Solidification estimates for random walks on supercritical percolation clusters

TL;DR

This paper establishes uniform estimates for the probability that a random walk on the infinite cluster of various percolation models in high dimensions is absorbed by porous interfaces, extending previous results to non-elliptic settings.

Contribution

It provides the first uniform control estimates for random walks on supercritical percolation clusters in a broad non-elliptic framework, including models with correlations.

Findings

Uniform absorption probability controls for random walks on percolation clusters

Extension of previous Brownian motion results to non-elliptic models

Applicability to models with algebraically decaying correlations

Abstract

We consider the simple random walk on the infinite cluster of a general class of percolation models on $\mathbb{Z}^d$, $d\geq 3$, including Bernoulli percolation as well as models with strong, algebraically decaying correlations. For almost every realization of the percolation configuration, we obtain uniform controls on the absorption probability of a random walk by certain "porous interfaces" surrounding the discrete blow-up of a compact set $A$. These controls substantially generalize previous results obtained in arXiv:1706.07229 for Brownian motion in $\mathbb{R}^d$ and in arXiv:2012.05230 for random walks on $\mathbb{Z}^d$ equipped with uniformly elliptic edge weights to a manifestly non-elliptic framework.

Equations

$${\mathcal{S}}_{\infty }=\text{ the unique infinite cluster of the subgraph of Zd induced by S,}$$

$$A_N=(NA)\cap {\mathbb{Z}}^d.$$

$$(a_N{)}_{N\ge 0}$$

$$a_N/\exp \left\{\log (b_N{)}^{\frac{1}{1+{\Delta }_{\mathrm{S}}/2}}\right\}\nearrow \infty ,$$

$$b_N\ge \tilde{c}(A)N,\text{for all }N\ge 1,$$

$$\begin{split}\mathscr{U}_{\ell_{\ast},N}^{\omega}=&\;\mbox{the collection of sets $U_{0}\subseteq B(0,b_{N})\cap\mathscr{S}_{\infty}$ for which the}\\[-2.15277pt] &\;\mbox{local density of $U_{1}=\mathscr{S}_{\infty}\setminus U_{0}$ for any box $B(x,2^{\ell})\cap\mathscr{S}_{\infty}$ }\\[-4.30554pt] &\;\mbox{centered at a point $x\in A_{N}\cap\mathscr{S}_{\infty}$ is at most $\frac{1}{2}$ when $\ell\leq\ell_{\ast}$}\end{split}$$

$$\begin{split}\mathscr{S}^{\omega}_{U_{0},\epsilon,\chi}=&\;\mbox{the collection of bounded subsets $\Sigma\subseteq\mathscr{S}_{\infty}$ such that for all $x\in S$}\\ &\;\mbox{$P_{x}^{\omega}[\text{Random walks enters $\Sigma$ before going to distance $\epsilon$}]\geq\chi$},\end{split}$$

$$\underset{N\to \infty }{\lim }\tilde{\sup }\underset{x\in A_N\cap {\mathcal{S}}_{\infty }}{\sup }{P}_x^{\omega }[H_{\Sigma }=\infty ]=0,\text{for Pu-a.e. ω∈{0,1}Zd}$$

$$\underset{N\to \infty }{\mathrm{liminf}}\tilde{\inf }\frac{{\mathrm{Cap}}^{\omega }(\Sigma )}{{\mathrm{Cap}}^{\omega }(A_N\cap {\mathcal{S}}_{\infty })}\ge 1,\text{for Pu-a.e. ω∈{0,1}Zd},$$

$$\mathrm{Res}={\mathrm{Res}}^{\omega }(U_0,I,J,L,{\ell }_{\ast })=\{x\in {\mathcal{S}}_{\infty }:\sum _{\ell \in {\mathcal{A}}_{\ast }}\text{mathbbm}{1}_{\{{\tilde{\sigma }}_{\ell }^{\omega }(x)\in [\tilde{\alpha },1-\tilde{\alpha }]\}}\ge J\},$$

$$\underset{N\to \infty }{\lim }\underset{{\ell }_{\ast }}{\sup }\underset{U_0\in {\mathcal{U}}_{{\ell }_{\ast },N}^{\omega }}{\sup }\underset{x\in A_N\cap {\mathcal{S}}_{\infty }}{\sup }{P}_x^{\omega }\left[H_{{\mathrm{Res}}^{\omega }(U_0,I(J_N),J_N,L(J_N),{\ell }_{\ast })}=\infty \right]=0,$$

$${\Psi }_x:\Omega \to \Omega ,({\Psi }_x\omega )(y)=\omega (y+x)\text{ for all ω∈Ω and x,y∈Zd}.$$

$$\mathcal{S}=\mathcal{S}(\omega )=\left\{x\in {\mathbb{Z}}^d:\omega (x)=1\right\}\subseteq {\mathbb{Z}}^d.$$

$${\mu }_{\{x,y\}}=\{\begin{array}{ll}1,& \{x,y\}\in E({\mathcal{S}}_{\infty }),\\ 0,& \text{otherwise},\end{array}{\mu }_x=\sum _{y\in {\mathcal{S}}_{\infty }}{\mu }_{\{x,y\}},x\in {\mathcal{S}}_{\infty },$$

$${\mathbb{P}}^u[A_1\cap A_2]\le {\mathbb{P}}^{u^{\prime }}[A_1]{\mathbb{P}}^{u^{\prime }}[A_2]+C\exp \left(-c\min \left\{(u^{\prime }-u{)}^{\beta }{s}^{\gamma },e^{(\log L{)}^{\zeta }}\right\}\right),$$

$${\mathbb{P}}^{u^{\prime }}[B_1\cap B_2]\le {\mathbb{P}}^u[B_1]{\mathbb{P}}^u[B_2]+C\exp \left(-c\min \left\{(u^{\prime }-u{)}^{\beta }{s}^{\gamma },e^{(\log L{)}^{\zeta }}\right\}\right).$$

$${\Delta }_{\mathrm{S}}={\Delta }_{\mathrm{S}}(u)>0$$

$${\mathbb{P}}^u\left[{\mathcal{S}}_R\cap B(0,R)\ne \varnothing \right]\ge 1-e^{-f_{\text{S}}(u,R)},\text{ and}$$

$$\mathbb{P}^{u}\left[\begin{array}[]{c}\text{for all $x,y\in\mathscr{S}_{\scriptscriptstyle{R/10}}\cap B(0,R)$,}\\ \text{$x$ is connected to $y$ in $\mathscr{S}\cap B(0,2R)$}\end{array}\right]\geq 1-e^{-f_{\text{S}}(u,R)}.$$

$$\eta (u)\stackrel{\mathrm{def}}{=}{\mathbb{P}}^u\left[0\in {\mathcal{S}}_{\infty }\right],$$

$$\Omega_{0}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{\omega\in\Omega\,:\,\mbox{$\mathscr{S}_{\infty}(\omega)$ is nonempty and connected}\},$$

$$u\ge \left(1+R^{-{\chi }_{\text{P}}}\right)\cdot \hat{u},$$

$$\mathbb{P}^{u}\left[A_{1}\cap A_{2}\right]\leq\mathbb{P}^{\widehat{u}}\left[A_{1}\right]\cdot\mathbb{P}^{\widehat{u}}\left[A_{2}\right]+e^{-f_{\text{P}}(L)},\$$

$$\mathbb{P}^{\widehat{u}}\left[B_{1}\cap B_{2}\right]\leq\mathbb{P}^{u}\left[B_{1}\right]\cdot\mathbb{P}^{u}\left[B_{2}\right]+e^{-f_{\text{P}}(L)},\$$

$$\mathbb{P}^{u}[\mathscr{S}\cap K=\varnothing]=e^{-u\mathrm{Cap}(K)},\quad\mbox{for all $\varnothing\neq K\subset\subset\mathbb{Z}^{d}$},$$

$$\mathrm{P}^{v}[K\subseteq\mathscr{S}]=e^{-v\mathrm{Cap}(K)},\quad\mbox{for all $\varnothing\neq K\subset\subset\mathbb{Z}^{d}$}.$$

$${\mathcal{L}}_{{\mathcal{S}}_{\infty }}f(x)=\frac{1}{{\mu }_x}\sum _{y\in {\mathcal{S}}_{\infty }}{\mu }_{\{x,y\}}(f(y)-f(x)),x\in {\mathcal{S}}_{\infty },$$

$${\tau }_r=\inf \{t\ge 0:|X_t-X_0{|}_{\infty }\ge r\}.$$

$$q_t^{\omega }(x,y)$$

$$q_{t,U}^{\omega }(x,y)$$

$$c_{\text{hk}1}{t}^{-d/2}\exp \left(-c_{\text{hk}2}\frac{|x-y{|}_1^2}{t}\right)\le q_t^{\omega }(x,y)\le c_{\text{hk}3}{t}^{-d/2}\exp \left(-c_{\text{hk}4}\frac{|x-y{|}_1^2}{t}\right)$$

Full text

1. 1 Introduction

2. 2 Notation and model assumptions on percolation clusters

3. 2.1 Notation

4. 2.2 Model assumptions and examples

5. 2.3 Random walks on the infinite percolation cluster

6. 3 Uniform large-scale behavior over very large regions

7. 4 Solidification of porous interfaces on percolation clusters

8. 4.1 Local density functions

9. 4.2 Resonance set

10. 4.3 Proof of Theorem 4.1

11. A Proof of Proposition 3.1

1. Introduction

In this article, we study the simple random walk on the infinite connected component $\mathscr{S}_{\infty}$ of a typical realization of a supercritical percolation configuration on $\mathbb{Z}^{d}$, $d\geq 3$, for a large class of percolation models. The latter contains Bernoulli bond- or site-percolation, as well as models with strong, algebraically decaying correlations fulfilling certain structural assumptions taken from [2, 23], including the vacant set of random interlacements, the level sets of the Gaussian free field, and the range or vacant set of a random walk loop soup.

As our main result, we consider the trace on $\mathscr{S}_{\infty}$ of the discrete blow-up $A_{N}$ of a compact set $A\subseteq\mathbb{R}^{d}$ with non-empty interior, $A_{N}\cap\mathscr{S}_{\infty}$, and prove uniform absorption estimates for a random walk started in $A_{N}\cap\mathscr{S}_{\infty}$ by a class of porous interfaces $\Sigma\subseteq\mathscr{S}_{\infty}$ surrounding $A_{N}$. Estimates of this type were first developed in [34] in the context of Brownian motion and later extended in [18] to random walks on $\mathbb{Z}^{d}$, $d\geq 3$, with uniformly elliptic weights $\lambda\in[\lambda_{\min},1]^{\mathbb{E}^{d}}$ attached to the edges $\mathbb{E}^{d}$ (here $\lambda_{\min}\in(0,1)$). Together with related lower bounds on the capacity of the porous interfaces $\Sigma$, these solidification estimates have been instrumental as a substitute for a Wiener-type criterion in deriving large deviation upper bounds for the probability of various events characterized by atypical isolation-type phenomena of macroscopic bodies, see in particular [16, 17, 18, 35, 34, 49] (but see also Remark 4.14 for more on these applications). While in [18] the aforementioned solidification estimates could be obtained for every fixed choice of edge weights $\lambda\in[\lambda_{\min},1]^{\mathbb{E}^{d}}$, leveraging quenched heat kernel bounds for random walks on $(\mathbb{Z}^{d},\mathbb{E}^{d},\lambda)$, the present set-up is markedly more delicate. This is due to an effect of local degeneracies of the percolation cluster, rendering the use of uniform controls even in the simpler case of Bernoulli bond percolation infeasible. Informally, while many global properties of $\mathscr{S}_{\infty}$ exhibit the same behavior as on $\mathbb{Z}^{d}$ (including chemical distances, volume regularity, quenched invariance principles, and Gaussian heat kernel bounds for the random walk, see, e.g., [5, 6, 7, 9, 21, 32, 44], as well as [4] for the scaling limit of the Gaussian free field on a large class of percolation clusters), the presence of regions of small volume or irregular heat kernel qualitatively changes certain characteristics of the cluster. These include, in the case of Bernoulli percolation, the maximal effective resistance between points of the largest connected component of the cluster in large boxes, see [1], or the phases of the polymer model on $\mathscr{S}_{\infty}$, see [20, 36]. In some cases, a behavior comparable to the full lattice is retained for $p$ close enough to one, see for instance [43] for the maximum of the Gaussian free field on a two-dimensional Bernoulli bond percolation cluster. However, removing the effect of locally bad behavior would require a very fine understanding of the local geometry of $\mathscr{S}_{\infty}$ (see Remark 4.15 of the same reference). Our main contribution may therefore be seen as a structural property of $\mathscr{S}_{\infty}$ for a large class of models. Notably, our methods are applicable throughout the entire supercritical phase of Bernoulli bond percolation, as well as the (strongly) supercritical phase of a wide class of (possibly correlated) site percolation models. A major challenge inherent in our present set-up is to verify quantitatively that the influence of atypically bad regions that are present in $\mathscr{S}_{\infty}$ is not felt by the random walk on sufficiently many scales. Crucially, our controls must be strong enough to deploy the construction of a certain generic resonance set, which is hard to avoid for a random walk starting in $A_{N}\cap\mathscr{S}_{\infty}$.

We now state our results in more detail. Consider $\mathbb{Z}^{d}$, $d\geq 3$, and a one-parameter family of probability measures $\mathbb{P}^{u}$, $u\in(a,b)$, on $\{0,1\}^{\mathbb{Z}^{d}}$ governing a site percolation model on $\mathbb{Z}^{d}$ fulfilling the regularity properties P1–P2, D, and S1–S2, where we declare sites with label one to be open (see Subsection 2.2 for a precise definition). We will turn to bond percolation models later in the introduction. The parameters $a,b$ are chosen to guarantee that the model is in a (strongly) supercritical phase. We denote the random set of open vertices by $\mathscr{S}\subseteq\mathbb{Z}^{d}$ and write

[TABLE]

which exists with $\mathbb{P}^{u}$-probability one under our standing assumptions. The set of assumptions is taken from [2], which itself is adapted from a slightly more restrictive set of assumptions that appeared in [23], and includes a large class of pertinent site percolation models, see Remark 2.2. Informally, assumptions P1, P2, and D correspond to ergodicity of the lattice shift operator, monotonicity in the percolation parameter $u$, and the availability of certain sprinkled decoupling inequalities (see (2.4), (2.5)), respectively. The properties S1 and S2 guarantee the local uniqueness of the infinite cluster in large boxes, and the continuity of the density of the infinite cluster $u\mapsto\mathbb{P}^{u}[0\in\mathscr{S}_{\infty}]$. For illustrative purposes of this introduction, one may restrict the attention to the family $(\mathbb{P}^{u})_{u\in(p_{c},1)}$ governing supercritical Bernoulli site percolation at varying levels $u$, with $p_{c}$ denoting the corresponding percolation threshold. Let $A\subseteq\mathbb{R}^{d}$ be compact with non-empty interior and denote its discrete blow-up by

[TABLE]

We consider below a class of bounded sets $U_{0}\subseteq\mathscr{S}_{\infty}$ with the exterior boundary $S=\partial_{\mathscr{S}_{\infty}}U_{0}$ (taken with respect to $\mathscr{S}_{\infty}$) of $U_{0}$ being a sort of “hard” interface at a certain distance from $A_{N}$. Informally, $S$ can be thought of as being located at sup-norm distance $2^{\ell_{\ast}}$ of the set $A_{N}\cap\mathscr{S}_{\infty}$ for an integer $\ell_{\ast}\geq 0$. To deal with the presence of “sizeable, far-away bad regions” of $\mathscr{S}_{\infty}$, we introduce a “growth condition” limiting the possible values of the distance parameter $2^{\ell_{\ast}}$ and the size of $U_{0}$. More precisely, we will assume the following:

[TABLE]

where $\Delta_{\mathrm{S}}=\Delta_{\mathrm{S}}(u)$ is a parameter, possibly depending on $u$, entering the definition of S1, see (2.6). In fact, we can assume without loss of generality that

[TABLE]

with a constant $\widetilde{c}(A)$ depending only on the set $A$. Heuristically, $a_{N}$ and $b_{N}$ should be viewed as lower and upper bounds for the distance $2^{\ell_{\ast}}$, respectively, and the condition in (1.3) is chosen in such a way that good controls on the heat kernel of the random walk and volume regularity hold uniformly across all boxes of sizes essentially between $a_{N}$ and $b_{N}$ and whose centers lie in a sup-norm ball of radius $b_{N}$, centred at the origin, for almost every realization of $\omega$, when $N$ is sufficiently large. With these preparations we formally introduce for any given “typical” percolation configuration $\omega\in\{0,1\}^{\mathbb{Z}^{d}}$ (so that $\mathscr{S}_{\infty}$ is non-empty and unique) and an integer $\ell_{\ast}\geq 0$ the class

[TABLE]

(here, $B(z,R)$ stands for the closed sup-norm ball in $\mathbb{Z}^{d}$ centered at $z\in\mathbb{Z}^{d}$ with radius $R\geq 0$). We then define the porous interfaces attached to a given $U_{0}\in\mathscr{U}^{\omega}_{\ell_{\ast},N}$, which may be viewed as deformations of the hard interface $S$. The former come with a certain integer “distance” parameter $\epsilon$ (limiting the sup-norm distance to $S$) and “strength” $\chi\in(0,1)$ (measuring how much the deformation is felt for a walk starting in $S$), and the set of all such interfaces is given by

[TABLE]

where $P^{\omega}_{x}$ is the canonical measure governing a (continuous-time, constant-speed) simple random walk on $\mathscr{S}_{\infty}$, starting from $x\in\mathscr{S}_{\infty}$ (see Subsection 2.3 for a precise definition).

