1 Jun 2013 Keywords Classiﬁcation 82C41

arXiv:1306.0076v1 [math.PR] 1 Jun 2013 Keywords Classification 82C41. 35K08, Subject Mathematics 2000 AMS htatpclevrneths‘a’rgosta edt ec be to need model. p that regions the has ‘bad’ in environment, has degeneracy environment the typical some of a is that realization there fixed when a especially task, for o is, behavior that the Studying law, resu probability. in early convergence a the the of established of of form possibly distribution or robust the environment, most the is, of The that realizations law, 43]. annealed the 42, for 40, limits [27, 1980s the to unn title Running model conductance random nainepicpe o admwlsin walks random for principles Invariance Introduction 1 ∗ † eerhprilyspotdb S rnsNFGatDMS-12 Grant NSF Grants NSF by supported partially Research eerhprilyspotdb h rn-nAdfrScien for Grant-in-Aid the by supported partially Research h eeatmxn ie utemr,i h nfrl litccs,w case, asymp elliptic boundaries. existing uniformly general resu improve the more similar to with in a domains Furthermore, us for establish allows principles We invariance time. which mixing box, area. relevant this a the in in mo problems model Brownian open conductance reflecting important a to the diffusively super a of rescaled in a below when on from converges walk uniformly etc., random bounded the conductances rando random that amongst in prove or walks we random particular, In for principles boundary. invariance quenched establish we i iihe ometnintermadmkn ulueo two-side of use full making and theorem extension form Dirichlet a Via hnQn Chen Zhen-Qing unhdivrac rnil,Drcltfr,ha kern heat form, Dirichlet principle, invariance quenched : : unhdivrac rnilsfrrno ei ihaboundary. a with media random for principles invariance Quenched o admWlsadElpi Diffusions Elliptic and Walks Random for nRno ei ihBoundary with Media Random in unhdIvrac Principles Invariance Quenched , ∗ ai .Croydon A. David d ue4 2013 4, June dmninlrvril admevrnet aeback date environments random reversible -dimensional Abstract 1 icRsac B 2407ad()25247007. (A) and 22340017 (B) Research tific rmr 03,6F7 eodr 31C25, Secondary 60F17; 60K37, Primary : lgtysrne ttmn novn some involving statement stronger slightly admwl vrgdoe h possible the over averaged walk random 67,adNSCGat11128101. Grant NNSFC and 06276, h admwlsudrtequenched the under walks random the f and nrle.Nvrhls,oe h last the over Nevertheless, ontrolled. hsi eas ti fe h case the often is it because is This oe ob uhmr difficult more much a be to roved t nti racnendscaling concerned area this in lts aah Kumagai Takashi rtclproaincluster percolation critical in hc a enone been has which tion, afsae quarter-space, half-space, l ueciia percolation, supercritical el, niomnswt a with environments m etkre estimates, kernel heat d rsn quenched present e oi siae for estimates totic tfrterandom the for lt † Z2 Figure 1: A section of the unique infinite cluster for supercritical percolation on + with parameter p = 0.52. decade significant work has been accomplished in this direction. Indeed, in the important case of the random walk on the unique infinite cluster of supercritical (bond) percolation on Zd, building on the detailed transition density estimates of [6], a Brownian motion scaling limit has now been established [15, 46, 52]. Additionally, a number of extensions to more general random conductance models have also been proved [3, 8, 17, 45]. Whilst the above body of work provides some powerful techniques for overcoming the technical challenges involved in proving quenched invariance principles, such as studying ‘the environment viewed from the particle’ or the ‘harmonic corrector’ for the walk (see [16] for a survey of the recent developments in the area), these are not without their limitations. Most notably, at some point, the arguments applied all depend in a fundamental way on the translation invariance or ergodicity under random walk transitions of the environment. As a consequence, some natural variations of the problem are not covered. Consider, for example, supercritical percolation in a half-space Z Zd−1, or possibly an orthant of Zd. Again, there is a unique infinite cluster (Figure 1 shows + × a simulation of such in the first quadrant of Z2), upon which one can define a random walk. Given the invariance principle for percolation on Zd, one would reasonably expect that this process would converge, when rescaled diffusively, to a Brownian motion reflected at the boundary. After all, as is illustrated in the figure, the ‘holes’ in the percolation cluster that are in contact with the boundary are only on the same scale as those away from it. However, the presence of a boundary means that the translation invariance/ergodicity properties necessary for applying the existing arguments are lacking. For this reason, it has been one of the important open problems in this area to prove the quenched invariance principle for random walk on a percolation cluster, or amongst random

2 conductances more generally, in a half-space (see [15, Section B] and [16, Problem 1.9]). Our aim is to provide a new approach for overcoming this issue, and thereby establish invariance principles within some general framework that includes examples such as those just described. The approach of this paper is inspired by that of [19, 20], where the invariance principle for random walk on grids inside a given Euclidean domain D is studied. It is shown first in [19] for a class of bounded domains including Lipschitz domains and then in [20] for any bounded domain D that the simple random walk converges to the (normally) reflecting Brownian motion on D when the mesh size of the grid tends to zero. Heuristically, the normally reflecting Brownian motion is a continuous Markov process on D that behaves like Brownian motion in D and is ‘pushed back’ instantaneously along the inward normal direction when it hits the boundary. See Section 2 for a precise definition and more details. The main idea and approach of [19, 20] is as follows: (i) show that random walk killed upon hitting the boundary converges weakly to the absorbing Brownian motion in D, which is trivial; (ii) establish tightness for the law of random walks; (iii) show any sequential limit is a symmetric Markov process and can be identified with reflecting Brownian motion via a Dirichlet form characterization. In [19, 20], (ii) is achieved by using a forward- backward martingale decomposition of the process and the identification in (iii) is accomplished by using a result from the boundary theory of Dirichlet form, which says that the reflecting Brownian motion on D is the maximal Silverstein’s extension of the absorbing Brownian motion in D; see [19, Theorem 1.1] and [23, Theorem 6.6.9]. For quenched invariance principles for random walks in random environments with a boundary, step (i) above can be established by applying a quenched invariance principle for the full-space case. For step (ii), i.e. establishing tightness, the forward-backward martingale decomposition method does not work well with unbounded random conductances. To overcome this difficulty, as well as for the desire to establish an invariance principle for every starting point, we will make the full use of detailed two-sided heat kernel estimates for random walk on random clusters. In particular, we provide sufficient conditions for the subsequential convergence that involve the Hölder continuity of harmonic functions (see Section 2.2). This continuity property can be verified in examples by using existing two-sided heat kernel bounds. We remark that the corrector-type methods for full- space models, such as the approach of [15], often require only upper bounds on the heat kernel. Using the Hölder regularity, we can further show that any subsequential limit of random walks in random environments is a conservative symmetric Hunt process with continuous sample paths. In step (iii), we can identify the subsequential limit process with the reflecting Brownian motion by a Dirichlet form argument (see Theorem 2.1). In summary, our approach for proving quenched invariance principles for random walks in random environments with a boundary encompasses two novel aspects: a Dirichlet form extension argument and the full use of detailed heat kernel estimates. The full generality of the random conductance model to which we are able to apply the above argument is presented in Section 3. As an illustrative application of Theorem 2.1, though, we state here a theorem that verifies the conjecture described above concerning the diffusive behavior of the random walk on a supercritical percolation cluster on a half-space, quarter-space, etc. We recall that the variable speed random walk (VSRW) on a connected (unweighted) graph is the continuous time Markov process that jumps from a vertex at a rate equal to its degree to a uniformly chosen neighbor (see Section 3 for further details). In this setting, similar results to that stated can be

3 obtained for the so-called constant speed random walk (CSRW), which has mean one exponential holding times, or the discrete time random walk (see Remark 3.18 below).

Theorem 1.1. Fix d , d Z such that d 1 and d := d + d 2. Let be the unique infinite 1 2 ∈ + 1 ≥ 1 2 ≥ C1 cluster of a supercritical bond percolation process on Zd1 Zd2 , and let Y = (Y ) be the associated + × t t≥0 VSRW. For almost-every realization of , it holds that the rescaled process Y n = (Y n) , as C1 t t≥0 defined by n −1 Yt := n Yn2t, started from Y n = x n−1 , where x x Rd1 Rd2 , converges in distribution to X ; t 0 , 0 n ∈ C1 n → ∈ + × { ct ≥ } where c (0, ) is a deterministic constant and X ; t 0 is the (normally) reflecting Brownian ∈ ∞ { t ≥ } motion on Rd1 Rd2 started from x. + × As an alternative to unbounded domains, one could consider compact limiting sets, replacing Y n in the previous theorem by the rescaled version of the variable speed random walk on the largest percolation cluster contained in a box [ n,n]d Zd, for example. As presented in Section 4.1, − ∩ another application of Theorem 2.1 allows an invariance principle to be established in this case as well, with the limiting process being Brownian motion in the box [ 1, 1]d, reflected at the boundary. − Consequently, we are able to refine the existing knowledge of the mixing time asymptotics for the sequence of random graphs in question from a tightness result [13] to an almost-sure convergence one (see Corollary 4.4 below). Note that the analytical part of our results (namely Section 2) holds for a large class of possibly non-smooth domains, called uniform domains, which include (global) Lipschitz domains and the classical van Koch snowflake planar domain as special cases. Although in the percolation setting we only consider relatively simple domains with ‘flat’ boundaries, this is mainly for technical reasons so that deriving the percolation estimates in Section 3.1 required for our proofs is manageable. Indeed, in the case when we restrict to uniformly elliptic random conductances, so that controlling the clusters of extreme conductances is no longer an issue, we are able to derive from Theorem 2.1 quenched invariance principles in any uniform domain. These are discussed in Section 4.2. Moreover, in Section 4.3 we explain how the same techniques can be applied in the uniformly elliptic random divergence form setting. (We refer to [41, 38] and the references therein for the history of homogenization for diffusions in random environments.) The remainder of the paper is organized as follows. In Section 2, we introduce an abstract framework for proving invariance principles for reversible Markov processes in a Euclidean domain. This is applied in Section 3 to our main example of a random conductance model in half-spaces, quarter-spaces, etc. The details of the other examples discussed above are presented in Section 4. Our results for the random conductance model depend on a number of technical percolation estimates, some of the proofs of which are contained in the appendix that appears at the end of this article. Finally, the appendix also contains a proof of a generalization of existing quenched invariance principles that allows for arbitrary starting points (previous results have always started the relevant processes from the origin, which will not be enough for our purposes).

Finally, in this paper, for a locally compact separable metric space E, we use Cb(E) and C∞(E) to denote the space of bounded continuous functions on E and the space of continuous functions on E that vanish at inﬁnity, respectively. The space of continuous functions on E with compact

4 support will be denoted by C (E). For real numbers a, b, we use a b and a b for max a, b and c ∨ ∧ { } min a, b , respectively. { } 2 Framework

The following definition is taken from Väisälä[55], where various equivalent definitions are discussed. An open connected subset D of Rd is called uniform if there exists a constant C such that for every x,y D there is a rectifiable curve γ joining x and y in D with length(γ) C x y ∈ ≤ | − | and moreover min x z , z y C dist(z, ∂D) for all points z γ. Here dist(z, ∂D) is the {| − | | − |} ≤ ∈ Euclidean distance between the point z and the set ∂D. Note that a uniform domain is called inner uniform in [35, Definition 3.6]. For example, the classical van Koch snowflake domain in the conformal mapping theory is a uniform domain in R2. Every (global) Lipschitz domain is uniform, and every non-tangentially accessible domain defined by Jerison and Kenig in [37] is a uniform domain (see (3.4) of [37]). How- ever, the boundary of a uniform domain can be highly nonrectifiable and, in general, no regularity of its boundary can be inferred (besides the easy fact that the Hausdorff dimension of the boundary is strictly less than d). It is known (see Example 4 on page 30 and Proposition 1 in Chapter VIII of [39]) that any uniform domain in Rd has m(∂D) = 0 and there exists a positive constant c> 0 such that

m(D B(x, r)) c rn for all x D and 0 < r 1, (2.1) ∩ ≥ ∈ ≤ where m denotes the Lebesgue measure in Rd. Let D be a uniform domain in Rd. Suppose (A(x)) is a measurable symmetric d d matrix- x∈D × valued function such that c−1I A(x) cI for a.e. x D, (2.2) ≤ ≤ ∈ where I is the d-dimensional identity matrix and c is a constant in [1, ). Let ∞ 1 (f, g) := f(x) A(x) g(x)dx for f, g W 1,2(D), (2.3) E 2 ∇ · ∇ ∈ ZD where W 1,2(D) := f L2(D; m) : f L2(D; m) . ∈ ∇ ∈ An important property of a uniform domain D Rd is that there is a bounded linear extension ⊂ operator T : W 1,2(D) W 1,2(Rd) such that Tf = f a.e. on D for f W 1,2(D). It follows that → ∈ ( ,W 1,2(D)) is a regular Dirichlet form on L2(D; m) and so there is a continuous diffusion process E X = (X ,t 0; P ,x D) associated with it, starting from -quasi-every point. Here a property is t ≥ x ∈ E said to hold -quasi-everywhere means that there is a set D having zero capacity with respect E N ⊂ to the Dirichlet form ( ,W 1,2(D)) so that the property holds for points in c. According to [35, E N Theorem 3.10] and (2.1) (see also [12, (3.6)]) X admits a jointly continuous transition density function p(t,x,y) on R D D and + × × c x y 2 c x y 2 c t−d/2 exp 2| − | p(t,x,y) c t−d/2 exp 4| − | (2.4) 1 − t ≤ ≤ 3 − t 5 for every x,y D and 0 < t 1. Here the constants c , , c > 0 depend on the diffusion ∈ ≤ 1 · · · 4 matrix A(x) only through the ellipticity bound c in (2.2). Consequently, X can be refined so that it can start from every point in D. The process X is called a symmetric reflecting diffusion on D. We refer to [21] for sample path properties of X. When A = I, X is the (normally) reflecting Brownian motion on D. Reflecting Brownian motion X on D in general does not need to be semimartingale. When ∂D locally has finite lower Minkowski content, which is the case when D is a Lipschitz domain, X is a semimartingale and admits the following Skorohod decomposition (see [22, Theorem 2.6]): t X = X + W + ~n(X )dL , t 0. (2.5) t 0 t s s ≥ Z0 Here W is the standard Brownian motion in Rd, ~n is the unit inward normal vector field of D on ∂D, and L is a positive continuous additive functional of X that increases only when X is on the boundary, that is, L = t 1 dL for t 0. Moreover, it is known that the reflecting t 0 {Xs∈∂D} s ≥ Brownian motion spends zero Lebesgue amount of time at the boundary ∂D. These together with R (2.5) justify the heuristic description we gave in the introduction for the reflecting Brownian motion in D.

2.1 Convergence to reﬂecting diﬀusion

In this subsection, D is a uniform domain in Rd and X is a reﬂecting diﬀusion process on D associated with the Dirichlet form ( ,W 1,2(D)) on L2(D; m) given by (2.3). Denote by (XD, PD,x D) E x ∈ the subprocess of X killed on exiting D. It is known (see, e.g., [23]) that the Dirichlet form of XD on L2(D; m) is ( ,W 1,2(D)), where E 0 W 1,2(D) := f W 1,2(D) : f = 0 -quasi-everywhere on ∂D . 0 ∈ E Suppose that D ; n 1 is a sequence of Borel subsets of D such that each D supports { n ≥ } n a measure mn that converges vaguely to the Lebesgue measure m on D. The following result plays a key role in our approach to the quenched invariance principle for random walks in random environments with boundary.

