i (ω)e i , and for all i ∈ {1, . . . , d}, we set T i := Y e i . It remains to check that ℓ + Ỹ is stationary.

Using that Y 0 ≡ 0 one can decompose Y k as a sum of differences along the d canonical directions, i. e.

Y k = k 1 i 1 =1 Y k 1 +1-i 1 ,k 2 ,...,k d -Y k 1 -i 1 ,k 2 ,...,k d + k 2 i 2 =1 Y 0,k 2 +1-i 2 ,k 3 ,...,k d -Y 0,k 2 -i 2 ,k 3 ,...,k d + • • • + k d i d =1 Y 0,...,0,k d +1-i d -Y 0,...,0,k d -i d .
By stationarity of the increments and definition of {Z e i } i∈{1,...,d} , this yields

Y k = d i=1 k i Y e i , Y k (ω) -Y k = k 1 i 1 =1 Z e 1 (θ k 1 -i 1 ,k 2 ,...,k d ω) + k 2 i 2 =1 Z e 2 (θ 0,k 2 -i 2 ,k 3 ,...,k d ω) + • • • + k d i d =1 Z e d (θ 0,...,0,k d -i d ω).
Using Z 0 ≡ 0 and Dφ i = ζ i , this turns into

Y k (ω) -Y k = k 1 i 1 =1 d l=1 D 1 φ l (θ k 1 -i 1 ,k 2 ,...,k d ω)e l + k 2 i 2 =1 d l=1 D 2 φ l (θ 0,k 2 -i 2 ,k 3 ,...,k d ω)e l + • • • + k d i d =1 d l=1 D d φ l (θ 0,...,0,k d -i d ω)e d .
By definition of the difference operators D i , terms cancel two by two, and the sum simplifies to

Y k (ω) -Y k = d l=1 φ l (θ k ω)e l - d l=1 φ l (ω)e l .
This implies the desired property by the choice of Ỹ and

T i : For all k ∈ Z d , ℓ(θ k ω) + Ỹ (θ k ω) = ℓ(ω) + Ỹ (ω) + d i=1 k i T i .
Step 2. Proof that stationarity up to translation implies boundedness of {φ µ,i } i∈{1,...,d} .

On the one hand, since ℓ is stationary up to translation, there exist Ỹ ∈ L 2 (Ω, R d ) and

{T i } 1≤i≤d ∈ R d such that for all k ∈ Z d and almost all ω, ℓ(θ k ω) + Ỹ (θ k ω) = ℓ(ω) + Ỹ (ω) + d i=1 k i T i .
On the other hand, increment-stationarity implies there exists

Y k ∈ L 2 (Ω, R d ) such that ℓ(θ k ω) = ℓ(ω) + Y k (ω). Since ℓ is assumed to be non-degenerate, this implies that Y k (ω) = Ỹ (ω) -Ỹ (θ k ω) + d i=1 k i T i . Hence, Y k = d i=1 k i T i and Z e l takes the form Z e l (ω) = Ỹ (θ e l ω) -Ỹ (ω).
Recall that for all µ > 0 and i ∈ {1, . . . , d}, φ µ,i ∈ L 2 (Ω) is solution of: For all ψ ∈ L 2 (Ω),

µ φ µ,i ψ + Dψ • Dφ µ,i = Dψ • ζ i with ζ i = (Z e 1 • e i , . . . , Z e d • e i ). For all i ∈ {1, . . . , d}, set φi := Ỹ • e i ∈ L 2 (Ω), so that D φi = ζ i . We then have for all ψ ∈ L 2 (Ω), µ (φ µ,i -φi )ψ + Dψ • D(φ µ,i -φi ) = -µ φi ψ , whence the a priori estimate (φ µ,i -φi ) 2 ≤ φ2 i = ( Ỹ • e i ) 2 ,
which yields the desired uniform-in-µ boundedness estimate by the triangle inequality:

φ 2 µ,i ( Ỹ • e i ) 2 .
2.2. Proof of Theorem 1. Let ℓ be an increment-stationary random point set, and for all i ∈ {1, . . . , d} and µ > 0, let ζ i and φ µ,i be as in Proposition 1. We first derive an integral representation for φ µ,i in physical space, then treat the case d ≤ 2 in Step 2, and the case d > 2 in Step 3.

Step 1. Green representation formula for φ µ,i . In this step we derive a Green representation formula for φ µ,i , see [START_REF] Gloria | An optimal variance estimate in stochastic homogenization of discrete elliptic equations[END_REF]Lemma 2.6]. Equation (1.7) indeed admits an equivalent form in the physical space. Let φµ,i , ζi : Z d × Ω → R be the stationary extensions of φ µ,i and ζ i , that is, the random fields defined by φµ,i (k, ω) := φ µ,i (θ k ω) and ζi (k, ω) := ζ i (θ k ω), respectively. Then, φµ,i solves almost surely the elliptic PDE

µ φµ,i -△ φµ,i = ∂ * • ζi in Z d , (2.1)
where ∂ is the forward discrete gradient, ∂ * • the backward discrete divergence, and

-△ := -∂ * • ∂ the discrete Laplace operator on Z d . Let G µ : Z d → R denote the Green's function associated with the elliptic operator µ-△ on Z d , that is, the only solution in L 2 (Z d ) of µG µ (x) -△G µ (x) = δ(x),
where δ is such that δ(0) = 1 and δ(x) = 0 for all x = 0 (the existence and uniqueness of G µ follows from the Riesz representation theorem). 9

Testing equation (2.1) with y → G µ (yx) yields the desired Green representation formula φµ,i (x) =

ˆZd ∂G µ (y -x) • ζi (y)dy,
where ´Zd dy stands for the sum over y ∈ Z d .

Step 2. Case d ≤ 2.

In this step we prove that even in the case when { ζi (z)} z∈Z d is a field of independent and identically distributed (iid) variables, the family φ 2 µ,i may be unbounded in µ. Consider in particular the field Y k characterized by: Y 0 ≡ 0 and

Y e l • θ k = Y k+e l -Y k = a l (k)e l ,
where {a l (k)} l∈{1,...,d},k∈Z d are iid variables following the law of some a ∈ L 2 (Ω). Then, the random point set ℓ satisfies Hypothesis 1 for any α > 0 (it has finite correlation-length). In view of Step 1, we have

φ 2 µ,i = ( φµ,i (0)) 2 = ˆZd ˆZd ∂G µ (y) ⊗ ∂G µ (y ′ ) : ζi (y) ⊗ ζi (y ′ ) dydy ′ . (2.2)
Since ζi (y) = (a i (y)a )e i , the sum reduces by independence to

φ 2 µ,i = var [a] ˆZd (∂ i G µ (y)) 2 dy. (2.3)
Since ∂G µ converges locally to the gradient ∂G of the whole space Green's function of the discrete Laplace operator -△ on Z d as µ ↓ 0, the RHS of (2.3) cannot be bounded in dimensions d ≤ 2. If it were, this would imply that ∂G ∈ L 2 (Z d ), which is not true for d ≤ 2 (as can be seen in Fourier space [START_REF] Martinsson | Asymptotic expansions of lattice Green's functions[END_REF], or by comparison to the large scale behavior of the continuum Green's function). This qualitative behavior is enough for the proof of Theorem 1. To be more quantitative, one indeed expects φ 2 µ,i ∼ µ -1 2 for d = 1, and

φ 2 µ,i
∼ | ln µ| for d = 2. The proof of these estimates would require a more careful analysis of the Green's functions.

Step 3. Case d > 2. The starting point in dimensions d > 2 is again (2.2) which in view of Hypothesis 1 implies

φ 2 µ,i ˆZd ˆZd |∂G µ (y)||∂G µ (y ′ )| 1 1 + |y -y ′ | α dydy ′ .
Without loss of generality, we assume in addition that α < d. We shall use the following uniform-in-µ bounds on ∂G µ : ∂G µ L ∞ (Z d ) 1 (cf. [7, Corollary 2.3]), and for all exponents 1 ≤ p < ∞, all i ∈ N and d > 2,

ˆ2i <|y|≤2 i+1 |∂G µ (y)| p dy (2 i ) d (2 i ) p(1-d) , (2.4)
which is optimal in terms of scaling. This estimate is standard and relies on the L pregularity theory for the operator µ -△. For a proof, we refer to [7, Lemma 2.9], which treats in addition the variable-coefficients case using the perturbation approach by Meyers. Indeed, Steps 3-6 of that proof show that if for some p > 2 the operator has an L pregularity theory, then (2.4) holds, whereas Step 1 shows that µ -△ has an L p -regularity theory for all 1 < p < ∞. The case 1 ≤ p < 2 in (2.4) follows from the estimate for p = 2 by Hölder's inequality.

We now prove the boundedness of φ 2 µ,i if α > 2 by estimating the integrals using a doubly dyadic decomposition of Z d × Z d . Note that the exponent α = 2 is borderline in terms of integrability.

