Local Limit Theorems for the Random Conductance Model and Applications to the Ginzburg-Landau $\nabla\varphi$ Interface Model

We study a continuous-time random walk on $\mathbb{Z}^d$ in an environment of random conductances taking values in $(0,\infty)$. For a static environment, we extend the quenched local limit theorem to the case of a general speed measure, given suitable ergodicity and moment conditions on the conductances and on the speed measure. Under stronger moment conditions, an annealed local limit theorem is also derived. Furthermore, an annealed local limit theorem is exhibited in the case of time-dependent conductances, under analogous moment and ergodicity assumptions. This dynamic local limit theorem is then applied to prove a scaling limit result for the space-time covariances in the Ginzburg-Landau $\nabla\varphi$ model. This result applies to convex potentials for which the second derivative may be unbounded.


INTRODUCTION
1.1. The Model. We consider the graph G = (Z d , E d ) of the hypercubic lattice with the set of nearest-neighbour edges E d := {{x, y} : x, y ∈ Z d , |x − y| = 1} with d ≥ 2. We place upon G positive weights ω = {ω(e) ∈ (0, ∞) : e ∈ E d }. We will also write ω for the conductance matrix; ω(x, y) = ω(y, x) = ω({x, y}) if the edge {x, y} ∈ E d and ω(x, y) = 0 otherwise. We define two measures on Z d , Let (Ω, F) := (R E d + , B(R + ) ⊗E d ) be the measurable space of all possible environments. We denote by P an arbitrary probability measure on (Ω, F) and E the respective expectation. The measure space (Ω, F) is naturally equipped with a group of space shifts τ z : z ∈ Z d , which act on Ω as (τ z ω)(x, y) := ω(x + z, y + z), acting on bounded functions f : Z d → R, is reversible with respect to θ ω , and we call this process the random conductance model (RCM) with speed measure θ ω . We denote P ω x the law of this process started at x ∈ Z d and E ω x the corresponding expectation. There are two natural laws on the path space that are considered in the literature -the quenched law P ω x (·) which concerns P-almost sure phenomena, and the annealed law EP ω x (·). The random walk X chooses its next position with probability ω(x, y)/µ ω (x), after waiting an exponential time with mean θ ω (x)/µ ω (x) at the vertex x. The main results of this paper are statements about the heat kernel of X, which is defined as p ω θ (t, x, y) := P ω x (X t = y) θ ω (y) for t ≥ 0 and x, y ∈ Z d .
Perhaps the 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 vertices it visits. Another well-studied process, the variable speed random walk (VSRW) is recovered by setting θ ω ≡ 1, so called because as opposed to the CSRW, the waiting time at a vertex x does indeed depend on the location; it is an Exp µ ω (x) -distributed random variable.

Main Results on the Static RCM.
As our first main results we obtain quenched and annealed local limit theorems for the static random conductance model. A general assumption required in this context is stationarity and ergodicity of the environment. All results on the static RCM are restricted to dimension d ≥ 2. = P for all x ∈ Z d and P(A) ∈ {0, 1} for any A ∈ F such that τ x (A) = A for all x ∈ Z d . (iii) θ is stationary, i.e. θ ω (x + y) = θ τyω (x) for all x, y ∈ Z d and P-a.e. ω ∈ Ω.
In particular, the last condition in Assumption 1.1(iii) ensures that the process X is non-exlosive. During the last decade, considerable effort has been invested in the derivation of quenched invariance principles or quenched functional central limit theorems (QFCLT), see the surveys [10,24] and references therein. The following QFCLT for random walks under ergodic conductances is the main result of [4]. Theorem 1.2 (QFCLT). Suppose Assumption 1.1 holds. Further assume that there exist p, q ∈ (1, ∞] satisfying 1 p + 1 q < 2 d such that E ω(e) p < ∞ and E ω(e) −q < ∞ for any e ∈ E d . For n ∈ N, define X (n) t := 1 n X n 2 t , t ≥ 0. Then, for P-a.e. ω, X (n) converges (under P ω 0 ) in law towards a Brownian motion on R d with a deterministic non-degenerate covariance matrix Σ 2 .
Recently the moment condition in Theorem 1.2 has been improved in [8].
Remark 1.3. If we letΣ 2 denote the covariance matrix of the above Theorem in the case of the VSRW, the corresponding covariance matrix of the random walk X with speed measure θ ω is given by Σ 2 = E θ ω (0) −1Σ 2 -see [4, Remark 1.5].