Theorem 2.1. For each n N, let (Xn, Pn,x D ) be an m -symmetric Hunt process on D . ∈ x ∈ n n n Assume that for every subsequence n , there exists a sub-subsequence n and a continu- { j} { j(k)} ous conservative m-symmetric strong Markov process (X, P ,x D) such that the following three x ∈ conditions are satisﬁed: e e nj(k) (i) for every xn x with xn Dn , Px converges weakly in D([0, ), D) to Px; j(k) → j(k) ∈ j(k) nj(k) ∞ (ii) XD, the subprocess of X killed upon leaving D, has the same distribution as XD; e

(iii) the Dirichlet form ( ˜, ˜) of X on L2(D; m) has the properties that e E Fe ˜ ande ˜(f,f) C (f,f) for every f , (2.6) C ⊂ F E ≤ 0 E ∈C where is a core for the Dirichlet form ( ,W 1,2(D)) and C [1, ) is a constant. C E 0 ∈ ∞

6 It then holds that for every x x with x D , (Xn, Pn ) converges weakly in D([0, ), D) to n → n ∈ n xn ∞ (X, Px).

Proof. With both X and X being m-symmetric Hunt processes on D, it suffices to show that their corresponding (quasi-regular) Dirichlet forms on L2(D; m) are the same; that is ( , ) = E F ( ,W 1,2(D)). Conditione (iii) immediately implies that W 1,2(D) and E ⊂ F e e (f,f) C (f,f) for every f W 1,2(D). E ≤ 0E ∈ e Next, observe that since Xeis a diffusion process admitting no killings, its associated Dirichlet form is strongly local. Thus for every u , (u, u) = 1 µ (D), where µ is the energy measure ∈ F E 2 hui hui corresponding to u. By thee proof of [47, Proposition on page 389] , e e e e µ (dx) C u(x)A(x) u(x)dx cC u(x) 2dx on D for every u W 1,2(D). hui ≤ 0∇ ∇ ≤ 0|∇ | ∈ This in particular implies that e µ (∂D)=0 for u W 1,2(D). (2.7) hui ∈ On the other hand, by the strong local property of µ and the fact that XD has the same e hui distribution as XD, we have that every bounded function in – the collection of which we denote 1,2 F e by b – is locally in W0 (D) and e F e 1 (x)µ (dx)= 1 (x) u(x)A(x) u(x)dx for u . (2.8) e D hui D ∇ ∇ ∈ Fb This together with (2.7) implies that (u, u)= (u, u) for every bounded u W 1,2(D), and hence e E E e∈ for every u W 1,2(D). Furthermore, (2.8) implies that for u , u(x) 2dx < and so ∈ ∈ Fb D |∇ | ∞ u W 1,2(D). Consequently we have e W 1,2(D) and thus ( , ) = ( ,W 1,2(D)). R ∈ F ⊂ E F e E n nj(k) Remark 2.2. (i) Note that if (X )t≥0 is conservative for each n N and Px is tight, then t e e e∈ { nj(k) } X˜ is conservative. (ii) Theorem 2.1 can be viewed as a variation of [19, Theorem 1.1]. The difference is that in [19, Theorem 1.1], the constant C0 in (2.6) is assumed to be 1 but the limiting process X only need to be Markov and does not need to be continuous a priori, while for Theorem 2.1, the condition on the e constant C0 is weaker but we need to assume a priori that the limit process X is continuous.

2.2 Suﬃcient condition for subsequential convergence e

In this subsection, we give some sufficient conditions for the subsequential convergence of Xn ; in { } other words, sufficient conditions for (i) in Theorem 2.1. For simplicity, we assume that 0 D for ∈ n all n 1 throughout this section, though note this restriction can easily be removed. ≥ We start by introducing our first main assumption, which will allow us to check an equicontinuity property for the λ-potentials associated with the elements of Xn (see Proposition 2.4 { } below). In the statement of the assumption, we suppose that (δn)n≥1 is a decreasing sequence in [0, 1] with lim δ = 0 and such that x y δ for all distinct x,y D . (When δ 0, this n→∞ n | − |≥ n ∈ n n ≡ condition always holds. However, our assumption will give an additional restriction.) We denote by n B(x, r) the Euclidean ball of radius r centered at x, and τA(X ) the first exit time of the process Xn from the set A.

7 Assumption 2.3. There exist c1, c2, c3, β, γ (0, ), N0 N such that the following hold for all 1/2 1/2 ∈ ∞ ∈ n N , x B(0, c n ), and δn r 1. ≥ 0 0 ∈ 1 ≤ ≤ (i) For all x B(x , r/2) D , ∈ 0 ∩ n En τ (Xn) c rβ. x B(x0,r)∩Dn ≤ 2 n (ii) If hn is bounded in Dn and harmonic (with respect to X ) in a ball B(x0, r), then

x y γ h (x) h (y) c | − | h for x,y B(x , r/2) D . | n − n |≤ 3 r k nk∞ ∈ 0 ∩ n

Deﬁne for λ> 0 the λ-potential

∞ U λf(x)= En e−λtf(Xn) dt for x D . n x t ∈ n Z0 Proposition 2.4. Under Assumption 2.3 there exist C = C (0, ) and γ′ (0, ) such that λ ∈ ∞ ∈ ∞ the following holds for any bounded function f on D , for any n N and any x,y D such that n ≥ 0 ∈ n x B(0, c n1/2) and x y < 1/4: ∈ 1 | − | ′ U λf(x) U λf(y) C x y γ f . (2.9) | n − n |≤ | − | k k∞ In particular, we have

λ λ lim sup sup Un f(x) Un f(y) = 0. (2.10) δ→0 1/2 | − | n≥N0 x,y∈Dn∩B(0,c1n ): |x−y|<δ

1/2 Proof. The proof is similar to that of [11, Proposition 3.3]. Fix x0 B(0, c1n ) Dn, let 1 r 1/2 n n ∈ ∩ ≥ ≥ δn , and suppose x,y B(x , r/2). Set τ := τ (X ). By the strong Markov property, ∈ 0 r B(x0,r)∩Dn n τr λ n −λt n n −λτ n λ n n λ n U f(x) = E e f(X ) dt + E (e r 1)U f(X n ) + E U f(X n ) n x t x n τr x n τr 0 − Z h i h i = I1 + I2 + I3, and similarly when x is replaced by y. We have by Assumption 2.3(i) that

I f Enτ n c rβ f , | 1| ≤ k k∞ x r ≤ 2 k k∞ and by noting U λf 1 f that k n k∞ ≤ λ k k∞ I λEnτ n U λf c rβ f . | 2|≤ x r k n k∞ ≤ 2 k k∞ Similar statements also hold when x is replaced by y. So,

λ λ β n λ n n λ n U f(x) U f(y) 4c2r f ∞ + E U f(X n ) E U f(X n ) . (2.11) n − n ≤ k k x n τr − y n τr

8 n λ n d But z E U f(X n ) is bounded in R and harmonic in B(x0, r), so by Assumption 2.3(ii), the → z n τr second term in (2.11) is bounded by c ( x y /r)γ U λf . So by U λf 1 f again, we 3 | − | k n k∞ k n k∞ ≤ λ k k∞ have x y γ U λf(x) U λf(y) c rβ + λ−1 | − | f for x,y B(x , r/2). (2.12) n − n ≤ r k k∞ ∈ 0

1/2 1/2 2 Now, for distinct x,y D with x B(0, c n ) and (δn ) x y < 1/4 (note that since ∈ n ∈ 1 ≤ | − | x y δ for distinct x and y, the ﬁrst inequality always hold), let x = x and r = x y 1/2. | − | ≥ n 0 | − | Then δ r< 1/2 and y B(x , r/2) (because x y = r2 < r/2). Thus we can apply (2.12) to n ≤ ∈ 0 | 0 − | obtain

U λf(x) U λf(y) c x y β/2 + λ−1 x y γ/2 f n − n ≤ | − | | − | k k∞ −1 (β∧γ)/2 c(1 + λ ) x y f ∞. ≤ | − | k k So (2.9) holds with C = c(1 + λ−1) and γ′ = (β γ)/2. The result at (2.10) is immediate from ∧ (2.9).

We note that with an additional mild condition, we can further obtain equi-H¨older continuity of the associated semigroup. (The next proposition will only be used in the proof of Theorem 3.13 below.) Set B := B(0,R) D for R [2, ). Denote by Xn,BR the subprocess of Xn killed R ∩ n ∈ ∞ n,BR n,BR upon exiting BR, and Pt ; t 0 the transition semigroup of X . (When R = , we set n n,B∞ { ≥ } n ∞ (Pt )t≥0 := (Pt )t≥0, i.e. the semigroup of X itself.) For p [1, ], we use p,n,R to denote p ∈ ∞ k · k the L -norm with respect to mn on BR.

Proposition 2.5. Let R [2, ] and t > 0. Suppose there exist c > 0 and N N (that may ∈ ∞ 1 1 ∈ depend on R and t) such that for every g L1(B ,m ), ∈ R n P n,BR g c g , for all n N . k t k∞,n,R ≤ 1k k1,n,R ≥ 1 n,B n Suppose in addition that Assumption 2.3 holds with X R and BR in place of X and Dn, respectively. It then holds that there exist constants c (0, ) and N 1 (that also may depend on R ∈ ∞ 2 ≥ and t) such that ′ P n,BR f(x) P n,BR f(y) c x y γ f , t − t ≤ 2| − | k k2,n,R for every n N , f L 2(B ; m ), and m -a.e. x,y B with x y < 1/4. Here γ′ is the ≥ 2 ∈ R n n ∈ R/2 | − | constant of Proposition 2.4.

Proof. We follow [11, Proposition 3.4]. For notational simplicity, we drop the suﬃces n N and ≥ 0 BR throughout the proof. Using spectral representation theorem for self-adjoint operators, there n,R 2 exist projection operators Eµ = Eµ on the space L (BR; mn) such that ∞ ∞ ∞ 1 f = dE (f), P f = e−µt dE (f),U λf = dE (f). (2.13) µ t µ λ + µ µ Z0 Z0 Z0

Define ∞ −µt h = (λ + µ)e dEµ(f). Z0 9 Since sup (λ + µ)2e−2µt c, we have µ ≤ ∞ ∞ h 2 = (λ + µ)2e−2µt d E (f), E (f) c d E (f), E (f) = c f 2, k k2 h µ µ i≤ h µ µ i k k2 Z0 Z0 where for f, g L2, f, g is the inner product of f and g in L2. Thus h is a well defined function ∈ h i in L2. Now, suppose g L1. By the assumption, P g c g , from which it follows that P g ∈ k t k∞ ≤ k k1 k t k2 ≤ c g . Since sup (λ + µ)e−µt/2 c, using Cauchy-Schwarz we have k k1 µ ≤ ∞ h, g = (λ + µ)e−µt d E (f), g h i h µ i Z0 ∞ 1/2 ∞ 1/2 (λ + µ)e−µt d E (f),f) (λ + µ)e−µt d E (g), g ≤ h µ i h µ i Z0 Z0 ∞ 1/2 ∞ 1/2 c d E (f),f e−µt/2 d E (g), g ≤ h µ i h µ i Z0 Z0 = c f P g c′ f g . k k2k t/2 k2 ≤ k k2k k1 Taking the supremum over g L1 with L1 norm less than 1, this yields h c f . Finally, by ∈ k k∞ ≤ k k2 (2.13), ∞ λ −µt U h = e dEµ(f)= Ptf, a.e., Z0 and so the Hölder continuity of Ptf follows from Proposition 2.4.)

Let D(R+, D) be the space of right continuous functions on R+ having left limits and taking values in D that is equipped with the Skorohod topology. For t 0, we use X to denote the ≥ t coordinate projection map on D(R , D); that is, X (ω)= ω(t) for ω D(R , D). For subsequential + t ∈ + convergence to a diﬀusion, we need the following.

Assumption 2.6. (i) For any sequence x x with x D , Pn is tight in D(R , D). n → n ∈ n { xn } + (ii) For any sequence x x with x D and any ε> 0, n → n ∈ n Pn n lim lim sup xn (J(X , δ) > ε) = 0, δ→0 n→∞

where J(X, δ) := ∞ e−u 1 sup X X du. 0 ∧ δ≤t≤u | t − t−δ| R We need the following well-known fact (see, for example [7, Lemma 6.4]) in the proof of Propo- sition 2.8. For readers’ convenience, we provide a proof here.

Lemma 2.7. Let K be a compact subset of Rd. Suppose f and f , k N, are functions on K such k ∈ that lim f (y ) = f(y) whenever y K converges to y. Then f is continuous on K and f k→∞ k k k ∈ k converges to f uniformly on K.

Proof. We ﬁrst show that f is continuous on K. Fix x K. Let x be any sequence in K that 0 ∈ k converges to it. Since lim f (x) = f(x) for every x K, there is a sequence n N that i→∞ i ∈ k ∈

10 increases to inﬁnity so that f (x ) f(x ) 2−k for every k 1. Since lim f (x )= f(x ), | nk k − k |≤ ≥ k→∞ nk k 0 it follows that lim f(x ) f(x ) = 0. This shows that f is continuous at x and hence on K. k→∞ | 0 − k | 0 We next show that fk converges uniformly to f on K. Suppose not. Then there is ε> 0 so that for every k 1, there are n k and x K so that f (x ) f(x ) > ε. Since K is compact, ≥ k ≥ nk ∈ | nk nk − nk | by selecting a subsequence if necessary, we may assume without loss of generality that x x nk → 0 ∈ K. As lim f (x )= f(x ) by the assumption, we have lim inf f(x ) f(x ) ε. This k→∞ nk nk 0 k→∞ | 0 − nk |≥ contradicts to the fact that f is continuous on K.

Now, applying the argument in [7, Section 6], we can prove that any subsequential limit of n Pn the laws of X under xn is the law of a symmetric diffusion. For this, we need to introduce a projection map from D to D . For each n 1, let φ : D D be a map that projects each n ≥ n → n x D to some φ (x) D that minimizes x y over y D (if there is more than one such ∈ n ∈ n | − | ∈ n point that does this, we choose and fix one). If needed, we extend a function f defined on Dn to be a function on D by setting f(x)= f(φn(x)). Note that each Dn supports the measure mn that converges vaguely to m. This implies that for each x D and r > 0, there is an N 1 so that ∈ ≥ φ (x) B(x, r) for every n N. From this, one concludes that n ∈ ≥ φ (x ) x for every sequence x D that converges to x . (2.14) n n → 0 n ∈ 0

Proposition 2.8. Suppose that Assumptions 2.3 and 2.6 hold and that Xn, Pn,x D is { x ∈ n} conservative for sufficiently large n. For every subsequence n , there exists a sub-subsequence { j} n and a continuous conservative m-symmetric Hunt process (X, P ,x D) such that for { j(k)} x ∈ Pnj(k) D Px every xnj(k) x, xn (k) converges weakly in ([0, ), D) to . → j ∞ e e Proof. For notational simplicity, let us relabel the subsequence as n . We first claim that there e { } exists a (sub-)subsequence n such that U λ f converges uniformly on compact sets for each λ> 0 { j} nj and f C (D). Indeed, let λ be a dense subset of (0, ) and f a sequence of functions in ∈ b { i} ∞ { k} C (D) such that f 1 and whose linear span is dense in (C (D), ). For fixed m and i, b k kk∞ ≤ b k · k∞ λi by Proposition 2.4 and the Ascoli-Arzelàtheorem, there is a subsequence of Un fk that converges uniformly on compact sets. By a diagonal selection procedure, we can choose a subsequence n { j} λi such that Unj fk converges uniformly on compact sets for every m and i to a Hölder continuous λi function which we denote as U fk. Noting that 1 β λ U λ U β = (β λ)U λU β, U λ , U λ U β − , (2.15) n − n − n n k n k∞→∞ ≤ λ k n − n k∞→∞ ≤ λβ

λ λ a careful limiting argument shows that Unj f converges uniformly on compact sets, say to U f, for any λ> 0 and any continuous function f, and (2.15) holds as well for U λ . By the equi-continuity λ λ λ { } of Unj f , we also have Unj f(xnj ) U f(x) for each xnj Dnj that converges to x D. nj → ∈ ∈ nj We next claim that Px converges weakly, say to Px. Indeed, by Assumption 2.6(i), Px is nj { nj } tight, so it suﬃces to show that any two limit points agree. Let P′ and P′′ be any two limit points. Then, one sees that e

∞ ∞ ′ −λs λ ′′ −λs E e f(Xs)ds = U f(x)= E e f(Xs)ds , Z0 Z0 11 for any f C (D). So, by the uniqueness of the Laplace transform, ∈ b ′ ′′ E [f(Xs)] = E [f(Xs)] for almost all s 0 and hence for every s 0 since s Xs is right continuous. So the one- ≥ ′ ≥ ′′ → ′ dimensional distributions of Xt under P and P are the same. Set Psf(x) := E f(Xs). We have nj Ps f(xnj ) Psf(x) for every sequence xnj Dnj that converges to x. Recall the restriction → ∈ nj map φn introduced proceeding the statement of this theorem. It follows from (2.14) that Ps (f nj ◦ φ )(y ) P f(y) for every sequence y D that converges to y. Thus by Lemma 2.7, Ps (f nj nj → s nj ∈ ◦ φnj ) converges to Psf uniformly on compact subsets of D and Psf Cb(D) for every f Cc(D). ∈ nj ∈ For f, g Cc(D) and 0 sstopping time T , e ∈ c E [f(X ) ]= P f(X ), x D. x T +t |FT + t T ∈ See the proof of Theorem 2.3.1 on p.56 of [24]. From it, one gets the strong Markov property of X by a standard measure-theoretic argument (see p.57 of [24]). Since X is continuous and has inﬁnite lifetime, this in fact shows that X is a continuous conservative Hunt process. Finally, for f, g C (D), by the convergence of semigroups and vague convergence of measures, ∈ c it holds that, for every t> 0,

(P nj f)(x)g(x)m (dx) (P f)(x)g(x)m(dx). t nj → t ZDn ZD n Since X j is mn-symmetric, this readily yields the desired m-symmetry of X˜.