We write the integral as:

ˆZd ˆZd |∂G µ (y)||∂G µ (y ′ )| 1 1 + |y -y ′ | α dydy ′ = ˆ|y|≤2 |∂G µ (y)| ˆ|y-y ′ |≤2 |∂G µ (y ′ )| 1 1 + |y -y ′ | α dydy ′ (2.5a) + ˆ|y|≤2 |∂G µ (y)| ∞ j=1 ˆ2j <|y-y ′ |≤2 j+1 |∂G µ (y ′ )| 1 1 + |y -y ′ | α dydy ′ (2.5b) + ∞ i=1 ˆ2i <|y|≤2 i+1 |∂G µ (y)| ˆ|y-y ′ |≤2 |∂G µ (y ′ )| 1 1 + |y -y ′ | α dydy ′ (2.5c) + ∞ i=1 ˆ2i <|y|≤2 i+1 |∂G µ (y)| ∞ j=1 ˆ2j <|y-y ′ |≤2 j+1 |∂G µ (y ′ )| 1 1 + |y -y ′ | α dydy ′ . (2.5d)
By the uniform bound on ∂G µ L ∞ (Z d ) , the RHS term (2.5a) is of order 1. For the RHS term (2.5b), this uniform bound and the triangle inequality yield

(2.5b) 1 + ∞ j=1 ˆ2j -2<|y ′ |≤2 j+1 +2 |∂G µ (y ′ )| 1 1 + |y ′ | α dy ′ .
Estimate (2.4) for p = 1 then yields that (2.5b) 1 since α > 1. The proof of the boundedness of the RHS term (2.5c) is similar. The most subtle part is the RHS term (2.5d). We split the double sum into two parts: ∞ i=1 j≤i and ∞ i=1 j>i , and start with the latter. If j > i, and y, y ′ are such that 2 i < |y| ≤ 2 i+1 and 2 j < |yy ′ | ≤ 2 j+1 , then 2 j-1 < |y ′ | ≤ 2 j+2 . Hence, by (2.4) for q = 1,

ˆ2i <|y|≤2 i+1 |∂G µ (y)| ˆ2j <|y-y ′ |≤2 j+1 |∂G µ (y ′ )| 1 1 + |y -y ′ | α dydy ′ ≤ (2 j ) -α ˆ2i <|y|≤2 i+1 |∂G µ (y)|dy ˆ2j-1 <|y ′ |≤2 j+2 |∂G µ (y ′ )|dy ′ (2 i ) d-(d-1) (2 j ) d-(d-1)-α = (2 i )(2 j ) 1-α . Since α > 1, summing over j > i yields ˆ2i <|y|≤2 i+1 |∂G µ (y)| j>i ˆ2j <|y-y ′ |≤2 j+1 |∂G µ (y ′ )| 1 1 + |y -y ′ | α dydy ′ (2 i ) 2-α ,
and therefore, using that α > 2,

∞ i=1 ˆ2i <|y|≤2 i+1 |∂G µ (y)| j>i ˆ2j <|y-y ′ |≤2 j+1 |∂G µ (y ′ )| 1 1 + |y -y ′ | α dydy ′ 1. ( 2

.6)

We now treat the sum ∞ i=1 j≤i . If j ≤ i, and y, y ′ are such that 2 i < |y| ≤ 2 i+1 and 2 j < |yy ′ | ≤ 2 j+1 , then 2 i-1 < |y ′ | ≤ 2 i+2 . Let q > 1 be such that that d q > α (which is 11 possible since d > α), and p > 1 be the associated dual exponent, i. e. 1 p + 1 q = 1. Then, by Hölder's inequality with exponents (p, q), and (2.4) with exponents 1 and p, we have

ˆ2i <|y|≤2 i+1 |∂G µ (y)| ˆ2j <|y-y ′ |≤2 j+1 |∂G µ (y ′ )| 1 1 + |y -y ′ | α dydy ′ ≤ ˆ2i <|y|≤2 i+1 |∂G µ (y)|dy ˆ2i-1 <|y ′ |≤2 i+2 |∂G µ (y ′ )| p dy ′ 1 p × ˆ2j <|y-y ′ |≤2 j+1 1 1 + |y -y ′ | qα dy ′ 1 q (2 i ) d-(d-1) (2 i ) 1 p (d-p(d-1)) (2 j ) 1 q (d-qα) = (2 i ) 2-d(1-1 p ) (2 j ) d q -α .
Summing over j ≤ i and using that d qα > 0 then yields

ˆ2i <|y|≤2 i+1 |∂G µ (y)| j≤i ˆ2j <|y-y ′ |≤2 j+1 |∂G µ (y ′ )| 1 1 + |y -y ′ | α dydy ′ (2 i ) 2-d(1-1 p ) j≤i (2 j ) d q -α (2 i ) 2-d(1-1 p )+ d q -α = (2 i ) 2-α .
Since α > 2, this yields a bound for the first sum ∞ i=1 j≤i :

∞ i=1 ˆ2i <|y|≤2 i+1 |∂G µ (y)| j≤i ˆ2j <|y-y ′ |≤2 j+1 |∂G µ (y ′ )| 1 1 + |y -y ′ | α dydy ′ 1.
(2.7)

The combination of (2.6) and (2.7) shows that (2.5d) 1, which, combined with the estimates of (2.5a), (2.5b), and (2.5c), implies the uniform-in-µ bound

ˆZd ˆZd |∂G µ (y)||∂G µ (y ′ )| 1 1 + |y -y ′ | α dydy ′ 1,
as desired.

2.3. Proof of Corollary 1. For all x ∈ R d , consider the random field Φ x : Z d × Ω : (k, ω) → Φ(x + k, ω). By Theorem 2, there exists some random vector X(x, To conclude, it remains to prove the measurability of X : R d × Ω → R d (where R d is endowed with the Borel sets). It is enough to show that X is a Carathéodory function, since Carathéodory functions are equivalent to Borel functions. Recall that X is a Carathéodory function if for almost all ω ∈ Ω, x → X(x, ω) is continuous, and if for all x ∈ R d , ω → X(x, ω) is measurable. The measurability of X(x, •) follows form the definition of X. It remains to prove the continuity, which we do in the form of a Lipschitz estimate. There exists a set Ω ′ ∈ F of full measure such that for all x ∈ Q d , z, k ∈ Z d and ω ∈ Ω ′ , Φ(x + z, ω) + X(x + z, ω) = Φ(x, θ k ω) + X(x, θ k ω).

•) ∈ L 2 0 (Ω) such that (z, ω) → Φ x (z, ω) + X(x, ω) is stationary: For all z, k ∈ Z d and almost every ω ∈ Ω, Φ x (z + k, ω) + X(x, ω) = Φ x (z, θ k ω) + X(x, θ k ω), which we may rewrite by definition of Φ x as Φ(x + z + k, ω) + X(x, ω) = Φ(x + z, θ k ω) + X(x, θ k ω). (2.8) Since Φ x+k (k ′ ) = Φ x (k + k ′ ) for all k, k ′ ∈ Z d ,
(2.9)

The uniform Lipschitz assumption on Φ yields: There exists C < ∞ such that for all k ∈ Z d and x, h ∈ R d , |Φ(x + h + k, •) -Φ(x + k, •)| ≤ C|h|. Hence, substracting (2.9) once with x ; x + h and once with x implies that for all x, h ∈ Q d , k ∈ Z d , and ω ∈ Ω ′ , X(x + h, ω) -X(x, ω) -X(x + h, θ k ω) -X(x, θ k ω) ≤ C|h|.

By summation over k, this yields for all N ∈ N X(x + h, ω) -X(x, ω) -

1 #([-N, N ) ∩ Z) d k∈([-N,N )∩Z) d X(x + h, θ k ω) -X(x, θ k ω) ≤ C|h|.
By the ergodic theorem, and since X(x, •) = X(x + h, •) = 0, there exists some Ω ′′ ∈ F with full measure such that for all x ∈ Q d , h ∈ Q d , k ∈ Z d and ω ∈ Ω ′′ the limit N ↑ ∞ yields X(x + h, ω) -X(x, ω) ≤ C|h|.

By symmetry, this implies | X(x + h, ω) -X(x, ω)| ≤ C|h|, so that X| Q d ×Ω ′′ can be extended to a Lipschitz function on R d for all ω ∈ Ω ′′ . Hence, X is a Carathéodory function, as desired.

2.4. Proof of Proposition 3. We split the proof into two steps, using the stationarity of ℓ + Ỹ = Φ(Z d ) + Ỹ , and of Φ + X, respectively. Recall that T = ∂Φ .

Step 1. Stationarity of ℓ. Since ℓ is stationary up to translation, there exists some Ỹ ∈ L 2 0 (Ω) such that for all k ∈ Z d and almost every ω ∈ Ω, ℓ(θ k ω) + Ỹ (θ k ω) = ℓ(ω) + Ỹ (ω) + T k.

Every point of ℓ(θ k ω) + Ỹ (θ k ω) is mapped to a point of ℓ(ω) + Ỹ (ω) + T k. Hence for almost all ω ∈ Ω, there exists a function γ ω : Z d × Z d → Z d such that for all k ∈ Z d , γ ω (k, •) is bijective on Z d and satisfies X y (θ k ω) + Ỹ (θ k ω) = X γ(k,ω,y) (ω) + Ỹ (ω) + T k, where X y (ω) = Φ(y, ω) as before. Combined with the stationarity of the increment X y -X 0 , this turns into X y (θ k ω) + Ỹ (θ k ω) = X 0 (θ k ω) + Ỹ (θ k ω) + X y (θ k ω) -X 0 (θ k ω) = X y (ω) + Ỹ (ω) + T k + X γ(k,ω,0) (ω) -X 0 (ω).

(2.10)

Step 2. Stationarity of Φ and conclusion. 13

Proposition 2 .