Our main result appears in Theorem 4.1. It states that for a given $A\subseteq\mathbb{R}^{d}$ as above (1.2) and $\chi\in(0,1)$, uniformly over pairs of integers $(\epsilon,\ell_{\ast})$ with $\epsilon/2^{\ell_{\ast}}$ tending to zero with $N$, $2^{\ell_{\ast}}\in[a_{N},b_{N}]$ with $a_{N},b_{N}$ fulfilling (1.3) (and (1.4)), $U_{0}\in\mathscr{U}^{\omega}_{\ell_{\ast},N}$, and $\Sigma\in\mathscr{S}^{\omega}_{U_{0},\epsilon,\chi}$, a random walk starting in $A_{N}\cap\mathscr{S}_{\infty}$ asymptotically cannot avoid $\Sigma$, for almost every realization of the percolation configuration $\omega$. More precisely, we have

[TABLE]

(and $\widetilde{\sup}$ stands for the supremum over all $(\epsilon,\ell_{\ast})$ with $\epsilon/2^{\ell_{\ast}}$ tending to zero with $N$, $2^{\ell_{\ast}}\in[a_{N},b_{N}]$ with $a_{N},b_{N}$ fulfilling (1.3) (and (1.4)), $U_{0}\in\mathscr{U}^{\omega}_{\ell_{\ast},N}$, and $\Sigma\in\mathscr{S}^{\omega}_{U_{0},\epsilon,\chi}$). We show in Corollary 4.2 that (1.7) implies the capacity lower bound

[TABLE]

where $\widetilde{\inf}$ is over the same configuration as $\widetilde{\sup}$ in (1.7), and $\mathrm{Cap}^{\omega}(\cdot)$ denotes the random walk capacity on $\mathscr{S}_{\infty}$, see (2.17). On the full lattice $\mathbb{Z}^{d}$, $d\geq 3$ or on $\mathbb{R}^{d}$, $d\geq 3$, uniform lower bounds of the type considered in (1.8) are typically easier to obtain in the context of a convex set $A$ (or its discrete blow-up $A_{N}$), by considering certain a projection attached to $A$ resp. $A_{N}$ (we refer to the discussion in [34, Remark 3.6 (3)] for more on this in the continuum set-up). However, even if $A$ is convex, such a projection is typically ill-defined for $A_{N}\cap\mathscr{S}_{\infty}\subseteq\mathscr{S}_{\infty}$, which reinforces the generality of the methods presented here.

We now briefly comment on the idea of the proof and the challenges compared to [34] and [18]. The main difficulty in the proof of (1.7) is the construction of a certain resonance set which is attached to $\mathscr{U}^{\omega}_{\ell_{\ast},N}$, and is hard to avoid for a random walk starting in $A_{N}\cap\mathscr{S}_{\infty}$. In essence, the former set is characterized by the non-degeneracy of the local density of both $U_{0}$ and $U_{1}=\mathscr{S}_{\infty}\setminus U_{0}$ relative to the infinite cluster, on many properly separated scales. More precisely, for a given (typical) percolation configuration $\omega\in\{0,1\}^{\mathbb{Z}^{d}}$, integers $I,J,L\geq 1$ (which fulfill a certain compatibility condition, see (4.58)) representing the total number, strength, and separation of inspected scales, respectively, and $U_{0}\subseteq\mathscr{S}_{\infty}$, we define

[TABLE]

where $\widetilde{\alpha}$ depends only on the dimension $d$, $\widetilde{\sigma}^{\omega}_{\ell}(x)$ measures the volume of $U_{1}$ relative to the box $B(x,4\cdot 2^{\ell})\cap\mathscr{S}_{\infty}$, and $\ell$ is chosen among a subset $\mathscr{A}_{\ast}$ of integers $L\mathbb{N}$ (see (4.55) for the precise definition). A key obstacle to directly utilizing previous results such as [18] is that due to the inherent irregularity of the percolation cluster, even at sufficiently large scales, one can only control the behavior of local densities up to a certain degree of precision, quantified by an (error) parameter $\alpha\in(0,1)$. By decreasing the size of $\alpha$, thereby aiming at a more precise control on the relative volume at a certain scale, the size of the region where the density is uniformly well-behaved at that scale is in turn reduced (see (3.2)). As $N$ tends to infinity, we require $\alpha$ to depend on $N$ in such a way that this well-behaved region remains sufficiently large (see (4.97)). Furthermore, to uniformly control the accumulated error as the “resonance strength” $J$ increases, the precision parameter $\alpha$ must also depend on $J$. For that reason, a more quantitative approach in which $J=J_{N}$ slowly tends to infinity with $N$ is then necessary. The key insight is that for any $\omega\in\Omega_{\mathrm{reg}}\subseteq\{0,1\}^{\mathbb{Z}^{d}}$, a set of full $\mathbb{P}^{u}$-measure, and any $(J_{N})_{N\geq 0}$ with $J_{N}\nearrow\infty$, one has

[TABLE]

where the supremum over $\ell_{\ast}$ is taken subject to the compatibility condition (4.58), which depends also on $\omega\in\Omega_{\mathrm{reg}}$.

Let us emphasize that we have chosen to formulate the results above for (strongly) supercritical site percolation models under regularity assumptions P1–P2, D, and S1–S2, however our approach can be generalized straightforwardly to (strongly) supercritical bond percolation models, see Remark 4.14 (1). The most important case is Bernoulli bond percolation on $\mathbb{Z}^{d}$, $d\geq 3$, to which our methods apply as well. Moreover, we highlight that our approach does not rely on precise quantitative homogenization techniques, but only requires a well-behaved geometry of the underlying graph in terms of heat kernel bounds and volume regularity. In particular, the bounds (1.7) and (1.8) remain valid if the edges of the subgraph of $(\mathbb{Z}^{d},\mathbb{E}^{d})$ induced by $\mathscr{S}_{\infty}$ are equipped with any choice of uniformly elliptic weights, see Remark 4.14 (2) for more on this aspect. We also stress that the bounds derived in this work ought to be pertinent to obtain upper bounds on several atypical disconnection- or isolation-type events for analogues of the models considered in [16, 17, 18, 35, 34, 49], in the non-elliptic random environments considered here, and we return to questions of this type elsewhere [15].

The organization of this article is as follows. In Section 2, we introduce further notation and state the formal definitions of the assumptions P1–P2, D, and S1–S2 posed for the underlying percolation models, as well as quenched Gaussian heat kernel upper and lower bounds from [2] or [42], which will be instrumental for the development of the multi-scale techniques in subsequent sections. In Section 3 we quantify the (uniform) regularity properties of the percolation cluster. In particular, we single out a set $\Omega_{\mathrm{reg}}^{\alpha,\vartheta}$ (depending on parameters $\alpha,\vartheta\in(0,1)$ characterizing the degree of irregularity) of “good percolation configurations”, cf. (3.17), of full $\mathbb{P}^{u}$-measure which will guarantee the required large-scale uniformity to implement the multi-scale argument in the following section. The main quenched asymptotic absorption estimates (1.7) are proved in Section 4, see Theorem 4.1. In the Appendix A we give a proof of a volume regularity property for percolation clusters with long-range correlations, slightly adapted from [42].

Finally, we state our convention concerning constants that we use throughout the remainder of the text. We will denote by $C$, $c$, $c^{\prime}$, $\ldots$ positive constants with values that can change from place to place. Numbered constants $c_{0}$, $c_{1}$, $\ldots$ retain their value assigned to them at the place of their first occurrence. All constants may implicitly depend on the dimension. Dependence of constants on further parameters will be made explicit in the notation. In Sections 3 and  4, we will fix the parameter $u$ governing the site percolation measure $\mathbb{P}^{u}$, and allow all constants to implicitly depend on $u$.

2. Notation and model assumptions on percolation clusters

We begin this section by introducing further notation that will be used throughout the article. Following this, we present the main assumptions P1–P2, D, and S1–S2, which characterize the probability measures $(\mathbb{P}^{u})_{u\in(a,b)}$ for some $0<a<b<\infty$ governing the relevant percolation models and the structure of their respective (almost surely unique) infinite connected component. Our set of assumptions corresponds to the generic framework of [2], see also Remark 2.1 concerning the generality of the latter. Importantly, a broad class of percolation models on $\mathbb{Z}^{d}$, $d\geq 3$, with strong, algebraically decaying correlations, satisfies these assumptions. We provide some pertinent examples in Remark 2.2. Finally, we conclude this section with a brief discussion of some properties of random walks on infinite percolation clusters.

2.1. Notation

We let $\mathbb{N}=\{0,1,2,...\}$ stand for the set of natural numbers, and $\mathbb{Z}_{+}=\{1,2,...\}$ for the set of positive integers. For real numbers $s,t$, we let $s\wedge t$ and $s\vee t$ stand for the minimum and maximum of $s$ and $t$, respectively, and we denote by $\lfloor s\rfloor$ the integer part of $s$ and by $\lceil s\rceil$ the next integer greater or equal to $s$, when $s$ is non-negative. For two sequences $(a_{n})_{n\in\mathbb{N}}$ and $(b_{n})_{n\in\mathbb{N}}$ of positive real numbers, we write $a_{n}=o(b_{n})$ if $\lim_{n\to\infty}|a_{n}/b_{n}|=0$, $a_{n}\gg b_{n}$ if $\lim_{n\to\infty}|a_{n}/b_{n}|=\infty$, and $a_{n}=O(b_{n})$ if $\limsup_{n\to\infty}|a_{n}/b_{n}|<\infty$. We write $a_{n}\nearrow a$ if $a_{n}\leq a_{n+1}$ for every $n\in\mathbb{N}$ and $a_{n}\to a$ as $n\to\infty$ (with $a\in\mathbb{R}\cup\{+\infty\}$), and similarly $b_{n}\searrow b$ if $b_{n}\geq b_{n+1}$ for every $n\in\mathbb{N}$ and $b_{n}\to b$ as $n\to\infty$ (with $b\in\mathbb{R}\cup\{-\infty\}$).

Unless otherwise stated, we assume $d\geq 3$ throughout the article. Occasionally we may also allow $d=2$, especially when we provide examples satisfying the assumptions (see Examples (1), (5), (6)). In fact, part of our approach remains applicable when $d=2$, but substantial modifications to the set-up would be necessary due to the recurrence of the underlying random walk; see also the discussion preceding [34, (2.1)].

We denote by $|\cdot|$, $|\cdot|_{1}$, and $|\cdot|_{\infty}$ the Euclidean, $\ell^{1}$-, and $\ell^{\infty}$-norms on $\mathbb{R}^{d}$, respectively. For $x\in\mathbb{R}^{d}$ and $r\geq 0$, we write $B_{\mathbb{R}^{d}}(x,r)$ for the closed $\ell^{\infty}$-ball in $\mathbb{R}^{d}$ with radius $r$, centred at $x$. When $A$ is a subset of $\mathbb{R}^{d}$, we write $\mathring{A}$ for its interior. For $x\in\mathbb{Z}^{d}$ and $r\geq 0$, we furthermore write $B(x,r)=\{y\in\mathbb{Z}^{d}\,:\,|x-y|_{\infty}\leq r\}\subseteq\mathbb{Z}^{d}$ for the (closed) $\ell^{\infty}$-ball of radius $r$ and center $x$. If $x,y\in\mathbb{Z}^{d}$ fulfill $|x-y|=1$, we call them neighbors and write $x\sim y$. A function $\pi:\{0,...,n\}\rightarrow\mathbb{Z}^{d}$ is called a nearest-neighbor path (of length $n\geq 1$) if $\pi_{i}\sim\pi_{i+1}$ for all $0\leq i\leq n-1$. For $K\subseteq\mathbb{Z}^{d}$, we let $|K|$ stand for the cardinality of $K$, and we write $K\subset\subset L$ if $K\subseteq L$ and $|K|<\infty$. Moreover, if $K\subseteq L$ we write $\partial_{L}K=\{y\in L\setminus K\,:\,y\sim x\text{ for some }x\in K\}$ for the external boundary of $K$ relative to $L$, and $\partial_{L,\text{in}}K=\{y\in K\,:\,y\sim x\text{ for some }x\in L\setminus K\}$ for the internal boundary of $K$ relative to $L$. For $A\subseteq\mathbb{Z}^{d}$, we denote the set of edges in $A$ by $E(A)=\{\{x,y\}\,:\,x,y\in A,x\sim y\}$ and use $\mathbb{E}^{d}=E(\mathbb{Z}^{d})$ as a shorthand notation. For a set $D\subseteq\mathbb{R}^{d}$ we denote by $D_{N}=(ND)\cap\mathbb{Z}^{d}$ its discrete blow-up.

We now introduce some notation pertaining to the site-percolation models we study below. Consider the set $\Omega=\{0,1\}^{\mathbb{Z}^{d}}$ equipped with the $\sigma$-algebra $\mathscr{F}$ generated by the coordinate maps $\{\omega\mapsto\omega(x)\}_{x\in\mathbb{Z}^{d}}$. Let $(\Psi_{x})_{x\in\mathbb{Z}^{d}}$ be the group of lattice shifts on $\Omega$, defined by

[TABLE]

For a given $\omega\in\Omega$, sites $x\in\mathbb{Z}^{d}$ with $\omega(x)=1$ are declared open, and we denote the induced subset of open sites of $\mathbb{Z}^{d}$ by

[TABLE]

We view $(\mathscr{S},E(\mathscr{S}))$ as a subgraph of $(\mathbb{Z}^{d},\mathbb{E}^{d})$, retaining all edges connecting two vertices of $\mathscr{S}$ of $\ell^{1}$-distance one. For $r\in[0,\infty]$, we denote by $\mathscr{S}_{r}$ the set of vertices of $\mathscr{S}$ which are in connected components of $(\mathscr{S},E(\mathscr{S}))$ of $\ell^{1}$-diameter greater or equal to $r$. In particular, $\mathscr{S}_{\infty}$ is the set of vertices in infinite connected component(s) of $\mathscr{S}$. We also view $(\mathscr{S}_{r},E(\mathscr{S}_{r}))$, $r\in[0,\infty]$, as a subgraph of $(\mathbb{Z}^{d},E(\mathbb{Z}^{d}))$. For $x,y\in\mathscr{S}_{\infty}$, we define the weights

[TABLE]

and extend them to measures on $E(\mathscr{S}_{\infty})$ and $\mathscr{S}_{\infty}$, respectively.