The local limit theorem roughly describes how the transition probabilities of the random walk X can be rescaled in order to get the Gaussian transition density of the Brownian motion with covariance matrix Σ 2 , which appears as the limit process in the invariance principle in Theorem 1.2. The Gaussian heat kernel associated with that process will be denoted The quenched local limit theorem will require the following moment condition.
Assumption 1.4. There exist p, q, r ∈ (1, ∞] satisfying 1 r such that E µ ω (0) θ ω (0) p θ ω (0) + E ν ω (0) q + E θ ω (0) −1 + E θ ω (0) r < ∞. (1.4) Note that in the case of the CSRW or VSRW Assumption 1.4 coincides with the moment condition in Theorem 1.2. For x ∈ R d write ⌊x⌋ = (⌊x 1 ⌋, ..., ⌊x d ⌋) ∈ Z d . This result extends the local limit theorem in [5,Theorem 1.11] for the CSRW to the case of a general speed measure. In general, a local limit theorem is a stronger statement than a FCLT. In fact, already in the i.i.d. case, where the QFCLT does hold [2], we see the surprising effect that due to a trapping phenomenon the heat kernel may behave subdiffusively (see [9]), in particular a local limit theorem may fail in general. Nevertheless, it does hold, for instance, in the case of uniformly elliptic conductances, where P(c −1 ≤ ω(e) ≤ c) = 1 for some c ≥ 1, or for random walks on supercritical percolation clusters (see [7]). For sharp conditions on the tails of i.i.d. conductances at zero for Harnack inequalities and a local limit theorem to hold we refer to [11]. Hence, it is clear that some moment condition is necessary. In the case of the CSRW under general ergodic conductances the moment condition in Assumption 1.4 is known to be optimal, see [5,Theorem 5.4]. Local limit theorems have also been obtained in slightly different settings, see [17], where based on the arguments in [7] some general criteria for local limit theorems have been provided.
For the proof of Theorem 1.5 we adapt the techniques employed in [15] to the static, general speed measure case. The idea comes from [28] where it is introduced as a way of deriving Hölder regularity of solutions to parabolic PDEs in continuum. Bounds on the level sets of caloric functions, such as the heat kernel, are derived and used to prove an oscillations inequality, Theorem 2.5, which bounds the oscillations of the function on a time-space cylinder by those on a larger cylinder. By iterating the oscillations bound, a Hölder regularity result is deduced. This, along with the QFCLT, is precisely what is required to prove a local limit theorem. We stress that this approach to show Hölder regularity directly circumvents the need for a parabolic Harnack inequality, in contrast to the proofs in [5,7], which makes it significantly simpler. However, as a by-product we still obtain the following weak parabolic Harnack inequality. Write m θ = dt × θ ω . For any x 0 ∈ Z d , t 0 ∈ R and P-a.e. ω, there exists for some ǫ > 0. Then there exists γ = γ(ǫ, λ) (also depending on the law of ω and θ ω ) such that for any σ ′ ∈ [ 1 2 , λ), A natural example for a harmonic function satisfying the condition u(t, x) ≤ 1/θ ω (x) in Theorem 1.6 is the heat kernel p ω θ (t, 0, x). Next we provide an annealed local limit theorem under a stronger and non-optimal moment condition. Theorem 1.7 (Annealed local limit theorem). Suppose Assumption 1.1 holds. There exist exponents p, q, r 1 , Remark 1.8. In the case of the VSRW, i.e. θ ω = 1, the moment condition required in Theorem 1.7 is more explicitly given by for some p, q ∈ (1, ∞) such that 1/p + 1/q < 2/d and κ ′ = κ ′ (d, p, q, ∞) defined in Proposition 3.1 below. Similarly, in the case of the CSRW, θ ω = µ ω , the condition reduces to again for some p, q ∈ (1, ∞) such that 1/p + 1/q < 2/d and κ ′ = κ ′ (d, ∞, q, p) again as defined in Proposition 3.1 below.
In general, a QFCLT does imply an annealed FCLT. However, the same does not apply to the local limit theorem. In fact, as mentioned above, the proofs of the quenched local limit theorems in [5] and Theorem 1.5 rely on Hölder regularity estimates on the heat kernel, which involve some random constants depending on the exponential of the conductances. Those constants can be controlled almost surely, but naively taking expectations would require exponential moment conditions stronger than the polynomial moment conditions in Assumption 1.4. To derive the annealed local limit theorem given the corresponding quenched result, one might hope to employ the dominated convergence theorem, which requires that the integrand above can be dominated uniformly in n by a function of finite expectation. We achieve this using a quenched maximal inequality from [6]. It is precisely the form of the random constants in this inequality that allows us to anneal the result using only polynomial moments, together with a simple probabilistic bound.