 2 Let Φ : Z d × Ω → R d be an increment-stationary (discrete) random field for the group action {θ k } k∈Z d , that is, such that its discrete gradient ∂Φ = (∂ 1 Φ, . . . , ∂ d Φ), with ∂ i Φ := Φ(• + e i ) -Φ(•), is stationary: For all k, z ∈ Z d and almost every ω ∈ Ω,

  Hypothesis 2 with finite range of dependence and which is not stationary up to translation.• For all d > 2: If Φ : Z d ×Ω → R d is increment-stationary and satisfies Hypothesis 2

  the uniqueness of X (cf. uniqueness of φ i in Step 1 of the proof of Proposition 1) shows that X(x + k, •) = X(x, •) for all k ∈ Z d and x ∈ R d . Hence, (2.8) turns into: For all x ∈ R d , there exists a set of full measure Ω x such that for all z, k ∈ Z d and ω ∈ Ω x , we have Φ(x + z + k, ω) + X(x + z + k, ω) = Φ(x + z, θ k ω) + X(x + z, θ k ω).

Acknowledgements

The author is grateful to Xavier Blanc for his constructive comments on a early version of this paper and to the anonymous referee for insightful suggestions to improve readability. The author acknowledges financial support from ANR-10-JCJC-0106 AMAM and from the European Research Council under the European Community's Seventh Framework Programme (FP7/2014-2019 Grant Agreement QUANTHOM 335410).

A. GLORIA

Since Φ is stationary up to translation, there exists X ∈ L 2 0 (Ω) such that for all y, k ∈ Z d and almost every ω ∈ Ω,

Combined with (2.10) this yields

Since the RHS does not depend on y, the increment X y+k -X y does not depend on y either. By the ergodic theorem, and since the increment is stationary, this yields for almost every ω ∈ Ω and for all y, k ∈ Z d ,

Hence, for almost every ω ∈ Ω and all y ∈ Z d , X y (ω) = X 0 (ω) + T y, so that we obtain Φ(y, ω) = ∂Φ y + Φ(0, ω), ℓ(ω) = ∂Φ Z d + Φ(0, ω), as desired.


Antoine Gloria

Stefan Neukamm
email: neukamm@mis.mpg.de

Felix Otto
email: otto@mis.mpg.de

An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations

Keywords: stochastic homogenization, homogenization error, quantitative estimate. 2010 Mathematics Subject Classification: 35B27, 39A70, 60H25, 60F99

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L 2 -norm in probability of the H 1 -norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Green's function by Marahrens and the third author.

Introduction

We establish a linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations, which is optimal. Before we turn to the stochastic case, let us recall some standard analogous results in the periodic case. Let A be a uniformly elliptic and bounded symmetric matrix field on the unit torus T := (R/Z) d , D ⊂ R d be a smooth domain, and f be a smooth function. Let ε > 0. It is well-known that the unique weak solution u ε ∈ H 1 0 (D) of the linear elliptic equation

-∇ • A • ε ∇u ε = f in D, u ε = 0 on ∂D (1.1)
converges weakly in H 1 (D) as ε → 0 to the unique weak solution u hom ∈ H 1 0 (D) of the homogenized equation

-∇ • A hom ∇u hom = f in D,
u hom = 0 on ∂D.

(1.

2)

The homogenized matrix A hom is symmetric and characterized for all ξ ∈ R d by ξ • A hom ξ = ˆT(∇φ ξ + ξ) • A(∇φ ξ + ξ),

Date: September 3, 2014.

1
where φ ξ is the unique weak solution in H 1 (T) with vanishing mean of the periodic corrector equation in direction ξ:

-∇ • A(∇φ ξ + ξ) = 0 in T.

From a two-scale expansion, one formally expects for all x in the interior of D:

u ε (x) = u hom (x) + ε d j=1 φ j ( x ε )∇ j u hom (x) + o(ε), (1.3)
where the φ j are the correctors in the canonical directions e j (extended by periodicity to R d ). This identity cannot hold at the boundary since the correctors φ j do not satisfy the homogeneous Dirichlet boundary conditions: There is indeed a boundary layer. Yet, for all D compactly supported in D, Allaire and Amar [START_REF] Allaire | Boundary layer tails in periodic homogenization[END_REF]Theorem 2.3] proved the following rigorous version of (1.3) if the coefficients A are Hölder continuous:

u ε -u hom -ε d j=1 φ j ( • ε )∇ j u hom H 1 ( D) ≤ Cε, (1.4)
where the multiplicative constant C only depends on D, D, the Hölder exponent and norm of A, and on the C 3 -norm of u hom (which is smooth since f and D are smooth). This result relies on a previous work by Avellaneda and Lin [2, Theorem 5], who proved under similar assumptions that u ε -u hom L ∞ (D) ≤ Cε.

(1.5) These are bounds on the homogenization error. In the stochastic case, we consider a matrix field A that is stationary and ergodic, in place of periodic. We refer the reader to [START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients[END_REF] for details. In order to obtain quantitative estimates in the spirit of (1.4) and (1.5), one has to make assumptions on the statistics of A in addition to ergodicity. There are few results in the literature on quantitative estimates of the homogenization error for elliptic equations in divergence form in the stochastic case. In [START_REF] Yurinskiȋ | Averaging of symmetric diffusion in random medium[END_REF]Theorem 3.1], Yurinskiȋ proved for algebraically decaying correlations that for all d > 2, there exists some Hölder exponent γ > 0 and a function T of ε such that

u ε -u hom -ε d j=1 φ T (ε),j ( • ε )∇ j u hom 2 H 1 (D) 1/2 ≤ Cε γ , (1.6)
where φ T,j is the modified corrector, which is the stationary, almost sure solution to

T -1 φ T,j -∇ • A(∇φ T,j + e j ) = 0 in R d . (1.7)
This equation is an approximation of the corrector equation when T → ∞. This is the first quantitative result in stochastic homogenization. Note that a formal linearization in the case of small ellipticity contrast λ ↑ 1 yields γ = 1 for d > 2. Besides not covering dimension d = 2, the work by Yurinskiȋ does not allow to reach the scaling γ = 1, even in the case of small ellipticity constrast (and for domains D compactly included in D).

In this article, we simplify the above setting with three respects: (a) We treat finite-difference equations instead of continuum partial differential equations. More precisely, we replace the differential operator in (1.1) by the finite difference operator ∇ * ε a( • ε )∇ ε which acts on discrete functions defined on the scaled lattice εZ d , ε > 0. Here ∇ ε and ∇ * ε are finite difference approximations of the continuum 2 gradient ∇ and continuum divergence operator -∇•, and are defined for scalar fields v : εZ d → R and vector fields, g = (g 1 , . . . , g d ) : εZ d → R d as follows:

∇ ε,i v(y) = ε -1 (v(y + εe i ) -v(y)), ∇ * ε,i v(y) = ε -1 (v(y -εe i ) -v(y)), ∇ ε v = (∇ ε,1 v, . . . , ∇ ε,d v), ∇ * ε g = d i=1
∇ * ε,i g i .

(1.8)

The coefficients of the operator ∇ * ε a( • ε )∇ ε are given by a randomly chosen matrix field a : Z d → R d×d . We assume that a is diagonal and uniformly elliptic in the sense that

a(x) ∈ Ω 0 :=         a 1 0 . . . 0 a d    λ ≤ a j ≤ 1, j = 1, . . . , d     
for all x ∈ Z d . Above, λ > 0 is a (deterministic) ellipticity constant and fixed throughout the paper. (b) In order to avoid boundary layers, we consider a problem on the (discretized) unit torus T ε := (εZ/Z) d and assume w. l. o. g. that the grid size ε is the inverse of a large integer L := ε -1 ∈ N. The equation under consideration is the following finite difference equation with random coefficients

∇ * ε a( • ε )∇ ε u ε = f ε on T ε , Tε u ε = 0, (1.9)
where f ε : T ε → R is a deterministic r. h. s. with zero mean, and should be viewed as a discretization of a continuum r. h. s. f ∈ L 2 (T). Note that on T ε we have by periodicity the discrete integration by parts formula

Tε ∇ ε v • g = Tε v∇ * ε g for all v : T ε → R and g : T ε → R d .
(c) We consider the simplest possible statistics, namely independent and identically distributed (i. i. d.) coefficients. Since (1.9) is posed on the discretized unit torus T ε , ε -1 = L ∈ N, we in fact consider the periodic i. i. d. ensemble which is constructed as follows: For a fixed measure β on our matrix space Ω 0 ("single-site measure") and for fixed size L of the discrete, rescaled torus T L := (Z/LZ) d = 1 ε T ε , we shall throughout the paper denote by • the periodic i. i. d. ensemble on

Ω L = {a : Z d → Ω 0 | ∀z ∈ Z d a(• + Lz) = a} = Ω T L 0 . For a random variable, i. e. a measurable function ζ : Ω L → R, it is given by the product measure ζ = x∈T L ˆΩ0 ζ(a) β(da(x)).
(1.10) Evidently, our i. i. d. ensemble is stationary in the sense that for all shifts y ∈ Z d the random (periodic) tensor fields a and a(• + y) have the same distribution in Ω L . In the above setting the qualitative theory of stochastic homogenization applies (see e. g. Kozlov [START_REF] Kozlov | The averaging of random operators[END_REF], Papanicolaou and Varadhan [START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients[END_REF] and Künnemann [START_REF] Künnemann | The diffusion limit for reversible jump processes on Z d with ergodic random bond conductivities[END_REF]Theorem 4]): There exists a deterministic, symmetric, positive definite d × d matrix a hom (only depending on the single-site probability measure β) such that the following statement is true: Suppose that f ε converges (in a discrete H -1 -norm) to some function f ∈ L 2 (T) with ´T f = 0. Let 3 u ε be the unique (random) solution with vanishing mean to (1.9), and let u hom ∈ H 1 (T) be the unique solution to the continuum homogenized equation

-∇ • a hom ∇u hom = f in T, ˆT u hom = 0. (1.11)
Then u ε converges almost surely (weakly in the discrete H 1 -norm) to u hom .

