Moment bounds for the corrector in stochastic homogenization of a percolation model

We study the corrector equation in stochastic homogenization for a simplified Bernoulli percolation model on $\mathbb{Z}^d$, $d>2$. The model is obtained from the classical $\{0,1\}$-Bernoulli bond percolation by conditioning all bonds parallel to the first coordinate direction to be open. As a main result we prove (in fact for a slightly more general model) that stationary correctors exist and that all finite moments of the corrector are bounded. This extends a previous result in [GO1], where uniformly elliptic conductances are treated, to the degenerate case. With regard to the associated random conductance model, we obtain as a side result that the corrector not only grows sublinearly, but slower than any polynomial rate. Our argument combines a quantification of ergodicity by means of a Spectral Gap on Glauber dynamics with regularity estimates on the gradient of the elliptic Green's function.


Introduction
We consider the lattice graph (Z d , B d ), d > 2, where B d denotes the set of nearest-neighbor edges. Given a stationary and ergodic probability measure · on Ω -the space of conductance fields a : B d → [0, 1] -we study the corrector equation from stochastic homogenization, i.e. the elliptic difference equation ∇ * (a(∇φ + e)) = 0, Here, ∇ and ∇ * denote discrete versions of the continuum gradient and (negative) divergence, cf. Section 2, and e ∈ R d denotes a vector of unit length, which is fixed throughout the paper. The corrector equation (1) emerges in the homogenization of discrete elliptic equations with random coefficients: For random conductances that are stationary and ergodic (with respect to the shifts a(·) → a(· + z), z ∈ Z d , cf. Section 2), and under the assumption of uniform ellipticity (i.e. there exists λ 0 > 0 such that a ≥ λ 0 on B d almost surely), a classical result from stochastic homogenization (e. g. see [Koz87,Kue83]) shows that the effective behavior of ∇ * a∇ on large length scales is captured by the homogenized elliptic operator ∇ * a hom ∇ where a hom is a deterministic, symmetric and positive definite d × d matrix. It is characterized by the minimization problem e · a hom e = inf ϕ (e + ∇ϕ) · a(e + ∇ϕ) , where the infimum is taken over random fields ϕ that are · -stationary in the sense of ϕ(a, x + z) = ϕ(a(· + z), x) for all x, z ∈ Z d and · -almost every a ∈ Ω. Minimizers to (2) are called stationary correctors and are characterized as the stationary solutions to the corrector equation (1). Due to the lack of a Poincaré inequality for ∇ on the infinite dimensional space of stationary random fields, the elliptic operator ∇ * a∇ is highly degenerate and the minimum in (2) may not be obtained in general. In fact, it is known to fail generally for d = 2. The only existence result of a stationary corrector (in dimensions d > 2) has been obtained recently in [GO11] by Gloria and the third author under the assumption that the a's are uniformly elliptic, and that · satisfies a Spectral Gap Estimate, which is in particular the case for independent and identically distributed coefficients. They also show that |φ| p 1 for all p < ∞.
The goal of the present paper is to extend this result to the case of conductances with degenerate ellipticity. To be definite, consider the probability measure · λ constructed by the following procedure: Take the classical {0, 1}-Bernoulli-bond percolation on B d with parameter λ ∈ (0, 1] and declare all bonds parallel to the coordinate direction e 1 to be open. (3) (We adapt the convention to call a bond "open" if the associated coefficient is "1", while a bond is "closed" if the associated coefficient is "0". The parameter λ denotes the probability that a bond is "open"). As for d-dimensional Bernoulli percolation, · λ describes a random graph of open bonds, which is locally disconnected with positive probability. However, as a merit of the modification, any two vertices in the random graph are almost surely connected by some open path. As a main result we show that (1) admits a stationary solution, all finite moments of which are bounded: Theorem (main result). Let d > 2 and λ ∈ (0, 1]. There exits φ : Ω × Z d → R such that for · λ almost every a ∈ Ω we have • φ(a, ·) solves (1), • φ(a, · + z) = φ(a(· + z), ·) for all z ∈ Z d , and ∀p < ∞ : |φ| p 1 p λ ≤ C. Here C denotes a constant that only depends on p, λ and d.
The modified Bernoulli percolation model · λ fits into a slightly more general framework that we introduce in Section 2 below, cf. Lemma 1. The result above will then follow as a special case of Theorem 1 stated below.