Main Results on the Dynamic RCM.
Next we introduce the dynamic random conductance model. As for the static case, we consider d ≥ 2 only. We endow G with a family ω = {ω t (e) ∈ (0, ∞) : e ∈ E d , t ∈ R} of positive, time-dependent weights. Define, for t ∈ R, the measures µ ω t , ν ω t on Z d by We define the dynamic variable speed random walk starting in x ∈ Z d at s ∈ R to be the continuous-time Markov chain (X t : t ≥ s) with time-dependent generator acting on bounded functions f : Z d → R. Note that the counting measure, which is time-independent, is an invariant measure for X. We denote P ω s,x the law of this process started at x ∈ Z d at time s, and E ω s,x the corresponding expectation. For x, y ∈ Z d and t ≥ s, we denote p ω (s, t, x, y) the heat kernel of (X t ) t≥s , that is . Let now Ω be the set of measurable functions from R to (0, ∞) E d equipped with a σ-algebra F and let P be a probability measure on (Ω, F). Upon it we consider the d + 1-parameter group of translations (τ t,x ) (t,x)∈R×Z d given by The required ergodicity and stationarity assumptions on the time-dependent random environment are as follows.
Theorem 1.10 (Quenched FCLT and local limit theorem). Suppose Assumption 1.9 holds and that there exist p, q ∈ (1, ∞] satisfying for any e ∈ E d and t ∈ R. (i) For P-a.e. ω, X (n) , defined as X  Proof. The QFLCT in (i) has been proven in [3], for the quenched local limit theorem in (ii) we refer to [15].
Similarly as in the static case we establish an annealed local limit theorem for the dynamic RCM under a stronger, non-optimal but polynomial moment condition. Theorem 1.11 (Annealed local limit theorem). Suppose Assumption 1.9 holds. There exist exponents p, q ∈ (1, ∞) (specified more explicitly in Assumption 4.2 below) such that if E ω 0 (e) p < ∞ and E ω 0 (e) −q < ∞ for any e ∈ E d , the following holds. For all K > 0 and 0 < T 1 ≤ T 2 , (1.6) 1.4. Application to the Ginzburg-Landau ∇ϕ Model. A somewhat unexpected context in which one encounters (dynamic) RCMs is that of gradient Gibbs measures describing stochastic interfaces in statistical mechanical systems. One wellestablished model is the Ginzburg-Landau model, where an interface is described by a field of height variables {φ t (x), x ∈ Z d , t ≥ 0}, whose stochastic dynamics are given by the following infinite system of stochastic differential equations involving nearest neighbour interaction: Here {w(x), x ∈ Z d } is a collection of independent Brownian motions and the potential V ∈ C 2 (R, R + ) is even and convex. The formal equilibrium measure for the dynamic is given by the Gibbs measure . Investigating the fluctuations of the macroscopic interface has been quite an active field of research, see [19] for a survey.
We are interested in the decay of the space-time covariances of height variables under an equilibrium Gibbs measure. By the Helffer-Sjöstrand representation [22] (cf. also [18,21]) such covariances can be written in terms of the annealed heat kernel of a random walk among dynamic random conductances. More precisely, where the covariance and expectation are taken with respect to an ergodic Gibbs measure µ and p ω denotes the heat kernel of the dynamic RCM with time-dependent conductances given by Thus far, all applications of the aforementioned Helffer-Sjöstrand relation have been restricted to gradient models with strictly convex potential function, which corresponds to uniformly elliptic conductances in the random walk picture. However, recent developments in the degenerate setting will also allow some potentials that are convex but not strictly convex. As an example in this direction, we use the annealed local limit theorem in Theorem 1.11 to derive a scaling limit for the space-time covariances of the φ-field for a wider class of potentials.
Then for all h ∈ R there exists a stationary, shift-invariant, ergodic ϕ-Gibbs measure where k t is the heat kernel of a Brownian motion on R d with a deterministic nondegenerate covariance matrix.
Theorem 1.12 extends the scaling limit result of [1, Theorem 5.2] to hold for potentials V for which V ′′ may be unbounded above.