We are interested in estimates on the speed of convergence of u ε to u hom . The error u ε -u hom consists of two contributions that are of different nature, namely the homogenization error and the discretization error. The latter is purely deterministic and introduced by approximating the deterministic, continuum, homogenized equation (1.11) by the deterministic, homogenized, finite difference equation

∇ * ε a hom ∇ ε u 0,ε = f ε in T ε , Tε u 0,ε = 0, (1.12)
where f ε ∈ L 2 (T ε ) denotes a suitable approximation of the continuum r. h. s. f . The discretization error is well-studied and well-understood (see for instance [START_REF] Leveque | Finite difference methods for ordinary and partial differential equations[END_REF]).

In this paper we focus on the homogenization error that monitors the difference between the random, variable-coefficient equation (1.9) and the deterministic, constant-coefficient equation (1.12). The main result of this paper is the upcoming quantitative two-scale expansion estimate. We quantify the error in the discrete L 2 (T ε )-and H 1 (T ε )-norms, which are defined for v :

T ε → R as v L 2 (Tε) = ε d Tε v 2 , v H 1 (Tε) = ε d Tε (v 2 + |∇ ε v| 2 ).
Theorem 1. Let d ≥ 2 and a be i. i. d. coefficients. Then there exists a deterministic, symmetric, positive definite matrix a hom ∈ R d×d (only depending on β and d) with the following property. Given ε > 0 with ε -1 = L ∈ N, and a r. h. s.

f ε : T ε → R with Tε f ε = 0, let u ε : Ω L × T ε → R
and u 0,ε : T ε → R be the unique solutions to (1.9) and (1.12), respectively. Then

u ε -u 0,ε -ε d j=1 φ j ( • ε )∇ ε,j u 0,ε 2 H 1 (Tε) 1/2 ε ||f ε || L 2 (Tε) (ln 1 ε ) 1/2 for d = 2 1 for d > 2 , (1.13)
where φ 1 , . . . , φ d are the periodic correctors associated with the periodic i. i. d. ensemble

• via the periodic corrector equation (see (2.3) below). The multiplicative constant in (1.13) only depends on the constant of ellipticity λ, and the dimension d.

As a corollary we get:

Corollary 1. In the situation of Theorem 1 we have

u ε -u 0,ε 2 L 2 (Tε) 1/2 ε f ε L 2 (Tε) (ln 1 ε ) 1/2 for d = 2 1 for d > 2 , (1.14)
The multiplicative constant in this estimate only depends on λ and d.

Note that a bound similar to (1.14) (with however a suboptimal exponent) was recentlty obtained by Conlon and Spencer in [START_REF] Conlon | Strong convergence to the homogenized limit of elliptic equations with random coefficients[END_REF], in the case of the whole space and a massive term. This result complements the estimates of the quantity u ε -u ε obtained in [5, Theorem 2], and in their optimal form in [14, Corollary 2 and Corollary 3]. In particular, the annealed estimates of [14, Theorem 1] (see Lemma 3.2 below) on the gradient of the (periodic) Green's function, combined with the argument leading to [5, Theorem 2], allow one to prove that

u ε -u ε 2 L 2 (Tε) 1/2 ε f ε L 2 (Tε) (ln 1 ε ) 1/2 for d = 2 1 for d > 2 .
The combination of this estimate with Corollary 1 then yields an estimate of the difference between the solution u 0,ε of the problem with constant coefficients and the expectation u ε of the solution of the original problem -i. e. the systematic error:

u 0,ε -u ε L 2 (Tε) ε f ε L 2 (Tε) (ln 1 ε ) 1/2 for d = 2 1 for d > 2 .
It is rather surprising that we have to go through the H 1 -norm in order to control this systematic error, or conversely that we do not have to estimate this term in order to prove the (seemingly stronger) statement of Theorem 1. Compared to the work [START_REF] Yurinskiȋ | Averaging of symmetric diffusion in random medium[END_REF] by Yurinskiȋ, Theorem 1 covers dimension d = 2 and gives optimal estimates (in terms of scaling in ε) in any dimension, as can be seen by considering the regime of small ellipticity contrast. Note that for d = 1, the scaling is different: (1.14) is expected to be replaced by

u ε -u 0,ε 2 L 2 (Tε) 1/2 √ ε f ε L 2 (Tε) ,
as explicity checked in the continuum setting by Bourgeat and Piatnitski in [START_REF] Bourgeat | Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator[END_REF]. As opposed to these works, the present analysis heavily relies on the use of a spectral gap estimate in the probability space. We refer the reader to [6] for relevant references on the subject.

Remark 1.1. The reason why we consider the discrete setting is the following: In the proof of Theorem 1 we make extensive use of recent, quantitative results that we obtained in a series of paper [START_REF] Gloria | An optimal variance estimate in stochastic homogenization of discrete elliptic equations[END_REF][START_REF] Gloria | An optimal error estimate in stochastic homogenization of discrete elliptic equations[END_REF]6,[START_REF] Marahrens | Annealed estimates on the Green's function[END_REF] in the discrete setting. The extension of some results to the continuum setting is currently under investigation, see [START_REF] Gloria | Quantitative results on the corrector equation in stochastic homogenization of linear elliptic equations[END_REF].

Remark 1.2. (Rescaling). For the proof of Theorem 1 it is convenient to rescale the discretized torus T ε so that the grid size becomes 1. Recall that ε = 1 L for some integer L ∈ N. Hence, the L-rescaled version of T ε yields the discrete rescaled torus

T L = (Z/LZ) d = 1 ε T ε .
In analogy with (1.8) we introduce discrete derivatives ∇ and ∇ * acting on scalar fields v : T L → R and vector fields g = (g 1 , . . . , g d ) : T L → R d as follows:

∇ i v(x) = v(x + e i ) -v(x), ∇ * i v(x) = v(x -e i ) -v(x), ∇v = (∇ 1 v, . . . , ∇ d v), ∇ * g = d i=1
∇ * i g i .

(1.15)

5

In order to state Theorem 1 in its rescaled version, we set

u(x) := u ε (εx), u 0 (x) := u 0,ε (εx), f (x) := ε 2 f ε (εx) for x ∈ Z d .
So defined, u, u 0 and f are functions on the rescaled torus T L with vanishing mean, and the finite difference equations (1.9) and (1.12) turn into

∇ * a∇u = f on T L , T L u = 0, (1.16
)

∇ * a hom ∇u 0 = f on T L , T L u 0 = 0. (1.17)
Furthermore, the two-scale expansion in (1.13) takes the form

z := u -u 0 - d j=1 φ j ∇ j u 0 , (1.18)
and estimate (1.13) of Theorem 1 can be reformulated as

T L (z 2 + L 2 |∇z| 2 ) 1 2 Lµ 1 2 d (L)   T L f 2   1 2 , (1.19)
where we set for abbreviation

µ d (L) = ln L for d = 2, 1 for d > 2.
In fact we shall establish (1.19) (and thus (1.13)) in form of the estimate

T L (z 2 + L 2 |∇z| 2 ) 1 2 Lµ 1 2 d (L)   T L |∇ 2 u 0 | 2   1 2 , (1.20)
where ∇ 2 u 0 (x) denotes the discrete Hessian of u 0 at x and is given by the d × d matrix with entries -∇ * i ∇ j u 0 (x). Note that (1.20) indeed implies (1.19), since u 0 (as a solution to the constant-coefficient difference equations (1.17)) satisfies the a priori estimate

T L |∇ 2 u 0 | 2
T L f 2 up to a multiplicative constant that only depends on λ and d.

Notation. Throughout this article, we use the following notation:

• d ≥ 2 is the dimension; • (e 1 , . . . , e d ) denotes the canonical basis of Z d ; • T L = (Z/LZ) d denotes the discretized L-rescaled torus; • x mod L denotes the unique point in ([0, L) ∩ Z) d with x = (x mod L) + Lx ′ for some x ′ ∈ Z d ; •
and stand for ≤ and ≥ up to a multiplicative constant which only depends on the quantities specified in the context;

• when both and hold, we simply write ∼; • β denotes a single-site probability measure on Ω 0 , see Section 2.1; • • denotes the L-periodic i. i. d. ensemble on Ω L associated with β, see Section 2.1;

• cov [•; •] denotes the covariance associated with • ; 6

• we denote the (i, j)-th entry of a d × d-matrix b by b ij and write ":" for the inner product in R d×d , i. e. a : b

= d i,j=1 a ij b ij ; • for all L > 0, µ d (L) = ln L for d = 2 and µ d (L) = 1 for d > 2.

Assumptions on the ensemble and the notion of the corrector

In this section we introduce and motivate the requi assumptions on the ensemble, the definition of the corrector and the homogenized coefficients. We recall some recent quantitative estimates from stochastic homogenization that are at the basis of the proof of Theorem 1. Finally, we comment on the role played by the i. i. d. assumption.

2.1. Assumptions on the ensemble. Recall that • denotes the i. i. d. ensemble associated with the single-site measure β via (1.10). Our theory involves only two probabilistic ingredients: A covariance estimate and a Logarithmic Sobolev Inequality (LSI) for • . We start with the covariance estimate, which is explicitly used in the proof of this paper. We shall comment on the LSI in the following section. Like the classical partial derivatives { ∂ζ ∂a kk (y) } k=1,

cov ζ; ζ ≤ y∈T L ( ∂ζ ∂y ) 2 1 2 ( ∂ ζ ∂y ) 2 1 2 . ( 2
T L → R characterized by
∇ * a(∇φ j + e j ) = 0 on T L and

T L φ j = 0. (2.3)
Here e j denotes the unit vector in direction of the j-th coordinate axis, and the discrete derivatives are defined in (1.15). Clearly, for every j = 1, • • • , d, this defines a random scalar field φ j . Evidently, this random field is stationary in the sense that for any shift

y ∈ R d one has φ j (a(• + y), •) = φ j (a, • + y).
The periodic function φ j : Z d → R "corrects" the affine function x → x j such that the resulting function

Z d ∋ x → φ j (x) + x j is a- harmonic.
In this sense, (φ 1 , • • • , φ d ) provide a-harmonic coordinates for T L . We thus call the φ j 's the (periodic) corrector.