Relation to stochastic homogenization. Consider the decaying solution u ε : Z d → R to the equation where f : R d → R is smooth, compactly supported, and a is distributed according to · λ . Classical results of stochastic homogenization (see [PV79], [Koz79], [Kue83]) show that for almost every a ∈ Ω the (piecewise constant interpolation of the) rescaled function u ε ( · ε ) converges as ε ↓ 0 to the unique decaying solution u hom : R d → R of the deterministic elliptic equation Moreover, a formal two-scale expansion suggests that where φ j denotes the (stationary) solution of (1) for e = e j -the jth coordinate direction. The question how to quantify the errors emerging in this limiting process is rather subtle. Note that in the case of deterministic periodic homogenization, the good compactness properties of the d-dimensional "reference cell of periodicity" yield a natural starting point for estimates. In contrast, in the stochastic case the reference cell has to be replaced by the probability space (Ω, · ), which has infinite dimensions and thus most "periodic technologies" break down. Nevertheless, estimates for the homogenization error u ε − u hom and related quantities have been obtained by [Yur76,CI03,BP04,CS13], see also [CS10,AS] for recent results on fully nonlinear elliptic equations or equations in non-divergence form.
While the asymptotic result of stochastic homogenization holds for general stationary and ergodic coefficients (at least in the uniformly elliptic case), the derivation of error estimates requires a quantification of ergodicity. In a series of papers (see [GNO,GNO14,GO11,GO12]) two of the authors and Gloria developed a quantitative theory for the corrector equation (1) (and regularized versions) based on the assumption that the underlying statistics satisfies a Spectral Gap Estimate (SG) for a Glauber dynamics on the coefficient fields. This assumption is satisfied e.g. in the case of independent and identically distributed (i. i. d. coefficients). In [GO11,GNO] moment bounds for the corrector, similar to the one in the present paper, have been obtained. These bounds are at the basis of various optimal estimates; e.g. [GNO] contains a complete and optimal analysis of the approximation of a hom via periodic representative volume elements, and [GNO14] establishes optimal estimates for the homogenization error and the expansion in (4). While in the works mentioned above it is always assumed that the coefficients are uniformly elliptic, i.e. a ∈ [λ 0 , 1] B d for some fixed λ 0 > 0, in the present paper we derive moment bounds for a model with degenerate elliptic coefficients. As in [GO11,GNO], a crucial element of our approach is an estimate on the gradient of the elliptic Green's function associated with ∇ * a∇. The required estimate is pointwise in a, but (dyadically) averaged in space, and obtained by a self-contained and short argument, see Proposition 1 below. It extends the argument in [GO11] to the degenerate elliptic case. Since in the degenerate case the elementary inequality λ 0 |∇u| 2 ≤ ∇u · a∇u breaks down, we replace it by a weighted, integrated version (see Lemma 4 below). Compared to more sophisticated methods that e.g. rely on isoperimetric properties of the graph, an advantage of our approach is that it only invokes simple geometrical properties, namely spatial averages (on balls) of the inverse of the chemical distance between nearest neighbor vertices. We believe that our approach extends (although not in a straight-forward manner) to the case of standard supercritical Bernoulli percolation. This is a question that we study in a work in progress.
Connection to random walks in random environments (RWRE). Although, the main motivation of our work is quantitative homogenization, we would like to comment on the connection to invariance principles for (RWRE). In fact, there is a strong link between stochastic homogenization and (RWRE): The operator ∇ * (a∇) generates a stochastic process, namely the variable-speed random walk X = (X a (t)) t≥0 , which is a continuous-time random walk in the random environment a. In the early work [KV86] (see also [Kue83]) the authors considered general stationary and ergodic environments. For uniformly elliptic coefficients they prove an annealed invariance principal for X, saying that the law of the rescaled process √ εX a (ε −1 t) weakly converges to that of a Brownian motion with covariance matrix 2a hom . In [SS04] Sidoravicius and Sznitman prove a stronger quenched invariance principle for X, which says that the convergence even holds for almost every environment a. More recently, invariance principles have been obtained for more general environments, see [Bis11] and [Kum14] for recent surveys in this direction. Most prominently, supercritical bond percolation on Z d has been considered: Here, the annealed result is due to [DFGW89], while quenched results have been obtained in [SS04] for d ≥ 4 and in [BB07,MP07] for d ≥ 2. See also [ABDH, ADS] for recent related results on degenerate elliptic, possibly unbounded conductances.