Remark 2.9. Note that we did not use any special properties of the Euclidean metric in this section, so that all the arguments in this section can be extended to a metric measure space without any changes.

12 n Pn By Theorem 2.1, Remark 2.2 (i) and Proposition 2.8, we see that in order to prove (X , xn ) converges weakly to (X, P ) in D([0, ), D) as n , it suﬃces to verify condition (ii), (iii) in x ∞ → ∞ Theorem 2.1, vague convergence of the measure mn to m on D, Assumptions 2.3 and 2.6, and the conservativeness of (Xn, Pn ) for each n N. xn ∈ 3 Random conductance model in unbounded domains

In this section we will obtain, as a first application of our theorem, a quenched invariance principle for random walk amongst random conductances on half-spaces, quarter-spaces, etc. The assumptions we make on the random conductances include the supercritical percolation model, and random conductances bounded uniformly from below and with finite first moments. For the main conclusion, see Theorem 3.17. d Fix d , d Z . Let d := d + d , and define a graph (L, EL) by setting L := Z 1 Zd2 and 1 2 ∈ + 1 2 + × EL := e = x,y : x,y L, x y = 1 . Given EL, let (L, ) be the infinite connected { { } ∈ | − | } O ⊆ C∞ O cluster of (L, ), provided it exists and is unique (otherwise set (L, ) := ). O C∞ O ∅ Let µ = (µe)e∈EL be a collection of independent and identically distributed random variables on [0, ), defined on a probability space (Ω, P) such that ∞ P bond Zd p1 := (µe > 0) >pc ( ), (3.1) where pbond(Zd) (0, 1) is the critical probability for bond percolation on Zd. We assume that c ∈ there is c> 0 so that P (µ (0, c)) = 0, (3.2) e ∈ and E (µ ) < . (3.3) e ∞ This framework includes the special cases of supercritical percolation (where P (µe =1) = p1 = 1 P (µ = 0)) and the random conductance model with conductances bounded from below (that − e is, P (c µ < ) = 1 for some c > 0) and having finite first moments. For each x L, set ≤ e ∞ ∈ µx = y∼x µxy. Set := e EL : µ > 0 , := (L, ). P O1 { ∈ e } C1 C∞ O1 Note that for the random conductance model bounded from below, = L. C1 For each realization of , there is a continuous time Markov chain Y = (Y ) on with C1 t t≥0 C1 transition probabilities P (x,y)= µxy/µx, and the holding time at each x 1 being the exponential −1 ∈C distribution with mean µx . Such a Markov chain is sometimes called a variable speed random walk (VSRW). The corresponding Dirichlet form is ( ,L2( ; ν)), where ν is the counting measure E C1 on and C1 1 (f, g)= (f(x) f(y))(g(x) g(y))µ for f, g L2( ; ν). E 2 − − xy ∈ C1 x,y∈CX1,x∼y The corresponding discrete Laplace operator is f(x)= (f(y) f(x))µ . For each f, g that LV y − xy have finite support, we have P (f, g)= ( f)(x)g(x). E − LV xX∈C1 13 We will establish a quenched invariance principle for Y in Section 3.3, but we first need to derive some preliminary estimates regarding the geometry of and the heat kernel associated with Y . C1 3.1 Percolation estimates

In this section, we derive a number of useful properties of the underlying percolation cluster . C1 Most importantly, we introduce the concept of ‘good’ and ‘very good’ balls for the model and provide estimates for the probability of such occurring, see Deﬁnition 3.4 and Proposition 3.5 below. Since a variety of percolation models will appear in the course of this paper, let us now make explicit that the critical probability for bond/site percolation on an inﬁnite connected graph containing the vertex 0 is

p := inf p [0, 1] : P (0 is in an inﬁnite connected cluster of open bonds/sites) > 0 , c { ∈ p } where Pp is the law of parameter p bond/site percolation on the graph in question. Note in L bond Zd particular that the critical probability for bond percolation on is identical to pc ( ), see [32, Theorem 7.2] (which is the bond percolation version of a result originally proved as [33, Theorem A]). Recall := e EL : µ > 0 , := (L, ). O1 { ∈ e } C1 C∞ O1 That is non-empty almost-surely is guaranteed by [10, Corollary to Theorem 1.1] (this covers C1 d 3, and, as is commented there, the case d = 2 can be tackled using techniques from [36]). ≥ Now, suppose that µ is actually a restriction of independent and identically distributed (under P ) random variables (µe)e∈EZd , where EZd are the usual nearest-neighbor edges for the integer lattice Zd, and deﬁne

d ˜ := e EZd : µ > 0 , ˜ := (Z , ˜ ). O1 { ∈ e } C1 C∞ O1 For suﬃciently large K so that

q = q(K) := P(0 <µ < K−1)+ P(µ > K)

˜ := ˜ −1 , OR O(0,K )∪(K,∞) ˜ := e ˜ : e e′ = for some e′ ˜ , OS ∈ O1 ∩ 6 ∅ ∈ OR ˜ := n˜ ˜ . o O2 O1\OS

We will also deﬁne := ˜ EL, and set := (L, ) – the next lemma will guarantee that O2 O2 ∩ C2 C∞ O2 this set is non-empty almost-surely. To represent the set of ‘holes’, let := . Moreover, for x , let (x) be the connected H C1\C2 ∈C1 H component of containing x. The following lemma provides control on the size of these C1\C2 components. Since its proof is a somewhat technical adaptation to our setting of that used to establish [3, Lemma 2.3], which dealt with the whole Zd model, we defer this to the appendix.

14 Note, though, that in the percolation case (i.e. when µe are Bernoulli random variables) or the uniformly elliptic random conductor case, the proof of the result is immediate; indeed, for large enough K, we have that = , and so = . O2 O1 H ∅ Lemma 3.1. For sufficiently large K, the following holds. (i) All the connected components of are finite. Furthermore, there exist constants c , c such H 1 2 that: for each x L, ∈ P (x and diam( (x)) n) c e−c2n, ∈C1 H ≥ ≤ 1 d where diam denotes the diameter with respect to the ℓ∞ metric on Z . (ii) There exists a constant α such that, P-a.s., for large enough n, the volume of any hole inter- secting the box [ n,n]d L is bounded above by (log n)α. − ∩ In what follows, we will need to make comparisons between two graph metrics on ( , ), and C1 O1 the Euclidean metric. The first of these, d1, will simply be defined to be the shortest path metric on ( , ), considered as an unweighted graph. To define the second metric, d , we follow [8] by C1 O1 1 defining edge weights t(e) := C µ−1/2, A ∧ e where C < is a deterministic constant, and then letting d be the shortest path metric on A ∞ 1 ( , ), considered as a weighted graph (in [8], the analogous metric was denoted d˜). We note C1 O1 that the latter metric on satisfies C1 2 C−2 µ d (x,y) d (x, z) 1, A ∨ {y,z} 1 − 1 ≤ for any x,y,z 1 with y, z 1. Observe that, since the weights t(e) are bounded above by CA, ∈C { } ∈ O we immediately have that d1 is bounded above by CAd1, Hence the following lemma establishes both d1 and d1 are comparable to the Euclidean one. An easy consequence of this is the comparability of balls in the different metrics, see Lemma 3.3. The proofs of both these results are deferred to the appendix.

Lemma 3.2. There exist constants c , c , c such that: for R 1, 1 2 3 ≥

−c3R sup P (x,y 1 and d1(x,y) c1R) c2e , (3.4) x,y∈L: ∈C ≥ ≤ |x−y|≤R and also, for every x,y L, ∈ P x,y and d (x,y) c−1 x y c e−c3|x−y|. (3.5) ∈C1 1 ≤ 1 | − | ≤ 2

Lemma 3.3. There exist constants c , c , c , c such that: for every x L, R 1, 1 2 3 4 ∈ ≥ c P x B (x, c R) B (x, R) B (x,C R) B (x, c R) c e−c4R, { ∈C1}∩ C1 ∩ E 1 ⊆ 1 ⊆ 1 A ⊆ E 2 ≤ 3 where B (x, R) is a ball in the metric space ( , d ), B (x, R) is a ball in ( , d ), and B (x, R) is 1 C1 1 1 C1 1 E a Euclidean ball.

15 We continue by adapting a deﬁnition for ‘good’ and ‘very good’ balls from [3]. In preparation 0 0 0 0 for this, we deﬁne µ := 1 , set µ := L µ for x L, and then extend µ to a measure e {e∈O1} x y∈ {x,y} ∈ on L. Moreover, we set β := 1 2(1 + d)−1. − P

Definition 3.4. (i) Let CV ,CP ,CW ,CR,CD be fixed strictly positive constants. We say the pair (x, R) R is good if: ∈C1 × + B (x,C−1r) B (x, r) B (x,C r), r R, (3.6) 1 A ⊆ 1 ⊆ 1 D ∀ ≥ y z C−1R, y B (x,R/2), z B (x, 8R/9)c, (3.7) | − |≥ R ∀ ∈ 1 ∈ 1 C Rd µ0(B (x, R)), (3.8) V ≤ 1 diam( (y)) Rβ, y B (x, R) , (3.9) H ≤ ∀ ∈ E ∩C1 and the weak Poincaréinequality

2 f(y) fˇ µ0 C R2 f(y) f(z) 2 (3.10) − B1(x,R) y ≤ P | − | y∈BX1(x,R) y,z∈B1X(x,CW R): {y,z}∈O1 holds for every f : B (x,C R) R. (Here fˇ is the value which minimizes the left-hand 1 W → B1(x,R) side of (3.10)). (ii) We say a pair (x, R) R is very good if: there exists N = N such that (y, r) is good ∈C1 × + (x,R) whenever y B (x, R) and N r R. We can always assume that N 2. Moreover, if N M, ∈ 1 ≤ ≤ ≥ ≤ we will say that (x, R) is M-very good. (α) (iii) Let α (0, 1]. For x 1, we deﬁne Rx to be the smallest integer M such that (x, R) is α ∈ ∈ C (α) R -very good for all R M. We set Rx = 0 if x . ≥ 6∈ C1 The following proposition, which is an adaptation of [3, Proposition 2.8], provides bounds for (α) the probabilities of these events and for the distribution of Rx .

Proposition 3.5. There exist c1, c2,CV ,CP ,CW ,CR,CD (depending on the law of µ and the dimension d) such that the following holds. For x L, R 1, α (0, 1], ∈ ≥ ∈ β P (x , (x, R) is not good) c e−c2R , (3.11) ∈C1 ≤ 1 αβ P (x , (x, R) is not Rα-very good) c e−c2R . (3.12) ∈C1 ≤ 1 Hence αβ P x ,R(α) n c e−c2n . (3.13) ∈C1 x ≥ ≤ 1

Proof. That P (x , (3.6) does not hold) c e−c4R ∈C1 ≤ 3 is a straightforward consequence of Lemma 3.3.

16 For the second property, we have

P (x , (3.7) does not hold) ∈C1 = P x , y B (x,R/2), z B (x, 8R/9)c : y z R/3 ≤ −1 ∈C y∈BE (x,c5R) z∈B (y,C R) X EX R c e−c9R ≤ 8 where we apply Lemma 3.3 to deduce the ﬁrst inequality, and (3.4) to obtain the ﬁnal one. For (3.8), applying Lemma 3.3 again yields

P (x , (3.8) does not hold) ∈C1 = P x , µ0(B (x, R))

β P (x 1, (3.9) does not hold) P y 1, diam( (y)) > R , ∈C ≤ L ∈C H y∈BEX(x,R)∩

β −c18R which may be bounded above by c17e by applying Lemma 3.1. Finally, for the Poincar´einequality, we will apply [6, Proposition 2.12]. In particular, this result yields that if Q is a cube of side-length 2R contained in L, +(Q) is the largest connected C component of the graph (Q, ), and H(Q) is the event that O1 min (f(y) a)2 µ0 CR2 f(y) f(z) 2 a y + − ≤ + | − | y∈CX(Q) y,z∈CX(Q): {y,z}∈O1

17 β for every f : +(Q) R, then P(H(Q)c) c e−c20R . Furthermore, it is clear that if H(Q) holds C → ≤ 19 and also B (x, R) +(Q) B (x, c R), then (3.10) holds. This means that 1 ⊆C ⊆ 1 21

P (x 1, (3.10) does not hold) ∈C β c c e−c23R + P x B (x, R) +(Q) B (x, c R) , ≤ 22 { ∈C1}∩ 1 ⊆C ⊆ 1 21 where Q is chosen such that B (x, R) L Q. Noting as above that P(x +(Q)) c e−c25R, E ∩ ⊆ ∈C1\C ≤ 24 at the expense of adjusting constants, we may replace x by x +(Q) in the above { ∈C1} { ∈C1 ∩C } bound. On the event x +(Q) B (x, R) +(Q) c, it is elementary to check that { ∈ C1 ∩C } ∩ { 1 ⊆ C } B (x, R) B (x, R), which is impossible. Since +(Q) B (x, 2R), we have thus shown that 1 6⊆ E C ⊆ E

P (x 1, (3.10) does not hold) ∈C β c e−c27R + P ( x B (x, 2R) B (x, cR) c) . ≤ 26 { ∈C1} ∩ { E ⊆ 1 } β −c29R By applying Lemma 3.3 once again, this expression is bounded above by c28e , and so we have completed the proof of (3.11). Given (3.11), a simple union bound subsequently yields (3.12), and the inequality at (3.13) is a straightforward consequence of this.