A crucial ingredient in the proof of our main result is the following boundedness estimate on the moments of the corrector:

Lemma 2.2. (Gloria, Neukamm & Otto, [6]
). For j = 1, . . . , d we have

|φ j | 2 ln L for d = 2 1 for d > 2 , (2.4
)

|∇φ j | 4 1.
(2.5)

The multiplicative constants in these estimates only depend on λ and d.

We remark that the previous estimate indeed holds for the more general class of stationary ensembles which satisfy SG. For the question in which sense lim L↑∞ φ j exists, which is however is not relevant for this paper, we refer to [6, Chapter 6].

Homogenized coefficient.

Let us recall that a coefficient field

a ∈ Ω := {a : Z d → Ω 0 } = Ω Z d 0
can be seen as a description of a network of resistors: Suppose u :

Z d → R is an a-harmonic function; if u(x
) is interpreted as the potential at vertex x and a kk (x) as conductivity along the edge joining x to e k + x, then a kk (x)∇ k u(x) can be interpreted as the (stationary) current along this edge. In this sense, the merit of the homogenized coefficient a hom is that, almost surely, it relates the spatially averaged potential gradient ξ = ∇u to the spatially averaged current a hom ξ = a∇u. For our special a-harmonic function u = φ j + x j , the spatially averaged (over one period cell) potential gradient is given by

L -d T L (∇φ j + e j ) = e j ,
whereas the spatially averaged current is given by

L -d T L a(∇φ j + e j ).
Hence the expected value of the latter, that is, where the multiplicative constant depends only on d and λ.

L -d T L a(∇φ j + e j ) stationarity = a(0)(∇φ j (0) + e j ) =: a hom,L e j ( 2
As opposed to the other important ingredients for our result, the annealed estimates on the corrector and on the Green's function in Lemma 2.2 and in Lemma 3.2 below, the above ingredient relies on the i. i. d. property in a subtle way. In fact, if we were not dealing with the i. i. d. ensemble • , but with a more general infinite ensemble (that we call • for the purpose of the discussion in this paragraph only), the choice of the periodic ensemble • L (that we endow with the index L for the purpose of this discussion only) is more subtleand of practical importance: On the one hand, in view of Lemma 2.2 and of Lemma 3.2 below in particular, • L should satisfy a Logarithmic Sobolev Inequality (LSI) uniformly in L ↑ ∞. On the other hand, in view of Lemma 2.3, • L should be well-coupled to • in order to avoid a large systematic error. If the ensemble is not i. i. d. , the definition of • L through a "brutal" periodization, which is obtained by restricting a(• + y) to T L , with a random shift y ∈ T L to retain stationarity, seems both unnatural and difficult to control. It seems more natural and promising to us to define • L as the distribution of a under • conditioned on the T L -periodicity of a. However, this conditioning is singular; and we only expect control if the ensemble • can be characterized by a sufficiently short-range (translation invariant) Hamiltonian. In this case, we expect that Dobrushin-Shlosman criteria (which as uniform mixing conditions ensure near-independence of a(x) and a(y) for |x -y| ≫ 1 for all conditional measures) conveniently provide LSI for • L uniform in L. The extension of Lemma 2.3 to such a situation is investigated in a forthcoming work, see [START_REF] Gloria | Approximation of effective coefficients by periodization in stochastic homogenization[END_REF]. Let us also point out that the present proof relies on the covariance estimate of Lemma 2.1. It is not yet clear to us whether such a covariance estimate can survive beyond the i. i. d. case.

Proofs of Theorem 1 and Corollary 1

In this section we present the proofs of Theorem 1 and Corollary 1. Beforehand, we recall some auxiliary estimates on the elliptic Green's function that we need in the proof. 9

Structure of the proof and auxiliary estimates on the Green's function. The starting point to prove Theorem 1 is the same as in the proof of [START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients[END_REF]Theorem 3] by Papanicolaou and Varadhan. Recall that z is given by (1.18). By uniform ellipticity and L-periodicity of a, an integration by parts on T L yields

λ T L |∇z| 2 ≤ T L ∇z • a∇z = T L z ∇ * a∇z . (3.1)
As we shall see below, in Step 1 of the proof of Theorem 1, an application of ∇ * a∇ to z yields the decomposition

∇ * a∇z = ∇ * g + r 1 + r 2 , (3.2
) where the random vector field g : T L → R d , the deterministic scalar field r 1 : T L → R, and the random scalar field r 2 : T L → R are given by

g i = - d j=1 a ii φ j (• + e i ) ∇ i ∇ j u 0 , r 1 = (a hom,L -a hom ) : ∇ * ∇u 0 , r 2 = (b -a hom,L ) : ∇ * ∇u 0 . (3.3)
Above, a hom,L is defined via (2.6), a hom is defined via (2.7), and b : T L → R d×d denotes the matrix field with entries

b ij := a ii (• -e i ) ∇ i φ j (• -e i ) + δ(i -j) for i, j = 1, . . . , d. (3.4)
As we shall see below (in Step 3 of the proof of Theorem 1), the scalar fields r 1 and r 2 satisfy The most intricate estimate is the one of T L z r 2 , see Step 5 in the proof of Theorem 1 below. Since the term r 2 (x) has vanishing expectation for all x ∈ T L one may write it as a covariance

x∈T L z(x)r 2 (x) = x∈T L cov [z(x); r 2 (x)] .
In order to benefit from this, we appeal to the covariance estimate of Lemma 2.1 and to a vertical derivative calculus on coefficient fields that we introduced in [6]. The process of estimating the vertical derivatives of r 2 and z involves the (periodic) Green's function:

|∇ x G L (x, y; a)|, |∇ y G L (x, y; a)| (|x -y mod L| + 1) 2-d-γ . (3.7)
In the estimate the multiplicative constant only depends on λ and d. (Note that G L is symmetric, so that the estimate (3.7) does not depend on the variable with respect to which we differentiate.)

See Appendix A for the proof. The crucial other ingredient is the recent annealed estimate of [START_REF] Marahrens | Annealed estimates on the Green's function[END_REF] by Marahrens and the third author, which we recall below in a version for the Lperiodic Green's functions:

Lemma 3.2. (Marahrens & Otto,[14, Theorem 1]
). The periodic Green's function G L satisfies the following annealed estimates:

|∇G L (x, y)| 4 1 4 (|x -y mod L| + 1) 1-d , (3.8)
and

|∇ x ∇ y G L (x, y)| 4 1 4 (|x -y mod L| + 1) -d , (3.9
) where the multiplicative constants only depend on λ and d.

Let us mention that the proof of the annealed estimates on the derivatives of the Green's functions relies on a strengthened version of the spectral gap estimate (2.2), namely a Logarithmic-Sobolev Inequality. We refer the reader to [START_REF] Marahrens | Annealed estimates on the Green's function[END_REF] for details. Note that as for the variance estimate by Naddaf and Spencer in [START_REF] Naddaf | Estimates on the variance of some homogenization problems[END_REF], an optimal control of the fourth moment is enough for our quantitative expansion.

Proof of Theorem 1. We prove the estimate of Theorem 1 in its rescaled formulation (1.20). Note that the identity

T L d i,j=1 |∇ * i ∇ j u 0 | 2 = T L d i,j=1 |∇ i ∇ j u 0 | 2 = T L |∇ 2 u 0 | 2 (3.10)
follows from periodicity and the elementary identity

|∇ * i ∇ j u 0 (•)| = |∇ i ∇ j u 0 (• -e i )|.
The argument is divided into five steps. In the first step we derive the decomposition (3.2) with property (3.5). In Steps 2 and 3 we argue that it suffices to prove

T L |∇z| 2 µ d (L) T L |∇ 2 u 0 | 2 . (3.11)
In the remaining steps we prove (3.11) starting with the inequality λ

T L |∇z| 2 ≤ T L z ∇ * a∇z .
To that end we appeal to the representation formula (3.2) which is a sum of a term in divergence form ∇ * g, a deterministic term r 1 and a remainder with vanishing expectation r 2 . The first two terms are estimated in Step 4. The third term r 2 , 11 which has vanishing expectation, is controlled using the covariance estimate of Lemma 2.1 and will be treated in Steps 5a-5d.

Step 1. Derivation of the decomposition (3.2) with property (3.5).