The main difficulty in proving a quenched invariance principle compared to the annealed version is to establish a quenched sublinear growth property (see (8) below) for a corrector field χ. The latter is closely related to the function φ considered in Theorem 1, see the discussion below Corollary 1 for more details. In the uniformly elliptic case, sublinearity of χ is obtained by soft arguments from ergodic theory combined with a Sobolev embedding, see [SS04]. For supercritical Bernoulli percolation the argument is more subtle: For d ≥ 3 the proofs in [SS04, BB07, MP07] use heat-kernel upper bounds (as deduced by Barlow [Bar04]) or other "heatkernel technologies" (e.g. see [BP07,ABDH,ADS]) that require a detailed understanding of the geometry of the percolation cluster, and thus require the use of sophisticated arguments from percolation theory (e.g. isoperimetry, regular volume growth and comparison of chemical and Euclidean distances). Conceptually, the use of such fine arguments seems not to be necessary in the derivation of quenched invariance principles. Motivated by this in [BP07] and [ADS] different methods are employed with a reduced usage of heat-kernel technology. Our approach yields, as a side-result, an alternative way to achieve this goal: The quenched sublinear growth property can easily be obtained from the moment bound derived in Theorem 1. In fact, the estimate of Theorem 1 is stronger: As we explain in the discussion following Corollary 1, our moment bounds imply that the growth of χ is not only sublinear, but slower than any rate, see (9). Of course, the environment considered in the present paper, namely the modified percolation model · λ , is much simpler than supercritical Bernoulli percolation. Nevertheless, it shares some of the "degeneracies" featured by percolation; e.g. for every ball B ⊂ Z d with finite radius Poincaré's inequality x∈B u 2 (x) ≤ C(a, B) b∈B a(b)|∇u(b)| 2 fails with positive probability. Furthermore, in contrast to the above mentioned results, our argument requires only mild estimates on the Green's function. More precisely, as already mentioned, we require an estimate on the gradient of the elliptic Green's function, which -in contrast to quenched heat kernel estimates -can be obtained by fairly simple arguments, see Proposition 1. Of course, as it is well-known, estimates on the gradient of the elliptic Green's function can also be obtained from estimates on the associated heat kernel by an integration in time, and a subsequent application of Caccioppoli's inequality. In particular, heat-kernel estimates in the spirit of the one obtained by Barlow in the case of supercritical Bernoulli percolation, see [Bar04, Theorem 1], would be sufficient to make this program work. Yet, since the elliptic estimates that we require are less sensitive to the geometry of the graph, and thus can be obtained by simpler arguments, we opt for a self-contained proof that only relies on elliptic regularity theory. Another interesting, and -as we believe -advantageous property of our approach is that (thanks to the Spectral Gap Estimate) probabilistic and deterministic considerations are well separated, e.g. Proposition 1 is pointwise in a and does not involve the ensemble.
Structure of the paper. In Section 2 we gather basic definitions and introduce the slightly more general framework studied in this paper. We then present the main result in the general framework. Section 3.1 is devoted to the proof of the main result: we first discuss the general strategy of the proof and present several auxiliary lemmas needed for the proof of the main theorem -in particular, the coercivity estimates, see Lemmas 4 and 5, and an estimate for the gradient of the elliptic Green's function, see Proposition 1, which play a key role in our argument. The proof of the main result is given at the end of Section 3.1, while the auxiliary results are proven in Section 4.
Throughout this article, we use the following notation, see Section 2 for more details: • d is the dimension; • Z d is the integer lattice; • (e 1 , . . . , e d ) is the canonical basis of Z d ; • e ∈ R d , which appears in (1), denotes a vector of unit length and is fixed throughout the paper; • |A| denotes the number of elements in A ⊂ Z d (resp. A ⊂ B d ).