For $\omega,\omega^{\prime}\in\Omega$ we write $\omega\leq\omega^{\prime}$ if $\omega(x)\leq\omega^{\prime}(x)$ for all $x\in\mathbb{Z}^{d}$. An $\mathscr{F}$-measurable function $f:\Omega\rightarrow\mathbb{R}$ is called increasing if $\omega\leq\omega^{\prime}$ implies $f(\omega)\leq f(\omega^{\prime})$, and decreasing if the function $-f$ is increasing. An event $G\in\mathscr{F}$ is called increasing (resp., decreasing) if $\mathbbm{1}_{G}$ is increasing (resp., decreasing).

2.2. Model assumptions and examples

In this subsection, we state the fundamental assumptions P1–P2, D, and S1–S2 on the underlying percolation models of interest. The first instance of assumptions of this type goes back to [23]. Several geometric and probabilistic properties, such as a shape theorem, quenched invariance principle, and Gaussian bounds for heat kernels have been established for models fulfilling these assumptions in, e.g., [23, 38, 42], see also [3] for results on first-passage percolation (albeit under a slightly different assumption replacing D concerning the decoupling). We refer to the introduction of [42] for a summary of results. In fact, the statements in [23, 38, 42] were first established under a stronger decoupling assumption P3, but these results were proved in [2] to remain valid when the latter assumption is relaxed to the slightly weaker version D (see also Remark 2.1). These assumptions capture several essential properties of (correlated) percolation models under which the geometry of the infinite cluster resembles that of the integer lattice on large scales.

On $(\Omega,\mathscr{F})$ we consider a family of probability measures $\{\mathbb{P}^{u}\,:\,u\in(a,b)\}$ with $0<a<b<\infty$ satisfying:

• P1

(Ergodicity) For each $u\in(a,b)$ and $x\in\mathbb{Z}^{d}$, the family of lattice shifts $(\Psi_{x})_{x\in\mathbb{Z}^{d}}$ on $\Omega$ is measure-preserving and ergodic with respect to $\mathbb{P}^{u}$.

• P2

(Monotonicity) For $a<u<u^{\prime}<b$, and any increasing event $G\in\mathscr{F}$, $\mathbb{P}^{u}[G]\leq\mathbb{P}^{u^{\prime}}[G]$.

• D

(Decoupling) There exist $\beta,\gamma,\zeta>0$ such that for $i\in\{1,2\}$, $L,s\in\mathbb{Z}_{+}$, $x_{1},x_{2}\in\mathbb{Z}^{d}$ with $|x_{1}-x_{2}|_{\infty}=sL$, and $a<u<u^{\prime}<b$, the following hold:

• (a)

if $A_{i}\in\sigma(\{\omega\mapsto\omega(y)\}\,:\,y\in B(x_{i},L))$ are increasing events, then

[TABLE]

• (b)

if $B_{i}\in\sigma(\{\omega\mapsto\omega(y)\}\,:\,y\in B(x_{i},L))$ are decreasing events, then

[TABLE]

• S1

(Local uniqueness) There exists a function $f_{\text{S}}:(a,b)\times\mathbb{Z}_{+}\to\mathbb{R}$ such that for any $u\in(a,b)$,

[TABLE]

• S2

(Continuity) The percolation density $\eta:(a,b)\rightarrow[0,1]$, defined by

[TABLE]

is strictly positive and continuous.

It is a standard consequence of S1 via a union bound argument that the set $\mathscr{S}_{\infty}$ is $\mathbb{P}^{u}$-a.s. non-empty and connected (see, e.g., [23, (2.8)] or [42, Remark 1.9(2)]), and we refer to it as the infinite cluster of the site percolation induced by $\omega$. We denote

[TABLE]

so $\mathbb{P}^{u}[\Omega_{0}]=1$ for any $u\in(a,b)$.

Remark 2.1*.*

Before the assumption D was introduced in [2], several results had been obtained with D replaced by the following stronger decoupling condition:

• P3

(Stronger decoupling) For $i\in\{1,2\}$, let $A_{i}\in\sigma(\{\omega\mapsto\omega(y)\}\,:\,{y\in B(x_{i},10L)})$ be decreasing events, and let $B_{i}\in\sigma(\{\omega\mapsto\omega(y)\}\,:\,{y\in B(x_{i},10L)})$ be increasing events, where $L\in\mathbb{Z}_{+}$ and $x_{1},x_{2}\in\mathbb{Z}^{d}$. There exist $R_{\text{P}},L_{\text{P}}<\infty$ and $\epsilon_{\text{P}},\chi_{\text{P}}>0$ such that for any integer $R\geq R_{\text{P}}$ and $a<\widehat{u}<u<b$ with

[TABLE]

and assuming $|x_{1}-x_{2}|_{\infty}\geq RL$, one has

[TABLE]

and

[TABLE]

where $f_{\text{P}}:\mathbb{Z}_{+}\rightarrow\mathbb{R}$ fulfills $f_{\text{P}}(L)\geq e^{(\log L)^{\epsilon_{\text{P}}}}$ for all $L\geq L_{\text{P}}$.

In fact, P3 implies D by [2, Remark 6.1(1)], and the motivation to introduce D in [2] is due to the fact that the models considered in the latter satisfy D but fail to satisfy P3. We refer to Remarks 6.1–6.3 in [2] for more discussions on the assumption.

Remark 2.2* (Examples).*

We first collect some examples that satisfy P1–P3 and S1–S2, which can be found in [23, Section III].

1. (1)

Bernoulli site percolation in $d\geq 2$. In this case, $\mathbb{P}^{u}$ is the product measure of a Bernoulli distribution with parameter $u$ standing for the probability of any given site to be open. Formally, one has $\mathbb{P}^{u}[\omega(x)=1]=1-\mathbb{P}^{u}[\omega(x)=0]=u$ with i.i.d. $\{\omega\mapsto\omega(x)\}_{x\in\mathbb{Z}^{d}}$. The assumptions P1–P3 and S1–S2 hold for $u\in(a,b)$ with $p_{c}(d)<a<b<1$, where $p_{c}(d)$ is the critical parameter for site percolation. We refer to [29] for a thorough treatment of Bernoulli percolation. 2. (2)

Random interlacements in $d\geq 3$. This model was introduced in [45]. In this case, $\mathbb{P}^{u}$ is characterized by the equation

[TABLE]

where $\mathrm{Cap}(K)$ denotes the capacity of $K$ (see (2.17) and below). The assumptions P1–P3 and S1–S2 hold for $u\in(a,b)$ for any $0<a<b<\infty$, by [39, 45, 46]. 3. (3)

Vacant set of random interlacements in $d\geq 3$. In this case, $\mathbb{P}^{u}=\mathrm{P}^{1/u}$ (the transformation $u\mapsto 1/u$ is implemented to align with the monotonicity in P2) and $\mathrm{P}^{v}$ is characterized by the equation

[TABLE]

Owing to the series of papers [25, 26, 24, 28], the equality of certain critical parameters governing the percolation phase transition has been established in [28, Theorem 1.1]. This shows in combination with [22, 45, 46, 51] that S1 holds for the entire supercritical phase, and the assumptions P1–P3 and S1–S2 are satisfied for $u\in(a,b)$ with $\frac{1}{u_{*}(d)}<a<b<\infty$, where $u_{*}(d)$ is the critical parameter for the percolation phase transition of the vacant set of random interlacements. 4. (4)

Level-sets of the Gaussian Free Field in $d\geq 3$. In this case, $\mathbb{P}^{u}=\mathrm{P}^{h_{*}(d)-u}$, where $\mathrm{P}^{h}$ for $h\in\mathbb{R}$ is the law of the upper level-set $E^{\geq h}_{\mathrm{GFF}}$ under the measure governing a Gaussian Free Field and $h_{*}(d)$ is the critical threshold for the percolation phase transition (see [11, 40]). Using the equality of the critical parameters characterizing the percolation phase transition of $E^{\geq h}_{\mathrm{GFF}}$, see [27], S1 holds for the entire supercritical phase and assumptions P1–P3 and S1–S2 are satisfied for $u\in(a,b)$ with $0<a<b<\infty$.

We also collect some further models in the literature satisfying P1–P3 and S1–S2.

1. (5)

Level-sets of the Gaussian membrane model in $d\geq 5$. Here, $\mathbb{P}^{u}=\mathrm{P}^{\overline{h}(d)-u}$, where $\mathrm{P}^{h}$ for $h\in\mathbb{R}$ is the law of the upper level-set $E^{\geq h}_{\mathrm{GMM}}$ under the measure governs a Gaussian membrane model (see [19] for its detailed description) and $\overline{h}(d)$ is a critical parameter such that $E^{\geq h}_{\mathrm{GMM}}$ is in a strongly percolative regime for $h<\overline{h}(d)$ (see [19, (5.4)] for its precise definition). It is known by the proof of Theorem 5.1 in [19] that $-\infty<\overline{h}(d)\leq h_{*}(d)$ (with $h_{*}(d)$ the percolation threshold for $E^{\geq h}_{\mathrm{GMM}}$) and the assumptions P1–P3 and S1–S2 hold for $u\in(a,b)$ with $0<a<b<\infty$. 2. (6)

Level-sets of fractional Gaussian fields in $d\geq 2$. As in the previous example, here one has $\mathbb{P}^{u}=\mathrm{P}^{\overline{h}(d)-u}$, where $\mathrm{P}^{h}$ governs for $h\in\mathbb{R}$ the law of the upper level-set $E^{\geq h}_{\mathrm{FF}}$ of a fractional Gaussian field in $d\geq 2$, with covariances given by the Green function of an isotropic, $\alpha$-stable random walk, with $\alpha\in(0,2\wedge d)$, see [10, 14] for a precise definition of these Gaussian fields. As before, $\overline{h}(d)$ is the critical parameter such that $E^{\geq h}_{\mathrm{FF}}$ is in a strongly percolative regime for $h<\overline{h}(d)$. As outlined in [19, Remark 3.4(2)], the phase transition for $E^{\geq h}_{\mathrm{FF}}$ follows analogously to that of the membrane model, and P1–P3 and S1–S2 hold for $u\in(a,b)$ with $0<a<b<\infty$. 3. (7)

Level-sets of the Ginzburg–Landau $\nabla\phi$-model in $d\geq 3$. Here, $\mathbb{P}^{u}=\mathrm{P}^{\overline{h}(d)-u}$, where $\mathrm{P}^{h}$ for $h\in\mathbb{R}$ is the law of the level-set $E^{\geq h}_{\mathrm{GL}}$ under the measure governing a Ginzburg–Landau $\nabla\phi$-model (see [41] for its detailed description) and $\overline{h}(d)$ is the threshold for the strongly percolative regime (see (4.27) in [41] for its precise definition). It is known by Remark 4.7 and the proof of Theorems 4.8 and 4.9 in [41] that $-\infty<\overline{h}(d)\leq h_{*}(d)$ (with $h_{*}(d)$ again denoting the corresponding percolation threshold) and the assumptions P1–P3 and S1–S2 hold for $u\in(a,b)$ with $0<a<b<\infty$.

It is plausible but still open at the moment that $\overline{h}(d)=h_{*}(d)$ for each of the models in Examples (5)–(7) (incidentally, we refer to [33] for a result on the subcritical sharpness for a general class of Gaussian models). If the two critical parameters $h_{\ast}(d)$ and $\overline{h}(d)$ are proved to coincide for any specific model in Examples (5)–(7), our results will thus apply to the entire supercritical phase of the respective model.

The subsequent examples satisfy P1–P2, D, and S1–S2 as established in [2], and we refer to the latter for a more detailed descriptions of the models.

1. (8)

Range of the random walk loop soup in $d\geq 3$. In this case, $\mathbb{P}^{u}$ is the law of the trace on $\mathbb{Z}^{d}$ of a Poisson point process of loops $\mathscr{L}^{u}$ with intensity measure $u\mu$, and $\mu$ is a certain loop measure on the set of equivalence classes of based loops $\mathscr{L}$. It follows from [2, Theorem 1.1] and [12] that the assumptions P1–P2, D, and S1–S2 hold for $u\in(a,b)$ with $\alpha_{c}<a<b<\infty$, where $\alpha_{c}$ is the critical parameter for percolation phase transition. 2. (9)

Vacant set of the random walk loop soup in $d\geq 3$. In this case, $\mathbb{P}^{u}=\mathrm{P}^{\overline{\alpha}-u}$, where $\mathrm{P}^{\alpha}$ for $\alpha>0$ is the law of the vacant set $\mathscr{V}^{\alpha}$ of random walk loop soup and $\overline{\alpha}(d)$ is the critical parameter for the strong percolating regime, which can be defined similarly as in Examples (5)–(7); see also [2, Theorem 1.3] for its positivity. By Remark 6.6 in [2], the assumptions P1–P2, D, and S1–S2 are satisfied for $u\in(a,b)$ with $0<a<b<\overline{\alpha}$. It is plausible but open at the moment that $\overline{\alpha}(d)=\alpha_{*}(d)$.

Although we have formulated our assumptions for site percolation models, our main result could similarly be stated for appropriate bond percolation models under suitable modifications of the assumptions above. In particular, our analysis extends directly to supercritical Bernoulli bond percolation on $\mathbb{Z}^{d}$, $d\geq 3$, see Remark 4.14 (1) for further discussion.

2.3. Random walks on the infinite percolation cluster

Unless otherwise stated, we consider a fixed configuration $\omega\in\Omega_{0}$ throughout this entire subsection (recall (2.8)). We write $\Gamma(\mathbb{R}_{\geq 0},\mathbb{Z}^{d})$ for the space of right-continuous, piecewise constant functions from $\mathbb{R}_{\geq 0}=[0,\infty)$ to $\mathbb{Z}^{d}$ that have finitely many jumps on any bounded interval. We let $X=(X_{t})_{t\geq 0}$ be the canonical process on $\Gamma(\mathbb{R}_{\geq 0},\mathbb{Z}^{d})$, which is distributed under $P_{x}^{\omega}$, $x\in\mathscr{S}_{\infty}$, as a continuous-time, constant-speed simple random walk on $\mathscr{S}_{\infty}(\omega)$ starting from $x$, where $\mathscr{S}_{\infty}(\omega)\subseteq\mathbb{Z}^{d}$ is the infinite percolation cluster introduced in Subsection 2.1. Formally, $X$ can be viewed as a Markov process with generator

[TABLE]

with $\mu_{x},\mu_{\{x,y\}}$ as in (2.3), which waits an exponential time with mean one at each vertex and jumps to a uniformly chosen nearest neighbor.

Let $(\theta_{t})_{t\geq 0}$ stand for the family of canonical time-shifts on $\Gamma(\mathbb{R}_{\geq 0},\mathbb{Z}^{d})$, i.e., $\theta_{t}w(\cdot)=w(t+\cdot)$ for $w\in\Gamma(\mathbb{R}_{\geq 0},\mathbb{Z}^{d})$ and $t\geq 0$. Let $\zeta_{1}=\inf\{t\geq 0\,:\,X_{t}\neq X_{0}\}$ denote the time of the first jump of $X$ (with the convention $\inf\varnothing=\infty$). Given $U\subseteq\mathscr{S}_{\infty}$, we introduce stopping times (with respect to the canonical right-continuous filtration $(\mathscr{F}_{t})_{t\geq 0}$ generated by $(X_{t})_{t\geq 0}$) $H_{U}=\inf\{t\geq 0\,:\,X_{t}\in U\}$, $\widetilde{H}_{U}=\inf\{t>\zeta_{1}\,:\,X_{t}\in U\}$, and $T_{U}=\inf\{t\geq 0\,:\,X_{t}\notin U\}$, which are the entrance, hitting, and exit times of $U$. We also introduce for $r\geq 0$ the stopping time

[TABLE]

We now introduce the heat kernel $q^{\omega}_{t}$ and the heat kernel $q^{\omega}_{t,U}$ killed upon leaving $U\subseteq\mathscr{S}_{\infty}$ of the random walk $X$ as

[TABLE]

Both $q_{t}^{\omega}(\cdot,\cdot)$ and $q_{t,U}^{\omega}(\cdot,\cdot)$ are symmetric in the exchange of their arguments.