Example 1.13. The moment condition in (1.7) on the potential V is satisfied for any V with V ′′ having polynomial growth (see Proposition 5.10 below). Hence, Theorem 1.12 applies, for instance, to the anharmonic crystal potential V (x) = x 2 + λx 4 (λ > 0), for which the decay of spatial correlations is discussed in [12].
1.5. Notation. We finally introduce some further notation used in the paper. We write c to denote a positive, finite constant which may change on each appearance. Constants denoted by c i will remain the same. For a number p ∈ [1, ∞] we write p * := p/(p − 1) ∈ [1, ∞] for its Hölder conjugate. We endow the graph G = (Z d , E d ) with the natural graph distance d, i.e. d(x, y) is the minimal length of a path between x and y. Denote B(x, r) := {y ∈ Z d : d(x, y) ≤ r} the closed ball with centre x and radius r. For a non-empty, finite, connected set A ⊆ Z d , we denotes by ∂A := {x ∈ A : d(x, y) = 1 for some y ∈ A c } the inner boundary and by ∂ + A := {x ∈ A c : d(x, y) = 1 for some y ∈ A} the outer boundary of A. We write A = A ∪ ∂ + A for the closure of A. The graph is given the counting measure, i.e. the measure of A ⊆ Z d is the number |A| of elements in A. For f : Z d → R we define the operator ∇ by where for each non-oriented edge e ∈ E d we specify one of its two endpoints as its initial vertex e + and the other one as its terminal vertex e − . Further, the corresponding adjoint operator ∇ * F : Notice that in the discrete setting the product rule reads where av(f )(e) := 1 2 (f (e + ) + f (e − )). We denote inner products as follows; for f, g : Z d → R and a weighting function φ : . The corresponding weighted norm is denoted f l 2 (Z d ,φ) . Given a bounded Lipschitz function F on l 2 (Z d , φ), its Lipschitz semi-norm will be written The Dirichlet form associated with the operator L ω θ is acting on bounded f, g : Z d → R. For non-empty, finite B ⊆ Z d and p ∈ (0, ∞), space-averaged ℓ p -norms on functions f : B → R will be used, , we define norms averaged over space and time: Furthermore, we will work with two varieties of weighted norms In Section 5, we employ also the following notation. We write Λ ⋐ Z d for Λ a finite subset of Z d . Λ * denotes the set of all directed edges in Λ, i.e. Λ * = {{x, y} ∈ E d : x, y ∈ Λ}. Write P(S) for the family of Borel probability measures on a topological space S. Given a measure µ ∈ P(S), E µ [X] denotes the expectation of a random variable X under µ and var µ (X) its variance. We denote the covariance of two random variables X, Y under µ, Cov µ X, Y .
1.6. Structure of the Paper. Section 2 is devoted to the proof of the quenched local limit theorem for general speed measures -Theorem 1.5. The annealed local limit theorems for the static and dynamic RCM in Theorem 1.7 and Theorem 1.11 are shown in Section 3 and Section 4, respectively. Finally, the application to the Ginzburg-Landau interface model is discussed in Section 5.

LOCAL LIMIT THEOREM FOR THE STATIC RCM UNDER GENERAL SPEED MEASURE
For the purposes of this section, we work with space-time cylinders defined as follows. For any x ∈ Z d and t 0 ∈ R let I τ := [t 0 − τ n 2 , t 0 ] and B σ := B(x, σn) for σ ∈ (0, 1], τ ∈ (0, 1]. We write Note that throughout this section we assume the dimension d ≥ 2.

Maximal Inequality.
We first derive a maximal inequality for caloric functions under a general speed measure using a De Giorgi iteration scheme. We require the following energy estimate, cf. [6, Lemma 3.7].
which can be verified by distinguishing several cases. Thus, a summation by parts gives for any t ∈ [s 1 , Further, by the product rule (1.8) , where we used that av(η) 2 ≤ av(η 2 ) by Jensen's inequality. By combining the last two inequalities we get , therefore by Hölder's inequality Finally, since ξ(s 1 ) = 0, applying integration by parts and Jensen's inequality for any s ∈ (s 1 , s 2 ]. Thus, by multiplying both sides of (2.2) with ξ(t) and integrating the resulting inequality over [s 1 , s] for any s ∈ I, the assertion (2.1) follows.
We will also need a modification of the Sobolev inequality derived in [4].