General framework
In the first part of this section, we introduce the general framework following the presentation of [GNO]: We introduce a discrete differential calculus, the random conductance model, and finally recall the standard definitions of the corrector and the modified corrector.

Lattice and discrete differential calculus
We consider the lattice graph ( . . , d } denotes the set of nearest-neighbor bonds. We write ℓ p (Z d ) and ℓ p (B d ), 1 ≤ p ≤ ∞, for the usual spaces of p-summable (resp. bounded for p = ∞) functions on Z d and B d . For u : Here We denote by ∇ * the adjoined of ∇, so that we have for F : Furthermore, the discrete integration by parts formula reads and holds whenever the sums converge.

Random conductance field
To each bond b ∈ B d a conductance a(b) ∈ [0, 1] is attached. Hence, a configuration of the lattice is described by a conductance field a ∈ Ω, where Ω := [0, 1] B d denotes the configuration space. Given a ∈ Ω we define the chemical distance between vertices x, y ∈ Z d by We equip Ω with the product topology (induced by [0, 1] ⊂ R) and the usual product σ-algebra, and describe random configurations by means of a probability measure on Ω, called the ensemble.
The associated expectation is denoted by · .
Our assumptions on · are the following: There exists a modulus of integrability Λ : [1, ∞) → [0, ∞) such that the distance of neighbors is finite on average in the sense that There exists a constant ρ > 0 such that for all ζ ∈ L 2 (Ω) we have where ∂ζ ∂b denotes the vertical derivative as defined in Definition 1 below. For technical reasons we need to strengthen (A2): where a e i ,0 denotes the conductance field obtained by "deleting" the bond {0, e i } (i. e. Let us comment on these properties. A minimal requirement needed for qualitative stochastic homogenization in the uniformly elliptic case is stationarity and ergodicity of the ensemble. The basic example for such an ensemble are i. i. d. coefficients which means that · is a B dfold product of a "single edge" probability measure on [0, 1]. The assumption (A3) is weaker than assuming i. i. d., but stronger than ergodicity. Indeed, in [GNO] it is shown that any i. i. d. ensemble satisfies (A3) with constant ρ = 1. Moreover, it is shown that (A3) can be seen as a quantification of ergodicity. From the functional analytic point of view the spectral gap estimate is a Poincaré inequality where the derivative is taken in vertical direction, see below. (The terminology "vertical" versus "horizontal" is motivated from viewing a ∈ Ω as a "height"-function defined on the "horizontal" plane B d ). We recall from [GNO] the definition of the vertical derivative: Property (A2) is a crucial assumption on the connectedness of the graph. In particular it implies that almost surely every pair of vertices can be connected by a path with finite intrinsic length. However, (A2) and (A2+) do not exclude configurations with coefficients that vanish with non-zero probability, as it is the case for · λ -the model considered in the introduction: Proof. Evidently, · λ can be written as the (infinite) product of probability measures attached to the bonds in B d . These "single-bond" probability measures only depend on the direction of the bond. Hence, · λ is stationary. Another consequence of the product structure is that · λ satisfies (A3) with constant ρ = 1 (see [GNO,Lemma 7] for the argument). It remains to check (A2+). By stationarity and symmetry we may assume that e i = e d . Consider the (random) set Clearly, each j ∈ L(a) yields an open path connecting 0 and e d , for instance the "U-shaped" path through the sites 0, je 1 , je 1 + e d and e d . Hence, dist a e d ,0 (0, e d ) ≤ 2dist(0, L(a)) + 1 almost surely, where dist(0, L(a)) := min j∈L(a) |j|. Consequently, it suffices to prove that (2dist(0, L(a)) + 1) p 1 p λ < ∞ for any p ≥ 1. Note that due to the definition a e d ,0 ({0, e d }) = 0 and thus dist(0, L(a)) ∈ N. Hence, The sum on the right-hand side converges, since 0 < λ ≤ 1 by assumption. This completes the proof.

Main result
We are interested in stationary solutions to the corrector equation (1). Note that we tacitly identify the vector e ∈ R d with the translation invariant vector field e(b) := e · (y b − x b ). For conciseness we write for all x, z ∈ Z d and · -almost every a ∈ Ω for the space of stationary random fields. Thanks to (A1) the expectation ϕ = ϕ(·, x) of a stationary random variable does not depend on x. Therefore, ϕ L 2 (Ω) := |ϕ| 2 1 2 defines a norm on (S, · L 2 (Ω) ).
• The conductances a may vanish with positive probability. Therefore, following [PV79], we regularize the equation by adding a 0th order term and consider for T > 0 the modified corrector equation Thanks to the regularization, (6) admits (for all T > 0) a unique solution in (S, · L 2 (Ω) ) as follows from Riesz' representation theorem.
We think about the modified corrector as an approximation for the stationary corrector and hope to recover a solution to (1) in the limit T ↑ ∞. This is possible as soon as we have estimates on (some) moments of φ T that are uniform in T -this is the main result of the paper: Theorem 1 (Moment bounds for the modified corrector). Let d > 2 and · satisfy Assumption 1 for some ρ and Λ. Let φ T denote the modified corrector as defined in Definition 2. Then for all T > 0 and 1 ≤ p < ∞ we have Here means ≤ up to a constant that only depends on p, Λ, ρ, and d.
Since the estimate in Theorem 1 is uniform in T we get as a corollary: Corollary 1. Let d > 2 and · satisfy Assumption 1 for some ρ and Λ. Then the corrector equation (1) has a unique stationary solution φ ∈ (S, · L 2 (Ω ) with φ = 0. Moreover, we have Here means ≤ up to a constant that only depends on p, Λ, ρ and d.
As mentioned in the introduction the corrector can be used to establish invariance principles for random walks in random environments. Suppose that · satisfies Assumption 1 for some ρ and Λ. Then, thanks to Corollary 1, for each coordinate direction e k there exist stationary correctors φ k ∈ (S, · L 2 (Ω ) with φ k = 0 that solve (1) with e = e k . Hence, we can consider the random vector field χ = (χ 1 , . . . , ) is a-harmonic, has finite second moments, and is shift covariant (i.e. χ(a, x + y) − χ(a, x) = χ(a(· + y), x). The field χ is precisely the "corrector" used e.g. in [KV86,SS04] to introduce harmonic coordinates for which the random walk in the random environment is a martingale. In particular, in [SS04] Sidovaricius and Sznitman use χ to prove a quenched invariance principle for the random walk in a random environment. A key step in their argument is to show that χ has sublinear growth, i.e.
This property has been established for supercritical bond percolation on Z d in dimension d ≥ 4 in [SS04] and for d ≥ 2 in [BB07,MP07]. The moment bounds established in our work (under the more restrictive Assumption 1) are stronger. Indeed, from Theorem 1 we get (8) in a stronger form by the following simple argument: For every θ ∈ (0, 1), p > d 1−θ and k = 1, . . . , d we have Hence, since φ k is stationary and p > d 1−θ , the maximal function estimate yields With the moment bounds of Corollary 1 we get for · satisfying Assumption 1:

Outline and Proof of Theorem 1
The proof of Theorem 1 is inspired by the approach in [GO11] where uniformly elliptic conductances are treated. The starting point of our argument is the following p-version of the Spectral Gap Estimate (A3), which we recall from [GNO, Lemma 2]: Lemma 2 (p-version of (SG)). Let · satisfy (A3) with constant ρ > 0. Then for p ∈ N and all ζ ∈ L 2p (Ω) with ζ = 0 we have where means ≤ up to a constant that only depends on p, ρ and d.
Applied to ζ = φ T (x = 0), this estimate yields a bound on stochastic moments of φ T in terms of the vertical derivatives ∂φ T (x=0) ∂b , b ∈ B d (see Definition 1). Heuristically, we expect the vertical derivative ∂φ T (x=0) ∂b to behave as the classical partial derivative ∂φ T (x=0) ∂a(b) . As we shall see, the latter admits the Green's function representation Here G T denotes the Green's function associated with ( 1 T + ∇ * a∇) and is defined as follows: Definition 3. For T > 0 the Green's function G T : Ω × Z d × Z d → R is defined as follows: For each a ∈ Ω and y ∈ Z d the function x → G T (a, x, y) is the unique solution in ℓ 2 (Z d ) to For uniformly elliptic conductances we have ∂φ T (x=0) up to a constant that only depends on the ratio of ellipticity. In the case of degenerate ellipticity this is no longer true. However, the discrepancy between the vertical and classical partial derivative of φ T can be quantified in terms of weights defined as follows: We introduce the weight function ω : For b ∈ B d and a ∈ Ω we denote by a b,0 the conductance field obtained by "deleting" the bond b (i. e. a b,0 (b ′ ) = a(b ′ ) for all b ′ = b and a b,0 (b) = 0), and introduce the modified weight ω 0 as Lemma 3. Assume that · satisfies (A1) and (A2+). For T > 0 let φ T denote the modified corrector. Then for all b ∈ B d we have Here means ≤ up to a constant that only depends on d.
To benefit from (10) (in the form of Lemma 3) we require an estimate on the gradient of the Green's function. As it is well known, the constant coefficient Green's function G 0 T (x) := G T (a = 1, x, 0) (which is associated with the modified Laplacian 1 T +∇ * ∇) satisfies the pointwise estimate We require an estimate that captures the same decay in x. It is known from the continuum, uniformly elliptic case, that such an estimate cannot hold pointwise in x and at the same time pointwise in a. In [GO11, Lemma 2.9], for uniformly elliptic conductances, a spatially averaged version of (14) is established, where the averages are taken over dyadic annuli. The constant in this estimate depends on the conductances only through their contrast of ellipticity. In the degenerate elliptic case, the ellipticity contrast is infinite. In order to keep the optimal decay in x, we need to allow the constant in the estimate to depend on a. For x 0 ∈ Z d , R > 1 and 1 ≤ q < ∞ consider the spatial average of the weight ω (cf. (12)) We shall prove the following estimate: Proposition 1. For R 0 > 1 and k ∈ N 0 consider Then for all 2d d+2 < p < 2 we have where means ≤ up to a constant that only depends on R 0 , d and p, and with β := 2 p * −1 p * −2 + p * and p * := dp d−p . The precise form of the constant C in (16) is not crucial. In fact, in the random setting, when Ω is equipped with a probability measure satisfying (A1) and (A2), we may view C as a random variable with controlled finite moments: Remark 1. Let · satisfy Assumption (A1). Then the spatial average introduced in (15) satisfies The proof of Proposition 1 relies on arguments from elliptic regularity theory, which in the uniformly elliptic case are standard. They typically involve the pointwise inequality where λ 0 > 0 denotes the constant of ellipticity. In the degenerate case, the conductances a may vanish on a non-negligible set of bonds and (17) breaks down. As a replacement we establish estimates which provide a weighted, integrated version of (17): Lemma 4. Let p > d + 1. For any function u : Z d → R and all a ∈ Ω we have (with the convention 1 where C(p, d) := x∈Z d (|x| + 1) 1−p and the inequality holds whenever the sums converge.
While Lemma 4 is purely deterministic, we also need the following statistically averaged version: Lemma 5. Let · be stationary, cf. (A1), and p > d + 1. Then for any stationary random field u and any bond b ∈ B d we have (with the convention 1 A last ingredient required for the proof of Theorem 1 is a Caccioppoli inequality in probability that yields a gain of stochastic integrability and helps to treat the ∇φ T -term on the right-hand side in (10). In the uniformly elliptic case, i. e. when 0 < λ 0 ≤ a ≤ 1, the Caccioppoli inequality holds for any integer exponents p (see [GO11, Lemma 2.7]). The inequality follows from combining the elementary discrete inequality with the estimate φ 2p The latter is obtained by testing the modified corrector equation (6) with φ 2p+1 T and uses the uniform ellipticity of a. In the degenerate elliptic case (21) is not true any longer. However, by appealing to Lemma 5 the following weaker version of (19) survives: for any factor 0 < θ < 1. Hence, we only gain an increase of integrability by exponents strictly smaller two. As a matter of fact, in the proof of our main result we only need the estimate in the following form: Lemma 6 (Caccioppoli estimate in probability). Let · satisfy (A1) and (A2). Let φ T denote the corrector associated with e ∈ R d , |e| = 1, T > 0. For every even integer p we have where means ≤ up to a constant that only depends on p, Λ and d.