Remark 3.6. It only requires a simple argument to check that if (x, R) is good and y satisﬁes ∈C1 d (x,y) C R, then 1 ≥ D C−1d (x,y) d (x,y) C d (x,y) D 1 ≤ 1 ≤ A 1 (cf. [8, Lemma 2.10(a)]).

Finally, we state a bound that allows us to compare ν withν ˜, which is the measure deﬁned similarly from the whole Zd model, i.e. uniform measure on ˜ . Its proof can be found in the C1 appendix.

Lemma 3.7. There exists a constant c such that if Q L is a cube of side length n, then ⊆ P ν˜(Q) ν(Q) nd−1(log n)d+1 cn−2. − ≥ ≤

3.2 Heat kernel estimates

Let D = n−1 , D = Rd1 Rd2 (here R := [0, )). Let Y be the VSRW on , and for a given n C1 + × + ∞ C1 realization of = (ω), ω Ω, write P ω for the law of Y started from x . Moreover, deﬁne C1 C1 ∈ x ∈C1 Z to be the trace of Y on , that is, the time change of Y by the inverse of A = t 1 ds. C2 t 0 (Ys∈C2) Speciﬁcally, writing a = inf s : A >t for the right-continuous inverse of A, we set t { s } R

Z = Ya , t 0. t t ≥ Note that unlike Y , the process Z may perform long jumps by jumping over the holes of . If C2 x (ω) then we have Z = Y = x, P ω-a.s., but otherwise Z = Ya . ∈C2 0 0 x 0 0

18 Given the percolation estimates of Section 3.1, we can follow [3, Section 4] to establish the following theorems, which correspond to Proposition 4.7(c) and Theorem 4.11 in [3]. We remark that the second of the two results will be used in this paper only for the proof of Theorem 3.12. Since it is the case that, given Proposition 3.5, the proofs are a simple modiﬁcation of those in [3], we omit them. For the statement of the ﬁrst result, we set

2 e−R /t if t > e−1R, Ψ(R,t)= (e−R log(R/t) if t < e−1R.

Proposition 3.8. Write τ Z = inf t : Z A , τ Y = inf t : Y A . There exist constants A { t 6∈ } A { t 6∈ } δ, c (0, ) and random variables (R ,x L) with i ∈ ∞ x ∈ δ P(R n,x ) c e−c2n , (3.14) x ≥ ∈C1 ≤ 1 such that the following holds: for x , t> 0 and R R , ∈C1 ≥ x ω Z P (τ

Theorem 3.9. There exist: constants δ, c (0, ); a set Ω Ω with P(Ω ) = 1; and random i ∈ ∞ 1 ⊂ 1 variables (S ,x L) satisfying S (ω) < for each ω Ω and x (ω), and x ∈ x ∞ ∈ 1 ∈C2 δ P(S n,x ) c e−c2n ; x ≥ ∈C2 ≤ 1 such that the following statements hold. (a) For x,y (ω) the transition density of Z, as deﬁned by setting qZ(x,y) := P ω(Z = y), ∈ C2 t x t satisﬁes

qZ(x,y) c t−d/2 exp( c x y 2/t), t x y S , t ≤ 3 − 4| − | ≥ | − |∨ x qZ(x,y) c t−d/2 exp( c x y 2/t), t x y 3/2 S . t ≥ 5 − 6| − | ≥ | − | ∨ x (b) Further, if x (ω), t S and B = B (x, 2√t) then ∈C2 ≥ x 2 qZ,B(x,y) c t−d/2 for y B (x, √t), t ≥ 7 ∈ 2 Z,B where B2(x, R) is a ball in the (unweighted) graph ( 2, 2), and q is the transition of Z killed Z,B ω Z C O on exiting B, i.e. qt (x,y) := Px (Zt = y,τB >t).

Applying Proposition 3.8, we can establish the following, which corresponds to [3, Proposition n n 5.13 (b)]. To state the result, we introduce the rescaled process Y = (Yt )t≥0 by setting

n −1 Yt := n Yn2t.

19 Proposition 3.10 (Tightness). Let K,T,r > 0. For P-a.e. ω, the following is true: if x n ∈ D ,n 1, x B (0, K) are such that x x, then n ≥ ∈ E n →

ω n lim lim sup Pnxn sup Ys > R = 0, (3.15) R→∞ n→∞ s≤T | | !

ω n n lim lim sup Pnx sup Ys Ys > r = 0. (3.16) δ→0 n 2 1 n→∞ |s1−s2|≤δ,si≤T | − | !

In particular, for P-a.e. ω, if x D ,n 1, x D are such that x x, under P ω , the family n ∈ n ≥ ∈ n → nxn of processes (Y n) ,n N is tight in D([0, ), D). t t≥0 ∈ ∞ Proof. Since the statement is slightly diﬀerent from [3, Proposition 5.13], we sketch the proof. Note that since x x B (0, K), then by setting M = K + 1 we have that nx B (0, nM) for all n → ∈ E n ∈ E n suitably large. Let nR > supn Rnxn . Then, by Proposition 3.8,

P ω sup Y n > R = P ω τ Y

2 ω n −c4R /T −R Pnxn sup Ys > R c3e e . s≤T | | ! ≤ ∨ Since lim sup R /n lim sup sup R /n < , P-a.s., due to the Borel-Cantelli argu- n nxn ≤ n x∈BE(0,Mn) x ∞ ment using (3.14), we obtain (3.15). We next prove (3.16). Write

ω p(x, T, δ, r)= Px sup Ys2 Ys1 > r , |s1−s2|≤δ,si≤T | − | ! so that ω n n 2 2 Pnxn sup Ys2 Ys1 > r = p(nxn,n T,n δ, nr). |s1−s2|≤δ,si≤T | − | ! Arguing similarly to the proof of [3, Proposition 5.13], we have

p(nx ,n2T,n2δ, 2nr) c exp( cnT 1/2)+ c(T/δ) exp( cr2/δ), n ≤ − − provided 1/2 −1 2/3 −1 −1 T n Rx , δ>n r, r n max Ry. (3.17) 3/2 3/4 ≥ ≥ y∈BE (nxn,n T ) Note that B (nx ,n3/2T 3/4) B (0, nM + n3/2T 3/4) for large n. If T, r and δ are ﬁxed, due to E n ⊂ E the Borel-Cantelli argument using (3.14), each of conditions in (3.17) holds when n is large enough. So, for P-a.e. ω, 2 2 2 lim sup p(nxn,n T,n δ, 2nr) c(T/δ) exp( cr /δ), n→∞ ≤ − and (3.16) follows. Using (3.15) and (3.16), we have tightness for P ω by [31, Corollary 3.7.4]. { nxn }

20 We can further establish the following theorems, which correspond to [17, Lemma 5.6, Propo- Y sition 6.1] and [3, Theorem 7.3]. We denote by (qt (x,y))x,y∈C1,t>0 the heat kernel associated with Y , i.e. for x,y 1, t> 0, ∈C Y ω qt (x,y) := Px (Yt = y). (We recall that the invariant measure of Y is the uniform measure ν on ). C1 Proposition 3.11. There exist c , c , c , γ (0, ) (non-random) and random variables (R ,x 1 2 3 ∈ ∞ x ∈ L) with P(R n,x ) exp( c nγ), (3.18) x ≥ ∈C1 ≤ − 1 such that if x,y , then ∈C1 qY (x,y) c t−d/2 for t (c 2d (x,y) R )1/4. t ≤ 2 ≥ 3 ∨ 1 ∨ x

Proof. Note that the corresponding result for Z-process is given in [3, Corollary 4.3]. We need to obtain similar result for Y -process. First, note that because we have because of Proposition 3.5, the proof of [3, Proposition 4.1 and Corollary 4.3] (with ε = 1/4 for simplicity) goes through once (4.7) in [3] is veriﬁed. To check [3, (3.7)], we use [8, Theorem 2.3], which can be proved almost identically in our case. Note that in [8, Theorem 2.3], the metric d˜ is used, but thanks to Remark 3.6, we can obtain the same estimates using the metric d1. Finally, using Cauchy-Schwarz, we obtain the desired inequality.

For G , we deﬁne ∂out(G) to be the exterior boundary of G in the graph ( , ), i.e. those ⊂C1 C1 O1 vertices of G that are connected to G by an edge in , and set cl(G) = G ∂out(G). We say C1\ O1 ∪ that a function h is Y –harmonic in A if h is deﬁned on cl(A) and h(x)=0 for x A. ⊂C1 LV ∈ Theorem 3.12 (Elliptic Harnack inequality). There exist random variables (R′ ,x L) with x ∈ ′ δ P(x ,R′ n) ce−c n , ∈C1 x ≥ ≤ ′ R and a constant CE such that if x0 1, R Rx0 and h : cl(B1(x0,R)) + is Y –harmonic on ′ ∈ C ≥ → B1 = B1(x0,R), then writing B1 = B1(x0, R/2),

sup h CE inf h. ′ ≤ B′ B1 1

Proof. Given Lemma 3.1, the proof is almost identical to that of [3, Theorem 7.3].

3.3 Quenched invariance principle

To prove a quenched invariance principle for Y (see Theorem 3.17 below), we will check the con- 2 ditions of Theorem 2.1 one by one. We choose δn = c∗/n, where c∗ is a constant that will be chosen later. First, since Xn is a continuous time Markov chain with holding time at x being an −1 exponential random variable of mean µx , it is conservative. Condition (ii) in Theorem 2.1 is a

21 consequence of the quenched invariance principle for the whole space (cf. [3]) and the fact that ˜ , which is a consequence of the uniqueness of the inﬁnite percolation clusters in the two C1 ⊆ C1 settings. Since in the original papers quenched invariance principles are uniformly stated in terms of the random walk started from the origin, whereas we require such to hold from an arbitrary ˜ω starting point, we state the following generalization of existing results. We will suppose that Py n refers to the quenched law of the VSRW Y˜ on ˜1 started from y ˜1, and Y˜ refers to the rescaled ˜ n −1 ˜ C ∈ C process deﬁned by setting Yt := n Yn2t. The proof of the result can be found in the appendix. Theorem 3.13. There exists a deterministic constant c (0, ) such that, for P-a.e ω, the laws of ˜ n ˜ω ∈ ∞ the processes Y under Pnxn , where nxn 1 and xn x, converge weakly to the laws of (Bct)t≥0, ∈C d → where (Bt)t≥0 is standard Brownian motion on R started from x.

Concerning Assumption 2.3(i), we will prove the following: there exist c , c (0, ) (non- ∗ 1 ∈ ∞ random) and N (ω) such that for all n N (ω), x ,x B (0,n1/2) D and c /n1/2 r′ 1, 0 ≥ 0 0 ∈ E ∩ n ∗ ≤ ≤ ω n ′2 E τ ′ (Y ) c r . (3.19) x BE (x0,r )∩Dn ≤ 1 Applying Proposition 3.11, we have the following: for all x ,x B (0,n3/2) and r n, if 0 ∈ E ∩C1 ≤ c r2 (c 2sup d (x, z) R )1/4, then 2 ≥ 3 ∨ z∈BE(x0,r)∩C1 1 ∨ x

ω Y 2 ω Y 0 P τ L c2r P Y 2 BE(x0, r) = q 2 (x, z)µ (dz) x BE(x0,r)∩ ≥ ≤ x c2r ∈ c2r ZBE (x0,r) d c 0 cc4r µ (BE(x0, r)) 1/2, (3.20) ≤ (c r2)d/2 ≤ d/2 d ≤ 2 c2 r where we used µ0(B (x , r)) c rd and we set c := (2cc )2/d c1/4. Now, using Lemma 3.3, E 0 ≤ 4 2 4 ∨ 3 there exists N (ω) N that satisﬁes P(N (ω) m) c e−c2m such that B (x, c R) B (x, R) 1 ∈ 1 ≥ ≤ 1 E 1 ⊂ 1 for x B (0,R) and R N (ω). On the other hand, x z x + x + x z 2n3/2 + r ∈ E ∩C1 ≥ 1 | − | ≤ | | | 0| | 0 − |≤ for z B (x , r), so taking r c n1/2 with c large enough, there exists N (ω) P(N (ω) m) ∈ E 0 ≥ ∗ ∗ 2 2 ≥ ≤ c e−c2m such that c r2 (2sup d (x, z))1/4 holds for n N (ω). Next, by (3.18), 1 2 ≥ z∈BE(x0,r)∩C1 1 ≥ 2

γ 3d/2 −c1n P sup Rx n n e . 3/2 ≥ ≤ x∈BE (0,n )∩C1 !

Summarizing, (3.20) holds for all x ,x B (0,n3/2) , c n1/2 r n and n N (ω) := 0 ∈ E ∩C1 ∗ ≤ ≤ ≥ 0 N2(ω) N3(ω), where N3(ω) := sup 3/2 Rx. Moreover, the random variable N0(ω) is ∨ x∈BE (0,n )∩C1 almost-surely ﬁnite; in fact, we have the following tail bound for it, which will be useful in Example 4.1 below: γ P (N (ω) m) c e−c2m . (3.21) 0 ≥ ≤ 1 Using the Markov property, we can inductively obtain

ω Y 2 k P τ L kc2r (1/2) , k N. x BE (x0,r)∩ ≥ ≤ ∀ ∈ So, ω Y 2 ω 2 Y 2 2 E τ L (k + 1)c2r P kc2r τ L < (k + 1)c2r 3c2r . x BE (x0,r)∩ ≤ x ≤ BE (x0,r)∩ ≤ Xk 22 n −1 1/2 1/2 For Y = n Y 2 , we therefore have: for n N (ω), x B (0,n ) D , x B (0,n ) D n t ≥ 0 0 ∈ E ∩ n ∈ E ∩ n and c /n1/2 r′ 1, ∗ ≤ ≤ ω n 1 ω Y 1 2 ′2 E τ ′ (Y ) E τ L 3c2r = 3c2r , nx BE (x0,r )∩Dn ≤ n2 nx BE (nx0,r)∩ ≤ n2 · ′ 2 where r = nr . Thus (3.19) holds, and so Assumption 2.3(i) holds with δn = c∗/n, β = 2. Regarding Assumption 2.3(ii) we observe that, using Theorem 3.12, the relevant condition can be obtained similarly to [9, Proposition 3.2]. (Note that Proposition 3.2 in [9] is a parabolic version, whereas we just need an elliptic version.) Indeed, taking (log n)2/δ as n in Theorem 3.12,

P ′ 2/δ d 2d −c′(log n)2 ′ 2 sup Rx (log n) c n e c /n . 2 x∈BE(0,cn )∩C1 ≥ ! ≤ ≤

Thus, by the Borel-Cantelli lemma, there exists N (ω) N such that 1 ∈ ′ 2/δ sup Rx (log n) , n N1(ω), 2 x∈BE (0,cn )∩C1 ≤ ∀ ≥ so the elliptic Harnack inequality holds for Y -harmonic functions on balls BE(x0,R) with x0 2 2/δ n −1 ∈ B (0, cn ), R (log n) . By scaling Y (t) = n Y 2 , the elliptic Harnack inequality holds E ≥ n t uniformly for Y n-harmonic functions on B (x ,R) with x B (0, cn), R (log n)2/δ/n. Given E 0 0 ∈ E ≥ the elliptic Harnack inequality, we can obtain the desired H¨older continuity in a similar way as in 2 1/2 the proof of Proposition 3.2 of [9]. Thus, setting δn := c∗/n, Assumption 2.3(ii) holds for R δn , 1/2 2/δ ≥ since δn (log n) /n. ≥ Next, we remark that part (i) of Assumption 2.6 is direct from Proposition 3.10, and part (ii) follows from Proposition 3.10 (especially (3.16) implies the condition). The following proposition gives the appropriate convergence for the sequence of measures −d (mn)n≥1 deﬁned by setting mn := n ν(n ). (Recall that ν is the invariant measure for Y , n · and so the measure mn is invariant for Y .)