For the rest of this section we fix p, q, r ∈ (1, ∞] such that Proof. The proof is based on an iteration argument and will be divided into two steps. First we will derive the estimate needed in a single iteration step, while the actual iteration is carried out in a second step. Step 1: Let 1/2 ≤ σ ′ < σ ≤ 1 and 0 ≤ k < l be fixed and set α := 1 + 1 p * − r * ρ . Note that, due to the discrete structure of the underlying space Z d , the balls B σ and B σ ′ may coincide. To ensure that B σ ′ B σ we assume in this step that (σ − σ ′ )n ≥ 1. Then, it is possible to define a spatial cut-off function η : . By Hölder's inequality, followed by applications of Hölder's and Young's inequalities, Note that by Jensen's inequality We use Hölder's inequality, the Sobolev inequality in Lemma 2.2, the fact that r * /ρ < 1 and Lemma 2.1 to obtain Further, again by (2.6) and Lemma 2.1, (2.9) Therefore, combining (2.5) with (2.7), (2.8) and (2.9) yields Introducing ϕ(l, σ ′ ) := u − l 2 + p * ,1,Q σ ′ ,θ and setting M := cÃ ω 1 (n) 1+ 1 α * the above inequality reads and holds for any 0 ≤ k < l and 1/2 ≤ σ ′ < σ ≤ 1.
The proof of the weak Harnack inequality in Section 2.4 below will require a stronger version of the maximal inequality, as follows.
2.2. Hölder Regularity. The next significant result allows us to control the oscillations of a caloric function on a time-space cylinder, by those on a larger cylinder. We denote the oscillation of a function u on a cylinder For n ≥ 4 we also set for abbreviation (2.12) such that for all n ≥ N 3 the following holds. There exists , which is continuous and increasing in all components, such that In the remainder of this subsection we will prove Theorem 2.5 by following the method in [15], originally used in [28] to prove Hölder regularity for parabolic equations in continuous spaces. Consider the function g : (0, ∞) → [0, ∞), which may be regarded as a continuously differentiable version of the function x → (− ln x) + , defined by 1 3 ] is the smallest solution of the equation 2c ln(1/c) = 1 − c. Note that g ∈ C 1 (0, ∞) is convex and non-increasing. Although g is not caloric, we can still bound its Dirichlet energy as follows.
By combining this inequality above with (2.18) and using that by Jensen's inequality, we get the claim.
For any finite B ⊂ Z d and any u : denote the weighted average of u over the subset B, and write (u) B := (u) B,1 .
By an application of Jensen's inequality, and the theorem is proven.
Thus, the sequence is regular.
(ii) Recall that γ ω is continuous and increasing in all components. Arguing as in (i) we get by the moment condition in Assumption 1.4 and again the spatial ergodic theorem in [23, Theorem 2.8 in Chapter 6] that for P-a.e. ω, Now we apply Lemma 2.10, which yields that forĀ ′ , for all r ≥ δ and n ≥ N 5 .
Proof. Set δ k := 4 −k √ t/2 and with a slight abuse of notation let . Now we apply Theorem 2.5 and Lemma 2.11-(ii), which gives that there exists N 5 = N 5 (ω, x, t, δ) such that for P-a.e. ω and all n ≥ N 5 , We iterate the above inequality on the chain Q 0 ⊃ Q 1 ⊃ · · · ⊃ Q k 0 to obtain Note that Hence, since γ k 0 ≤ c(δ/ √ t) ̺ , the claim follows from (2.25) and Lemma 2.12.
Proof of Theorem 1.5. Having proven the pointwise result Proposition 2.14, the full local limit theorem follows by extending over compact sets in x and t. This is done using a covering argument, exactly as in the proof of [15, Proposition 3.1], which is a slight modification of the proofs [17] and [7].

Weak Parabolic Harnack Inequality.
In this subsection, we adapt the techniques of [28] to prove Theorem 1.6. The proof requires the L 1 -version of the maximal inequality in Corollary 2.4 and the following auxiliary estimate on the level sets of a caloric function under the measure θ ω . Recall that m θ = dt × θ ω .