Now we are ready to prove our main result:
Proof of Theorem 1. It suffices to consider exponents p ∈ 2N that are larger than a threshold only depending on d -the threshold is determined by (25) below. Further, we only need to prove Indeed, in combination with the Caccioppoli estimate in probability, cf. Lemma 6, estimate (24) yields φ 2p T 1 2p p p+1 + 1. Since p p+1 < 1 the first term can be absorbed and the desired estimate follows.
We prove (24). For reasons that will become clear at the end of the argument we fix an exponent 2d d+2 < q < 2 such that d( This is always possible for p ≫ 1 and 0 < 2 − q ≪ 1, since lim q↑2,p↑∞ Our argument for (24) starts with the p-version of the spectral gap estimate, see Lemma 2, that we combine with Lemma 3: Now we wish to benefit from the decay estimate for ∇G T in Proposition 1, and therefore decompose B d into dyadic annuli: Let the dyadic annuli A k , k ∈ N 0 be defined as in Proposition 1 with initial radius R 0 = 2. Note that B d can be written as the disjoint union of A 0 , A 1 , A 2 , . . . . With the triangle inequality w. r. t. (·) p 1 p and Hölder's inequality in b-space with exponents ( p p−1 , p) we get Because 2d d+2 < q < 2 < 2p p−1 , the discrete ℓ q -ℓ 2p p−1 -estimate combined with the decay estimate of Proposition 1 yields Here and below, C denotes a generic, non-negative random variable with the property that C m 1 for all m < ∞, where means ≤ up to a constant that only depends on m, p, q, Λ and d. Combining (26) and (27) yields Next we apply a triple Hölder inequality in probability with exponents (θ, θ ′ , θ ′ ), where we choose θ = 2p+1 2p (so that 2pθ = 2p + 1). We have The first term is estimated by stationarity of ∇φ T and the assumption |e| = 1 as Combined with (28) we get In the last line we used that which holds since the exponent is negative, cf. (25). This proves (24).

Proofs of the auxiliary lemmas 4.1 Proof of Lemma 3
The argument for Lemma 3 is split into three lemmas.
Lemma 7. Let b ∈ B d be fixed. For T > 0 let φ T and G T denote the modified corrector and the Green's function, respectively. Then Proof of Lemma 7. For simplicity we write φ and G instead of φ T and G T .
Step 1. Argument for (32). We first claim that Indeed, since ∇ and ∂ ∂a(b) commute, an application of ∂ ∂a(b) to (11) yields We test this identity with G(·, x): Since ∂a(b ′ ) ∂a(b) is equal to 1 if b ′ = b and 0 else, the sum on the right-hand side reduces to ∇G(b, y)∇G(b, x) and we get (33a). An application of ∇ to (33a) yields (33b), and an application of ∇ to (33b) finally yields (32).
Step 3. Positivity of ∇∇G(b, b) and 1 − a(b)∇∇G(b, b). Let b = (x b , y b ) ∈ B d be fixed. An application of ∇ (w. r. t. the y-component) to (11) yields We test this equation with ∇G(·, b) and get  (∇∇G(b, b)) 2 . The latter can be seen by the following argument: The next lemma establishes a (quantitative) link between the vertical and classical partial derivative of φ T .
Lemma 8. Let b ∈ B d be fixed. For T > 0 let φ T and G T denote the modified corrector and the Green's function. Then Proof of Lemma 8. Fix a ∈ Ω and b ∈ B d . Set a 0 := a(b). We shall use the following shorthand notation where a b,a denotes the coefficient field obtained from a by setting a b,a (b ′ ) = a if b ′ = b and a b,a (b ′ ) := a(b ′ ) else. With that notation (31) and (32) turn into Since we have ∂φ T (x=0) ∂b ≤´1 0 |ϕ(a)| da, it suffices to shoŵ The positivity of g and (41) imply that ϕ is either strictly positive, strictly negative or that it vanishes identically. In the latter case, the claim is trivial. In the other cases we have From (41) we learn that h ′ = −2g. Since g > 0, h is decreasing. Combined with the identity h(a 0 ) = 0 we get On the other hand, we learn from integrating (42) that g(a ′ ) = g(a 0 ) 1+(a ′ −a 0 )g(a 0 ) . Hence, for a < a 0 the right-hand side in (45) turns into 2ˆa 0 a g(a ′ ) da ′ = −2 ln(1 + (a − a 0 )g(a 0 )), which in combination with (45) yields (44).
Lemma 3 is a direct consequence of (39), (30) and the following estimate: Lemma 9. Let G T denote the Green's function. Assume that (A1) is satisfied. Then for all T > 0, a ∈ Ω and b ∈ B d we have where means up to a constant that only depends on d.
Proof of Lemma 9.
Step 1. Reduction to an estimate for a b,0 . We claim that For the argument let a ∈ Ω and b ∈ B d be fixed. With the shorthand notation introduced in (40), the claim reads a 0 1 − a 0 g(a 0 ) ≤ (1 + g(0)) 2 .
For a 0 = 0 the statement is trivial. For a 0 > 0 consider the function with help of which the left-hand side in (47) can be written as g(a 0 ) f (a 0 ) . The function f is nonnegative and decreasing, as can be seen by combining the inequality 0 < g(a) < 1 a from Lemma 7 with the identity f ′ (a) = g(a)(g 2 (a) − 1 a 2 + g 2 (a) − 1 a g(a)) which follows from (42). The latter also implies that g(1) = g(0) 1+g(0) and thus f (1) = g(1)(1 − g(1)) = g(0) (1+g(0)) 2 . Hence, in the last step we used in addition that g(a 0 ) ≤ g(0) which is a consequence of (42).