Proposition 3.14. P-a.s., the measures (mn)n≥1 converge vaguely to m, a deterministic multiple of Lebesgue measure on D.

Proof. First note that if Q D is a cube of side length λ, then applying Lemma 3.7 in a Borel ⊂ Cantelli argument yields that, P-a.s., ν˜(nQ) ν(nQ) − 0. (3.22) nd → Next, consider a rectangle of the form R = [0, λ ] [0, λ ]. Since the full Zd model is ergodic 1 ×···× d under coordinate shifts, a simple application of a multidimensional ergodic theorem yields that, P-a.s., d ν˜(nR) c λ , nd → 1 i Yi=1 where c := P(0 ˜ ) (0, 1]. An inclusion-exclusion argument allows one to extend this result to 1 ∈ C1 ∈ any rectangle of the form [x ,x + λ ] [x ,x + λ ], where x 0 for i = 1,...,d. Clearly the 1 1 1 ×···× d d d i ≥

23 particular orthant is not important, so the result can be further extended to cover any rectangle R D, and, in particular, we have that, P-a.s., ⊂ ν˜(nQ) c λd. nd → 1 Applying (3.22), we obtain that the above limit still holds whenν ˜ is replaced by ν. The proposition follows.

Finally, in order to verify Condition (iii) in Theorem 2.1, we ﬁrst give a lemma.

Lemma 3.15. Let η be independent and identically distributed with E η < . Suppose { i}i≥1 | 1| ∞ an n is a sequence of real numbers with an M for all k,n (here M > 0 is some ﬁxed { k }k=1 | k | ≤ constant) such that the following two limits exist:

n n 1 1 a := lim an, lim an . n→∞ n k n→∞ n | k | Xk=1 Xk=1 It then holds that 1 n anη converges to aE[η ] almost surely as n . n k=1 k k 1 →∞ Proof. This can beP proved similarly to Etemadi’s proof of the strong law of large numbers (see [30]).

Proposition 3.16. If C := E(µ ) < , then P-a.s., 0 e ∞ ˜(f,f) lim sup (n)(f,f) C f(x) 2dx, f C2(D), (3.23) E ≤ E ≤ 0 |∇ | ∀ ∈ c n→∞ ZD 2 where Cc (D) is the space of compactly supported functions on D with a continuous second derivative, and n2−d (n)(f,f) := (f(x/n) f(y/n))2µ E 2 − x,y x,y∈C1: {x,yX}∈O1 2 n is the Dirichlet form on L (Dn; mn) corresponding to Y . In particular, condition (iii) in Theorem 2.1 holds.

Proof. The ﬁrst inequality of (3.23) is standard. Indeed,

˜ 1 1 nj(k) (f,f) = sup (f Ptf,f) = sup lim inf (f Pt f,f) E t>0 t − t>0 k→∞ t − 1 n j(k) (nj(k)) lim inf sup (f Pt f,f) = lim inf (f,f), ≤ k→∞ t>0 t − k→∞ E

2 2 where the ﬁrst inner product is in L (D; m), and the other two are in L (Dnj(k) ; mnj(k) ). Moreover, nj(k) to establish the second inequality we apply the local uniform convergence of Pt f to Ptf (cf. the proof of Proposition 2.8) and the vague convergence of mnj(k) to m (Lemma 3.14). We now prove the second inequality. Suppose Supp f B (0, M) D for some M > 0, then ⊂ E ∩ n2−d (n)(f,f)= (f(x/n) f(y/n))2µ , E 2 − x,y (x,y)X∈Hn,M

24 where Hn,M := (x,y) : x,y BE(0, nM) 1, x,y 1 . Clearly, this quantity increases { ′∈ ∩C { } ∈ O } when H is replaced by H := (x,y) : x B (0, nM) L, x,y EZd . Note that n,M n,M { ∈ E ∩ { } ∈ } #H′ c (nM)d. Moreover, set an := n2(f(x/n) f(y/n))2 and η := µ . Then, since n,M ∼ 1 (x,y) − (x,y) x,y f C2, 0 a M ′ for some M ′ > 0, and further, ∈ c ≤ (x,y) ≤

d −1 n −1 −d 2 lim (2c1(nM) ) a = c M f(x) dx, n→∞ (x,y) 1 |∇ | (x,y)∈H′ D Xn,M Z by applying Lemma 3.15 we obtain that, P-a.s.,

−d n 2 lim n a η(x,y) = 2C0 f(x) dx. n→∞ (x,y) |∇ | (x,y)∈H′ D Xn,M Z The result at (3.23) follows.

Putting together the above results, we conclude the following.

Theorem 3.17. For the random conductance model on L with independent and identically distributed conductances (µe)e∈EL satisfying (3.1), (3.2) and (3.3), there exists a deterministic constant c (0, ) such that, for P-a.e ω, the laws of the processes Y n under P ω , where nx ∈ ∞ nxn n ∈ C1 and x x, converge weakly to the laws of X ; t 0 , where X ; t 0 is the reﬂecting n → { ct ≥ } { t ≥ } Brownian motion on D started from x.

Remark 3.18. (i) The diffusion constant c is the same for the model restricted to L as for the full Zd model. (ii) When the conductance is not bounded from below, we cannot apply our theorem because Assumption 2.3(i) does not hold in general, and we do not know how to obtain the quenched invariance principle without this. Indeed, consider the realization of edge weights shown in Figure 2, where the conductance on x,y is 1 and it is O(n−α) on y, z where α> 2. One can easily compute ω α{ } 2 P { } P that Ex τBE (x,2)(Y ) c1n n . Let p0 = (µe = 0) and p1 = (µe = 1). Then the probability ≥ ≫ 4d−3 P −α P −α that such a trap configuration appears is p0 p1 (0 <µe n )= c2 (0 <µe n ). Now let ω α≤ P ≤ d/α Ωn := xn BE(0, n/2) such that Exn τBE(xn,2)(Y ) c3n . If we have (0 <µe x) c4x {∃ ∈ d ≥ } ≤ ≥ P −d n −c5 for small x> 0, then (Ωn) 1 (1 c4n ) 1 e for large n. In particular, lim supn Ωn ≥ − −n −1 ≥ − occurs with positive probability. Set X := n Y 2 . Then, for ω Ω , we have t n t ∈ n ω n ω ω α−2 E −1 τ (X ) = E τ (Y 2 ) E τ (Y 2 ) c3n . n xn BE (0,1)∩Dn xn BE (0,n)∩D0 n · ≥ xn BE (xn,2) n · ≥ Since Ωn occurs infinitely often with positive probability, Assumption 2.3(i) does not hold for any choice of β > 0 (by choosing x0 = 0, r = 1). (iii) There is another natural continuous time Markov chain on , namely with transition proba- C1 bility P (x,y)= µxy/µx and the holding time being the exponential distribution with mean one for each point. (Such a Markov chain is sometimes called a constant speed random walk (CSRW).) It is 2 a time change of the VSRW; the corresponding Dirichlet form is ( ,L ( 1; µ)), and the correspond- 1 E C ing discrete Laplace operator is Cf(x) = (f(y) f(x))µxy. For the whole space case, one L µx y − can deduce the quenched invariance principle of CSRW from that of VSRW by an ergodic theorem. P

25 z y x

Figure 2: An example trap structure.

(See [3, Section 6.2] and [8, Section 5]. Note that the limiting process degenerates if Eµ = .) e ∞ Since our state space L features a lack of translation invariance, we cannot use the ergodic theorem. So far, we do not know how to circumvent this issue to prove the quenched invariance principle for CSRW on L. (However, we do note that for the case of random walk on a supercritical percolation cluster, the CSRW and VSRW behave similarly, and the quenched invariance principle for the CSRW can be proved in a similar way as for the VSRW case. Moreover, the quenched invariance principle for the discrete time simple random walk on follows easily from that for the CSRW.) C1 (iv) To extend Theorem 3.17 to apply to more general domains, it will be enough to check the percolation estimates from which we deduced Assumptions 2.3 and 2.6 in these settings. Whilst we believe doing so should be possible, at least under certain smoothness assumptions on the domain boundary, we do not feel the article would benefit significantly by the increased technical complication of pursuing such results, and consequently omit to do so here. Instead we restrict our discussion of more general domains to the case of uniformly elliptic conductances, where the relevant estimates are straightforward to check (see Section 4.2 below). Similarly, given suitable full space quenched invariance principles and percolation estimates (namely the estimates given in Lemmas 3.1 – 3.3), our results should readily adapt to percolation models on other lattices. (v) Given the various estimates we have established so far, it is possible to extend the quenched invariance principle of Theorem 3.17 to a local limit theorem, i.e. a result describing the uniform convergence of transition densities. More specifically, the additional ingredient needed for this is an equicontinuity result for the rescaled transition densities on , which can be obtained by applying C1 an argument similar to that used to deduce Assumption 2.3(ii), together with the heat kernel upper bound estimate of Proposition 3.11. Since the proof of such a result is relatively standard (cf. [9, 25]), we will only write out the details in the compact box case (see Section 4.1 below), where convergence of transition densities is also useful for establishing convergence of mixing times.

4 Other Examples

4.1 Random conductance model in a box

The purpose of this section is to explain how to adapt the results of Section 3 to the compact space d d case. Set B(n) := [ n,n] Z , let EB = e = x,y : x,y B(n), x y = 1 be the set of − ∩ (n) { { } ∈ | − | } nearest neighbor bonds, and suppose µ = (µe)e∈EZd is a collection of independent random variables satisfying the assumptions made on the weights in Section 3, i.e. (3.1), (3.2) and (3.3). For each n and each realization of µ, let (n) be the largest component of B(n) that is connected by edges C1

26 n ω satisfying µe > 0, and let Y be the VSRW on 1(n). We will write Pn,x for the quenched law of n C Y started from x 1(n). The aim of this section is to show, via another application of Theorem n n∈C 2.1, that X = (Xt )t≥0, deﬁned by setting

n −1 n Xt := n Yn2t, converges as n to reflecting Brownian motion on D = [ 1, 1]d, for almost-every realization of →∞ − the random environment µ. We observe that, in the case of uniformly elliptic random conductances, this result was recently established using an alternative argument in [18]. Note also that, by applying a result from [26], the above functional scaling limit readily yields the corresponding convergence of mixing times (see Corollary 4.4 below for a precise statement). To prove the results described in the previous paragraph, we start by considering a decomposition of B(n). In particular, fix ε (0, 1) and for i = (i ,...,i ) 1, 1 d, let Bε(n) be the ∈ 1 d ∈ {− } i cube of side-length n(1 + ε) which has a corner at ni and contains 0. Within each of the 2d sets ⌊ ⌋ of the form Bε(n), the random walk on (n) reflects only at the faces of the box adjacent to the i C1 single corner vertex ni. As a consequence, we will be able to transfer a number of key estimates to the current framework from the unbounded case considered in Section 3 – note that the reason Bε for taking ε> 0 is so that the boxes i (n) overlap, which will allow us to ‘patch’ together results proved for different parts of the box. For the purpose of transferring results from Section 3, the following lemma will be useful. Its proof can be found in the appendix.

Lemma 4.1. There exist constants c , c such that if Q ,Q Zd are the cubes of side length 1 2 1 2 ⊆ + n(1 + ε) , 2n containing 0, respectively, +(Q ) is the largest connected component of the graph ⌊ ⌋ C 2 (Q , ), and is the unique inﬁnite component of (Zd , ), then 2 O1 C1 + O1 P +(Q ) Q = Q c e−c2n. C 2 ∩ 1 6 C1 ∩ 1 ≤ 1 In particular, if i (n) is the unique inﬁnite percolation cluster on the copy of Zd that has corner C1 + vertex ni and contains 0, then the above result implies that with probability at least 1 c e−c2n − 1 (or, by Borel-Cantelli, almost-surely for large n) we have that

(n) Bε(n)= i (n) Bε(n) ˜ , i 1, 1 d, (4.1) C1 ∩ i C1 ∩ i ⊆ C1 ∀ ∈ {− } where the inclusion is a consequence of the uniqueness of the inﬁnite percolation clusters in question. (This result is summarized by Figure 3.) We now check the conditions listed at the end of Section 2 one by one. Since in light of (4.1), most of these are straightforward adaptations of the arguments given in Section 3, we will only provide a brief description of how to do this. Firstly, as was the case in the L setting, since Xn is a continuous time Markov chain with holding time at x being exp(µx), it is conservative. Secondly, given (4.1), condition (ii) in Theorem 2.1 is a consequence of the quenched invariance principle for the whole space stated as Theorem 3.13. Moreover, since (n) agrees with ˜ up to a distance C1 C1 c log n of the boundary, at least for large n (see [13, Proposition 1.2]), by applying the full Zd version of Proposition 3.14, we have that the measures mn, deﬁned analogously to the previous section, P-a.s. converge weakly to (a suitably rescaled version of) Lebesgue measure on [ 1, 1]d. −

27 i (n) C1

(n) C1

(n) Bε(n) C1 ∩ i = i (n) Bε(n) C1 ∩ i

Figure 3: Schematic illustration of how, for large n, the largest cluster 1(n) (dark-grey) in B(n) i C (square with solid boundary) and the largest cluster 1(n) (light-grey) in the quarter-plane with Bε C corner in and containing 0 agree within i (n) (square with dashed boundary).

Similarly, the Dirichlet form comparison of (iii) can be obtained by following the same argument used to prove the corresponding result in Section 3 – Proposition 3.16. Applying (4.1), we are also able to deduce the following tightness result, which is analogous to Proposition 3.10, and from which Assumption 2.6 is readily obtained.

Proposition 4.2. For P-a.e. ω, if x n−1 (n), n 1 is such that x x D, then under n ∈ C1 ≥ n → ∈ P ω , the family of processes (Xn) ,n N is tight in D([0, ), [0, 1]d), and any convergent n,nxn t t≥0 ∈ ∞ subsequence has limit in C([0, ), [0, 1]d). ∞ Proof. Note that in the bounded case the limit corresponding to (3.15) is immediate, and hence it will suﬃce to check the limit corresponding to (3.16). To do this, the same argument can be applied, so long as one can check the following: for any r (0, ε), there exist c and random ∈ i variables (Rn,x B(n),n 1) with x ∈ ≥ δ P(Rn rn,x (n)) c e−c2n , x ≥ ∈C1 ≤ 1 such that if x (n), t> 0 and R Rn, then ∈C1 ≥ x ω Y n P τ

28 i n ˜n n to the part of 1(n) contained in this set, then set Rx = Rx. Else, set Rx = 3n. The required C n exponential decay for the distributional tail of Rx then follows from Proposition 3.8 and (4.1). Moreover, since the probability on the left-hand side of (4.2) is 0 for R 3n, the bound at (4.2) ≥ follows.

It remains to check Assumption 2.3. For part (i), we simply note that the combination of (3.19) (or more precisely, the exponential tail bound for N0 that appears as (3.21)) and (4.1) in a standard Borel-Cantelli argument implies the following: there exist c∗, c1 > 0 (non-random) and N0(ω) such that if n N (ω), then, for each x ,x n−1 (n) and c /n1/2 r′ 1, ≥ 0 0 ∈ C1 ∗ ≤ ≤ ω n ′2 E τ ′ (X ) c r , n,x BE(x0,r ) ≤ 1 as desired. For part (ii) of this assumption, ﬁrst note that we can obtain the elliptic Har- nack inequality uniformly for Xn-harmonic functions on B (x ,R), where x n−1 (n) and E 0 0 ∈ C1 (logn)2/δ/n R 1 for large n. (This can be proved similarly as before, namely when x B0(n), ≤ ≤ 0 ∈ i Theorem 3.12 can be applied by replacing by i (n) due to (4.1).) Given the elliptic Harnack C1 C1 inequality, we can obtain H¨older continuity in a similar way as in the proof of [9, Proposition 3.2], for example. Hence we have established the following.