It is proved in [2, Corollary 6.5] (see also [42, Theorem 1.15]) that if the assumptions P1–P2, D, and S1–S2 hold, then for any $\varepsilon>0$ there exists a family of random variables $\{\mathscr{T}(x,\varepsilon,\cdot)\}_{x\in\mathbb{Z}^{d}}$, and constants $c_{\text{hk}i}=c_{\text{hk}i}(\varepsilon)$, $i=1,\ldots,6$, such that for $\mathbb{P}^{u}$-almost every $\omega$, $x\in\mathscr{S}_{\infty}(\omega)$, $\mathscr{T}(x,\varepsilon,\omega)$ is finite, and for $t\geq\mathscr{T}(x,\varepsilon,\omega)\vee|x-y|_{1}^{1+\varepsilon}$ and $y\in\mathscr{S}_{\infty}(\omega)$,

[TABLE]

holds. Moreover, for all $z\in\mathbb{Z}^{d}$ and $r\geq 1$,

[TABLE]

where $\Delta_{\text{S}}$ is defined in (2.6). We will apply the result above with $\varepsilon=\frac{1}{2}$ and abbreviate $c_{\text{hk}i}(\frac{1}{2})$, $\mathscr{T}(\cdot,\frac{1}{2},\omega)$ by $c_{\text{hk}i}$, $\mathscr{T}(\cdot,\omega)$ for $i=1,\ldots,6$, respectively.

We conclude this section by introducing some potential-theoretic notions for the random walk on $\mathscr{S}_{\infty}$. For the remainder of this section, we assume that $\omega$ is in a subset of $\Omega_{0}$ of full $\mathbb{P}^{u}$-measure under which (2.13) holds, and that $d\geq 3$. We define for $x,y\in\mathscr{S}_{\infty}(\omega)$ the Green function of the random walk on $\mathscr{S}_{\infty}$ by

[TABLE]

which is $\mathbb{P}^{u}$-a.s. finite by (2.13) and symmetric (in particular, $X$ is transient on $\mathscr{S}_{\infty}$). We note in passing that one has in fact quantitative bounds on the decay of $g^{\omega}$ for $\mathbb{P}^{u}$-a.e. $\omega$, see [42, Theorem 1.17], although we will not need such strong controls. For $\varnothing\neq A\subset\subset\mathscr{S}_{\infty}$, we define the equilibrium measure of $A$ on $\mathscr{S}_{\infty}$ by

[TABLE]

and denote its (finite) total mass, the capacity of $A$, by

[TABLE]

(when $\omega=\boldsymbol{1}$, the constant function assigning $1$ to every $x\in\mathbb{Z}^{d}$, we drop $\omega$ from the notation, and note that this coincides with the classical simple random walk capacity on $\mathbb{Z}^{d}$). We record for later use that one has for $\varnothing\neq A\subset\subset\mathscr{S}_{\infty}$ the last-exit decomposition

[TABLE]

see, e.g., [8, Proposition 7.10].

3. Uniform large-scale behavior over very large regions

We will demonstrate in this section that both the heat kernel and the relative volume density of the cluster exhibit controlled behavior over many scales, uniformly over sizeable boxes. This will be instrumental in Section 4 for the development of a “one-step estimate” (see Proposition 4.6). We begin by quantifying the mixing property of the relative volume density, in which a convergence rate which is faster than any polynomial is obtained. The proof builds on the framework developed in [42] and is detailed in Appendix A. The spatial uniformity we need will then follow readily from the quantitative control on the volume regularity and the previously stated heat kernel estimates (2.13)–(2.14). Finally, we conclude the section with a heuristic overview of the results and explain how they will be applied in the proof of the solidification estimates.

We assume $d\geq 2$ throughout this section and work under the following assumption throughout the remainder of the present article, unless stated otherwise.

Main Assumption**.**

The family of measures $\{\mathbb{P}^{u}\,:\,u\in(a,b)\}$ on $(\Omega,\mathscr{F})$ satisfies assumptions P1–P2, D and S1–S2.

For $x\in\mathbb{Z}^{d}$, we denote $\mathbb{P}^{u}[~\cdot\mid x\in\mathscr{S}_{\infty}]$ by $\mathbb{P}^{u}_{x}[~\cdot~]$ (by P1 and S2, $\mathbb{P}^{u}[x\in\mathscr{S}_{\infty}]=\eta(u)>0$).

Proposition 3.1** ($d\geq 2$, Main Assumption).**

For $\alpha\in(0,1)$, there exist positive constants $\kappa_{\mathrm{d}i}(\alpha)$, $i=1,2$, such that for $R\geq 1$,

[TABLE]

(with $\Delta_{\mathrm{S}}$ defined in (2.6)). Moreover, the function $\alpha\mapsto\kappa_{\mathrm{d}2}(\alpha)$ is increasing on $(0,1)$.

The proof of the proposition is a rather straightforward modification of the proof of Lemma 3.3 in [42]; the necessary modifications are given in Appendix A for the convenience of the reader.

We now define for a given integer $R\geq 1$ the radius of a sup-norm ball on which the behavior of the relative volume density of the graph $\mathscr{S}_{\infty}$ and the heat kernel are uniformly controlled for all sub-balls at scale $R$ (when $R$ is large enough). For each positive integer $R\geq 1$ and $\alpha\in(0,1)$, we introduce

[TABLE]

Here, the constants $\kappa_{\text{d}2}(\alpha)$ and $c_{\text{hk}6}$ are from (3.1) and (2.14), respectively, and in particular $\alpha\mapsto\kappa_{\mathrm{reg}}(\alpha)$ is increasing on $(0,1)$. The scale $r_{\alpha,R}$ will be used to characterize a (very) large sup-norm ball centered at the origin, over which certain uniform controls (see Propositions 3.2 and 3.4 below) can be obtained. In the proof of our main result, Theorem 4.1, we will actually require $\alpha$ to tend to zero slowly with $N$, i.e. by considering a sequence $(\alpha_{N})_{N\geq 0}$ with $\alpha_{N}\searrow 0$, which allows the inspection of many properly separated scales, see (4.97).

We turn to the main results of this section. The following proposition states that for any small approximation error $\alpha\in(0,1)$, and $\mathbb{P}^{u}$-almost every realization of $\omega$, there exists a minimal scale $R_{\mathrm{den}}(\omega,\alpha)$ such that for all $R$ above the scale, the relative volume densities of the cluster $\mathscr{S}_{\infty}$ measured within boxes of radius $R$ are deviate from $\eta(u)$ by at most $\alpha$, uniformly over $B(0,r_{\alpha,R})$.

Proposition 3.2** ($d\geq 2$, Main Assumption).**

For $0<\alpha<1$ and $\omega\in\Omega$, consider

[TABLE]

(with $\inf\varnothing=\infty$). One has, for every $n\in\mathbb{Z}_{+}$, the upper bound

[TABLE]

In particular, $R_{\mathrm{den}}(\,\cdot\,,\alpha)$ is a $\mathbb{P}^{u}$-a.s. finite random variable, and for all $R\geq R_{\mathrm{den}}(\omega,\alpha)$, one has

[TABLE]

for all $x\in\mathscr{S}_{\infty}\cap B(0,r_{\alpha,R})$.

Proof.

For $n\geq 1$,

[TABLE]

and by applying a union bound and using the definition of $r_{\alpha,R}$ in (3.2), we have

[TABLE]

proving (3.4). The claim (3.5) follows directly from the definition of $R_{\mathrm{den}}(\omega,\alpha)$. ∎

For $\alpha\in(0,1)$, we denote

[TABLE]

It is clear from Proposition 3.2 that $\Omega_{\mathrm{den}}^{\alpha}$ has full $\mathbb{P}^{u}$-measure, for every $\alpha\in(0,1)$.

We now derive an analogous statement concerning uniform controls on the heat kernel of the random walk on $\mathscr{S}_{\infty}$.

Lemma 3.3** ($d\geq 2$, Main Assumption).**

Let $0<\alpha<1$ and $\omega\in\Omega$. The random variable defined by

[TABLE]

(with $\inf\varnothing=\infty$) satisfies that

[TABLE]

In particular, $R_{\mathrm{hk}}(\,\cdot\,,\alpha)$ is a $\mathbb{P}^{u}$-a.s. finite random variable.

Proof.

Recall the definition of the random variables $\mathscr{T}(\cdot)$ above (2.13), and the convention following (2.14). For $n\geq 1$

[TABLE]

Then we can apply a union bound to obtain

[TABLE]

which concludes the proof. ∎

The first part of following proposition follows from the previous lemma by the definition of $R_{\mathrm{hk}}$ in (3.8). The second part is a consequence of the first part via a standard argument; see [8, Proposition 5.26] or [7, Lemma 5.8].

Proposition 3.4** ($d\geq 2$, Main Assumption).**

Let $\alpha\in(0,1)$.

1. (i)

For $\mathbb{P}^{u}$-almost every $\omega$, $R_{\mathrm{hk}}(\omega,\alpha)$ defined in (3.8) is finite, and for all $R\geq R_{\mathrm{hk}}(\omega,\alpha)$, the heat kernel satisfies ($c_{\text{hk}i}$* for $i=1,\ldots,4$ defined at the end of Section 2.3)*

[TABLE]

for any $x\in\mathscr{S}_{\infty}\cap B(0,2r_{\alpha,R})$, $y\in\mathscr{S}_{\infty}$, and $t\geq R\vee|x-y|^{3/2}_{1}$. 2. (ii)

For any $\vartheta\in(0,1)$, there exist constants $c_{\text{khk}1}(\vartheta)>0$, $0<c_{\text{khk}2}(\vartheta)<1$ and a random variable $R_{\mathrm{khk}}(\cdot,\alpha,\vartheta)$ such that for $\mathbb{P}^{u}$-almost every $\omega\in\Omega$, $R_{\mathrm{khk}}(\omega,\alpha,\vartheta)<\infty$, and for all integers $R\geq R_{\mathrm{khk}}(\omega,\alpha,\vartheta)$, the heat kernel $q_{t,B(x_{0},R)\,\cap\,\mathscr{S}_{\infty}}^{\omega}(\cdot,\cdot)$ of the random walk killed upon exiting $B(x_{0},R)\cap\mathscr{S}_{\infty}$ (recall (2.12)) satisfies

[TABLE]

where $x_{0}\in\mathscr{S}_{\infty}\cap B(0,r_{\alpha,R})$, $x,y\in\mathscr{S}_{\infty}\cap B(x_{0},(1-\vartheta)R)$ and $c_{\text{khk}2}(\vartheta)R^{2}\leq t\leq R^{2}$.

For $\alpha,\vartheta\in(0,1)$, let

[TABLE]

and, with $\Omega_{0}$ and $\Omega_{\mathrm{den}}^{\alpha}$ as in (2.8) and (3.7) respectively,

[TABLE]

It is clear that $\Omega_{\mathrm{reg}}^{\alpha,\vartheta}\in\mathscr{F}$ and $\mathbb{P}^{u}\bigl[\Omega_{\mathrm{reg}}^{\alpha,\vartheta}\bigr]=1$.

Propositions 3.2 and 3.4 show that for $\omega\in\Omega_{\mathrm{reg}}^{\alpha,\vartheta}$ the relative volume density of $\mathscr{S}_{\infty}$ and heat kernels exhibit well-controlled behavior at any scale $R\geq\widetilde{R}=R_{\mathrm{den}}\vee R_{\mathrm{hk}}\vee R_{\mathrm{khk}}$, uniformly over the ball $B(0,r_{\alpha,R})$. This is similar in spirit to the notion of “good” and “very good” boxes introduced in [7, Definition 1.7] to control degeneracies of the supercritical cluster of Bernoulli (bond) percolation, and developed in [42] in the set-up of assumptions P1–P3 and S1–S2 (see also [42, Definition 4.1] for the notion of “regular” and “very regular” boxes). For the application we have in mind, it is expedient to directly use the controls on the heat kernel of [42] and to require quite precise volume concentration.

We will typically use $\Omega_{\mathrm{reg}}^{\alpha,\vartheta}$ to argue as follows. When the “lowest accessible scale” satisfies $R_{\min}\geq\widetilde{R}$, we obtain uniform estimates over the ball $B(0,r_{\alpha,R_{\min}})$. We choose $\alpha$ to depend on $N$ so that $r_{\alpha_{N},R_{\min}}$ is comparable to or larger than the macroscopic scale. In other words, we can examine the macroscopic sup-norm ball $B(0,r_{\alpha_{N},R_{\min}})$ at any location down to the microscopic scale $R_{\min}$, while retaining good control over the underlying graph structure (see Fig. 2 for an illustration). Remarkably, the superpolynomial separation between the microscopic and macroscopic scales is sufficient to implement the multi-scale analysis in the proof of the solidification estimates.

4. Solidification of porous interfaces on percolation clusters

In this section, we state and prove in Theorem 4.1 the solidification estimates of porous interfaces on typical realizations of supercritical site percolation clusters under our standing Main Assumption. We begin by stating our main result in a precise way, after providing some additional notation, and then present the proof. The latter consists of three steps, which are similar to [34] in the context of Brownian motion and to [18] in the context of random walks on $(\mathbb{Z}^{d},\mathbb{E}^{d})$ equipped with uniformly elliptic edge weights. In the first step, we study how a random walk on the supercritical cluster enters sets of well-balanced local density, relative to a given set $U_{0}\subseteq\mathscr{S}_{\infty}$ (more precisely, the density will be defined in terms of the complement $U_{1}=\mathscr{S}_{\infty}\setminus U_{0}$). Secondly, we introduce a certain “resonance set” which is naturally attached to $U_{0}$, and show that it is atypical for a random walk to avoid this set, in an asymptotic sense. The precise formulation appears in Theorem 4.8. Lastly, we essentially use the resonance set as a substitute for a Wiener-type criterion to show that, again in an asymptotic sense, the porous interfaces cannot be avoided by a random walk.

Compared with the uniformly elliptic case, the presence of local degeneracies introduces substantial difficulties in implementing this strategy. For instance, in the uniformly elliptic setting the local density of $U_{1}$ (crucial to the estimate of hitting probabilities of the resonance set) is deterministic and enjoys good regularity properties similar to the continuous case, while in our case, it is intrinsically dependent on the percolation configuration $\omega\in\{0,1\}^{\mathbb{Z}^{d}}$. This is addressed by only considering scales above a certain threshold, utilizing the large-scale regularity properties of the infinite cluster $\mathscr{S}_{\infty}$ from Section 3. Our analysis in this section shows that, for almost every $\omega$, there exists a sufficient “number of scales” to that can be used to show a decay of the probability to avoid the resonance set. Here, the smallest scale corresponds to the choice of $R_{\min}$ as in the discussion below (3.17), which must be sufficiently “large” to guarantee the aforementioned good regularity properties, and the maximal, “very large” scale exceeds $r_{\alpha,R_{\min}}$.