Lemma 2.15. Suppose Assumptions 1.1 and 1.4 hold. For any x 0 ∈ Z d , t 0 ∈ R and P-a.e. ω, there exists N 7 = N 7 (ω, x 0 ) ∈ N such that for all n ≥ N 7 the following holds. Assume there exists λ ∈ (0, 1) such that Then, for any σ 1 ∈ (0, λ) and σ 2 ∈ (λ, 1), there exists h = h(d, λ) ∈ (0, 1) (also depending on the law of ω and θ ω ) such that For any 0 < t 1 ≤ t 2 the same arguments as in Lemma 2.6 give (2.28) Set Λ t := {x ∈ B 1 : u(t, x) ≥ 1} . Then, the assumption in (2.26) can be rewritten as By the mean value theorem there exists τ ∈ [t 0 − n 2 , t 0 − σ 1 n 2 ] such that Take t 1 = τ and t 2 ∈ [t 0 − σ 1 n 2 , t 0 ] arbitrary. Then, by (2.28) we have and, using that g(s) = 0 for all s ≥ 1, we get by the definition of Λ τ and (2.29), Substituting the above into (2.30) yields Hence, Since Assumptions 1.1 and 1.4 hold, we can choose N 7 (ω, x 0 ) such that for all n ≥ N 7 (ω, x 0 ), Proof of Theorem 1.6. Without loss of generality, assume ǫ = 1 (otherwise, replace u by u/ǫ). Let G(s) := g( s+k h ) for s ∈ R, with 0 < k < h to be specified later. We write W := G(u). Then ∂ t W = G ′ (u) ∂ t u and by Taylor expansion )) ω(x, y) u(t, y) − u(t, x) for someū x, y ∈ R. Since the latter term in the above is non-negative, this implies So we can apply Corollary 2.4, which gives that for 1 2 ≤ σ ′ < σ 1 < σ 2 ≤ 1 with σ 1 < λ < σ 2 and large enough n, Let η be a linear cut-off function between B σ 1 and B 1 such that ∇η l ∞ (E d ) ≤ 1/n. Then, by the same arguments as in Lemma 2.6, where we have used that , so that by the Poincaré inequality in Proposition 2.8, Hence, noting that θ Returning to (2.31), we use Jensen's inequality, the fact that W ≤ g( k h ) and β n ≤ 1 to obtain for n ≥ N 1 (ω, x 0 ), where we have chosen N 1 (ω, x 0 ) large enough to control the random constants and such that (1 − γ) Kn < 1 4 . Now, choose γ > 0 small enough such that 2γ < h and with c as above. Take k = γ. The weak Harnack inequality now follows by contradiction. Suppose there exists (t,x) ∈ Q σ ′ (n) such that u(t,x) < γ. Then

Proof of Theorem 1.7.
Here we anneal the results of Section 2 to derive the annealed local limit theorem for the static RCM under a general speed measure stated in Theorem 1.7. This will require a stronger moment condition. For any p, q, r 1 , n d p ω θ (n 2 t, 0, ⌊nx⌋) < ∞.
Before we prove Proposition 3.2 we remark that it immediately implies the annealed local limit theorem.
Proof of Theorem 1.7. The statement follows from the corresponding quenched result in Theorem 1.5 and Proposition 3.2 by an application of the dominated convergence theorem.
As a by-product we obtain an annealed on-diagonal estimate on the heat kernel.
The rest of this section is devoted to the proof of Proposition 3.2. We start with a consequence of the maximal inequality in Proposition 3.1.
Another ingredient in the proof of Proposition 3.2 will be the following version of the maximal ergodic theorem, which we recall for the reader's convenience. (3.5) Proof for any p, q, r ∈ (1, ∞] satisfying (1.3). After an application of Hölder's inequality it suffices to show E sup n≥1 ν ω 4κ ′ q,B(n) < ∞ and similar moment bounds on the other terms. Now suppose that E ν ω (0) 4κ ′ ∨q ′ < ∞ for any q ′ > q. Then, if 4κ ′ > q, given Assumption 1.1, we can apply Lemma 3.5 to deduce In the case 4κ ′ ≤ q < q ′ , we have by Jensen's inequality followed by Lemma 3.5, The other terms involving θ ω r,B(n) etc. can be treated similarly.
Proof of Theorem 1.11. The statement follows from the corresponding quenched result, see Theorem 1.10-(ii) above, together with Proposition 4.3 by an application of the dominated convergence theorem. Note that the moment condition in Assumption 4.2 is stronger than the one required in Theorem 1.10.
As in the static case Proposition 4.3 also directly implies an annealed on-diagonal heat kernel estimate (cf. Corollary 3.3 above).
For the proof of Proposition 4.3 we also require a maximal ergodic theorem for space-time ergodic environments.