Theorem 4.3. There exists a constant c (0, ) such that, for P-a.e. ω, the process Xn under ∈ ∞ P ω , where nx and x x [ 1, 1]d, converges in distribution to (B ) , where (B ) n,nxn n ∈C1 n → ∈ − ct t≥0 t t≥0 is the reflecting Brownian motion on [ 1, 1]d started from x. − Next, for p [1, ], define the Lp mixing time of the VSRW on (n) to be ∈ ∞ C1 1/p 1 tp ( (n)) := inf t> 0: sup qn(x,y) 1 p πn(dy) < , (4.3) mix 1  t  C x∈C1(n) C1(n) | − | ! 4  Z  where we denote by qn the transition density of the VSRW with respect to its (uniqu e) invariant probability measure πn. The above result then has the following corollary. Note that in the percolation setting, the obvious adaptation of this result to discrete time gives a refinement of the first statement of [13, Theorem 1.1].

Corollary 4.4. Fix p [1, ]. For P-a.e. ω, we have that ∈ ∞ n−2tp ( (n)) c−1tp ([ 1, 1]d), mix C1 → mix − where c is the constant of Theorem 4.3, and tp ([ 1, 1]d) is the mixing time of reflecting Brownian mix − motion on [ 1, 1]d (defined analogously to (4.3)). − Proof. First note that a simple rescaling yields that, P-a.s., πn converges weakly to a rescaled version of Lebesgue measure on [ 1, 1]d. The P-a.s. Hausdorff convergence of n−1 (n) (equipped with − C1 Euclidean distance) to [ 1, 1]d is a straightforward consequence of this. To establish the corollary − by applying [26, Theorem 1.4], it will thus be enough to extend the weak convergence result of Theorem 4.3 to a uniform convergence of transition densities (so as to satisfy [26, Assumption 1]).

29 According to [26, Proposition 2.4] (cf. [25, Theorem 15]) and the quenched invariance principle mentioned above, it is enough to show [26, (2.11)], namely, for any 0

To prove this, ﬁrst we have the following H¨older continuity, which can be checked similarly to Assumption 2.3(ii):

n n γ n −1 q 2 (nx,ny) q 2 (nx,nz) c y z q 2 (nx, ) , x,y,z n (n). (4.5) n t − n t ≤ 1| − | k n t · k∞ ∀ ∈ C1

For 0 < a t< 1, say, a compact version of Proposition 3.11 and scaling gives that ≤ n 2 −d/2 −d/2 q 2 (nx, ) c (n) (n t) c a k n t · k∞ ≤ 3|C1 | ≤ 3 for large n. For t 1, Cauchy-Schwarz and monotonicity of qn (nx,nx) implies qn (nx, ) c . ≥ n2t k n2t · k∞ ≤ 4 In particular, n q 2 (nx, ) c (a) (4.6) k n t · k∞ ≤ 2 uniformly in x D , t a, for large n, P-a.s. Thus, for t [a, b], the right-hand side of (4.5) is ∈ n ≥ ∈ bounded from above by c (a) y z γ . Taking n and then δ 0, we obtain (4.4). 3 | − | →∞ → Finally, as a corollary of the heat kernel continuity derived in the proof of the previous result, we obtain the following local central limit theorem. We let g : [ 1, 1]d (n) be such that g (x) n − →C1 n is a closest point in (n) to nx in the -norm. (If there is more than one such point, we choose C1 | · |∞ one arbitrarily.)

Corollary 4.5. Let q ( , ) be the heat kernel of the reﬂecting Brownian motion on [ 1, 1]d. For t · · − P-a.e. ω and for any 0

Proof. Given the above results, the proof is standard. By (4.5) and (4.6) and the Ascoli-Arzel`a theorem (along with the Hausdorﬀ convergence of n−1 (n) to [ 1, 1]d), there exists a subsequence C1 − of qn (g ( ), g ( )) that converges uniformly to a jointly continuous function on [a, b] [ 1, 1]d nt n · n · × − × [ 1, 1]d. Using Theorem 4.3, it can be checked that this function is the heat kernel of the limiting − process. Since the limiting process is unique, we have the convergence of the full sequence of qn (g ( ), g ( )). The uniform convergence in (4.7) is then another consequence of (4.5) and (4.6). nt n · n ·

4.2 Uniformly elliptic random conductances in uniform domains

When the conductances are uniform elliptic, i.e. bounded from above and below by ﬁxed positive constants, we can obtain quenched invariance principles for a much wider class of domains than those considered in the examples presented so far. In particular, let D be a uniform domain

30 in Rd. For each n 1, let Dˆ be the largest connected component of of nD Zd, and set ≥ n ∩ E ˆ = e = x,y : x,y Dˆ n, x y = 1 . Suppose µ = (µe)e∈E is a collection of independent Dn { { } ∈ | − | } Zd random variables such that

P(C µ C ) = 1, e EZd , 1 ≤ e ≤ 2 ∀ ∈ n where C1,C2 are non-random positive constants. Let Y be either VSRW or CSRW on Dˆ n. More- −1 ˆ n −1 n n n over, set Dn := n Dn and deﬁne Xt := n Yn2t. It is then the case that X = (Xt )t≥0 converges as n to a (constant time change of) reﬂecting Brownian motion on D, for P-almost-every → ∞ realization of the random environment µ. Since checking the details for this case is much simpler than for previous settings, we will not provide a full proof of the result described in the previous paragraph, but merely indicate how to establish the key estimates. Indeed, in the uniformly elliptic case, we have

c Rd n−d B(x, R) D c Rd, c Rd n−dµ(B(x, nR) Dˆ ) c Rd 1 ≤ | ∩ n|≤ 2 3 ≤ ∩ n ≤ 4 for all large n and all n−1 R diam (D), x D, where c are non-random positive constants. ≤ ≤ ∈ i Furthermore, the weak Poincar´einequality (3.10) holds both for the counting measure and n−dµ uniformly for n−1 R diam (D), in the sense that the constants do not depend on n. Given ≤ ≤ these, we can apply [28] (and the natural relations between heat kernels of discrete and continuous time Markov chains) to deduce

c t−d/2 exp( c x y 2/t) qn(x,y) c t−d/2 exp( c x y 2/t), x y t diam (D), 5 − 6| − | ≤ t ≤ 7 − 8| − | | − |≤ ≤ n −d ω n −d −1 ω n where qt (x,y) is deﬁned as n Px (Xt = y) for the VSRW and n µ(ny) Px (Xt = y) for the CSRW. Given these heat kernel estimates, it is then straightforward to verify the conditions required for the quenched invariance principle by applying similar arguments to those of Sections 3 and 4.1.

4.3 Uniform elliptic random divergence form in domains

In this section, we explain how Theorem 2.1 can be applied in the random divergence form setting. Let D be a uniform domain in Rd. Assume that we have a random divergence form as follows. There P exists a probability space (Ω, ) with shift operators (τx)x∈Rd that are ergodic, and a symmetric d d matrix Aω(x) for each x Rd and ω Ω such that Aω(x)= Aτxω(0) and × ∈ ∈ P(c I Aω(x) c I) = 1, x Rd, 1 ≤ ≤ 2 ∀ ∈ where c , c (0, ) are deterministic constants. For n 1, let 1 2 ∈ ∞ ≥ 1 n(f,f)= f(x)Aω(x) f(x)dx. E 2 ∇ ∇ ZnD n n 1,2 Let (Yt )t≥0 be the diﬀusion process associated with the regular Dirichlet form ( ,W (nD)) on 2 n −1 n E L (nD; dx), and set Xt := n Yn2t. (A natural setting would be to take D to be a cone, in which n n n case nD = D, so that the process Y is always on the cone.) It is then the case that X = (Xt )t≥0 converges as n to a reﬂecting Brownian motion on D with some strictly positive covariance → ∞

31 matrix B, for P-almost-every realization of the random environment µ. (Note that B is determined by the invariance principle on the whole space Rd.) It can be easily veriﬁed that Dirichlet form on L2(D; dx) associated with Xn is

1 n2−d n(f ,f )= f(x)A(nx) f(x)dx, E n n 2 ∇ ∇ ZD −1 n where fn(x) := f(n x). In view of Section 2.1, the transition density function of X has estimates (2.4) with constants c , , c independent of n. In this case, the quenched invariance principle 1 · · · 4 on Rd is proved in [48]. (To be precise, in the paper the author assumed C2 smoothness for the coefficients. However, this was to apply the Itôformula, and could be avoided by using the Fukushima decomposition instead.) Given these, one can easily verify the conditions required for the quenched invariance principle in D. (Note that because of the uniform ellipticity, Condition (iii) in Theorem 2.1 is trivial in this case. Moreover, one can extend the invariance principle of [48] to arbitrary starting points by applying the argument of Theorem 3.13.) We remark that homogenization of reflected SDE/PDE on half-planes has been studied for periodic coefficients in [5, 14, 53] etc., and for random divergence forms in [51]. (Note that their equations contain additional reflection terms, though the precise framework varies in each paper.) Homogenization for random divergence forms without extra reflection terms on bounded C2 domains is discussed in [41, Section 14.4]. Although we can only handle symmetric cases, our results hold for general uniform domains.

A Appendix

A.1 Proofs for percolation estimates

The aim of this section is to verify the percolation estimates stated as Lemmas 3.1-3.3, 3.7 and 4.1. For the purpose of proving Lemma 3.1, it will be useful to note that for large K the collection of edges ˜ stochastically dominates ˜ , the edges of a bond percolation process on EZd with O2 O3 probability p3 = p3(K), where the parameter p3 can be chosen to satisfy limK→∞ p3 = p1 (see [3, Proposition 2.2]). In fact, the proof of this result from [3] further shows that, for a given value of K (that is suitably large), it is possible to couple all the relevant random variables in such a way that ˜ ˜ almost-surely. We will henceforth assume that this is the case, where K is fixed large O3 ⊆ O2 enough to ensure that p > pbond(Zd). We will also define := ˜ EL and := (L, ). 3 c O3 O3 ∩ C3 C∞ O3 Note that is non-empty by the uniqueness of infinite supercritical bond percolation clusters on C3 L.

Proof of Lemma 3.1. First observe that . It follows that there exists an inﬁnite O3 ⊆ O2 ⊆ O1 connected component of (L, ) such that . For such a (at the moment, we do not C O2 C3 ⊆C⊆C1 C know its uniqueness), we have that L . Denote by (x) the connected component C1\C ⊆H3 ⊆ \C3 G of L containing x (if x , we set (x)= ). To prove part (i) of the lemma, it suﬃce to show \C3 ∈C3 G ∅ that there exist constants c , c such that: for each x L, 1 2 ∈ P (diam( (x)) n) c e−c2n. (A.1) G ≥ ≤ 1

32 For this, we will follow the renormalization argument used in the proof of [17, Proposition 2.3], making the adaptations necessary to deal with the boundary issues that arise in our setting. We start by coupling a ﬁnite range-dependent site percolation model with our bond percolation process. For L N, x Zd, deﬁne ∈ ∈ Q (x) := L(x + e + + e ) + [0,L]d Zd, Q˜ (x) := L(x + e + + e ) + [ L, 2L]d Zd, L 1 · · · d1 ∩ 3L 1 · · · d1 − ∩ d where e1,...,ed are the standard basis vectors for Z . Given these sets, Let GL(x) be the event such that

there exists an ˜ -crossing cluster for Q (x) in Q˜ (x), i.e. there is a ˜ -connected cluster • O3 L 3L O3 in Q˜3L(x) that, for all d directions, joins the ‘left face’ to the ‘right face’ of QL(x),

all paths along edges of ˜ that are contained in Q˜ (x) and have diameter greater than L • O3 3L are connected to the (necessarily unique) crossing cluster.

We will say that x Zd is ‘white’ if G (x) holds and ‘black’ otherwise, and make the important ∈ L observation that if two neighboring vertices are white, then their crossing clusters must be con- bond Zd nected. Since p3 > pc ( ), we can apply (the bond percolation version of) [49, Theorem 5] for d = 2 and [50, Theorem 3.1] for d 3 to deduce that ≥

lim P (GL(x)) = 1, (A.2) L→∞

d (cf. [4, (2.24)]). Moreover, although (1GL(x))x∈Z are not independent, the dependence between these random variables is of ﬁnite range. Thus, by [44, Theorem 0.0], one can suppose that, for d d Z suitably large L, the collection (1GL(x))x∈Z dominates a site percolation process on of density arbitrarily close to 1. Noting that for any inﬁnite connected graph G with maximal vertex degree ∆ the critical site and bond percolation probabilities satisfy

psite(G) 1 (1 pbond(G))∆−1, c ≤ − − c ([34, Theorem 3]), we have that psite(L) is bounded above by 1 (1 pbond(Zd))2d−1 < 1. Hence, c − − c by taking L suitably large, it is possible to assume that there is a non-zero probability of 0 being contained in an inﬁnite cluster of white vertices in L. From this result, a standard ergodicity argument with respect to the shift x x + e + + e allows one to check that, P-a.s., there 7→ 1 · · · d1 exists at least one inﬁnite connected cluster of white sites in L. In particular, writing (x) for the D connected cluster of white sites containing a particular vertex x L (taking (x) := if G (x) ∈ D ∅ L does not occur), we obtain that the set

:= x L : (x) is inﬁnite (A.3) D∞ { ∈ D } is non-empty, P-a.s. Let (x) be the connected component of L containing x (we set this to be the empty set C \D∞ if x ). The next step of the proof is to check that: for x L, ∈ D∞ ∈ P (diam( (x)) n) c e−c4n. (A.4) C ≥ ≤ 3

33 To do this, we start by introducing some notions of set boundaries that will be useful. For x , 6∈ D∞ the inner boundary of (x) is the set C ∂in (x) := y (x) : y is adjacent to a vertex in L (x) . C { ∈C \C } It is simple to check from its construction that all the vertices in this set are black. Since L (x) \C ⊇ = , then L (x) contains at least one inﬁnite connected component, say. The outer D∞ 6 ∅ \C D boundary of is given by D ∂out := y L : y is adjacent to a vertex in . (A.5) D { ∈ \D D} With being disjoint from (x), we can also deﬁne the part of its outer boundary visible from D C (x) by setting C ∂visC(x) := y ∂out : there exists a path from y to (x) in L that is disjoint from . (A.6) D ∈ D C D We claim the following relationship between the various boundary sets:

∂visC(x) = ∂out ∂in (x). (A.7) D D⊆ C To verify the equality, first suppose that there exists a vertex y ∂out (x), then y L (x) and ∈ D\C ∈ \C we can find a vertex z L (x) such that y and z are adjacent. This implies that y and z are ∈D⊆ \C in the same connected component of L (x), which is a contradiction because y by definition. \C 6∈ D Hence ∂out (x), and so (noting that (x) = (x)) D⊆C C \D C ∂out = y (x) : y is adjacent to a vertex in ∂visC(x) . D { ∈C D} ⊆ D Since the opposite inclusion is trivial, we obtain the equality at (A.7). From ∂out (x), the D⊆C inclusion at (A.7) is also clear. We proceed by applying the conclusion of the previous paragraph to show that = L (x). D \C First, the boundary connectivity result of [54, Lemma 2] implies that ∂visC(x) is -connected. D ∗ Combining this with (A.7), we obtain that ∂out is a -connected set of black vertices (recall D ∗ that the vertices of ∂in (x) are black). Secondly, note that if P is the law of a parameter p site C p percolation process on Z and ∗ is the corresponding -connected component of closed vertices d C ∗ containing 0, then for suitably large p we have that