Throughout this section, we assume $d\geq 3$, and require the family of probability measures $\{\mathbb{P}^{u}\,:\,u\in(a,b)\}$ on $(\Omega,\mathscr{F})$ to satisfy the Main Assumption. We also fix

[TABLE]

We recall (2.8) and assume, without loss of generality, throughout this section that $\omega\in\Omega_{0}$. Consider a bounded, non-empty set $U_{0}\subseteq\mathscr{S}_{\infty}$, its complement $U_{1}=\mathscr{S}_{\infty}\setminus U_{0}$ in $\mathscr{S}_{\infty}$, and also the boundary $S=\partial_{\mathscr{S}_{\infty}}U_{0}=\partial_{\mathscr{S}_{\infty},\text{in}}U_{1}$ relative to $\mathscr{S}_{\infty}$. To measure the local density of $U_{0}$ in balls of dyadic scale, we define for $\omega\in\Omega_{0}$, a non-negative integer $\ell$, and $x\in\mathbb{Z}^{d}$, the local density measure by

[TABLE]

(with the convention that $0/0=0$, and we will also use this convention in (4.3)–(4.5)). Note that if $x\in\mathscr{S}_{\infty}$, $B(x,R)\cap\mathscr{S}_{\infty}\neq\varnothing$ for $R\geq 1$. For $\omega\in\Omega_{0}$, two local density functions associated with $U_{1}$ are defined as (with $U_{1}$ implicit in the notation)

[TABLE]

For $\omega\in\Omega_{0}$ and a given function $f:\mathbb{Z}^{d}\to\mathbb{R}$ we write $(f)_{x,\ell}^{\omega}$ for the average of $f$ on $B(x,2^{\ell})\cap\mathscr{S}_{\infty}$, namely

[TABLE]

Given $\omega\in\Omega_{0}$, and integers $N\geq 1$, $\ell_{*}\geq 0$, we define (recall (1.3) and (4.1))

[TABLE]

We furthermore define for fixed $U_{0}\in\mathscr{U}_{\ell_{*},N}^{\omega}$, $\epsilon\in\mathbb{N}$, and $\chi\in(0,1)$, the class of porous interfaces associated with $U_{0}$

[TABLE]

(recall the notation $H_{\Sigma}$ for the entrance time of the random walk into $\Sigma$, and $\tau_{\epsilon}$ for the first time the random walk moves at sup-distance $\epsilon$ from its starting point, see (2.10)). The parameter $\epsilon$ controls at which distance to $S$ the porous interface is located, while $\chi$ is a measure for how markedly it is felt for a random walk starting from any point in $S$ (see also Figure 1 for an illustration of the sets under consideration).

We now state the uniform controls for the probability that a random walk on $\mathscr{S}_{\infty}$ is absorbed by porous interfaces. This corresponds to the extension of the solidification estimates from [34] and [18] to our case. Recall the growth condition (1.3) governing the two sequences $(a_{N})_{N\geq 0}$ and $(b_{N})_{N\geq 0}$.

Theorem 4.1** ($d\geq 3$, Main Assumption, (4.1), (1.3)).**

Let $\chi\in(0,1)$, and let $(c_{N})_{N\geq 0}$ be another sequence of positive real numbers decreasing to [math]. For $\mathbb{P}^{u}$-almost every $\omega$, it holds that

[TABLE]

and moreover,

[TABLE]

We immediately obtain the following corollary concerning the capacity:

Corollary 4.2** ($d\geq 3$, Main Assumption, (4.1), (1.3)).**

Under the same assumptions as Theorem 4.1, we have the following: For $\mathbb{P}^{u}$-almost every $\omega$, it holds that

[TABLE]

Proof.

The proof of Corollary 4.2 assuming Theorem 4.1 follows similarly as that of [18, Corollary 4.2], and we only sketch it for completeness. Indeed, consider $\omega$ in a set of full $\mathbb{P}^{u}$-measure such that (4.8) holds. Then, for $N\geq N_{0}(\omega)$ and $\epsilon/2^{\ell_{\ast}}\leq c_{N}$, $a_{N}\leq 2^{\ell_{\ast}}\leq b_{N}$, $U_{0}\in\mathscr{U}_{\ell_{\ast},N}^{\omega}$, and $\Sigma\in\mathscr{S}^{\omega}_{U_{0},\epsilon,\chi}$, we have

[TABLE]

having also used the symmetry of $g^{\omega}(\cdot,\cdot)$. ∎

We briefly explain the restriction inherent in Theorem 4.1 compared with the uniformly elliptic counterpart [18, Theorem 4.1]. In our set-up, we introduce the sequences $(a_{N})_{N\geq 0}$ and $(b_{N})_{N\geq 0}$ as in (1.3) to control the growth rate of $2^{\ell_{*}}$ in the present theorem. The additional restriction for $(\epsilon,\ell_{*})$ over the first supremum requires $2^{\ell_{*}}$ to fulfill $a_{N}\leq 2^{\ell_{*}}\leq b_{N}$. One should view $a_{N}$ heuristically as determining a lower bound on the “smallest spatial scale” inspected, and $b_{N}$ as the corresponding “largest spatial scale”. Such a restriction is natural, since one cannot expect good regularity properties of the cluster to hold uniformly in small boxes (essentially of size $a_{N}$) outside a certain box of size $b_{N}$. Heuristically, the growth condition (1.3) is chosen to ensure that such regions are not inspected.

4.1. Local density functions

In this subsection, we investigate how a random walk enters regions in which certain local densities are well-balanced on multiple scales. Recall from (4.3)–(4.4) that the definition of the local density functions depends on a set $U_{1}\subseteq\mathscr{S}_{\infty}$ (with $U_{0}=\mathscr{S}_{\infty}\setminus U_{1}$), a non-negative integer $\ell$, and a realization $\omega$ of the environment. We assume throughout this subsection that

[TABLE]

(recall $\Omega_{0}$ in (2.8)) and will impose further conditions on $\ell$ and the set of $\omega$ in the statements.

We begin with two properties of the local density functions, which are analogous to the continuum set-up [34, Lemma 1.1] and the discrete set-up with uniformly elliptic conductances [18, Lemma 4.3]. Recall that $\eta(\cdot)$ is assumed to be strictly positive in S2, $\alpha$ is introduced in Proposition 3.1 to represent the deviation between the relative volume of the cluster in a box and its expectation, $R_{\mathrm{den}}(\omega,\alpha)$ is defined in (3.3) as the minimal scale for spatial volume regularity, $\Omega_{\mathrm{den}}^{\alpha}$ is defined in (3.7), and the sequence $(r_{\alpha,N})_{N\geq 1}$ is defined in (3.2) and refers to the radius of a sup-norm ball in which one has uniformly good regularity properties (see Propositions 3.2 and 3.4).

Lemma 4.3** ($d\geq 3$, Main Assumption, (4.12)).**

Let $\alpha\in(0,\frac{1}{3})$, $\omega\in\Omega_{\mathrm{den}}^{\alpha}$. Assume $2^{\ell}>2^{\ell^{\prime}}\geq R_{\mathrm{den}}(\omega,\alpha)$ and $r_{\alpha,2^{\ell^{\prime}}}-2^{\ell}-2^{\ell^{\prime}}>0$.

1. (i)

For any $x,x+y\in\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell}})$ which are connected by a path in $\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell}})$, the local density functions are Lipschitz-continuous in the sense that

[TABLE]

where $\widehat{d}_{\omega}$ denotes the graph distance on $\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell}})$. 2. (ii)

For $x\in\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell^{\prime}}}-2^{\ell}-2^{\ell^{\prime}})$, we have

[TABLE]

(see (4.5) for notation).

Before we state the proof of the lemma above, we briefly comment on the choices of the scales $\ell,\ell^{\prime}$, as well as the of coefficients $c_{\mathrm{lip}}$ and $c_{0}$.

Remark 4.4*.*

1. (1)

We have $R_{\mathrm{den}}(\omega,\alpha)<\infty$ since $\omega\in\Omega_{\mathrm{den}}^{\alpha}$. There exists $\ell>\ell^{\prime}$ such that $r_{\alpha,2^{\ell^{\prime}}}-2^{\ell}-2^{\ell^{\prime}}>0$, because $r_{\alpha,n}=\exp(\kappa_{\text{reg}}(\alpha)(\log n)^{1+\Delta_{\mathrm{S}}})\gg n$ as $n\to\infty$, so that we can first choose a large enough $\ell^{\prime}$ leaving enough room for the choice of $\ell$. 2. (2)

We highlight that for a given $\alpha\in(0,\frac{1}{3})$, $c_{\mathrm{lip}}$ and $c_{0}$ are not optimal and could be chosen slightly smaller (depending on $\alpha$), but since this plays no role for our analysis, we work with the choices in (4.13) and (4.14).

Proof of Lemma 4.3.

Let $\ell,\ell^{\prime}$ satisfy the conditions in the statement of the lemma. We begin with the proof of (i). Let $x,x+y\in\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell}})$ and assume that there is a nearest-neighbor path $\pi$ in $\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell}})$ of length $n=\widehat{d}_{\omega}(x,x+y)$ connecting the two points (otherwise, $\widehat{d}_{\omega}(x,x+y)=\infty$ and the claim is immediate). By the definition of $\sigma_{\ell}^{\omega}(\cdot)$ in (4.3), for $0\leq i\leq n-1$,

[TABLE]

Note that we have the elementary inequality for $a_{1},a_{2},b_{1},b_{2}\in\mathbb{R}_{+}$,

[TABLE]

We then apply (4.16) to (4.15), and use $\frac{|B(\pi_{i},2^{\ell})\cap U_{1}|}{|B(\pi_{i},2^{\ell})\cap\mathscr{S}_{\infty}|}\leq 1$, obtaining that

[TABLE]

Both of the numerators in the above expression can be bounded by

[TABLE]

(with $\Delta$ denoting the symmetric difference between sets). Applying the latter to (4.17) yields

[TABLE]

(the application of (3.5) is possible since $2^{\ell}\geq R_{\mathrm{den}}(\omega,\alpha)$ and $\pi_{i}\in\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell}})$, for every $0\leq i\leq n-1$). Lastly,

[TABLE]

The Lipschitz continuity (4.13) then follows since $\alpha<\frac{1}{3}$.

We turn to the proof of (ii). Let $x\in\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell^{\prime}}}-2^{\ell}-2^{\ell^{\prime}})$. Note that

[TABLE]

where

[TABLE]

(with the convention $0/0=0$). It is clear by definition of $\psi_{\ell,\ell^{\prime}}^{\omega,x}$ that

[TABLE]

Moreover, we have for $z\in B(x,2^{\ell}-2^{\ell^{\prime}})$,

[TABLE]

(in both inequalities the use of (3.5) is possible since $y\in B(z,2^{\ell^{\prime}})\subseteq B(x,2^{\ell})\subseteq B(0,r_{\alpha,2^{\ell^{\prime}}}-2^{\ell^{\prime}})$ and $z\in B(x,2^{\ell}-2^{\ell^{\prime}})\subseteq B(0,r_{\alpha,2^{\ell^{\prime}}}-2\cdot 2^{\ell^{\prime}})$). Note that by (4.22), $\psi_{\ell,\ell^{\prime}}^{\omega,x}(z)=0$ for $z\notin B(x,2^{\ell}+2^{\ell^{\prime}})$, while for $z\in B(x,2^{\ell}+2^{\ell^{\prime}})$, we have

[TABLE]

(noting again that the use of (3.5) is possible since $y\in B(x,2^{\ell})\subseteq B(0,r_{\alpha,2^{\ell^{\prime}}}-2^{\ell^{\prime}})$ and $z\in B(x,2^{\ell}+2^{\ell^{\prime}})\subseteq B(0,r_{\alpha,2^{\ell^{\prime}}})$), which shows that $\psi_{\ell,\ell^{\prime}}^{\omega,x}(z)\leq\tfrac{1+\alpha}{1-\alpha}$ for all $z\in\mathbb{Z}^{d}$. We will use (4.23) to bound $(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega}$ from below by restricting $\psi_{\ell,\ell^{\prime}}^{x}$ on $B(x,2^{\ell}-2^{\ell^{\prime}})$, and bound the same quantity from above by using the upper bound (4.24) of $\psi_{\ell,\ell^{\prime}}^{x}$ on all its support $B(x,2^{\ell}+2^{\ell^{\prime}})$. As for the lower bound, we recall (4.20) and see that

[TABLE]

using a simple union bound in the last step. As we have

[TABLE]

it follows that

[TABLE]

Similarly, for the upper bound, we obtain

[TABLE]

Using inequality $(a+b)^{p}\leq 2^{p-1}(a^{p}+b^{p})$ for $a,b\in\mathbb{R}_{+}$ and $p\geq 1$, one obtains

[TABLE]

Finally, we note that $\frac{1+\alpha}{(1-\alpha)^{2}}\leq 3$ for $\alpha<\frac{1}{3}$. Combining (4.25) and (4.27) yields the claim (ii). ∎

Lemma 4.5**.**

Let $(\Gamma,\mathscr{A},\mathrm{P})$ be a probability space and $Y$ be a $[0,1]$-valued random variable with finite expectation $\mu=\mathrm{E}[Y]$. Then for $0\leq\delta\leq\mu\wedge(1-\mu)$, at least one of the equations below holds:

[TABLE]

We refer to [34, Lemma 1.2] for the proof of the previous lemma. Applying this result with $\mathrm{P}=\mu_{x,\ell}^{\omega}$, $Y=\sigma_{\ell^{\prime}}^{\omega}$, and

[TABLE]

we obtain that for $0\leq\delta\leq(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega}\wedge(1-(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega})$, at least one of the equations below holds:

[TABLE]

We will use the observation to prove the following proposition, which is analogous to [34, Proposition 1.3] and [18, Proposition 4.5], and shows that a random walk has a non-degenerate probability of entering the set where the local density $\sigma_{\ell^{\prime}}^{\omega}$ takes values close to its average over a ball before exiting the latter.

In addition to the condition concerning regular volumes that appeared in Lemma 4.3, we will impose further conditions on $\ell$ in the next proposition for the regularity of random walks on the percolation cluster. These, together with other technical conditions, are encapsulated in the minimal scale $\ell_{\min}(\omega,\alpha,\delta)$ (see (4.33)). A similar notion of a minimal scale for such regularity properties was previously introduced in [18, Proposition 4.5]; however, our definition additionally depends on the particular realization $\omega$ and the degree of volume irregularity $\alpha$, reflecting the inherent degeneracies present in the percolation cluster. Recall the scales $R_{\mathrm{hk}}(\omega,\alpha)$ and $R_{\mathrm{khk}}(\omega,\alpha,\vartheta)$ defined in Proposition 3.4, the set of regular configurations, $\Omega_{\mathrm{reg}}^{\alpha,\vartheta}$, defined in (3.17), and the stopping time $\tau_{r}$ defined in (2.10).

Proposition 4.6** ($d\geq 3$, Main Assumption, (4.12)).**

Let $\alpha\in(0,\frac{1}{3})$. For $0<\delta\leq(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega}\wedge(1-(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega})\wedge\frac{1}{4}$, there exists a minimal scale $\ell_{\min}(\omega,\delta,\alpha)$ (defined in (4.33)), a positive constant $\delta^{\prime}(\delta)\in(0,1)$ (defined in (4.31)) such that for scales $\ell>\ell^{\prime}\geq\ell_{\min}(\omega,\delta,\alpha)\in\mathbb{N}$ and, $\omega\in\Omega_{\mathrm{reg}}^{\alpha,\delta^{\prime}(\delta)}$, we have

[TABLE]

for all $x\in\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell^{\prime}}})$, where here and in the following we use the abbreviation $\{\sigma_{\ell^{\prime}}^{\omega}\in[(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega}-\delta,(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega}+\delta]\}$ for $\{y\in\mathbb{Z}^{d}\,:\,\sigma_{\ell^{\prime}}^{\omega}(y)\in[(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega}-\delta,(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega}+\delta]\}$.

Proof.

Let $x\in\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell^{\prime}}})$ and $0<\delta\leq(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega}\wedge(1-(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega})\wedge\frac{1}{4}$. By the definition of $\mu_{x,\ell}^{\omega}$ (see (4.2)), we can take

[TABLE]

so that

[TABLE]

holds for all $2^{\ell}\geq R_{\mathrm{den}}(\omega,\alpha)$.

