Green kernel asymptotics for two-dimensional random walks under random conductances

We consider random walks among random conductances on $\mathbb{Z}^2$ and establish precise asymptotics for the associated potential kernel and the Green function of the walk killed upon exiting balls. The result is proven under a general set of assumptions, where examples include uniformly elliptic conductances, random walks on supercritical percolation clusters and ergodic degenerate conductances satisfying a moment condition. We also provide a similar result for the time-dynamic random conductance model. As an application we present a scaling limit for the variances in the Ginzburg-Landau $\nabla \phi$-interface model.


INTRODUCTION
We consider the Euclidean lattice (Z d , E d ) with d ≥ 2. The edge set E d of this graph is given by the set of all non-oriented nearest neighbour bonds, that is E d := {{x, y} : x, y ∈ Z d , |x − y| = 1}.We will also write x ∼ y if {x, y} ∈ E d .Consider a family of non-negative weights ω = {ω(e) ∈ [0, ∞) : e ∈ E d } ∈ Ω, where Ω = [0, ∞) E d is the set of all possible configurations.We also refer to ω(e) as the conductance of the edge e.With an abuse of notation, for x, y ∈ Z d we set ω(x, y) = ω(y, x) = ω({x, y}) if {x, y} ∈ E d and ω(x, y) = 0 otherwise.We call an edge e ∈ E d open if ω(e) > 0.
We equip Ω with a σ-algebra F. Further, we will denote by P a probability measure on (Ω, F), and we write E to denote the expectation with respect to P. Let us further define the measure µ ω on Z d by µ ω (x) := y∼x ω(x, y).
Throughout the paper we assume that the conductance are ergodic and P-a.s.there exists a unique infinite cluster C ∞ (ω) of open edges.For instance, in the case of i.i.d.conductances this is fulfilled if P[ω(e) > 0] > p c , where p c = p c (d) denotes the critical probability for bond percolation on Z d .Then, we define the measure (1.1) We now introduce the random conductance model (RCM).Given a speed measure θ ω : Z d → [0, ∞) with θ ω (x) > 0 for all x ∈ C ∞ (ω), we consider a continuous time continuous time Markov chain, X = X t : t ≥ 0 , on C ∞ (ω) with generator L ω θ acting on bounded functions f : C ∞ (ω) → R as Then, the Markov chain, X, is reversible with respect to the speed measure θ ω , and regardless of the particular choice of θ ω the jump probabilities of X are given by p ω (x, y) := ω(x, y)/µ ω (x), x, y ∈ C ∞ (ω).The various random walks corresponding to different speed measures will be time-changes of each other.
The maybe most natural choice for the speed measure is θ ω ≡ µ ω , for which we obtain the constant speed random walk (CSRW) that spends i.i.d.Exp(1)-distributed waiting times at all visited vertices.Another frequently arising choice for θ ω is the counting measure, i.e. θ ω (x) ≡ 1 for all x ∈ Z d , under which the random walk waits at x an exponential time with mean 1/µ ω (x).Since the law of the waiting times depends on the location, X is also called the variable speed random walk (VSRW).
For any θ ω we denote by P ω x the law of the process X starting at the vertex x ∈ C ∞ (ω).For x, y ∈ C ∞ (ω) and t ≥ 0 let p ω t (x, y) be the transition densities of X with respect to the reversible measure (or the heat kernel associated with L ω θ ), i.e.
In dimension d ≥ 3 the Green's function of X is defined by Recall that, for every x ∈ C ∞ (ω), the function y → g ω (x, y) is a fundamental solution of L ω θ u = −½ {x} /θ ω (x).Further, for any d ≥ 2, the Green's function of the random walk killed upon exiting a finite set A ⊂ Z d is given by where τ A := inf{t > 0 : X t ∈ A}.In d ≥ 3 the following results on g ω are known.
The constant σ V > 0 in definition of ḡ in Theorem 1.1(ii) and the matrix Σ in Theorem 1.2 represent the diffusivity of the Brownian motion appearing as the scaling limit in the invariance principle for the VSRW (see [2, Theorem 1.1] and [3]).In the proof of Theorem 1.1 the result is first established for the trace process of the random walk X on a smaller cluster, for which Gaussian heat kernel estimates and a local limit theorem hold.Then the result for the original walk can be deduced due to the fact that g ω does not depend on the choice of the speed measure θ ω .Similarly, Theorem 1.2 can be derived also by integration using the Gaussian upper bounds and the local limit theorem for the heat kernel established in [4,5].