Let us show that (3.2) holds with g, r 1 and r 2 given by (3.3). By the defining equation (1.9) for u,

∇ * a∇z (1.18) = ∇ * a∇u -∇ * a∇u 0 - d j=1 ∇ * a∇(φ j ∇ j u 0 ) (1.9) = f -∇ * a∇u 0 =: I - d j=1 ∇ * a∇(φ j ∇ j u 0 ) =: II . (3.12)
We shall use the following discrete Leibniz rule: For all

ζ 1 , ζ 2 : T L → R, ∇ i (ζ 1 ζ 2 ) = (∇ i ζ 1 ) ζ 2 + ζ 1 (• + e i )∇ i ζ 2 , ∇ * i (ζ 1 ζ 2 ) = (∇ * i ζ 1 ) ζ 2 + ζ 1 (• -e i )∇ * i ζ 2 .
(3.13) For the first term this yields

I = d i=1 (∇ * i a ii ) ∇ i u 0 + d i=1 a ii (• -e i ) ∇ * i ∇ i u 0 ,
while for the second term we obtain

II = d i,j=1 ∇ * i a ii ∇ i (φ j ∇ j u 0 ) (3.13) = d i,j=1 ∇ * i a ii (∇ i φ j ∇ j u 0 + φ j (• + e i )∇ i ∇ j u 0 ) (3.13) = d i,j=1 ∇ * i ( a ii ∇ i φ j )∇ j u 0 + d i,j=1 a ii (• -e i )∇ i φ j (• -e i )∇ * i ∇ j u 0 + d i,j=1 ∇ * i a ii φ j (• + e i )∇ i ∇ j u 0 .
Because of the periodic corrector equation (2.3), the first term of the r. h. s. turns into

d i,j=1 ∇ * i ( a ii ∇ i φ j ) ∇ j u 0 = - d j=1 ∇ * j a jj ∇ j u 0 .
Hence, the terms in I + II that involve the first derivative of u 0 cancel, so that

I + II = d i,j=1 a ii (• -e i ) δ(i -j) + ∇ i φ j (• -e i ) ∇ * i ∇ j u 0 + d i,j=1 ∇ * i a ii φ j (• + e i )∇ i ∇ j u 0 . (3.14)
The last term on the r. h. s. is precisely -∇ * g, the first term is b : ∇ 2 u 0 , cf. (3.3) and (3.4). The claim (3.2) then follows from (3.12), (3.14) and identity (1.17) which can be written in the form f = a hom : ∇ * ∇u 0 since a hom is constant.

To conclude this step, we prove (3.5). The first identity simply follows from the Lperiodicity of u 0 . The second identity can be seen as follows: By the definition of b, the stationarity of a and φ j , and the definition of a hom,L via (2.6) we have

b ij (•) = a ii (0) δ(i -j) + ∇ i φ j (0) = a ij hom,L , so that b : ∇ 2 u 0 = b : ∇ 2 u 0 = a hom,L : ∇ 2 u 0 as desired.
Step 2. Reduction to an estimate for ∇z.

We claim that (1.20) (and thus the statement of Theorem 1, see Remark 1.2) follows from (3.11). Indeed, by the discrete Poincaré inequality

T L z 2 L 2 T L |∇z| 2 + L d z2 we only need to prove that z2 L 2-d µ d (L) T L |∇ 2 u 0 | 2 .
Since the spatial means of u and u 0 vanish by definition, we have z = -L -d

T L d j=1 φ j ∇ j u 0 , so that z2 = L -2d x∈T L x ′ ∈T L d i=1 φ i (x)∇ i u 0 (x) d j=1 φ j (x ′ )∇ j u 0 (x ′ )
.

We expand the square on the r. h. s.. Since u 0 is deterministic we get

z2 = L -2d x∈T L x ′ ∈T L d i,j=1 ∇ i u 0 (x)∇ j u 0 (x ′ ) φ i (x)φ j (x ′ ) .
By Cauchy-Schwarz' inequality, stationarity of the correctors, and the bounds of Lemma 2.2,

φ i (x)φ j (x ′ ) ≤ max k=1,...,d φ 2 k µ d (L).
Hence,

z2 µ d (L)   L -d x∈T L |∇u 0 |   2 .
The desired estimate then follows from Jensen's and Poincaré's inequalities.

Step 3. Reduction based on the decomposition (3.2).

In this step we argue that the desired estimate reduces to the following statement: Suppose that the functions g, r 1 and r 2 of the decompositon (3.2) satisfy the following estimates 13 in addition to (3.5):

T L |g| 2 µ d (L) T L |∇ 2 u 0 | 2 , (3.15)
T L r 2 1 ln d L L d T L |∇ 2 u 0 | 2 , (3.16)
T L z r 2 T L |∇z| 2 1 2   T L |∇ 2 u 0 | 2   1 2 + µ d (L) T L |∇ 2 u 0 | 2 , (3.17)
then estimate (1.20) (and thus Theorem 1) holds.

By

Step 2 we just have to check (3.11). Indeed, by combining (3.1), the decomposition (3.2), the triangle inequality, an integration by parts, and (3.5) we get

λ T L |∇z| 2 ≤ T L ∇z • g + T L (z -z)r 1 + T L z r 2 .
The first term is estimated by (3.15) and the Cauchy-Schwarz inequality:

T L ∇z • g µ 1 2 d (L) T L |∇z| 2 1 2   T L |∇ 2 u 0 | 2   1 2
.

The second term is estimated by (3.16), the Cauchy-Schwarz inequality, the Poincaré inequality for functions on T L with zero mean, and the elementary estimate

ln d L L d-1
1 for d > 1:

T L (z -z)r 1 T L |∇z| 2 1 2   T L |∇ 2 u 0 | 2   1 2
. The combination of (3.17) with the previous three inequalities yields

T L |∇z| 2 µ 1 2 d (L) T L |∇z| 2 1 2   T L |∇ 2 u 0 | 2   1 2 + µ d (L) T L |∇ 2 u 0 | 2 ,
which implies (3.11) by Young's inequality.

Step 4. Proof of the estimates (3.15) and (3.16).

Estimate (3.15) follows from the definition of g, cf. (3.3), the bound (2.4) of Lemma 2.2 on the second moment of the stationary φ j and identity (3.10). Similarly, (3.16) follows from the definition of r 1 and the optimal bound on the error |a hom,La hom | of Lemma 2.3.

In the last step we prove (3.17). Since this step is rather long, we subdivide it further.

Step 5a. Application of the covariance estimate.

Since by (3.5), r 2 (x) = 0 for all x ∈ T L , we have

x∈T L z(x)r 2 (x) = x∈T L cov [z(x); r 2 (x)] (3.3) = x∈T L d i,j=1 ∇ * i ∇ j u 0 (x) cov z(x); a ij hom,L -b ij (x) (3.10) ≤   T L |∇ 2 u 0 | 2   1 2   x∈T L d i,j=1 cov z(x); a ij hom,L -b ij (x) 2   1 2
.

With the covariance estimate in Lemma 2.1 the r. h. s. is bounded by

  T L |∇ 2 u 0 | 2   1 2    x∈T L d i,j=1   y∈T L ∂z(x) ∂y 2 1 2 ∂b ij (x) ∂y 2 1 2   2    1 2
.

Hence, for (3.17) it suffices to prove that

   x∈T L   y∈T L ∂z(x) ∂y 2 1 2 ∂b ij (x) ∂y 2 1 2   2    1 2 ≤ T L |∇z| 2 1 2 + µ d (L)   T L |∇ 2 u 0 | 2   1 2
for i, j = 1, . . . , d.

(

To estimate the vertical derivatives we need to identify ∂z ∂y and ∂b ij ∂y . This is done by appealing to the elliptic equations (1.16) and (2.3) and vertical differential calculus. Since the basic argument is simple, but polluted due to the discrete nature of the vertical and spatial derivatives, we will first present a formal calculation where the vertical derivative ∂ ∂y is replaced by the classical partial derivative ∂ ∂a kk (y) (defined for differentiable functions on Ω L ). The rigorous argument is then carried out in Step 5c and Step 5d below.

Step 5b. Formal derivation of formulas for the vertical derivatives.

We first (formally) identify ∂u ∂a kk (y) and ∂φ i (x) ∂a(x) . Applying ∂ ∂a kk (y) to the elliptic equations (1.16) and (2.3) yields

∇ * a(x)∇ ∂u(x) ∂a kk (y) = -∇ * ∂a(x) ∂a kk (y) ∇u(x), ∇ * a(x)∇ ∂φ j (x) ∂a kk (y) = -∇ * ∂a(x) ∂a kk (y) (∇φ j (x) + e j ), 15
using that ∂ ∂a kk (y) and ∇ commute. Since ∂a(x) ∂a kk (y) = (e k ⊗ e k )δ(x -y mod L) for all x, y ∈ T L , the Green representation formula yields

∂u(x) ∂a kk (y) = -∇ y k G L (x, y)∇ k u(y), ∂φ j (x) ∂a kk (y) = -∇ y k G L (x, y)(∇ k φ j (y) + δ(k -j)). (3.19)
Next we identify ∂b ij ∂a kk (y) . We apply ∂ ∂a kk (y) to the definition (3.4) of b ij and use the identity above in the form of

∂∇ i φ j (x) ∂a kk (y) = -∇ x i ∇ y k G L (x, y)(∇ k φ j (y) + δ(k -j)). (3.20)
Rearranging the terms yields the identity

∂b ij (x + e i ) ∂a kk (y) = δ(k -i)δ(x -y mod L)(∇ i φ j (y) + δ(i -j)) -a ii (x)∇ x i ∇ y k G L (x, y)(∇ k φ j (y) + δ(k -j)).
(3.21)

Likewise, for the identification of ∂z(x) ∂a kk (y) we apply ∂ ∂a kk (y) to (1.18):

∂z(x) ∂a kk (y) = ∂u(x) ∂a kk (y) - d j=1 ∂φ j (x) ∂a kk (y) ∇ j u 0 (x) (3.19) = -∇ y k G L (x, y)   ∇ k u(y) - d j=1 (∇ k φ j (y) + δ(k -j))∇ j u 0 (x)   .
Since we want to make ∇z appear, we substitute ∇u by the following expression ∇u(y) = ∇z(y)

+ d j=1
(∇φ j (y) + e j )∇ j u 0 (y) + d j=1 φ j (y)∇∇ j u 0 (y), (3.22) which can formally be obtained by applying ∇ to (1.18) and using the continuum Leibniz rule ∇(φ j ∇ j u 0 ) = ∇φ j ∇ j u 0 + φ j ∇∇ j u. We then get

∂z(x) ∂a kk (y) = -∇ y k G L (x, y) ∇ k z(y) + d j=1 (∇ k φ j (y) + δ(k -j))(∇ j u 0 (y) -∇ j u 0 (x)) + d j=1 φ j (y)∇ k ∇ j u 0 (y) . (3.23)
Let us stress the fact that the expression in the brackets on the r. h. s. is ∇z(y) plus terms that vanish if u 0 is affine. This will be crucial in order to obtain an optimal estimate.