We will apply Proposition 3.4 (with $\vartheta=\delta^{\prime}(\delta)$), Proposition 3.2, and Lemma 4.3. To this end, we define

[TABLE]

We now let $\ell>\ell^{\prime}\geq\ell_{\min}(\omega,\delta,\alpha)$ and show that (4.30) holds. Assume we are in the situation where (4.29a) is fulfilled. By (4.32), it follows that

[TABLE]

We recall the notation related to random walks introduced at the start of Subsection 2.3, up to (2.12). We have

[TABLE]

where the application of (3.15) is possible since $x\in\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell^{\prime}}})\subseteq\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell}})$ and $z\in\mathscr{S}_{\infty}\cap B(x,(1-\delta^{\prime})2^{\ell})$, and the use of (3.5) is valid since $x\in\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell^{\prime}}})\subseteq\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell}})$. Similarly, for all $y\in B(x,2^{\ell}(1-\delta^{\prime}))\cap\mathscr{S}_{\infty}$, we also have

[TABLE]

It then follows from the strong Markov property that the random walk starting from $x$ enters $\{\sigma_{\ell^{\prime}}^{\omega}>(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega}+\delta\}$ and then $\{\sigma_{\ell^{\prime}}^{\omega}<(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega}-\delta\}$ before exiting $B(x,2^{\ell})$ with probability at least $C(\delta)^{2}$. Note that the trajectory of this walk must lie in $B(x,2^{\ell})\cap\mathscr{S}_{\infty}$, so by the Lipschitz continuity (using $\ell^{\prime}\geq\ell_{\min}$) of $\sigma_{\ell^{\prime}}^{\omega}$ in the subgraph $\mathscr{S}_{\infty}\cap B(0,r_{\alpha,2^{\ell}})$ with Lipschitz constant smaller than $\delta$, it must enter $\{(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega}-\delta<\sigma_{\ell^{\prime}}^{\omega}<(\sigma_{\ell^{\prime}}^{\omega})_{x,\ell}^{\omega}+\delta\}$ before exiting $B(x,2^{\ell})$.

It remains to consider the situation (4.29b). Note that

[TABLE]

We can argue using the killed heat kernel bound again to show

[TABLE]

This completes the proof. ∎

Based on the previous propositions, we are now in a position to introduce “properly separated” scales $2^{\ell_{0}}>\ldots>2^{\ell_{J}}$ that will enter the definition of the resonance sets in the next subsection. Our goal is to show that starting from a point $x$ of well-balanced relative volume (see (4.43)), there is positive probability for a random walk to enter the region where all local densities $\widetilde{\sigma}_{\ell_{j}}^{\omega}(\,\cdot\,)$, $0\leq j\leq J$, lie in some interval $[\widetilde{\alpha},1-\widetilde{\alpha}]$ (with $\widetilde{\alpha}$ a dimension-dependent constant, see (4.45)), before travelling to a point at sup-distance $\frac{3}{2}\cdot 2^{\ell_{0}}$ from $x$. This crucially enters the definition of the resonance set in the next subsection.

To this end, we fix an integer $J\geq 1$ and define

[TABLE]

We intentionally reuse the symbols $\alpha$ and $\delta$ to denote quantities that now depend explicitly on the integer $J$, as defined above. This dependence will be fixed throughout the remainder of the article. The reason for these specific choices will become clear in (4.46). We say the scales $\ell_{0}>\ldots>\ell_{J}$ are properly separated if for all $j\in\{0,\ldots,J-1\}$, it holds that

[TABLE]

where

[TABLE]

(recall $c_{0}$ in (4.14)). As a consequence of (4.37a), we have

[TABLE]

We will also consider a sequence of intervals for $j=0,\ldots,J$, given by

[TABLE]

and introduce the stopping times

[TABLE]

We recall the function $\delta^{\prime}(\,\cdot\,)$ and the minimal scale $\ell_{\min}(\omega,\delta,\alpha)$ from Proposition 4.6, given by (4.31) and (4.33), respectively.

Proposition 4.7** ($d\geq 3$, Main Assumption, (4.12)).**

Let $J\geq 1$, $\omega\in\Omega_{\mathrm{reg}}^{\alpha(J),\delta^{\prime}(\delta(J))}$ be fixed. Assume $(\ell_{j})_{j=0}^{J}$ to be properly separated in the sense of (4.37), and furthermore that $\tfrac{2}{3}r_{\alpha(J),2^{\ell_{J}}}>2^{\ell_{0}}>\ldots>2^{\ell_{J}}\geq 2^{\ell_{\min}(\omega,\delta(J),\alpha(J))}$. Define

[TABLE]

Then $P_{x}^{\omega}[\mathscr{C}]\geq c_{2}(J)$ holds for $x\in\mathscr{S}_{\infty}\cap B(0,r_{\alpha(J),2^{\ell_{J}}}-\frac{3}{2}2^{\ell_{0}})$ satisfying

[TABLE]

Moreover, on the set of trajectories constituting the event $\mathscr{C}$, one has

[TABLE]

and

[TABLE]

Proof.

We first record the following observations on the parameters $\alpha(J)$ and $\delta(J)$ defined in (4.36), and the intervals $I_{j}$ from (4.40).

1. (i)

As $\alpha(J)$ is a decreasing function of $J$, $\alpha(J)\leq\alpha(1)=\frac{1}{38}(<\frac{1}{3})$ and $\delta\leq\frac{1}{38\cdot 4}(<\frac{1}{4})$. 2. (ii)

It is immediate from the expression of $\alpha(J)$ that $\alpha(J)<1-\frac{2}{(\frac{10}{9})^{1/J}+1},$ which implies

[TABLE] 3. (iii)

Denote the left and right endpoint of $I_{j}$ by $L_{j}$ and $R_{j}$ respectively, for $j\in\{0,\ldots,J\}$. As $L_{j}$ is decreasing and $R_{j}$ is increasing in $j$, using observation (ii) and $\alpha(J)\leq\frac{1}{38}$ in observation (i), we know that for $j\in\{0,\ldots,J\}$

[TABLE]

and

[TABLE]

which shows $I_{j}\subseteq[\frac{1}{3},\frac{2}{3}]$.

Consider $x\in\mathscr{S}_{\infty}\cap B(0,r_{\alpha(J),2^{\ell_{J}}}-\frac{3}{2}2^{\ell_{0}})$ satisfying (4.43). We will repeatedly apply Proposition 4.6 and show by induction on $j\in\{0,\ldots,J\}$ that

[TABLE]

(note that $\mathscr{C}=\mathscr{C}_{J}$), which will conclude the first part.

When $j=0$, $I_{0}=[\frac{1}{2}-\delta(J),~\frac{1}{2}+\delta(J)]$. Since $2^{-\ell_{\min}(\omega,\delta(J),\alpha(J))}\leq\frac{\delta(J)}{8}<\delta(J)$, we find that $P_{x}^{\omega}[\gamma_{0}=0]=1$. Assume now that (4.49) is satisfied for some $0\leq j<J$. Firstly, we want to investigate the range of $X_{\gamma_{j}}$ to make sure Lemma 4.3(ii) is applicable with $X_{\gamma_{j}}$ in place of $x$ therein, for all $j\in\{1,\ldots,J\}$. On $\mathscr{C}_{j}$, by the definition of $\gamma_{j}$, we have

[TABLE]

Note that by (4.37a),

[TABLE]

It then follows from (4.50) that

[TABLE]

Together with (4.37b), we know that Lemma 4.3(ii) is applicable with $X_{\gamma_{j}}$ for all $j\in\{1,\ldots,J\}$. Denote $(\sigma_{\ell_{j+1}}^{\omega})_{X_{\gamma_{j}},\ell_{j}}^{\omega}$ by $(\beta_{j+1}^{\omega})^{\prime}$. On $\mathscr{C}_{j}$, by Lemma 4.3(ii) and (4.39), we have

[TABLE]

As $\sigma^{\omega}_{\ell_{j}}(X_{\gamma_{j}})\in I_{j}$ by the definition of $\gamma_{j}$, it is clear by our choice of $\delta=\frac{\alpha}{4}$ that

[TABLE]

(recall that $L_{j}$ and $R_{j}$ are the left and right endpoint of $I_{j}$ respectively). It then holds on $\mathscr{C}_{j}$ that

[TABLE]

We can now apply the strong Markov property at time $\gamma_{j}$ and find that

[TABLE]

The first part of the proposition is concluded.

We now turn to (4.44) and (4.45). First, (4.44) follows by the definition of stopping times (4.41) and a calculation using triangle inequalities similar to (4.51), and we refer to [34, (1.40)] for details. For the lower bound in (4.45),

[TABLE]

Replacing $U_{1}$ by $U_{0}$ in the above calculations then yields the upper bound (see (4.4)). The proof is therefore complete. ∎

4.2. Resonance set

In this subsection, our goal is to show that a simple random walk on the percolation cluster starting from $x\in A_{N}\cap\mathscr{S}_{\infty}$ is very unlikely to avoid a certain resonance set, defined in (4.59). The main result is stated in Theorem 4.8.

We begin by providing the definition of the resonance set. To describe it, we first introduce several parameters: the integer $\ell_{*}\geq 0$ controls from above the scales under consideration; the integer $J\geq 1$ represents the “strength” of resonance, the integer $L\geq L(J)$ (with $L(J)$ as in (4.38)) determines the separation between successive scales, and the integer $I$ denotes the number of scales examined. The resonance set also depends crucially on the realization of the environment $\omega$, which we choose in a set of $\mathbb{P}^{u}$-measure one, and which is characterized by good regularity properties holding for all possible choices of the parameter $J$. Let

[TABLE]

Recall the set $\Omega_{\mathrm{reg}}^{\alpha,\vartheta}$ in (3.17) and let

[TABLE]

It is clear that this is a set of full $\mathbb{P}^{u}$-measure. For $\omega\in\Omega_{\mathrm{reg}}$, we say that $\ell_{*}$ is $(I,J,L,N)$-compatible, if with $\ell_{0}$ as in (4.54), one has

[TABLE]

(with $b_{N}$ as in (1.3)).

The resonance set can be seen informally as the set of points such that there are at least $J$ members among the collection (indexed by $\mathscr{A}_{*}$) of $(J+1)I$ local densities $\widetilde{\sigma}^{\omega}_{\ell}$ (see (4.3)) attaining values in $[\widetilde{\alpha},1-\widetilde{\alpha}]$; more precisely, we set

[TABLE]

For $N,J,I\geq 1$, $L\geq L(J)$, we define

[TABLE]

(with the convention $\sup\varnothing=-\infty$) where the first supremum is over all $\ell_{*}$ which are $(I,J,L,N)$-compatible.

Theorem 4.8** ($d\geq 3$, Main Assumption, (4.1)).**

Let $\omega\in\Omega_{\mathrm{reg}}$. For any sequence of positive integers $(J_{N})_{N\geq 0}$ with $J_{N}\nearrow\infty$, we have

[TABLE]

where $I(\cdot)$ is an explicit function defined in (4.82) and $L(\cdot)$ is as in (4.38).

Proof.

We begin by considering $J\geq 2$, $L\geq L(J)$, $I,N\geq 1$ fixed. We will later allow them to depend on $N$ in a subsequent part of the proof. Note that the set of $\ell_{*}$ which are $(I,J,L,N)$-compatible is always non-empty, by an argument analogous to the one in Remark 4.4 (1). Suppose $\ell_{*}\geq 0$ is $(I,J,L,N)$-compatible, $U_{0}\in\mathscr{U}_{\ell_{*},N}^{\omega}$, and $x\in A_{N}\cap\mathscr{S}_{\infty}$.

We first note that for $\ell\leq\ell_{0}$, if $U_{0}$ has positive relative volume in $B(z,2^{\ell})$, i.e. $\sigma_{\ell}^{\omega}(z)<1$ for some $z\in\mathscr{S}_{\infty}$, then $B(z,2^{\ell})\cap U_{0}\neq\varnothing$, so that $z$ must be in $B(0,b_{N}+2^{\ell})$ as $U_{0}\subseteq B(0,b_{N})$ (see (4.6)). The compatibility condition (4.58b) yields

[TABLE]

This shows that for $\ell\leq\ell_{0}$,

[TABLE]

Now we introduce the concept of $I^{\prime}$-families and related definitions. For an integer $1\leq I^{\prime}\leq I$, an $I^{\prime}$-family consists of the initial point $x$, stopping times $(S_{i})_{i=0}^{I^{\prime}}$ with respect to the canonical filtration $(\mathscr{F}_{t})_{t\geq 0}$, a random subset $\mathscr{L}$ of $\mathscr{A}$, and integer-valued random variables $(\widehat{\ell}_{i})_{i=1}^{I^{\prime}}$, satisfying

[TABLE]

The “canonical” $I^{\prime}$-family as defined in [34, (2.12)] also exists in our case, if we replace the local density function $\widehat{\sigma}_{\ell}$ by $\sigma^{\omega}_{\ell}$, and the conditions $\widehat{\sigma}_{\ell}(X_{S_{i}})=\frac{1}{2}$ by $\sigma_{\ell}^{\omega}(X_{S_{i}})\in[\frac{1}{2}-\frac{1}{2^{\ell_{\min}}},\frac{1}{2}+\frac{1}{2^{\ell_{\min}}}]$. Given a general $I^{\prime}$-family as above, we also define for $1\leq i\leq I^{\prime}$ the stopping times

[TABLE]

“intermediate labels” and “labels”

[TABLE]

Finally, we will need for $1\leq k\leq J$ the $(\mathscr{L}_{*},k)$-resonance set

[TABLE]

and the quantity (which also depends on $N,J,L,\omega$, but we omit the dependence since they are all fixed during the discussion of this quantity) for $1\leq k\leq J$, $I^{\prime}\geq 1$

[TABLE]

where the first supremum is taken over all $I^{\prime}$-families and $\Gamma_{k}(I^{\prime})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}1$ whenever $I^{\prime}\leq 0$. The following analogue of [18, Lemma A.1] is the main ingredient of the proof.

Lemma 4.9** (fix $N,J,I\geq 1$, $L\geq L(J)$, $\ell_{*}\geq 0$ $(I,J,L,N)$-compatible).**

It holds that

[TABLE]

and for $1\leq k<J$, $1\leq I^{\prime}\leq I,N\geq 1$, $\Delta=\lfloor\sqrt{I^{\prime}}\rfloor$,

[TABLE]

Proof.

We only sketch the proof. As (4.68) follows by exactly the same observation as in [18, Lemma A.1], we briefly explain the derivation of (4.69). It is important to note that

[TABLE]

by (4.64d) and (4.63). In addition, the compatibility condition (4.58b) implies (4.37b) for all $\ell_{0}-(I+1)(J+1)L<\ell\leq\ell_{0}$ (with $\ell_{j+1}$ as $\ell$). Therefore, together with (4.64d), Proposition 4.7 is applicable to $X_{S_{i}}$ for all $1\leq i\leq I^{\prime}$. The remaining part of the proof of (4.69) is then similar to the proof of [18, Lemma A.1]. ∎

We set

[TABLE]

where in the first case, the supremum in $\ell_{*}$ is over all $(I,J,L,N)$-compatible $\ell_{*}\geq 0$. Using the “canonical” $I$-family, one has that

[TABLE]

We want to show that the right-hand side of (4.72) is sufficiently small as long as $I$ is much larger than $J$ and it will be convenient to find an appropriate functional term characterizing the dependence of $I$ on $J$. We can first bound from above the right hand side of (4.69) by its supremum over $U_{0}\in\mathscr{U}_{\ell_{*},N}^{\omega}$ and $(I,J,L,N)$-compatible $\ell_{*}\geq 0$ and then take the same suprema on the left hand side to obtain a recursive relation for $\widetilde{\Gamma}_{\cdot}(\,\cdot\,)$:

[TABLE]

Lemma 4.10** ($J\geq 2$).**

Let $2\leq k\leq J$. Define

[TABLE]

where $c_{3}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}2\vee\big(-\log(1-c_{2}(1))\big)$. For any $\epsilon\in(0,1)$, if $I\geq I_{0}(\epsilon,k)$, then

[TABLE]

($c_{2}(J)$ is the constant in Proposition 4.7).