Remark 1.3.We expect that for any d ≥ 2 a local limit theorem for the killed Green's function can be deduced from the invariance principle for the random walk (cf.[3]) and the elliptic Harnack inequality (cf.[4]).
In the present paper we study the case d = 2, which is genuinely different and requires separate consideration.This is mainly due to the fact that under suitable conditions the random walk X is recurrent in d = 2, so the Green kernel g ω (x, y) as in (1.3) is ill-defined.Instead, we consider the potential kernel Note that a ω (x, y) = a τxω (0, y − x) and for every we have the following relation between the killed Green kernel and the potential kernel, (see Lemma 2.11 below).We obtain precise asymptotics of the potential kernel and the killed Green kernel under some assumptions on the geometry of C ∞ (ω) (see Assumption 2.1 below) and some assumptions on the decay and the regularity on the heat kernel (see Assumption 2.3 below).Those assumptions are satisfied in a number of relevant situations such as uniformly elliptic i.i.d.conductances, supercritical i.i.d.percolation or ergodic conductances satisfying an integrability condition (see Section 2.2 below).Throughout the paper, for x = (x 1 , x 2 ) ∈ R 2 we write |x| = |x 1 | + |x 2 |.For x ∈ Z 2 we denote by B(x, r) := {y ∈ Z 2 : |x − y| < r} balls in Z 2 centred at x with respect to the graph distance.We will also write ∂B(x, r) := {y ∈ Z 2 : |x − y| = r}.Further, we choose a function Theorem 1.4.Let d = 2 and suppose that Assumptions 2.1 and 2.3 are satisfied.Then, a ω is well-defined, and for any annulus ) Using relation (1.4) we can deduce from Theorem 1.4 the following asymptotics for the killed Green kernel.(i) P 0 -a.s., for any z ∈ C ∞ (ω), ) Remark 1.6.(i) Independence of speed measures.Similarly as the Green kernel in d ≥ 3, the potential kernel a ω does not depend on the speed measure θ ω .In view of (1.4) the same applies to the killed Green kernel g ω A .Thus, the constant ḡ does not depend on θ ω either, where the matrix Σ 2 (depending on θ ω ) appearing in the definition of ḡ typically coincides with the covariance matrix of the Brownian motion appearing as the limit process in the invariance principle for X.
As a further consequence, it suffices to verify Assumptions 2.1 and 2.3 for only one choice of θ ω to conclude that Theorems 1.4 and 1.5 hold for all possible speed measures.In this sense, Theorems 1.4 and 1.5 (as well as Theorems 1.1 and 1.2) are stable under time-changes.
(ii) Classical random walks.More precise asymptotics have been established for the Green and potential kernel for classical random walks on Z d by Fourier analysis (see e.g.[21,20]).In this setting the potential kernel a is also identified by , where T 0 := min{t > 0 : X t = 0}.
(see [21,Theorem 16.1].We expect that the corresponding quenched identification also holds for random walks among random conductances under suitable assumptions.
(iii) Annealed Green kernel estimates.A careful analysis of the proofs of Theorems 1.4 and 1.5 shows that if we assume in addition E 0 θ ω (0 where N 1 and N 2 are the random constants in Assumption 2.3 below, then the convergence in (1.5) and (1.6) also holds in L1 (P 0 ) and the function Possible applications of Theorem 1.5 appear in the study of the maximum of discrete Gaussian free field on supercritical percolation clusters 1 .
In Section 3 we state the corresponding asymptotics for the quenched and annealed potential kernel of the VSRW under time-dynamic conductances (see Theorems 3.3 and 3.5 below).The latter is relevant in the context of the Ginzburg-Landau model for stochastic interfaces (see [17]).In fact, by the Helffer-Sjöstrand representation (cf.[17,19,15]) the variance of the difference of the interface can be expressed in terms of the annealed potential kernel for a particular choice of random dynamic conductances linked to the potential function of the interface, see Section 3.2.Then, Theorem 3.5 allows to deduce scaling limits for such variances, see Theorem 3.6.