P ( ∗ n) c e−c6n, (A.8) p C ≥ ≤ 5 (see [1, Theorem 7.3] and [2, Proposition 7.6]). In particular, it is easy to check from this that all -connected components of closed vertices in the site percolation process with this choice of p are ∗ P d finite, p-a.s. Hence, because (1GL(x))x∈Z dominates a site percolation process whose parameter can be made arbitrarily close to 1 by taking L suitably large, it must be the case that, for large L, ∂out is P-a.s. a finite set. Since is infinite, it readily follows that L is also finite. Now, D D \D suppose is a connected component of L (x) distinct from and such that = . By D1 \C D D1 ∩ D∞ 6 ∅ the definition of , it holds that (y) = for some y such that (y) is an infinite set. Since D∞ D1 ∩ D 6 ∅ D (y) is an infinite connected component of L (x), it must be the case that is infinite. D1 ∪ D \C D1

34 However, this contradicts the finiteness of L , and so no such can exist. Thus it must be the \D D1 case that if is a connected component of L (x) distinct from , then = . Since, D1 \C D D1 ∩ D∞ ∅ similar to the results of the previous paragraph, we have that ∂out ∂in (x), the set (x) D1 ⊆ C D1 ∪C must be a connected component of L . By the definition of (x), this implies that = , \D∞ C D1 ∅ which yields = L (x) as required. D \C An immediate corollary of the equality = L (x) is that ∂in (x)= ∂out , which, as we have D \C C D already established, is a finite -connected component of black vertices. We will use these results ∗ to finally prove (A.4). Note first that diam( (x)) = diam(∂in (x)). Hence, writing ∂B(x,m) for C C the vertices of L at an ℓ∞ distance m from x,

∞ P (diam( (x)) n) P diam(∂in (x)) n, ∂in (x) ∂B(x,m) = C ≥ ≤ C ≥ C ∩ 6 ∅ m=0 X∞ P (diam( ∗(y)) n m) , ≤ C ≥ ∨ m=0 X y∈∂BX(x,m) where ∗(y) is the -connected component of black vertices in Zd containing y. By again comparing C ∗ d (1GL(x))x∈Z to a site percolation process, it is possible to apply (A.8) to deduce that the tail of −c6(n∨m) the probability in the above sum is bounded above by c5e . The estimate at (A.4) follows. We now return to the problem of deriving the estimate at (A.1). For x Zd, deﬁne the set ′ d d ′ ∈ d Q (x) := L(x + e + + e )+ [0,L) Z , so that (Q (x)) Zd is a partition of Z . For x L, let L 1 · · · d1 ∩ L x∈ ∈ a(x) be the closest element of L, with respect to ℓ distance, to the x′ Zd such that x Q′ (x′). 1 ∈ ∈ L (Only when x is within a distance L of the boundary of L does x′ = a(x).) It is then easy to check 6 that if diam( (x)) L, then G ≥ ′ (x) ′ Q˜ (x ). (A.9) G ⊆∪x ∈C(a(x)) 3L (cf. [17, (3.7)]). In particular, this implies that, if diam( (x)) L, then it must be the case that G ≥ diam( (x)) 3Ldiam( (a(x))). Consequently, (A.1) follows from (A.4), and thus the proof of part G ≤ C (i) is complete. Given part (i), we observe

P x [ n,n]d L : # (x) (log n)d+1 ∃ ∈ − ∩ H ≥ d d+1 (2n + 1) sup P x 1, # (x) (log n) ≤ x∈L ∈C H ≥ d (d+1)/d (2n + 1) sup P x 1, diam( (x)) (log n) ≤ x∈L ∈C H ≥ (d+1)/d (2n + 1)dc e−c2(log n) . ≤ 1 Since this is summable in n, part (ii) follows by a Borel-Cantelli argument.

We now work towards the proof of Lemma 3.2.

Proof of Lemma 3.2. To establish the bound in (3.4), let us start by recalling/adapting some definitions from the previous proof. In particular, for x Zd, define Q (x) and Q˜ (x) as in the proof ∈ L 3L of Lemma 3.1. Moreover, let G (x) be defined similarly, but with ˜ replaced by ˜ , and redefine L O3 O1

35 x being ‘white’ to mean that this version of GL(x) holds (and say x is ‘black’ otherwise). Note that the statement (A.2) remains true with this deﬁnition of GL(x), and the dependence between

d the random variables (1GL(x))x∈Z is only finite range, and so we can suppose that it dominates a dominates a site percolation process on Zd of density arbitrarily close to 1. Now, fix x,y L, and recall the definition of a(x) from the proof of Lemma 3.1. If n is the ℓ ∈ 1 distance between a(x) and a(y), then there exists a nearest neighbor path a0, . . . , an in L such that a = a(x) and a = a(y). We claim that if x and y are both contained in , then there exists a 0 n C1 path from x to y along edges of whose vertices all lie in O1 n ∗ Q˜3L(b), (A.10) ∪i=0 ∪b∈C (ai) ∗ ∗ where (a) := a if a is white, otherwise (a) := ∗(a) ∂out ∗(a), where ∗(a) is the - C { } C C ∪ C C ∗ connected component of black sites in L containing a (∂out ∗(a) is the outer boundary of ∗(a), C C defined similarly to (A.5)). This is essentially [4, Proposition 3.1] rewritten for L instead of Zd. The one slight issue with modifying the proof of this result to our situation is that, unlike the Zd case, the outer boundary in L of a finite connected cluster of vertices, say, is no longer -connected C ∗ in general and so it is not possible to run around it in quite the same way. However, this problem is readily overcome by applying [54, Lemma 2], which implies that for each x , the part of the 6∈ C outer boundary of that is visible from x, ∂vis(x) (cf. (A.6)), is -connected. A simple estimate C C ∗ of the number of vertices in the set at (A.10) yields

n n ∗ d (x,y) (3L + 1)d # (a ) (3L + 1)d 1 + 3d# ∗(a ) . 1 ≤ C i ≤ C i Xi=0 Xi=0 (We take ∗(a) := if a is white.) Clearly ∗(a) ˜∗(a), where ˜∗(a) is the -connected component C ∅ C ⊆ C C ∗ of black sites in Zd containing a. Consequently, we obtain that

n d d ∗ P (x,y 1 and d1(x,y) cR) P (3L + 1) 1 + 3 # ˜ (ai) cR . ∈C ≥ ≤ C ≥ ! Xi=0 Moreover, applying [29, Lemma 2.3] as in the proof of [4, Theorem 1.1], we may replace the summands on the right-hand side by independent ones, each with the same distribution as the term they are replacing. Recalling from (A.8) the exponential bound for the size of a -connected ∗ vacant cluster in a site percolation process of parameter p close to 1, one readily obtains from this the bound at (3.4). (It is useful to also note that n c x y /L.) ≤ | − | We proceed next with the proof of the second bound. In this direction, let us begin by defining a metric d on related to the process Z introduced in Section 3.2, that is, the time change of Z C2 Y with time in cut out. Assume that K is large enough so that the conclusions of Lemma 3.1 H hold. Define a set of edges E′ by supposing, for x,y , x,y E′ if and only if x,y Z ∈ C2 { } ∈ Z { } 6∈ O2 and also there exists a path x = z , z ,...,z = y such that z ,...,z and z , z 0 1 k 1 k−1 ∈ H { i−1 i} ∈ O1 for i = 1,...,k. Thus the jumps of Z will be on edges in either or E′ . Set E := E′ , and O2 Z Z O2 ∪ Z let d be the graph distance on ( , E ). Our first goal will be to prove that there exist constants Z C2 Z c , c , c such that: for every x,y L, 1 2 3 ∈ P x,y and d (x,y) c−1 x y c e−c3|x−y|, (A.11) ∈C2 Z ≤ 1 | − | ≤ 2 36 where x y is the Euclidean distance between x and y. | − | For proving (A.11), we suppose that the definition of GL(x) reverts to that given in the proof of Lemma 3.1, i.e. in terms of ˜ . Also, define G′ (x) to be the event that there are no edges of O3 L the set ˜ ˜ connecting two vertices of Q˜ (x), so that if G (x) G′ (x) holds, then so do the O1\O3 3L L ∩ L defining properties of G (x) when ˜ is replaced by ˜ . Clearly, for fixed L, P (G′ (x)c) 0 as L O3 O2 L → p p (i.e. K ). Hence, for any δ, by first choosing L and then K large, we can ensure 3 → 1 → ∞ P (G (x) G′ (x)) 1 δ. For the remainder of this proof, we redefine x being ‘white’ to mean L ∩ L ≥ − that G (x) G′ (x) holds, and say x is ‘black’ otherwise. L ∩ L Similarly to above, the finite range dependence of the random variables in question means that it is possible to suppose that (1 ′ ) Zd stochastically dominates a collection (η(x)) Zd of GL(x)∩GL(x) x∈ x∈ independent and identical Bernoulli random variables whose parameter p is arbitrarily close to 1. Let be the vertices of L that are contained in an infinite connected component of x L : η(x)= C∞ { ∈ 1 (cf. (A.3)). By arguments from the proof of Lemma 3.1, we have that if p is large enough, then } this set is non-empty and its complement in L consists of finite connected components, P-a.s. Now, as in the proof of [17, Lemma 3.1], we ‘wire’ the holes of by adding edges between every pair of C∞ sites that are contained in a connected component of L or its outer boundary, and denote the \C∞ induced graph distance by d′. By proceeding almost exactly as in [17], it is then possible to show that, for suitably large L and K: for x,y L, ∈ 1 P d′(x,y) x y e−|x−y|. (A.12) ≤ 2| − | ≤ (The one modification needed depends on the observation that, similarly to what was deduced in the proof of Lemma 3.1, the inner boundary of any connected component of L is -connected \C∞ ∗ and consists solely of vertices with η(x) = 0.) Finally, a minor adaptation of (A.9) yields, for x ∈C1 with diam( (x)) L, H ≥ ′ (x) ′ Q˜ (x ), H ⊆∪x ∈C(a(x)) 3L where (a(x)) is now the connected component of L containing a(x). It follows that if x,y , C \C∞ ∈C2 then d (x,y) d′(a(x), a(y)), (cf. [17, (3.10)]). Therefore, since it also holds for x y 3L that Z ≥ | − |≥ a(x) a(y) c x y /L, the bound at (A.11) can be obtained from (A.12). | − |≥ | − | Finally, note that, since µ [K−1, K] for every e such that e e′ = for some e′ , e ∈ ∈ O1 ∩ 6 ∅ ∈ O2 it holds that t(e) [C K−1/2, K1/2] for such edges. Moreover, for every other e , we have ∈ A ∧ ∈ O1 t(e) 0. As a consequence, the metric d is bounded below by a constant multiple of d on . ≥ 1 Z C2 Applying this, as well as setting ∂out (x)= x for x , it follows that H { } 6∈ H P x,y and d (x,y) c−1 x y ∈C1 1 ≤ 4 | − | P ′ ′ −1 x,y 1 and inf dZ (x ,y ) c5 x y ≤ ∈C x′∈∂outH(x),y′∈∂outH(y) ≤ | − | P x′,y′ and d (x′,y′) c−1 x y ≤ ∈C2 Z ≤ 5 | − | x′,y′: |x−x′|,|y−y′|≤|x−y|/4 X +2P (x and diam( (x)) + 1 x y /4) . ∈C1 H ≥ | − | From this, the bound at (3.5) is a straightforward consequence of Lemma 3.1(i) and (A.11).

37 Given Lemma 3.2, the proof of Lemma 3.3 is straightforward.

Proof of Lemma 3.3. Since d C d , the inclusion B (x, R) B (x,C R) always holds. We will 1 ≤ A 1 1 ⊆ 1 A thus concern ourselves with the other two inclusions only. First, by the inequality at (3.5),

P x and B (x,C R) B (x, c R) ∈C1 1 A 6⊆ E 2 P x,y 1 and d1(x,y) CAR ≤ ∈C ≤ y6∈B (x,c R) EX 2 c e−c6R, ≤ 5 for suitably large c2. Secondly,

P (x and B (x, c R) B (x, R)) P (x,y and d (x,y) R) , ∈C1 C1 ∩ E 1 6⊆ 1 ≤ ∈C1 1 ≥ y∈BEX(x,c1R)

−c8R and applying (3.4) yields a bound of the form c7e , thereby completing the proof.

Next, the comparison of measures stated as Lemma 3.7.

Proof of Lemma 3.7. First note that any point x Q that is contained in ˜ must lie in a ∈ C1\C1 connected component of (Q , ) that meets the inner boundary of Q, which we denote here by \C1 O1 ∂inQ. Moreover, we recall that any points x Q that are contained in must also be contained ∈ C1 in ˜ . It follows that C1 ν˜(Q) ν(Q) # (x), − ≤ in F x∈X∂ Q where (x) is the connected component of L containing x. Now, similarly to (A.1), we have F \C1 that P (diam( (x)) n) c e−c2n, F ≥ ≤ 1 uniformly in x L. Since #∂inQ is bounded above by c nd−1, the lemma follows. ∈ 3 Finally, we prove Lemma 4.1.

Proof of Lemma 4.1. First, observe that

P +(Q ) C 2 6⊆ C1 P +(Q ) ε Q + P diam( +(Q )) ε1/d(2n + 1), +(Q ) . ≤ C 2 ≤ | 2| C 2 ≥ C 2 6⊆ C1 −c n As in the proof of Proposition 3.5, the ﬁrst term here is bounded above by c1e 2 . The second term is bounded above by

d P Zd c3n sup the diameter of the connected component of + 1 containing x is c4n . x∈Q2 \C ≥ That this admits a bound of the form c e−c6n follows from (A.1) (replacing by ). 5 C3 C1 Consequently, to complete the proof, it will suﬃce to show that

P +(Q ) Q Q c e−c8n. C 2 ∩ 1 ⊂C1 ∩ 1 ≤ 7 38 For the event on the left-hand side of the above to hold, it must be the case that there exists an open path in (Q , ) of diameter at least (1 ε)n that is not part of +(Q ). Moreover, as in the 2 O1 − C 2 previous paragraph, we have that, with probability at least 1 c e−c2n, diam( +(Q )) ε1/d(2n+1). − 1 C 2 ≥ However, by (the bond percolation version of) [49, Theorem 5], we have that with probability at least 1 c e−c10n, there is a unique open cluster in (Q , ) of diameter at least ε1/d(2n+1). Hence, − 9 2 O1 by taking ε suitably small, the result follows.

A.2 Arbitrary starting point quenched invariance principle

This section contains the proof of Theorem 3.13. For it, we note that the full Zd model also satisﬁes the conclusions of Propositions 3.10 and 3.11, as well as Assumption 2.3 (in the same sense as we checked for the L model in Section 3.3).