APPLICATIONS TO THE GINZBURG-LANDAU ∇ϕ MODEL
In this section we connect the annealed local limit theorem for the dynamic RCM with a stochastic interface model, the Ginzburg-Landau ∇ϕ model. The survey [19] provides a nice review of this class of models. The Ginzburg-Landau ∇ϕ model describes a hypersurface (interface) embedded in d + 1-dimensional space, R d+1 , which separates two pure thermodynamical phases. The interface is represented by a field of height variables φ = {φ(x) ∈ R : x ∈ Γ}, which measure the vertical distances between the interface and Γ ⊆ Z d , a fixed d-dimensional reference hyperplane. The Hamiltonian H represents the energy associated with the field of height variables φ. In general, for Γ = Z d or Γ ⋐ Z d , Note that boundary conditions ψ = {ψ(x) : x ∈ ∂ + Γ} are required to define the sum in the case Γ ⋐ Z d , i.e. we set φ(x) = ψ(x) for x ∈ ∂ + Γ. The sum in (5.1) is merely formal when Γ = Z d . The dynamics of the ∇ϕ model are governed by the following infinite system of SDEs Similarly, if Γ ⋐ Z d , we define the finite volume process by subject to the boundary conditions φ Γ,ψ t (y) = ψ(y), y ∈ ∂ + Γ. The evolution of φ t is designed such that it is stationary and reversible under the equilibrium ϕ-Gibbs measure µ ψ Γ or µ (see (5.4) below). We denote by P µ the law of the process φ t started under the distribution µ (and by E µ the corresponding expectation).
Most of the mathematical literature on the ∇ϕ model treats the case of a suitably smooth, even and strictly convex interaction potential V such that V ′′ is bounded above. However, we will relax these conditions; throughout the rest of this section we work with V as in the following assumption.
Assumption 5.1. The potential V ∈ C 2 (R) is even and there exists c − > 0 such that Note that under Assumption 5.1, the coefficients of the SDE (5.2) are not necessarily globally Lipschitz continuous. However, it is still possible to construct an almost surely continuous solution φ t , see Proposition 5.3. The assumption that the potential has second derivative bounded away from zero is required for the existence of an equilibrium ϕ-Gibbs measure. For Γ ⋐ Z d , the finite volume ϕ-Gibbs measure for the field of heights φ ∈ R d is defined as with boundary condition ψ ∈ R ∂ + Γ , where dφ Γ is the Lebesgue measure on R Γ and Z ψ Γ is a normalisation constant. Note that the condition (5.3) implies Z ψ Γ < ∞ for every Γ ⋐ Z d and hence µ ψ Γ ∈ P(R Γ ) is a probability measure. In the infinite volume case Γ = Z d , (5.4) has no rigorous meaning but one can still define Gibbs measures as follows.
Definition 5.2. A probability measure µ ∈ P R Z d is a ϕ-Gibbs measure if its conditional probability on F Γ c = σ{φ(x) : x / ∈ Γ} satisfies the DLR (Dobrushin-Lanford- for all Γ ⋐ Z d . In order to study the properties of solutions to the system of SDES (5.2), it is necessary to restrict to a suitable class of initial configurations. Let S := {(φ(x)) x∈Z d : φ(x) ≤ a +|x| n , for some a ∈ R, n ∈ N} denote the configurations of heights with at most polynomial growth. Proposition 5.3. Given any initial configuration φ 0 ∈ S, there exists a unique solution to the system of SDEs (5.2) such that for any x ∈ Z d the process φ t (x) is almost surely continuous and for all t > 0 the configuration φ t ∈ S almost surely. Any Gibbs measure on S is stationary and reversible with respect to the process φ t .
Proof. The proof follows by similar arguments as for the Ising model case of [26,Theorem 4.2.13]. The key observations are that equation (4.2.5) there holds for our Hamiltonian and the relation (4.2.12b) holds for our interaction potential V , Brascamp-Lieb inequalities state that for Γ ⋐ Z d , covariances under the aforementioned ϕ-Gibbs measure µ ψ Γ are bounded by those under µ ψ, G Γ , the Gaussian finite volume ϕ-Gibbs measure determined by the quadratic potential V * (x) = c − 2 x 2 . Proposition 5.4 (Brascamp-Lieb inequality for exponential moments). Let Γ ⋐ Z d , for every ν ∈ R Γ , we have Proof. See [19,Theorem 4.9]. Note that the condition V ′′ (x) ≤ c + , x ∈ R, for some c + > 0, is not needed for the proof.