The rest of the paper is organised as follows.In Section 2 we discuss in detail the sufficient conditions for the statements of Theorems 1.4 and 1.5 to hold and we present the proofs.In Section 3 we discuss the corresponding results for the twodimensional dynamic random conductance model.Throughout the paper we write c to denote a positive constant which may change on each appearance.Constants denoted c i will be the same through the paper.

GREEN KERNEL ASYMPTOTICS IN THE TWO-DIMENSIONAL STATIC RCM
2.1.Assumptions.From now on we will consider the case d = 2 only.For any z ∈ Z 2 we denote by τ : Ω → Ω the space shift by z defined by (ii) For P-a.e. ω, there exists a unique infinite cluster, C ∞ (ω), of open edges.Let the measure P 0 be defined as in (1.1).We write d ω (x, y) for the graph metric on C ∞ (ω).(iii) For every r ≥ 1, let h ω (r) be the size of the biggest 'hole' in B(0, r) ∩ C ∞ (ω), i.e.
(i) Near-diagonal estimate.There exists N 3 (ω) and a constant c 9 such that if t ≥ N 3 (ω) we have for all y ∈ C ∞ (ω), (2.1) (ii) Hölder regularity in space.There exists N 4 (ω) such that and positive constants c 10 and ̺ such that for R ≥ N 4 (ω) and √ T ≥ R the following holds.Setting T 0 := T + 1 and R 2 0 := T 0 we have for any p ω s (0, y).
(iii) Local limit theorem in zero.For some symmetric and positive-definite matrix (iv) For any annulus Proof.Note that by Lemma 2.2 there exists N < ∞ such that |λ n (x)| ≥ cn for any x ∈ K provided that n ≥ N .Hence, for such x and t ∈ (n α , n β ), Consider the CSRW with speed measure θ ω = µ ω .Then C ∞ (ω) = Z 2 for every ω and Assumptions 2.1 obviously hold.In the uniformly elliptic setting, the Gaussian bounds on the heat kernel in Lemma 2.5 follow from the results in [14].The parabolic Harnack inequality has also been proven in [14].Finally, a local limit theorem has been stated in [ 2 denotes the critical threshold for bond percolation in Z 2 .Then, it is well known that Assumption 2.1 (ii) holds.For Assumption 2.1 (iii) see [9, Lemma 5.4], and Assumption 2.1 (iv) follows from [9, Lemma 5.3], which is based on arguments in [7] (those results are stated for 'holes' and balls w.r.t. the maximum norm rather than the equivalent | • |-norm used in the present paper).Consider again the CSRW, i.e. θ ω = µ ω .Upper Gaussian heat kernel bounds have been obtained in [7].A parabolic Harnack inequality and a local limit theorem have been established as the main results in [9].Example 2.9 (I.i.d.conductances bounded away from zero).Let {ω(e) : e ∈ E 2 } be i.i.d. with P[ω(e) ≥ 1] = 1 and consider the VSRW with speed measure θ ω = 1.Then Assumption 2.1 follows again immediately.Gaussian bounds, a parabolic Harnack inequality and a local limit theorem have been shown in [8].
The integral can be decomposed into p ω t (0, λ n (x)) dt.
By the local limit theorem in Assumption 2.3 (iii) for any δ > 0 there exists n δ such that |s p ω s (0, 0) − ḡ/2| ≤ δ for all s ≥ n δ .Hence, for n such that n α > n δ , Moreover, Since α is chosen to be arbitrarily small, it remains to show that For that purpose, for any β ∈ (α, 2) we write integral as where we used again Assumption 2.3 (i) in the last step.Finally, by Assumption 2.3 (iv) this yields and, since β can be chosen arbitrarily close to 2, we obtain (2.6).Lemma 2.11.P 0 -a.s., for any finite set A ⊂ Z 2 we have for all x, y ∈ C ∞ (ω),

Proof of
Proof.Recall that, for any y ∈ C ∞ (ω) fixed, h(x) = a ω (y, x) is a fundamental solution of L ω θ u = ½ {y} /θ ω (y) and under P ω x , the process (M t : t ≥ 0) defined by Since A is finite, by the dominated convergence theorem and by the monotone convergence theorem which gives the claim.