Proof of Theorem 3.13. To begin with, we recall the quenched invariance principle of [3, Theorem 1.1(a)] for the VSRW started at the origin: there exists a deterministic constant c > 0 such P ˜ n ˜ω that, for 1-a.e ω, the laws of the processes Y under P0 converge weakly to the law of (Bct)t≥0, d where (Bt)t≥0 is standard Brownian motion on R started from 0. Here, P1 is the conditional law P( 0 ˜1). Moreover, we note that by proceeding as in [3, Remark 5.16], one can check that the ·| ∈ C P P ˜ω ˜ω result remains true if 1 is replaced by , and P0 is replaced by Px0 , where x0 is chosen to be the (not necessarily uniquely defined) closest point to the origin in the infinite cluster ˜ . C1 Given the above result, the argument of this paragraph applies for P-a.e. ω. Fix x Rd, ε> 0, ∈ and define τ n := inf t> 0 : Y˜ n B (x, ε) , t ∈ E n o τ B := inf t> 0 : B B (x, ε) . { ct ∈ E } A standard argument gives us that, jointly with the convergence of the previous paragraph, τ n converges in distribution to τ B. Letting T > 0 be a deterministic constant, it follows that the n ω n laws of the processes (Y˜ n ) under P˜ ( τ T ) converge weakly to the law of (B ) , τ +t t≥0 x0 ·| ≤ c(τ+t) t≥0 started from 0 and conditional on τ T . Consequently, the Markov property gives us that for ≤ every bounded, continuous function f : C([0, ), Rd) R, ∞ → ω n ω n n B B B E˜ f(Y˜ )P Y˜ n dy τ T E f(B )P B B dy τ T , (A.13) ny x0 τ ∈ | ≤ → y c· 0 cτ ∈ | ≤ ZBE (x,ε) ZBE (x,ε) B B where Px is the law of the standard Brownian motion B started from x, and Ex is the corresponding expectation. Furthermore, it is elementary to check for such f that, as ε 0, → B B B B E f(B )P B B dy τ T E f(B ). (A.14) y c· 0 cτ ∈ | ≤ → x c· ZBE (x,ε) Suppose that the following also holds for every sequence of starting points x n−1 ˜ such that n ∈ C1 x x, every finite collection of times 0

˜ω ˜ n ω ˜ n n ˜ω ˜ n lim lim sup Enyf(Yt )Px Yτ n dy τ T Enx f(Yt ) = 0, (A.15) ε→0 0 n n→∞ BE(x,ε) ∈ | ≤ − Z

39 ˜ n ˜ n ˜ n ˜ n where Yt := (Yt1 , Yt2 ,..., Ytk ). Combining (A.13), (A.14) and (A.15) readily yields that the finite- ˜ n ω B dimensional distributions of Y (under Pnxn ) converge to those of Bc· (under Px ). Moreover, from ˜ n ω the full plane version of Proposition 3.10, we have that the laws of Y under Pnxn are tight. These two facts yields the desired quenched invariance principle. To complete the argument of the previous paragraph and the proof of the theorem, it remains to check the limit at (A.15) holds (simultaneously over sequences of starting points x x, times n → t = (t1,...,tk), and functions f) for P-a.e. ω. In fact, using the independent increments property of Y˜ n and some standard analysis, it will suffice to check the result for k = 2 and for functions f ˜ n ˜ n ˜ n ˜ n ˜ n ñ of the form f(Yt1 , Yt2 ) = f1(Yt1 )f2(Yt2 ). Writing the semigroup of Y as P , we have for such a function f that

ω n n ω n n ω n n ˜ ˜ ˜ ˜ n ˜ ˜ ˜ Enyf(Yt1 , Yt2 )Px0 Yτ dy τ T Enxn f(Yt1 , Yt2 ) BE(x,ε) ∈ | ≤ − Z

n ω n n n ˜ ˜ n ˜ = Pt1 g(y)Px0 Yτ dy τ T Pt1 g(xn) BE (x,ε) ∈ | ≤ − Z ˜n ˜n sup Pt1 g(y) Pt1 g(xn) , (A.16) ≤ −1 ˜ − y∈BE(x,ε)∩n C1

where g := f (P˜n f ) (which is a bounded, continuous function). Take R > 2 with x 1 × t2−t1 2 ∈ B (0, R/2) n−1 ˜ . For each λ> 1, it holds that E ∩ C1

n n,BλR n n,BλR n P˜ g = P˜ (g1B ) + (P˜ P˜ )(g1B )+ P˜ (g1Bc ), t t λR t − t λR t λR where B := B (0,s) n−1 ˜ . We have s E ∩ C1

n n,BλR n z n z n c (P˜ P˜ )(g1B )(z)+ P˜ (g1Bc )(z) g ∞P˜ (˜τ t)+ g ∞P˜ (Y˜ B ) | t − t λR t λR | ≤ k k ω BλR ≤ k k ω t ∈ λR 2 g P˜z(˜τ n t). ≤ k k∞ ω BλR ≤ So, setting B := B (x, ε) n−1 ˜ , for large n we have x,ε E ∩ C1 sup Pñ g(y) Pñ g(x ) t1 − t1 n y∈Bx,ε n,BλR n,BλR z n sup P˜ (g1B )(y) P˜ (g1B )(xn) + 4 g ∞ sup P˜ (˜τ t) ≤ t1 λR − t1 λR k k ω BλR ≤ y∈Bx,ε z∈Bx,2ε =: I + 4 g I . 1 k k∞ 2 n Now let us note that Assumption 2.3 holds for X killed on exiting BλR when Dn is replaced by BλR (which can be verified similarly to the discussion in Section 3.3 for the L case; for this, it is useful to note that the killing does not have any effect since points in BλR/2 are suitably far away n d Y˜ from the boundary of BλR). Moreover, applying the scaling relationq ˜t (x,y)= n qtn2 (nx,ny), by Proposition 3.11 (for the full Zd model) we have, P n,BλR g c t−d/2 g whenever we k t k∞,n,λR ≤ 1 k k1,n,λR also have tn2 c (sup R 1 sup 2d (x,y))1/4, where R is defined as in Proposition ≥ 2 x∈BλR x ∨ ∨ x,y∈BλR 1 x 3.11. Hence a simple Borel-Cantelli argument using the tail estimate of that proposition to control supx∈BλR Rx and Lemma 3.2 to control supx,y∈BλR 2d1(x,y) yields that the above bound holds true

40 ′ for all large n, P-a.s. Hence, by applying Proposition 2.5, we have I C (2ε)γ g for all 1 ≤ t k k2,n,λR x B and all large n, P-a.s. By applying the full Zd version of Proposition 3.14, i.e. the vague ∈ R/2 convergence of mn to a multiple of Lebesgue, it follows that for P-a.e. ω: for t > 0, R > 2 and λ> 1, lim lim sup sup I1 = 0. ε→0 n→∞ x∈BR/2

d For I2, we will apply the full Z version of the exit time bound of Proposition 3.8. In particular, this result implies that for P-a.e. ω: for t> 0, R> 2 and λ> 1,

2 2 2 −c3λ R /t lim lim sup sup I2 lim lim sup c2Ψ(c3(λR/2 2ε)n,tn )= c2e . ε→0 n→∞ ≤ ε→0 n→∞ − x∈BR/2

(Note that a Borel-Cantelli argument that depends on the tail estimate for Rx of Proposition 3.8 is hidden in the inequality.) Letting λ , this converges to 0. Thus, we deduce that for ﬁxed R> 2 →∞ and P-a.e.-ω that: for every sequence of starting points x n−1 ˜ such that x x B (0, R/2), n ∈ C1 n → ∈ E for every 0

Acknowledgement. Parts of this work were completed during visits by the ﬁrst and second named authors to the Research Institute for Mathematical Sciences, Kyoto University in early 2012 and early 2013, respectively. Both authors kindly thank RIMS for the hospitality and ﬁnancial support.

References

[1] M. Aizenman and D. J. Barsky, Sharpness of the phase transition in percolation models, Comm. Math. Phys. 108 (1987), no. 3, 489–526.

[2] M. Aizenman and C. M. Newman, Tree graph inequalities and critical behavior in percolation models, J. Statist. Phys. 36 (1984), no. 1-2, 107–143.

[3] S. Andres, M. T. Barlow, J.-D. Deuschel, and B. M. Hambly, Invariance principle for the random conductance model, Probab. Theory Related Fields, to appear.

[4] P. Antal and A. Pisztora, On the chemical distance for supercritical Bernoulli percolation, Ann. Probab. 24 (1996), no. 2, 1036–1048.

[5] G. Barles, F. Da Lio, P.-L. Lions, and P. E. Souganidis, Ergodic problems and periodic homogenization for fully non-linear equations in half-space type domains with Neumann boundary conditions, Indiana Univ. Math. J. 57 (2008), no. 5, 2355–2376.

[6] M. T. Barlow, Random walks on supercritical percolation clusters, Ann. Probab. 32 (2004), no. 4, 3024–3084.

[7] M. T. Barlow and R. F. Bass, The construction of Brownian motion on the Sierpi´nski carpet, Ann. Inst. H. Poincar´eProbab. Statist. 25 (1989), no. 3, 225–257.

41 [8] M. T. Barlow and J.-D. Deuschel, Invariance principle for the random conductance model with unbounded conductances, Ann. Probab. 38 (2010), no. 1, 234–276.

[9] M. T. Barlow and B. M. Hambly, Parabolic Harnack inequality and local limit theorem for percolation clusters, Electron. J. Probab. 14 (2009), no. 1, 1–27.

[10] D. J. Barsky, G. R. Grimmett, and C. M. Newman, Percolation in half-spaces: equality of critical densities and continuity of the percolation probability, Probab. Theory Related Fields 90 (1991), no. 1, 111–148.

[11] R. F. Bass, M. Kassmann, and T. Kumagai, Symmetric jump processes: localization, heat kernels and convergence, Ann. Inst. Henri Poincar´eProbab. Stat. 46 (2010), no. 1, 59–71.

[12] I. Benjamini, Z.-Q. Chen and S. Rohde, Boundary trace of reﬂecting Brownian motions. Probab. Theory Related Fields, 129 (2004), 1-17.

[13] I. Benjamini and E. Mossel, On the mixing time of a simple random walk on the super critical percolation cluster, Probab. Theory Related Fields 125 (2003), no. 3, 408–420.

[14] A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic methods in periodic media, Ed. North Holland, 1978.

[15] N. Berger and M. Biskup, Quenched invariance principle for simple random walk on percolation clusters, Probab. Theory Related Fields 137 (2007), no. 1-2, 83–120.

[16] M. Biskup, Recent progress on the random conductance model, Probab. Surv. 8 (2011), 294–373.

[17] M. Biskup and T. M. Prescott, Functional CLT for random walk among bounded random conductances, Electron. J. Probab. 12 (2007), no. 49, 1323–1348.

[18] S. Buckley and B. M. Hambly, An invariance principle for random walk in a box in the random conductance model, in preparation.

[19] K. Burdzy and Z.Q. Chen, Discrete approximations to reﬂected Brownian motion. Ann. Probab. 36 (2008), 698-727.

[20] K. Burdzy and Z.-Q. Chen, Reﬂecting random walk in fractal domains. To appear in Ann. Probab.

[21] Z.-Q. Chen, On reﬂecting diﬀusion processes and Skorokhod decompositions. Probab. Theory Related Fields, 94 (1993), 281-316.

[22] Z.-Q. Chen, Reﬂecting Brownian motions and a deletion result for Sobolev spaces of order (1; 2). Potential Analysis, 5 (1996). 383-401.

[23] Z.-Q. Chen and M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory. Princeton University Press, 2012.

[24] K. L. Chung and J. B. Walsh, Markov Processes, Brownian Motion, and Time Symmetry, Second Edition. Springer, 2005.

[25] D. A. Croydon and B. M. Hambly, Local limit theorems for sequences of simple random walks on graphs, Potential Anal. 29 (2008), no. 4, 351–389.

[26] D. A. Croydon, B. M. Hambly, and T. Kumagai, Convergence of mixing times for sequences of random walks on ﬁnite graphs, Electron. J. Probab. 17 (2012), no. 3, 32.

42 [27] A. De Masi, P. A. Ferrari, S. Goldstein, and W. D. Wick, An invariance principle for reversible Markov processes. Applications to random motions in random environments, J. Statist. Phys. 55 (1989), no. 3-4, 787–855.

[28] T. Delmotte, Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev. Mat. Iberoamericana 15 (1999), no. 1, 181–232.

[29] J.-D. Deuschel and A. Pisztora, Surface order large deviations for high-density percolation, Probab. Theory Related Fields 104 (1996), no. 4, 467–482.

[30] R. Durrett, Probability: theory and examples, second ed., Duxbury Press, Belmont, CA, 1996.

[31] S. N. Ethier and T. G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1986, Characterization and convergence.

[32] G. R. Grimmett, Percolation, second ed., Grundlehren der Mathematischen Wissenschaften [Funda- mental Principles of Mathematical Sciences], vol. 321, Springer-Verlag, Berlin, 1999.

[33] G. R. Grimmett and J. M. Marstrand, The supercritical phase of percolation is well behaved, Proc. Roy. Soc. London Ser. A 430 (1990), no. 1879, 439–457.

[34] G. R. Grimmett and A. M. Stacey, Critical probabilities for site and bond percolation models, Ann. Probab. 26 (1998), no. 4, 1788–1812.

[35] P. Gyrya and L. Saloﬀ-Coste, Neumann and Dirichlet Heat Kernels in Inner Uniform Domains. Ast´erisque 336, 2011.

[36] T. E. Harris, A lower bound for the critical probability in a certain percolation process, Proc. Cambridge Philos. Soc. 56 (1960), 13–20.

[37] D. Jerison and C. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. in Math. 46 (1982), 80–147.

[38] V. V. Jikov, S. M. Kozlov and O. A. Ole˘ınik, O. A., Homogenization of diﬀerential operators and integral functionals, Translated from the Russian by G. A. Iosif′yan, Springer-Verlag, Berlin, 1994.

[39] A. Jonsson and H. Wallin, Function Spaces on Subsets of Rn. Mathematical Reports, Vol. 2, Part 1. Acad. Publ., Harwood, 1984.

[40] C. Kipnis and S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Comm. Math. Phys. 104 (1986), no. 1, 1–19.

[41] T. Komorowski, C. Landim, and S. Olla, Fluctuations in Markov processes: time symmetry and martingale approximation, Grundlehren der mathematischen Wissenschaften 345, Springer, 2012.

[42] S. M. Kozlov, The averaging method and walks in inhomogeneous environments, Uspekhi Mat. Nauk 40 (1985), no. 2(242), 61–120, 238.

[43] R. K¨unnemann, The diﬀusion limit for reversible jump processes on Zd with ergodic random bond conductivities, Comm. Math. Phys. 90 (1983), no. 1, 27–68.

[44] T. M. Liggett, R. H. Schonmann, and A. M. Stacey, Domination by product measures, Ann. Probab. 25 (1997), no. 1, 71–95.

43 [45] P. Mathieu, Quenched invariance principles for random walks with random conductances, J. Stat. Phys. 130 (2008), no. 5, 1025–1046.

[46] P. Mathieu and A. Piatnitski, Quenched invariance principles for random walks on percolation clusters, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2085, 2287–2307.

[47] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123 (1994), no. 2, 368–421.

[48] H. Osada, Homogenization of diﬀusion processes with random stationary coeﬃcients, Probability theory and mathematical statistics (Tbilisi, 1982), Lecture Notes in Math., vol. 1021, Springer, Berlin, 1983, pp. 507–517.

[49] M. D. Penrose and A. Pisztora, Large deviations for discrete and continuous percolation, Adv. in Appl. Probab. 28 (1996), no. 1, 29–52.

[50] A. Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Related Fields 104 (1996), no. 4, 427–466.

[51] R. Rhodes, Stochastic homogenization of reﬂected stochastic diﬀerential equations, Electron. J. Probab. 15 (2010), Paper no. 31, 989–1023.

[52] V. Sidoravicius and A.-S. Sznitman, Quenched invariance principles for walks on clusters of percolation or among random conductances, Probab. Theory Related Fields 129 (2004), no. 2, 219–244.

[53] H. Tanaka, Homogenization of diﬀusion processes with boundary conditions, Stoc. Anal. Appl., Adv. Probab. Related Topics 7 Dekker, New York, 1984, 411–437.

[54] A. Timar, Boundary-connectivity via graph theory, Proc. Amer. Math. Soc. 141 (2013), 475–480.

[55] J. Väisälä, Uniform domains. Tohoku Math. J. 40 (1988), 101-118.

Zhen-Qing Chen Department of Mathematics, University of Washington, Seattle, WA 98195, USA. E-mail: [email protected]

David A. Croydon Department of Statistics, University of Warwick, Coventry, CV4 7AL, United Kingdom. E-mail: [email protected]

Takashi Kumagai Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. E-mail: [email protected]