Step 5c. Rigorous derivation of formulas for the vertical derivatives.

We now derive rigorous versions of (3.21) and (3.23), which will lead to the desired estimate (3.18). For the rigorous argument ∂ ∂a kk (y) has to be replaced by the discrete vertical derivative ∂ ∂y for which the Leibniz rule is not valid. The main ingredient is the following rigorous version of (3.19): For j = 1, . . . , d and x, y ∈ T L we have

∂φ j (x) ∂y = -∇ y G L (x, y) • [∇φ j (y) + e j ] y , (3.24)
∂∇ i φ j (x) ∂y = -∇ x i ∇ y G L (x, y) • [∇φ j (y) + e j ] y (3.25) ∂u(x) ∂y = -∇ y G L (x, y) • [∇u(y)] y , (3.26)
where [•] y denotes the commutator of the multiplication with a(y) and ∂ ∂y , i. e.

[F ] y := ∂(a(y)F ) ∂y a(y) ∂F ∂y = a(y) F ya(y)F y (3.27) for all random vectors F . By Jensen's inequality, the commutator satisfies the following estimate: For all 1 ≤ q < ∞, |[F ] y | q ≤ 2 q |F | q , (3.28) which we will use in the sequel for q = 2 and q = 4. Note that for all x, y ∈ T L , since the coefficients are i. i. d.,

∂(a(x)F ) ∂y -a(x) ∂F ∂y = [F ] y δ(x -y mod L). (3.29)
Here comes the argument for (3.24). By (2.3) we have 0 = ∇ * a(x)(∇φ j (x) + e j ) -∇ * a(x)(∇φ j (x) + e j ) y = ∇ * a(x)(∇φ j (x) + e j )a(x)(∇φ j (x) + e j ) y = ∇ * a(x) ∂∇φ j (x) ∂y + a(x) ∇φ j (x) + e j ya(x)(∇φ j (x) + e j ) y

Using then (3.27) and (3.29) for F = ∇φ j (x) + e j , this turns into

∇ * a(x)∇ ∂φ j (x) ∂y = -∇ * [∇φ j (x) + e j ] y δ(x -y mod L) ,
from which (3.24) follows by the Green representation formula and an integration by parts. Identity (3.25) follows from applying ∇ x to (3.24). The argument for (3.26) is similar to the one for (3.24) and left to the reader.

We are now in position to derive the rigorous versions of (3.21) and (3.23), and start with b ij . We claim that

∂b ij (x + e i ) ∂y = δ(x -y mod L)e i -a ii (x)∇ x i ∇ y G L (x, y) • [∇φ j (y) + e j ] y , (3.30)
Indeed, using again (3.27) for F = ∇φ j (x) + e j , we have ∂(a(x)(∇φ j (x) + e j )) ∂y = a(x)∇ ∂φ j (x) ∂y + a(x) ∇φ j (x) + e j ya(x)(∇φ j (x) + e j ) y = a(x) ∂∇φ j (x) ∂y + δ(x -y mod L)[∇φ j (y) + e j ] y , which, combined with (3.25), yields (3.30). 17

We then turn to z and claim that ∂z(x) ∂y

= -∇ y G L (x, y) • F 1 (y) + F 2 (y, x) + F 3 (y) , (3.31)
where

F 1 (y) := [∇z(y)] y , F 2 (y, x) := d j=1 ∇ j u 0 (y) -∇ j u 0 (x) [∇φ j (y) + e j ] y , F 3 (y) := d i,j=1 ∇ i ∇ j u 0 (y)[φ j (y + e i )e i ] y .
We first apply ∂ ∂y to (1.18) and use (3.24) and (3.26):

∂z(x) ∂y = -∇ y G L (x, y) • [∇u(y)] y - d j=1 ∇ j u 0 (x) [∇φ j (y) + e j ] y . (3.32)
We then wish to substitute the term ∇u(y) by an expression that involves ∇z(y). To that end we apply ∇ to (1. ∇ i ∇ j u 0 (y) φ j (y + e i )e i .

Combined with (3.32) the desired identity (3.31) follows.

Step 5d. Estimates of the vertical derivatives of b ij and z.

We claim that ∂b ij (x) ∂y

I 1 + I 2 , (3.34)
where

I 1 := (|y -x mod L| + 1) 2-d-γ |∇z(y)| 2 1/2 + µ d (L)|∇ 2 u 0 (y)| , I 2 := (|y -x mod L| + 1) 1-d |∇u 0 (x) -∇u 0 (y)|,
for some γ > 0 depending only on λ and d.

We start with (3.33). By the Cauchy-Schwarz inequality in probability, (3.30) turns into

|[L. H. S. of (3.33)]| δ(x-e i -y mod L) + |∇ x i ∇ y G L (x-e i , y)| 4 1/4 × |[∇φ j (y) + e j ] y | 4 1/4 .
The first term of the r. h. s. is estimated by the annealed estimate (3.9) of Lemma 3.2.

For the second term we appeal to (3.28) with q = 4, and to the bound (2.5) in Lemma 2.2 on the quartic moment of ∇φ j :

|[∇φ j (y) + e j ] y | 4 ≤ 2 4 ∇φ j (y) + e j 4 |∇φ j | 4 + 1 1. (3.35)
The desired estimate (3.33) follows.

We then turn to (3.34). Based on (3.31), we first bound the l. h. s. in (3.34) by the sum of three terms:

|[L. H. S. of (3.34)]| |∇ y G L (x, y)| 2 |F 1 (y)| 2 1/2 + |∇ y G L (x, y)| 2 |F 2 (y, x)| 2 1/2 + |∇ y G L (x, y)| 2 |F 3 (y)| 2 1/2 .
(3.36)

We estimate the first and third terms on the r. h. s. by appealing to the quenched estimate of Lemma 3.1 and Jensen's inequality in probability:

|∇ y G L (x, y)| 2 |F 1 (y)| 2 1/2 + |∇ y G L (x, y)| 2 |F 3 (y)| 2 1/2 (|x -y mod L| + 1) 2-d-γ |∇z(y)| 2 1/2 + max j=1,...,d φ 2 j 1/2 |∇ 2 u 0 (y)| .
Due to (2.4) in Lemma 2.2 the second moment of φ j is bounded by µ d (L), and thus we can control the r. h. s. by I 1 . It remains to estimate the second term in (3.36). By appealing to the definition of F 2 , the Cauchy-Schwarz inequality in probability, and (3.35) we have

|∇ y G L (x, y)| 2 |F 2 (y, x)| 2 1/2 |∇ y G L (x, y)| 4 1/4 |∇u 0 (x) -∇u 0 (y)|.
In view of the annealed estimate (3.8) of Lemma 3.2 the r. h. s. is controlled by I 2 as desired.

Step 5e. Proof of (3.18) and thus (3.17

 x∈T L   y∈T L (|y -x mod L| + 1) 2(1-d)-γ |∇z(y)| 2 1/2 + µ d (L)|∇ 2 u 0 (y)|   2   1/2 +   x∈T L   y∈T L (|y -x mod L| + 1) 1-2d |∇u 0 (x) -∇u 0 (y)|   2   1/2 .
Since γ > 0, the discrete convolution kernel x → (|x mod L|+1) 2(1-d)-γ has ℓ 1 (T L )-norm bounded independently of L (even for d = 2), i. e. T L (|x mod L| + 1) 2(1-d)-γ 1. Hence, by the convolution estimate w. r. t. the ℓ 2 (T L )-norm, the first term on the r. h. s. is controlled by the r. h. s. of (3.18). It remains to treat the second sum and suffices to 19 Step 1. Quenched estimate on G L for d > 2.

show that   x∈T L   y∈T L (|y -x mod L| + 1) 1-2d |∇u 0 (x) -∇u 0 (y)|   2   1/2 µ d (L)   T L |∇ 2 u 0 | 2   1/2 . ( 3
Let G ′ L (t, y) denote the periodic, parabolic Green's function considered in [6, Definition 2]; it is characterized as follows: For all x ∈ Z d the function

G ′ L is the unique C ∞ (R, ℓ ∞ (Z d )) solution of ∂ t G ′ L (t, x) -∇ • a∇G ′ L (t, x) = 0, G ′
L (0, x) = δ(x mod L).

(A.4)

Since both the initial data and the coefficient field are L-periodic, G ′ L is in fact of class C ∞ (R, ℓ 1 (T L )). The elliptic Green's function G L can be formally obtained by integrating in time the mean free version of the parabolic Green's function:

G L (x) = ˆ∞ 0 (G ′ L (t, x) -L -d )dt.
To turn this into a rigorous argument one needs to prove that t → G ′ L (t, x) -L -d ∈ L 1 (0, +∞). This is a direct consequence of [6, eq. (219) in the proof of the discrete version of Lemma 24]: For all α > 0 we have Step 2. Quenched estimate on ∇G L for d > 2.