Proof of Theorem 1.5.It suffices to consider the case z = 0, otherwise we may replace ω by τ z ω.
(i) Note that a ω (0, 0) = 0 and we have by (2.7), Further, notice that, for every n, X τ B(0,n) = ny n = λ n (y n ) for some y n contained in the annulus and the claim follows from Theorem 1.4.
Similarly, setting N xy := |x − y|, we may write Combining the above estimates gives The corresponding lower bound follows by the same arguments.

GREEN KERNEL ASYMPTOTICS IN THE TWO-DIMENSIONAL DYNAMIC RCM
3.1.Setting and results.In this section we consider the dynamic random conductance model.Let now Ω be the set of measurable functions from R to (0, ∞) E 2 equipped with a σ-algebra F and let P be a probability measure on (Ω, F).We will refer to ω t (e) as the time-dependent conductance of the edge e ∈ E 2 at time t ∈ R. A space-time shift by (s, z) ∈ R × Z 2 is the map τ : Ω → Ω, (τ s,z ω) t ({x, y}) := ω t+s ({x + z, y + z}) The set {τ t,x : (t, x) ∈ R × Z 2 } together with the operation τ t,x • τ s,y = τ t+s,x+y defines the group of space-time shifts.For any fixed realization ω ∈ Ω, it is a timeinhomogeneous Markov chain, X = (X t : t ≥ 0), on Z 2 with time-dependent generator acting on bounded functions f : Z 2 → R as We denote by P ω s,x the law of the process starting in x ∈ Z 2 at time s ≥ 0 and by p ω s,t (x, y) := P ω s,x X t = y for x, y ∈ Z 2 and t > s ≥ 0 the transition density with respect to the counting measure.Assumption 3.1.Assume that P satisfies the following condition.
(i) P 0 < ω t (e) < ∞ = 1 for all e ∈ E 2 and t ∈ R. (ii) P is ergodic and stationary with respect to space-time shifts, that is P • τ −1 t,x = P for all x ∈ Z 2 , t ∈ R, and P[A] ∈ {0, 1} for any A ∈ F such that P[A△τ t,x (A)] = 0 for all x ∈ Z 2 , t ∈ R.
The static model where the conductances are constant in time and ergodic with respect to space shifts is included as a special case.Assumption 3.2.For P-a.e. ω, the heat kernel satisfies the following conditions.
(iii) Local limit theorem in zero.For some symmetric and positive-definite matrix (iv) For any annulus As in the static case Assumptions 3.2 (i) and (iv) follows follow from typical Gaussian upper bounds on the heat kernel, see Lemma 2.5 above.In the uniformly elliptic case, i.e.P c −1 < ω t (e) < c = 1 for some c ∈ [1, ∞), such bounds have been established in [15,Proposition 4.2] and a local limit theorem has been stated in [1,Theorem 1.6].The latter has been extended to dynamic degenerate conductances in [12], where also a Hölder regularity statement as in Assumption 3.2 has been shown.
Proof.This follows by similar arguments as in the proof of Theorem 1.4 above.
We shall also state a corresponding annealed result.For abbreviation we write pt (x, y) := E p ω 0,t (x, y) for the averaged transition density.
Assumption 3.4.The averaged heat kernel satisfies the following conditions.
(i) Near-diagonal estimate: There exists c 19 < ∞ such that pt (0, y) ≤ c 19 t −1 for all y ∈ Z 2 and t > 0. (ii) There exists c 20 such that either of the following two conditions holds.
(iii) Assumption 3.2 (iii) and (iv) hold with p ω replaced by p.
In the uniformly elliptic case Assumption 3.4 (i), (iii) and (iv) follow again from the results in [15] and [1].An annealed gradient estimate on the heat kernel as in Assumption 3.4 (ii.b) has been established in [  which may serve as an replacement for (2.4).