Proof.

We prove the lemma by induction. For $k=2$, the claim follows from (4.73) because $\widetilde{\Gamma}_{1}(I)=0$. From now on we assume the claim holds for some $2\leq k\leq J$ and we want to show that for any $\epsilon>0$, if $I\geq I_{0}(\epsilon,k+1)$, then

[TABLE]

Due to (4.73), we have

[TABLE]

To bound the right hand side of (4.76) by $\log(1-c_{2}(J))+\epsilon$, it suffices to have the following four estimates when $I\geq I_{0}(\epsilon,k+1)$:

[TABLE]

As $c_{3}\geq 2$, $\epsilon\in(0,1)$, and $(k+1)2^{k+1}\geq 2$, using the definition (4.74), one has

[TABLE]

Hence (4.77a) holds for $I\geq{I_{0}(\epsilon,k+1)}$. As for (4.77b), we note that $\log(1-c_{2}(J))<0$ and $I^{-1/2^{k}}(\sqrt{I}-1)\geq 1$, since $I\geq I_{0}(\epsilon,2)\geq 256$, using $k\geq 2$. We now focus on (4.77c) and (4.77d).

Now note that (4.77c) holds if and only if

[TABLE]

If

[TABLE]

then (4.78) holds for all $I\geq 1$ as $-xe^{-x}\geq-e^{-1}$ for $x\geq 0$. We can therefore assume without loss of generality that

[TABLE]

and thus (4.78) holds if

[TABLE]

where $W_{-1}(\cdot)$ is a real branch of the Lambert $W$-function (see, e.g., [13] for its definition). It is proved in [13, Theorem 1] that $W_{-1}(-e^{-u-1})>-1-\sqrt{2u}-u$ for $u>0$, so

[TABLE]

as $u-\sqrt{2u}+1>0$ for $u>0$. Applying the inequality (4.80) with $-u=\log\frac{\epsilon/4}{(1+\frac{k-1}{2})2^{k}}+1$, we obtain

[TABLE]

Hence (4.79) holds when $I\geq I_{0}(\epsilon,k+1)$. Combined with (4.78), (4.77c) holds when $I\geq I_{0}(\epsilon,k+1)$.

To show that (4.77d) holds when $I\geq I_{0}(\epsilon,k+1)$, it suffices to show that

[TABLE]

Indeed, suppose (4.81) holds, then $I\geq I_{0}(\epsilon,k+1)$ implies $I\geq(I_{0}(\epsilon/2,k)+k)^{2}$ so that $\lfloor\sqrt{I}\rfloor-k+1\geq I_{0}(\epsilon/2,k)$. By the induction hypothesis,

[TABLE]

Then (4.77d) holds because

[TABLE]

We now show (4.81). To that end, observe that

[TABLE]

This completes the induction. ∎

For the following result, we let $I$ depend on $J$ in the following explicit way:

[TABLE]

where $I_{0}$ is defined in (4.74).

Corollary 4.11**.**

$\widetilde{\Gamma}_{J}(I(J))\to 0$* as $J\to\infty$.*

Proof.

Let $\epsilon=-\frac{1}{2}\log(1-c_{2}(J))$ ($<1$ for $J$ large enough). On one hand, by Lemma 4.10, we have

[TABLE]

for $I\geq I_{0}(-\frac{1}{2}\log(1-c_{2}(J)),J)$. On the other hand, we have

[TABLE]

provided that $I\geq(\frac{J}{c_{2}(J)})^{2^{J-1}}$. Combining the two cases and recalling the definition of $I(J)$ in (4.82), we have

[TABLE]

The result then follows by (4.83). ∎

With this, we conclude the proof of (4.61) as follows. Let $(J_{N})_{N\geq 0}$ be a positive sequence with $J_{N}\nearrow\infty$. Inserting the choices $I=I(J_{N}),J=J_{N},L=L(J_{N})$ into (4.72) and letting $N$ tend to infinity, we observe

[TABLE]

This proves (4.61) and completes the proof of the theorem. ∎

4.3. Proof of Theorem 4.1

We start by introducing an analogue of [18, Lemma 4.9].

Lemma 4.12** ($d\geq 3$, Main Assumption).**

Let $\omega\in\Omega_{\mathrm{reg}}^{1/3,1/9}$ and $U_{0}\in\mathscr{U}_{\ell_{\ast},N}^{\omega}$. Let $\epsilon\in\mathbb{N}$, $\chi\in(0,1)$, and $\Sigma\in\mathscr{S}_{U_{0},\epsilon,\chi}^{\omega}$. There exists a sufficiently large constant $c_{4}(\omega)$ such that if $\ell\geq c_{4}(\omega)$ with $\epsilon\leq\frac{1}{4}\cdot 2^{\ell}$, $x_{0}\in\mathscr{S}_{\infty}\cap B(0,r_{\frac{1}{3},4\cdot 2^{\ell}})$ with $\widetilde{\sigma}_{\ell}^{\omega}(x_{0})\in[\widetilde{\alpha},1-\widetilde{\alpha}]$, and $y\in\mathscr{S}_{\infty}$ with $|y-x_{0}|\leq\frac{1}{4}\cdot 2^{\ell}$, then

[TABLE]

Proof.

Define $\widetilde{U}_{0}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}U_{0}\cap B(x_{0},4\cdot 2^{\ell})$ and $\widetilde{U}_{1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}U_{1}\cap B(x_{0},4\cdot 2^{\ell})$, and note that

[TABLE]

where we used that both $\widetilde{U}_{0}$ and $\widetilde{U}_{1}$ have at least cardinality $\frac{1}{3}\widetilde{\alpha}\eta(u)(2^{\ell+3}+1)^{d}$, combined with the killed heat kernel bound (3.15) (and setting $\vartheta=\frac{1}{9}$, $R=\frac{9}{2}\cdot 2^{\ell}$ so that $(1-\vartheta)R=4\cdot 2^{\ell}$, and $t=R^{2}$), and $c_{4}(\omega)$ large enough. On the event under the probability on the left-hand side of (4.86), $X_{\cdot}$ enters both $U_{0}$ and $U_{1}$ before exiting $B(x_{0},\frac{9}{2}\cdot 2^{\ell})\cap\mathscr{S}_{\infty}$, which means

[TABLE]

where we recall that $S=\partial_{\mathscr{S}_{\infty}}U_{0}$. Then, upon using the strong Markov property, we obtain

[TABLE]

since $\epsilon\leq\frac{1}{4}\cdot 2^{\ell}$. ∎

With these preparations in place, we now turn to the proof of our main result.

Proof of Theorem 4.1.

Let $(a_{N})_{N\geq 0}$, $(b_{N})_{N\geq 0}$ be sequences fulfilling (1.3) and $(c_{N})_{N\geq 0}$ decreasing to zero, as in the statement of the theorem. Then $a_{N}=s(b_{N})\cdot u_{N}$ for an increasing sequence $(u_{N})_{N\geq 0}$ with $u_{N}\nearrow\infty$, where

[TABLE]

Let $\omega\in\Omega_{\mathrm{reg}}\cap\Omega_{\mathrm{reg}}^{1/3,1/9}$. We claim that we can choose a sequence of positive integers $(J_{N}^{\omega})_{N\geq 0}$, which depends on the realization $\omega$, with $J_{N}^{\omega}\nearrow\infty$ and a natural number $N_{0}(\omega)$ large enough such that for each $N\geq N_{0}(\omega)$, we have

[TABLE]

where for simplicity, we write $I_{N}^{\omega}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}I(J_{N}^{\omega})$ and $L_{N}^{\omega}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}L(J_{N}^{\omega})$, with $I(\,\cdot\,)$ and $L(\,\cdot\,)$ defined in (4.82) and (4.38), respectively, and we recall that, compatibility is defined in (4.58), and $c_{4}(\omega)$ is the constant defined in Lemma 4.12.

We first show how the proof is finished assuming the existence of $(J_{N}^{\omega})_{N\geq 0}$. To that end, consider $\epsilon/2^{\ell_{*}}\leq c_{N}$, $2^{\ell_{\ast}}\in[a_{N},b_{N}]$, $\Sigma\in\mathscr{S}_{U_{0},\epsilon,\chi}^{\omega}$ with $U_{0}\in\mathscr{U}_{\ell_{*},N}^{\omega}$ for $N\geq N_{0}$, and let $x_{0}\in\mathrm{Res}=\mathrm{Res}^{\omega}(U_{0},I^{\omega}_{N},J^{\omega}_{N},L^{\omega}_{N},\ell_{\ast})$. Then

[TABLE]

We can apply the strong Markov property at successive exit times of the sup-norm ball $B(x_{0},5\cdot 2^{\ell})$, $\ell\in\mathscr{A}_{*}$ and find (because $5\cdot 2^{\ell^{\prime}}\leq\frac{1}{4}2^{\ell}$ for $\ell^{\prime}<\ell\in\mathscr{A}_{*}$) upon repeated use of Lemma 4.12 that

[TABLE]

where we used that $x_{0}\in\mathrm{Res}$ in the last step. Indeed, for $\ell^{\prime}<\ell\in\mathscr{A}_{*}$,

[TABLE]

In the last step, Lemma 4.12 is applicable for any $\ell\in\mathscr{A}_{\ast}$ with $\widetilde{\sigma}^{\omega}_{\ell}(x_{0})\in[\widetilde{\alpha},1-\widetilde{\alpha}]$ by (4.91), (4.90d), and

[TABLE]

where the second inclusion follows by a similar argument as in (4.62), and in the last line we used that $\ell\geq\ell_{0}-(I^{\omega}_{N}+1)(J^{\omega}_{N}+1)L^{\omega}_{N}$, the fact that $r_{\alpha,N}$ is increasing in $\alpha$ (see (3.2)), and the observation that $\alpha(J)\leq\alpha(1)<\frac{1}{3}$ (see (i) in the proof of Proposition 4.7).

Therefore, for $x\in A_{N}\cap\mathscr{S}_{\infty}$ and $N\geq N_{0}(\omega)$,

[TABLE]

We then take supremum over $x\in A_{N}\cap\mathscr{S}_{\infty},\Sigma\in\mathscr{S}_{U_{0},\epsilon,\chi}^{\omega}$, $U_{0}\in\mathscr{U}_{\ell_{*},N}^{\omega}$, $\epsilon/2^{\ell_{*}}\leq c_{N}$, $a_{N}\leq 2^{\ell_{*}}\leq b_{N}$, and let $N\to\infty$ to obtain

[TABLE]

where we have used the result of Theorem 4.8 in the last step. Finally, (4.9) follows in the same way as [16, Lemma 2.1].

We are left with constructing $(J_{N}^{\omega})_{N\geq 0}$ fulfilling (4.90a)–(4.90d). The following elementary lemma will be helpful for that purpose.

Lemma 4.13**.**

Let $(x_{n})_{n\geq 1}$ be a sequence of real numbers with $x_{n}\nearrow\infty$, and $f:\mathbb{Z}_{+}\to\mathbb{R}$. Then there exists a sequence of positive integers $(y_{n})_{n\geq 1}$ with $y_{n}\nearrow\infty$ and an $n_{0}$ such that $f(y_{n})\leq x_{n}$ for $n\geq n_{0}$. Moreover, if $(y^{\prime}_{n})_{n\geq 1}$ is another sequence of positive integers with $y_{n}^{\prime}\leq y_{n}$, then $f(y_{n}^{\prime})\leq x_{n}$ for $n\geq n_{0}$.

Proof.

Let $\tilde{f}(m)=\max_{1\leq j\leq m}f(j)$. We can assume without loss of generality that $\tilde{f}(m)\to\infty$ as $m\to\infty$ since otherwise the claim is obvious. The first part of the lemma follows by considering $y_{n}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\max\{m\in\mathbb{Z}_{+}:\tilde{f}(m)\leq x_{n}\}$ (with the convention $\max\varnothing=1$, but note that since $x_{n}\nearrow\infty$, we see that there exists $n_{0}\in\mathbb{Z}_{+}$ such that $\tilde{f}(1)\leq x_{n}$ for $n\geq n_{0}$). The second claim follows from $f(y^{\prime}_{n})\leq\tilde{f}(y^{\prime}_{n})\leq\tilde{f}(y_{n})$, for $n\geq n_{0}$, since $\tilde{f}$ is increasing. ∎

In what follows, we define $J_{N}^{j,\omega}$ with $1\leq j\leq 4$ and set $J_{N}^{\omega}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}J_{N}^{1,\omega}\wedge J_{N}^{2,\omega}\wedge J_{N}^{3,\omega}\wedge J_{N}^{4,\omega}$, where each of the sequences $(J_{N}^{j,\omega})_{N\geq 0}$ is chosen to fulfill one of the conditions in (4.90a)–(4.90d). By the second part of Lemma 4.13, the sequence $(J_{N}^{\omega})_{N\geq 0}$ then satisfies (4.90a)–(4.90d).

For (4.90a), note that $2^{(I(\cdot)+2)(\cdot+1)L(\cdot)}$ is a function defined on positive integers. By Lemma 4.13, we can choose $(J_{N}^{1,\omega})_{N\geq 0}$ as a sequence of integers such that $J_{N}^{1,\omega}\nearrow\infty$ and (4.90a) holds for $(J_{N}^{1,\omega})_{N\geq 0}$. We can argue similarly for (4.90b) and (4.90d), where in (4.90d) we require

[TABLE]

It remains to construct a sequence $(J_{N}^{3,\omega})_{N\geq 0}$ fulfilling (4.90c) (see again (4.58) for the definition of compatibility). For the first compatibility condition (4.58a), we first choose a sequence $(\widetilde{J}^{3,\omega}_{N})_{N\geq 0}$, similarly as before by Lemma 4.13, with

[TABLE]

(see (4.33) for the definition of $\ell_{\min}$). To fulfill the second compatibility condition (4.58b), we choose another sequence $(\widehat{J}_{N}^{3,\omega})_{N\geq 0}$ with $\widehat{J}_{N}^{3,\omega}\leq J^{1,\omega}_{N}$ such that (recall that $\kappa_{\text{reg}}(\alpha)$ enters the definition of $r_{\alpha,R}$; see (3.2))

[TABLE]

and assume that we have chosen $N_{0}^{\prime}(\omega)$ large enough such that for $N\geq N_{0}^{\prime}(\omega)$, (4.90a) holds. Then for $N\geq N_{0}^{\prime}(\omega)$, if $a_{N}\leq 2^{\ell_{*}}\leq b_{N}$, we have (with $\widehat{I}^{3,\omega}_{N}=I(\widehat{J}_{N}^{3,\omega})$ and $\widehat{L}^{3,\omega}_{N}=L(\widehat{J}_{N}^{3,\omega})$)

[TABLE]

(with $s(\cdot)$ defined in (4.89)). This implies

[TABLE]

Hence,

[TABLE]

for $N\geq N_{0}^{\prime}(\omega)$, which tends to infinity as $N\to\infty$. In other words, for $N$ large enough (depending on $\omega$), we have

[TABLE]

so (4.58b) holds. We finally set $J^{3,\omega}_{N}=\widetilde{J}^{3,\omega}_{N}\wedge\widehat{J}^{3,\omega}_{N}$, and $(J_{N}^{3,\omega})_{N\geq 0}$ satisfies (4.90c). ∎

We conclude this section with a discussion of generalizations of our main result and pertinent applications.