Proof of Lemma 4 and Lemma 5
Proof of Lemma 4. Fix for a moment a ∈ Ω.
denote a shortest open path that connects x b and y b , i.e.
Thanks to the triangle inequality and the Cauchy-Schwarz inequality we have Hence, using the convention 1 ∞ = 0, we conclude that for all b ∈ B d and a ∈ Ω: We drop the "a" in the notation from now on. Summation of (49 Since π(b) is a shortest path, and because a ≤ 1, we have dist( Combined with the previous estimate we get Proof of Lemma 5. Fix b ∈ B d . For L ∈ N consider the indicator function With the convention 1 for all a ∈ Ω. In the following we drop "a" in the notation. We recall (49) in the form of From a ≤ 1 and dist(x b , y b ) < 2L for χ L = 0, cf. (50), we learn that π(b) is contained in the box Q 2L (x b ). Hence, (52) turns into We take the expectation on both sides and appeal to stationarity: Using 1 + d − p < 0 we get

Proof of Proposition 1 -Green's function estimates
We first establish an estimate for the Green's function itself: Lemma 10. Let d ≥ 2 and consider u, f ∈ ℓ 1 (Z d ) with Then for all 2d d+2 < p < 2, R ≥ 1 and x 0 ∈ Z d we have Here,ū := Proof of Lemma 10. W. l. o. g. we assume Z d |f | = 1 and R ∈ N. To shorten the notation we write B R and Q R for B R (x 0 ) and Q R (x 0 ), respectively. Let M (u) denote a median of u on B R , i. e.
By Jensen's inequality we have |ū − M (u)| ≤ 1 Then v M satisfies Since u ∈ ℓ 1 (Z d ) (by assumption) and v M ∈ ℓ ∞ (Z d ) (by construction), we may integrate by parts: Set p * = pd d−p and q * := p * p * −1 . By construction we have |{v M = 0} ∩ B R | = |{v M ≤ 0} ∩ B R | ≥ 1 2 |B R |. Hence, the Sobolev-Poincaré inequality yields Lemma 4 combined with Hölder's inequality with exponents ( 2 2−p , 2 p ) yields so that Next we use Chebyshev's inequality in the form of With (57) we get which upgrades by symmetry to Since p > 2d d+2 (by assumption), we have p * 2 > 1 and the "wedding cake formula" for M := CR 2−d yields A careful Caccioppoli estimate combined with the previous lemma yields: Lemma 11. Let d ≥ 2, x 0 ∈ Z d and R ≥ 1. Consider f ≥ 0 and u related as Then for 2d d+2 < p < 2 we have where u − := max{−u, 0} denotes the negative part of u, C := C(a, Q 2R (x 0 ), p 2−p ), α := 2 p * −1 p * −2 and p * := dp d−p . Here stands for ≤ up to a constant that only depends on p and d Proof of Lemma 11. Step 1. Caccioppoli estimate. We claim that for every cut-off function η that is supported in B 2R−1 (x 0 ) (so that in particular ∇η = 0 outside of Q 2R (x 0 )) we have (60) Indeed, we get with Lemma 4 (using an argument similar to (56)):