3.2.Application to stochastic interface models.We briefly outline an application of Theorem 3.5 in the context of the Ginzburg-Landau ∇φ interface model, see [17].The interface is described by a field of height variables {φ t (x) : x ∈ Z d , t ≥ 0}, whose stochastic dynamics by the following infinite system of stochastic differential equations involving nearest neighbour interaction: Here φ is the height of the interface at time t = 0, {w(x) : x ∈ Z d } is a collection of independent Brownian motions and the potential V ∈ C 2 (R, R + ) is even and strictly convex, i.e. c − ≤ V ′′ ≤ c + for some 0 < c − < c + < ∞.Then the formal equilibrium measure for the dynamic is given by the Gibbs measure Z −1 exp(−H(φ)) x dφ(x) on R Z d with formal Hamiltonian given by H(φ) = 1 2 x∼y V (φ(x)−φ(y)).In dimension d ≥ 3 this can be made rigorous by taking the thermodynamical limit.In any lattice dimension d ≥ 1 one considers the gradient process (∇ e φ t , : e ∈ E d , t ≥ 0) instead.Then, for every u ∈ R d describing the tilt of the interface, the gradient process admits a unique shift invariant ergodic ∇φ-Gibbs measure m u , see [18].
By the so-called Helffer-Sjöstrand representation (cf.[15,19,17]) the variances in the ∇φ model can be written in terms of the annealed potential kernel of a random walk among dynamic random conductances.More precisely, for any x ∈ Z d , var mu φ 0 (x) − φ t (0) = 2 ā(0, x), (3.5)where ā denotes the annealed potential kernel (with expectations taken w.r.t.m u ) associated with the dynamic RCM with conductances given by ω t (x, y) := V ′′ φ t (y) − φ t (x) , {x, y} ∈ E d , t ≥ 0. (3.6) As an immediate consequence from Theorem 3.5 we get the following scaling limit.Proof.The conductances in (3.6) are stationary ergodic under any Gibbs measure µ, and they are uniformly elliptic since the potential function V is assumed to be strictly convex.Hence, Theorem 3.5 applies and implies the result by (3.5).

Theorem 1 . 5 .
The result will follow from Theorem 1.4 and the following relation between the potential kernel and the Green's function of the random walk killed upon exiting a finite set (cf. [20, Proposition 4.6.2(b)]for the case of a simple random walk in discrete time).
[14,9,4]k 2.4.The Hölder-continuity of the heat kernel stated in Assumption 2.3 is a standard consequence from a parabolic Harnack inequality (cf.e.g.[14,9,4]).In conjunction with the near-diagonal estimate in (2.1) it also ensures the existence of the potential kernel a ω , cf. [19, Proof of Lemma 5.2].Next, observe that (2.2) follows immediately from Gaussian-type upper bounds.Suppose that Assumption 2.1 (iii) holds and that, for P-a.e. ω, there exist N 5 (ω) and constants c i such that for any given t with t ≥ N 5 (ω) and all y ∈ C ∞ (ω) the following hold.
(i) If c 11 |y| ≤ t then p ω t (0, y) ≤ c 12 t −1 exp −c 13 c 11 |λ n (x)| .Remark 2.6.Suppose that Assumption 2.1 and Assumption 2.3 (i) hold.Further, assume that for every x ∈ Z 2 there exists N 6 (ω, x) with stretched-exponential tails such that if t ≥ N 6 (ω, x) we have a near-diagonal bound p ω t (x, y) ≤ c 15 t −1 for all y ∈ C ∞ (ω) and a mean-displacement estimate E ω x |X t − x| ≤ c 16 t 1/2 .Then, for t ∈ (n α , n β ) one can show by a symmetry argument that p ω t (0, λ n (x)) ≤ o(t −1 ), which implies Assumption 2.3 (iv).In some situations such a mean-displacement estimate can already be obtained from near-diagonal estimates by a so-called Bass-Nash argument (e.g.[8]).2.2.Examples.We list a number of relevant examples for which Assumptions 2.1 and 2.3 are satisfied and therefore the results of Theorems 1.4 and 1.5 hold.Recall that, provided Assumption 2.1 holds, Gaussian type upper bounds imply Assumption 2.3 (i) and (iv) by Lemma 2.5.Further recall that the Hölder regularity of the heat kernel in Assumption 2.3 (ii) is a consequence from a parabolic Harnack inequality (cf.Remark 2.4 above).(Uniformly elliptic i.i.d.conductances).Suppose that {ω(e) : e ∈ E 2 } are i.i.d. with P