0 ≤ G ′ L (t
We now deduce (3.7) from (A.5) for d > 2. Recall that it suffices to prove the estimate for y = 0 and that we write G L (x) = G L (x, 0) for brevity. Fix a radius 2 ≤ R ∼ 1 ≪ L. For |x mod L| ≤ 2R the r. h. s. of (3.7) is of order 1 so that the estimate directly follows from the combination of (A.

Lemma 2 . 1 .

 21 Let • denote the periodic i. i. d. ensemble (see (1.10)). Then we have for any two random variables ζ, ζ:

. 1 )

 1 Here cov ζ; ζ := (ζ -ζ )( ζζ ) denotes the covariance of ζ and ζ. Furthermore, for a site y ∈ T L , the random variable ∂ζ ∂y is defined by ( ∂ζ ∂y )(a) := ζ(a) -ˆΩ0 ζ(a) β(da(y)).

T L r 1 = 0 and r 2

 12 (x) = 0 for all x ∈ T L . (3.5) By combining (3.1) and (3.2), the bound (1.20) follows from estimates of the terms T L ∇z • g , T L (z -z) r 1 and T L z r 2 , an estimate of the spatial mean z := L -d T L z, and the discrete Poincaré inequality on the torus.

Definition 3 . 1 .Lemma 3 . 1 .

 3131 The L-periodic Green's function G L : T L × T L × Ω L → R is defined as follows. For all y ∈ T L and a ∈ Ω L the function G L (•, y; a) is the unique L-periodic mean free solution to∇ * a∇G L (•, y; a) = δ(• -y) -L -d on T L , (3.6) 10where δ is the Dirac mass at zero.When no confusion occurs we use the shorthand notation G L (x, y) for G L (x, y; a). We shall use both quenched (i. e. pointwise deterministic) and annealed (i. e. statistically averaged) estimates on |∇G L |. The pointwise estimates rely on the De Giorgi-Nash-Moser Hölder regularity theory (and are standard in the continuum case): There exists γ > 0 depending only on λ and d such that for all a ∈ Ω L and L ∈ N, the Green's function G L (•, •; a) satisfies the following quenched estimate:

  18) and get with the help of the discrete Leibniz rule (3.13) ∇u(y) = ∇z(y) + ∇u 0 (y) + d j=1 ∇ j u 0 (y)∇φ j (y) + d i,j=1 ∇ i ∇ j u 0 (y) φ j (y + e i )e i = ∇z(y) + d j=1 ∇ j u 0 (y)(∇φ j (y) + e j ) + d i,j=1

2 1/ 2 (

 2 |y -x mod L| + 1)

  ). By combining (3.34) and (3.33) we get [L. H. S. of (3.18)] 

2 , 2 . 3 . 1 .A. 2 .

 22312 .37) By the definition of the discrete gradient and periodicity we have for i = 1, . . . , d   x∈T L |∇u 0 (x + e i ) -∇u 0 (x)| 2which combined with the triangle inequality and periodicity yields  x∈T L |∇u 0 (x + z) -∇u 0 (x)| 2 all Z d .We then use the triangle inequality in the form of  L| + 1) 2(1-d) |∇u 0 (x) -∇u 0 (x + z)| (|z mod L| + 1) mod L| + 1) 4(1-d) x∈T L |∇u 0 (x) -∇u 0 (x + z)| 2 (|z mod L| + 1)Evaluating the sum in z on the r. h. s. yields the claimed estimate (3.37), recalling that µ d (L) = ln L for d = 2 and µ d (L) = 1 for d > 2. This proves (3.18), and therefore the desired estimate (3.17) by Step 5a. Proof of Corollary 1. Estimate (1.14) is a direct consequence of (1.13) and the estimate max j=1,...,d |φ j | 2 ln L for d = 2 1 for d > 2 of Lemma 2.2.20 Proof of Lemma 3.1. It suffices to consider the case y = 0, since G L (x, y; a) = G L (x -y, 0; a(• -y)) and due to the fact that the asserted estimate depends on a only through its ellipticity constant λ. We shall write G L (x) := G L (x, 0; a) for brevity. We divide the proof into three steps. First we derive quenched estimates on the Green's function for d > 2 from the quenched estimates of [6, Lemma 24 and (219)] on the corresponding parabolic Green's functions. In the second step we prove the desired estimate for d > 2 using the De Giorgi-Nash-Moser Hölder estimate, see[START_REF] Mosconi | Discrete regularity for elliptic equations on graphs[END_REF] Theorem 5.2] in the discrete setting. We then deduce the estimate for d = 2 from the estimate for d = 3 using an argument by Avellaneda and Lin, see [2, Proof ofTheorem 13].

5 )

 5 with the discrete estimate |∇ i G L (x)| ≤ |G L (x + e i )| + |G L (x)|.Hence, it suffices to consider the case |x mod L| ≥ 2R. For all y ′ ∈ T L and r > 0, set B r (y ′ ) = {y ′′ ∈ T L , |y ′ -y ′′ mod L| ≤ r}. Using the elementary inequality|∇G L (x)| ≤ √ d osc B 1 (x) G L ,this turns into an estimate of the oscillation of G L on B 1 (x). Noting that G L satisfies∇ * a∇G L = L -d (A.6)

  ••• ,d , the function ∂ζ ∂y measures how sensitively ζ depends on the variable a(y) = {a kk (y)} k=1,••• ,d ∈ Ω 0 . For reasons explained in [6] we call these derivatives vertical. For obvious reasons, ∂ζ ∂y is called the discrete vertical derivative of ζ at y. We now introduce the important concept of the corrector. Since we only have to deal with the periodic (as opposed to the infinite) ensemble • , we can avoid discussing all technicalities. Indeed, for any realization of a according to • , that is, for any periodic coefficient field a ∈ Ω L and for any coordinate direction j = 1, • • • , d, there exists a unique scalar field φ j (a, •) :

	We note that for ζ = ζ, (2.1) turns into the Spectral Gap Estimate (SG) with constant 1,
	i. e.
	ζ 2 ≤	y∈T L	(	∂ζ ∂y	) 2	(2.2)
	for any random variable ζ with vanishing expectation ζ = 0. We refer to [6, Lemma 7]
	for a proof in the present context and note that (2.2) is extensively used in the proof of
	Lemma 2.2 as only probabilistic ingredient. This estimate is reminiscent of the Brascamp-

Lieb inequality used by Naddaf and Spencer in

[START_REF] Naddaf | Estimates on the variance of some homogenization problems[END_REF]

. In a slightly different context, a covariance estimate like (2.1) was established in [10,

Lemma 3]

. For the convenience of the reader, we present the elementary proof of Lemma 2.1. 7 2.2. Corrector.

  Theorem 4.6]. For us, this has the convenient side-effect that we don't have to give the technically more demanding, independent definition of a hom . For the latter, we refer to [6, Introduction] for instance. Moreover -and this is a second important ingredient for our result -the quantitative theory in [6] provides an optimal estimate of this "systematic error" in the case of the (infinite) i. i. d. ensemble.

			.6)
	is a good (and computable) proxy for a hom as L ↑ ∞. Qualitative homogenization theory ensures that it is indeed true that
	lim L↑∞	a hom,L = a hom ,	(2.7)
	see for instance [17,

Lemma 2.3. (Gloria, Neukamm & Otto, [6, Proposition 5]). Consider the i. i. d. ensemble. Then we have |a hom,La hom | L -d ln d L,

Appendix A. Proofs of the auxiliary lemmas A.1. Proof of Lemma 2.1. We adapt the arguments of [6, Proof of Lemma 7], as for the proof of [START_REF] Gloria | An optimal error estimate in stochastic homogenization of discrete elliptic equations[END_REF]Lemma 3] starting from [START_REF] Gloria | An optimal variance estimate in stochastic homogenization of discrete elliptic equations[END_REF]Lemma 2.3]. We first introduce a couple of notations: Let {y n } n=1,...,N , N := L d , be an enumeration of T L . For n = 1, • • • , N define the average we assume that both ζ and ζ have zero expectation. We split the proof into two steps.

Step 1. Martingale decomposition.

We claim that

Here comes the argument: Since

The identity (A.1) then follows from taking the expectation of (A.2) provided we prove that

This last identity can be seen as follows: By definition we have

Since ζ n does not depend on y 1 , . . . , y n , we have

Integrating both sides w. r. Step 2. Conclusion.

From [6, Step 2, Proof of Lemma 7], since • is an L-periodic i. i. d. measure, for all n ∈ {1, . . . , N } we have

Hence the claim follows from (A.1) and Cauchy-Schwarz' inequality.

21

in the set B |x mod L|/2 (x) (which does not contain 0 mod L), one may appeal to the De Giorgi-Nash-Moser Hölder estimate. In particular, by [15, Theorem 5.2], there exists 0 < γ < 1 depending only on λ and d such that

Combined with (A.5), this yields (3.7) for d > 2.

Step 3. Quenched estimate on ∇G L for d = 2.

Following [2, Proof of Theorem 13] we derive the estimates for d = 2 from the estimate for d = 3. To distinguish quantities in different dimensions we use the superscripts (2) and (3) ; e.g. T

L denotes the 2-dimensional torus. To a given two-dimensional, L-periodic coefficient field a (2) ∈ Ω

L we associate a three-dimensional coefficient-field a (3) ∈ Ω

In the following we use the shorthand notation G

(2)

L (x, 0; a (2) ) and G

L (x, x 3 ) := G

L ((x, x 3 ), (0, 0); a (3) ). It is elementary to check that G

L (x, x 3 ), and thus

L (x, x 3 ) for i = 1, 2.

The quenched estimate (3.7) for d = 3 then turns into

L (x)| ≤