Combined with the elementary identity
the equation for u, and the fact that −f uη 2 ≤ f u − η 2 (here we use f ≥ 0), the claimed estimate (60) follows.
As we shall see below in Step 3, there exists a cut-off function η with η = 1 in B R+1 (x 0 ) and Let us explain the right-hand side of this estimate. While the first term on the right-hand side would also appear in the continuum case (i.e. when Z d is replaced by R d ), the second term is an error term coming from discreteness. In fact, it is of lower order: A sharp look at (66) below shows that (62) holds with the vanishing factor R −ǫ (for some ǫ > 0 only depending on p and d) in front of the second term on the right-hand side. By combining this estimate with the Gagliardo-Nirenberg-Sobolev inequality on Z d , p , and two applications of Youngs' inequality, we find that for all δ > 0 there exists a constant C(δ) > 0 only depending on δ, p and d, such that We combine this estimate with (60) and absorb the first term on the right-hand side of the previous estimate into the left-hand side of (60). Since ∇(ηu) = ∇u in Q R (x 0 ) this yields (59).
Proof of Proposition 1.
Step 1. Argument for k ≥ 1. For brevity set R := 2 k−1 R 0 and recall that A k = Q 2R (0) \ Q R (0). We cover the annulus A k by boxes Q R Since the diameter of the annulus and the side length of the boxes are comparable, we may choose X R such that its cardinality is bounded by a constant only depending on d. Since in addition we have for x 0 ∈ X R the inequality C(a, Q R (x 0 ), p 2−p ) C(a, Q 3R (0), p 2−p ) (thanks to the third inclusion in (67)), it suffices to prove for each x 0 ∈ X R separately. We use the shorthand G T (x) := G T (a, x, 0) and setḠ T : Lemma 10 yields Thanks to the third inclusion in (67) we have 0 / ∈ B R (x 0 ), and thus u satisfies (58) with f = 1 T G T (with B 2R (x 0 ) replaced by B R (x 0 )). Hence, Lemma 11 yields Regarding the second term on the right-hand side we only need to show We note that ( Combined with (69) and the inequality 1 Step 2. Argument for k = 0. Fix a ∈ Ω. For brevity set G T (x) := G T (a, x, 0) and G T := 1 |B 2R 0 (0)| x∈B 2R 0 (0) G T (x). By the discrete ℓ 1 -ℓ p -estimate and the elementary inequality

As in
Step 1 an application of Lemma 10 yields Since R 2 0 ∼ R 1−d 0 and because the exponent of the constant satisfies p * 2 ≤ β 2 , the desired estimate follows.

Proof of Lemma 6
In order to deal with the failure of the Leibniz rule we will appeal to a number of discrete estimates, which are stated in Lemma 12 below. As already mentioned, we replace the missing uniform ellipticity of a by the coercivity estimate of Lemma 5 which makes use of the weight ω defined in (12). Morally speaking it plays the role of 1 λ 0 in (21). In view of Assumption (A2) all moments of ω are bounded, i. e. ω k 1, where means ≤ up to a constant that only depends on k, p, Λ and d. We split the proof of Lemma 6 into the following two inequalities: Here and below we write φ instead of φ T for simplicity. Note that due to stationarity the left-hand side of (72) and the right-hand side of (73) do not depend on b ∈ B d (resp. x ∈ Z d ). Therefore, we suppress these arguments in the following. We start with (72). We smuggle in ω by appealing to Hölder's inequality with exponent 2p+2 2p+1 and exploit that all moments of ω are bounded by Assumption (A2): We combine (20) in the form of |∇φ(b)| 2p+2 ( φ p (x b )+φ p (y b ) 2 ) 2 |∇φ(b)| 2 (where we use that p is even) with the discrete version of the Leibniz rule F p ∇F = 1 p+1 ∇(F p+1 ), see (77) in Corollary 2 below: |∇φ| 2p+2 ω −1 |∇(φ p+1 )| 2 ω −1 .
Now (72) follows from the coercivity estimate of Lemma 5. Next we prove (73). The discrete version of the Leibniz rule |∇(F p+1 )| 2 = (p+1) 2 (2p+1) ∇F ∇(F 2p+1 ) (see Lemma 12 (ii)) yields b ′ ={0,e i } i=1,...,d By stationarity and the modified corrector equation (6) where for the last inequality we use that φ 2(p+1) ≥ 0 and |e| = 1. By Corollary 2 and Young's inequality we get for any Since we may choose ǫ > 0 as small as we wish, the first term on the right-hand side can be absorbed into the left-hand side of (73) and the claim follows.

Proof of part (i
By dividing by |∇F (b)| one immediately finds the claimed result.
In the course of proving our main result we will use the discrete Leibniz rule, (i) in the above lemma, in the following form.
Corollary 2. For every scalar function F , every bond b and every even integer p we have Here means up to a constant that only depends on p.