The above inequality is pivotal in proving the following existence result. We shall also employ the massive Hamiltonian, for m > 0, ∂φ(x) 2 ≥ c − so by Brascamp-Lieb again the limit µ 0 = lim m↓0 µ 0 m exists. The distribution of φ + h where φ is µ 0 distributed is a shift-invariant ϕ-Gibbs measure on Z d under which φ(x) has mean h for all x ∈ Z d . Having shown that the convex set of shift-invariant ϕ-Gibbs measures of mean h is non-empty, there exists an extremal element of this set which is ergodic, see [20,Theorem 14.15]. Finally, by Proposition 5.3 this Gibbs measure is reversible and hence stationary for the process φ t .
Our aim is to investigate the decay of the space-time correlation functions under the equilibrium Gibbs measures. The idea, originally from Helffer and Sjöstrand [22], is to describe the correlation functions in terms of a certain random walk in a dynamic random environment (cf. also [18,21]). Let (X t ) t≥0 be the random walk on Z d with jump rates given by the random dynamic conductances ω t (e) := V ′′ (∇ e φ t ) = V ′′ (φ t (y) − φ t (x)), e = {x, y} ∈ E d . (5.9) Note that the conductances are positive by Assumption 5.1 and, since V is even, the jump rates are symmetric, i.e. ω t ({x, y}) = ω t ({y, x}). Further, let p ω (s, t, x, y), x, y ∈ Z d , s ≤ t, denote the transition densities of the random walk X. Then the Helffer-Sjöstrand representation (see [19,Theorem 4.2] or [18, Equation (6.10)]) states that if F, G ∈ C 1 b (S) are differentiable functions with bounded derivatives depending only on finitely many coordinates then for all t > 0, (φ t ) p ω (0, t + s, x, y) ds, (5.10) where µ is a stationary, ergodic, shift-invariant ϕ-Gibbs measure. Note that in d ≥ 3 the integral in (5.10) is finite due to an on-diagonal heat kernel estimate. Proof. Note that by Assumption 5.1, ω t (e) ≥ c − for all t ≥ 0 and e ∈ E d , which implies the Nash inequality, i.e. for any f : Z d → R, from which the statement follows by standard arguments, see [13] and [25].
A consequence of the above is the following variance estimate, an example of algebraic decay to equilibrium, in contrast to the exponential decay to equilibrium which would follow from a spectral gap estimate or Poincaré inequality. For this model, these inequalities hold on finite boxes but fail on the whole lattice. Corollary 5.9. Suppose d ≥ 3 and let µ ∈ P R Z d be any ergodic, shift-invariant, stationary ϕ-Gibbs measure.
We are now in a position to prove our main result on the scaling limit of covariances of the random heights in the ∇ϕ-model.
Proof of Theorem 1.12. Recall that the existence of a stationary, shift-invariant, ergodic ϕ-Gibbs measure µ has been shown in Theorem 5.6 above. Further, the environment ω defined in (5.9) satisfies Assumption 1.9 by the ergodicity of µ. Note also that ω t (e) ≥ c − > 0 for any e ∈ E d and t > 0 by Assumption 5.1. Thus we may set q = ∞ in Assumption 4.2, which then reduces to (1.7). The Helffer-Sjöstrand relation (5.10) gives Cov µ (φ 0 (0), φ t (x)) = ∞ 0 E µ p ω (0, t + s, 0, x) ds. which is the claim. Note that Theorem 1.11 gives uniform convergence of the integrand on any compact interval [0, T ] and Corollary 4.4 gives that g(s) = cs − d 2 is a dominating function, integrable on [T, ∞) since d ≥ 3. Therefore, by dominated convergence we are justified in interchanging the limit and the integral.
Having proven Theorem 1.12 we finally provide polynomial moment bounds on the heights φ under any ergodic, shift-invariant, stationary ϕ-Gibbs measure. This may be useful in verifying the moment condition of Theorem 1.12.
The proof will require the following comparison estimate for φ t and φ Ln t where L n := [−n, n] d ∩ Z d for n ∈ N. gives for all t > T ǫ,M , with constant c independent of M . However, φ t is stationary with respect to µ so (5.18) in fact holds for all t ≥ 0. We conclude by the monotone convergence theorem, letting M ↑ ∞, that E µ φ 0 (0) p < ∞.