Remark 4.14*.*

1. (1)

(Bond percolation models) We have chosen in the Main Assumption a set-up of site percolation models fulfilling P1–P2, D, and S1–S2 to include a broad class of examples of percolation clusters with strong, algebraically decaying correlations, see Remark 2.2. One can alternatively consider bond percolation models by setting $\Omega_{\mathrm{bond}}=\{0,1\}^{\mathbb{E}^{d}}$, $\mathscr{F}_{\mathrm{bond}}=\sigma(\omega_{b}\mapsto\omega_{b}(e)\,:\,e\in\mathbb{E}^{d})$ and considering a class of probability measures $\{\mathbb{P}^{u}\,:\,u\in(a,b)\}$ with real numbers $a<b$ on $(\Omega_{\mathrm{bond}},\mathscr{F}_{\mathrm{bond}})$. One then studies the subgraph $(\mathbb{Z}^{d},\mathscr{E})$ with $\mathscr{E}=\{e\in\mathbb{E}^{d}\,:\,\omega_{b}(e)=1\}$ and defines

[TABLE]

A particularly important example is supercritical Bernoulli bond percolation on $(\mathbb{Z}^{d},\mathbb{E}^{d})$, $d\geq 3$, where $\mathbb{P}^{u}$ is a product measure of a Bernoulli distribution with success parameter $u$ on the edges, and $u>p_{c}^{\mathrm{bond}}$. A simple inspection of the proof of Theorem 4.1 shows that our results remain valid if the quenched heat kernel bounds (2.13) (together with an integrability condition as in (2.14)) and the large-deviation estimates for the volume (3.1) hold for the corresponding infinite connected component $\mathscr{S}_{\infty}^{\mathrm{bond}}$. The analogues of (2.13) and (2.14) are the main results in [7] (see Theorem 1 and (0.5), p. 3026 therein), and (3.1) is found for instance in [37, Theorem 1.2] (or [21, Proposition 11] for an even more quantitative result). We can therefore obtain the equivalent of (4.8) in this situation, namely that for $\chi\in(0,1)$, $(c_{N})_{N\geq 0}$ a sequence of positive real numbers decreasing to [math], and $\mathbb{P}^{u}$-almost every $\omega_{b}\in\{0,1\}^{\mathbb{E}^{d}}$, one has

[TABLE]

with $(a_{N})_{N\geq 0}$ and $(b_{N})_{N\geq 0}$ as in (1.3) with any choice of $\Delta_{\mathrm{S}}>0$. In fact, the integrability for the random variables $\mathscr{T}(z)$ is stretched exponential in [7] (as compared to the superpolynomial rate in (2.14)), which should also straightforwardly lead to an improvement in condition in (1.3), but we do not pursue this extension since (1.3) seems strong enough for most applications we have in mind. 2. (2)

(Stability under perturbations with uniformly elliptic edge weights) As we remarked earlier, in [18] solidification estimates were obtained for random walks on $(\mathbb{Z}^{d},\mathbb{E}^{d},\lambda)$, where $\lambda\in[\lambda_{\min},1]^{\mathbb{E}^{d}}$ are uniformly elliptic edge weights, and $\lambda_{\min}\in(0,1)$. As we briefly explain, our results can be adapted in a direct way to include uniformly elliptic (potentially non-random) edge weights.

To that end, let $\lambda_{\min}\in(0,1)$ and $\lambda\in[\lambda_{\min},1]^{\mathbb{E}^{d}}$ be fixed. Under either our standing Main Assumption for site percolation models or for the supercritical Bernoulli bond percolation of the previous item, we define the induced weighted percolation configuration $(\mathscr{S}_{\infty},E(\mathscr{S}_{\infty}),(\lambda_{e})_{e\in E(\mathscr{S}_{\infty})})$ or $(\mathscr{S}^{\mathrm{bond}}_{\infty},\mathscr{E},(\lambda_{e})_{e\in\mathscr{E}})$, governed by $\mathbb{P}^{u}$, $u\in(a,b)$ (where in the Bernoulli bond percolation case, $(a,b)=(p_{c}^{\mathrm{bond}},1)$). We then consider the simple random walk on $(\mathscr{S}_{\infty},E(\mathscr{S}_{\infty}),(\lambda_{e})_{e\in E(\mathscr{S}_{\infty})})$ or $(\mathscr{S}^{\mathrm{bond}}_{\infty},\mathscr{E},(\lambda_{e})_{e\in\mathscr{E}})$, governed by $(P^{\omega,\lambda}_{x})_{x\in\mathscr{S}_{\infty}}$ with generator

[TABLE]

with

[TABLE]

in the site percolation case, and similarly for the case of Bernoulli bond percolation. We also define the heat kernel for the random walk, $q^{\omega,\lambda}_{t}$, and heat kernel for the walk killed upon leaving $U$, $q^{\omega,\lambda}_{t,U}$, as in (2.11) and (2.12) with $\mu^{\lambda}$ replacing $\mu$ and $P^{\omega,\lambda}_{x}$ replacing $P^{\omega}_{x}$ (again, similarly in the bond percolation case). By uniform ellipticity, we see that

[TABLE]

holds for every $x\in\mathscr{S}_{\infty}$. Using this observation one can see that the proof of the functional inequalities in [42, Section 4] hold by potentially adjusting the constants in the definition of (very) regular balls (see [42, p. 1874]), showing that (2.13) and (2.14) remain valid (again after possibly after adjusting the constants therein) with $q^{\omega,\lambda}_{t}$ replacing $q^{\omega}_{t}$. Leaving the control on the volumes in Proposition 3.2 unchanged, one can see that Theorem 4.1 and Corollary 4.2 remain valid for $P_{x}^{\omega,\lambda}$ replacing $P_{x}^{\omega}$. As a particular application (choosing the $(\lambda_{e})_{e\in\mathbb{E}^{d}}$ i.i.d. random variables under some environment measure $\widehat{\mathbb{Q}}$ and considering $\widehat{\mathbb{Q}}\otimes\mathbb{P}^{u}$ with $\mathbb{P}^{u}$ governing Bernoulli bond percolation and $u\in(p_{c}^{\mathrm{bond}},1)$), we obtain solidification estimates and capacity controls for i.i.d. edge weights $\mu_{\{x,y\}}^{\lambda}\in\{0\}\cup[\lambda_{\min},1]$. This is for instance the framework considered in [6],  [21], or [43]. 3. (3)

(Prospective applications) The solidification estimates and related capacity controls of [34] and [18] have been indispensable for the proofs of certain upper bounds on largely deviant disconnection or isolation-type events in the (strongly) supercritical phases of various long-range percolation models, as well as in the investigation of the picture emerging when conditioning on such events. As one case in point (but see also below for further examples), we consider as in Remark 2.2 (4) the level set $E^{\geq h}_{\mathrm{GFF}}$ of the Gaussian free field $(\varphi_{x})_{x\in\mathbb{Z}^{d}}$ on $\mathbb{Z}^{d}$, $d\geq 3$ when $h<h_{\ast}(d)$ and introduce for $N\geq 1$ the disconnection event

[TABLE]

where $S_{N}=\partial_{\mathbb{Z}^{d}}B(0,MN)$ with $M>0$, and $A\subseteq\mathring{B}_{\mathbb{R}^{d}}(0,M)$ is a compact set with non-empty interior fulfilling $\mathrm{Cap}_{\mathbb{R}^{d}}(A)=\mathrm{Cap}_{\mathbb{R}^{d}}(\mathring{A})$ (and $\mathrm{Cap}_{\mathbb{R}^{d}}$ denoting the Brownian capacity). By [47] and [35] one knows in combination with the sharpness of the phase transition of $E^{\geq h}_{\mathrm{GFF}}$ from [27] that

[TABLE]

Furthermore by [16] (again in combination with [27]) one has that conditionally on $\mathscr{D}_{N}^{h}$, the empirical mean $\frac{1}{N^{d}}\sum_{x\in\mathbb{Z}^{d}}\varphi_{x}\delta_{x/N}$ tends to follow a local profile given by $-(h_{\ast}-h)\mathscr{h}_{A}$ (with $\mathscr{h}_{A}$ the harmonic potential of $A$). While upper and lower bounds for (4.107) were developed in [47] in the case where $A=[-1,1]^{d}$, the upper bound implicit in (4.107) requires a capacity bound obtained from solidification estimates in the case where $A$ is non-convex. For the case of the Gaussian free field on $(\mathbb{Z}^{d},\mathbb{E}^{d},(\lambda_{e})_{e\in\mathbb{E}^{d}})$ with uniformly elliptic random edge weights $\lambda_{e}\in[\lambda_{\min},1]$, stationary and ergodic under an environment measure $\widehat{\mathbb{Q}}$, quenched upper and lower bounds for the disconnection event in (4.107) have been established in [18] with the help of a version of solidification estimates for uniformly elliptic edge weights.

We briefly explain how capacity lower bounds are used to obtain asymptotic upper bounds for the probability of disconnection events, and refer to [18, Section 7] for details. One considers a certain “effective disconnection event” $\widetilde{\mathscr{D}}^{h,\lambda}_{N}$, on which a coarse graining of low enough combinatorial complexity is performed. This results in a decomposition

[TABLE]

with $|\mathscr{K}_{N}|=o(\exp(N^{d-2}))$ and for any choice of $\kappa\in\mathscr{K}_{N}$ (corresponding to the coarse graining in effect) on the event $\mathscr{D}_{N,\kappa}^{\lambda}$ an interface $\Sigma_{\kappa}$ of “blocking boxes” is present (see [18, (7.44)–(7.45)]) that corresponds to the porous interfaces considered in Section 4. The challenge is then to bound uniformly over all possible sets of blocking boxes encoded by $\kappa\in\mathscr{K}_{N}$ the capacity $\mathrm{\mathrm{Cap}}^{\lambda}(\Sigma_{\kappa})$ from below by $\mathrm{\mathrm{Cap}}^{\lambda}(A^{\prime}_{N})$, where $A^{\prime}\subseteq\mathring{A}$ is compact. Together with a segmentation $U_{1}$ (depending also on the coarse-grained configuration $\kappa\in\mathscr{K}_{N}$) with $d_{\infty}(A^{\prime}_{N},U_{0})\geq cN$, one can then verify that $\Sigma\in\mathscr{S}^{\lambda}_{U_{0},10\widehat{L}_{0},c}$, where $\widehat{L}_{0}=o(N)$ is a “near macroscopic” scale (see [18, (7.31)] for its precise definition). This is where the capacity control

[TABLE]

see [18, (4.8)] is deployed, with the choices $\epsilon=10\widehat{L}_{0}$, $2^{\ell_{\ast}}\geq cN$, and $c_{N}=\frac{2^{\ell_{\ast}}}{N}\to\infty$.

It is instructive to compare this to our set-up. The capacity bound derived in (4.10) which ought to replace (4.109) in our context holds for $\mathbb{P}^{u}$-a.e. $\omega\in\Omega_{0}$, and involves the additional constraint $a_{N}\leq 2^{\ell_{\ast}}\leq b_{N}$, with $(a_{N})_{N\geq 0}$ and $(b_{N})_{N\geq 0}$ fulfilling (1.3). Since the application we have in mind requires $\widehat{c}N\leq 2^{\ell_{\ast}}\leq\widehat{C}N$, one can for instance choose $a_{N}=\widehat{c}N$ and $b_{N}=\widehat{C}N^{1+\nu}$ for any $\nu>0$ and see that (1.3) is fulfilled for any $\Delta_{\mathrm{S}}>0$.

It is therefore quite plausible that the bounds in (4.10) are pertinent to the investigation of disconnection-type events for the level-sets of the Gaussian free field on various percolation clusters, and we hope to return to this elsewhere [15]. We also highlight that similar disconnection-type questions have been investigated for the vacant set of random interlacements and the random walk (see [30] and [31] for lower bounds, and [48],  [34], and [17] for corresponding upper bounds). In a similar direction, in [49] large-deviation type upper and lower bounds were developed for the probability that the adequately thickened connected component of the boundary of a large box centered at the origin in either the vacant set of the random walk or random interlacements, or the upper level set of the GFF leaves in the box a macroscopic volume in its complement. In the derivation of the corresponding upper bounds, capacity-type controls are brought into play in much the same vein as for the previously mentioned disconnection-type estimates. Removing the thickening inherent in the above construction is highly non-trivial, and in fact relies on a more sophisticated control on a certain type of “bubble set” present when observing an excess of points disconnected from the boundary of an enclosing box. In the case of random interlacements on $\mathbb{Z}^{d}$, $d\geq 3$, results of this type have been developed in [50] (see also references therein). Remarkably, the construction of the bubble set is achieved in part by relying on a notion of resonance sets (on $\mathbb{Z}^{d}$, $d\geq 3$). Methods in this direction developed in the present article may thus also be useful to provide some insight into natural questions pertaining to the excess behavior of random interlacements or the GFF on the infinite cluster of Bernoulli (bond- or site-)percolation.

Appendix A Proof of Proposition 3.1

In this appendix, we prove a large-deviation type estimate for the volume density as stated in Proposition 3.1. The proof of this result is a slight generalization of [42, Lemma 3.3], and follows along the same lines.

We first introduce some further notation following [42]. Let $(\ell_{n})_{n\geq 0}$, $(r_{n})_{n\geq 0}$, and $(L_{n})_{n\geq 0}$ be sequences of positive integers, and consider the rescaled lattice $\mathbb{G}_{n}=L_{n}\mathbb{Z}^{d}$ as a graph with an edge between each pair of $\ell^{1}$-nearest neighbors in the latter. For $x\in\mathbb{Z}^{d}$ and $K,s\in\mathbb{Z}_{+}$, we define $Q_{K,s}(x)=x+\mathbb{Z}^{d}\cap[0,KL_{s})^{d}$ and let $\mathscr{C}_{K,s,r}(x)$ be the largest connected component in $\mathscr{S}_{r}\cap Q_{K,s}(x)$ for $r\in\mathbb{Z}_{+}$ (with ties broken arbitrarily), where we recall that $\mathscr{S}_{r}$ is the set of vertices of $\mathscr{S}$ which are in connected components of $\mathscr{S}$ with $\ell^{1}$-diameter greater or equal to $r$. We write $Q_{K,s}$ and $\mathscr{C}_{K,s,r}$ as a shorthand if $x=0$. We recall below the definition of $(\overline{D},\overline{I},n)$-bad events with respect to two generic sequences of events $\overline{D}=\{\overline{D}_{x,L_{0}}\,:\,L_{0}\geq 1,x\in\mathbb{G}_{0}\}$ and $\overline{I}=\{\overline{I}_{x,L_{0}}\,:\,L_{0}\geq 1,x\in\mathbb{G}_{0}\}$ from [42, Definition 2.1]. For our purposes, it will be sufficient to consider the specific events in (A.1) and (A.2).

We fix $\alpha\in(0,1)$ as in the statement of the proposition. Let $\eta_{1}(\alpha)=\sqrt{1-\frac{\alpha}{2}}\eta(u)$ and $\eta_{2}(\alpha)=(1+\frac{\alpha}{4})\eta(u)$ (with $\eta(u)$ as in (2.7)). We define the “seed events” similarly as in [42, Section 3.1]

[TABLE]

for $L_{0}\geq 1$ and $x\in\mathbb{G}_{0}$. We then recursively define for $\overline{E}\in\{\overline{D},\overline{I}\}$ the events $\overline{G}_{x,n,L_{0}}(\overline{E}^{\alpha})$ by $\overline{G}_{x,0,L_{0}}(\overline{E}^{\alpha})=\overline{E}_{x,L_{0}}^{\alpha}$ and

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We say that for $n\geq 0$, $x\in\mathbb{G}_{n}$ is $n$-bad if the event $\overline{G}_{x,n,L_{0}}(\overline{D}^{\alpha})\cup\overline{G}_{x,n,L_{0}}(\overline{I}^{\alpha})$ occurs, and $n$-good otherwise.

Heuristically, if all vertices in $\mathbb{G}_{s}\cap Q_{K,s}$ are $s$-good, we can iteratively remove “small” cubic regions at each scale, resulting in a subgraph $\mathscr{Q}_{K,s,0}$ of $\mathbb{G}_{0}$ in which all remaining vertices are [math]-good, see [42, Fig. 2] for a visualization. Since only small portions of the domain are removed at each step, the connectivity of $\mathscr{Q}_{K,s,0}$ is maintained, and its volume remains comparable to that of $Q_{K,s}$. Hence, we obtain a globally connected component that satisfies the two-sided volume constraints at small scales. This allows us to conclude that the largest connected component in $Q_{K,s}$ meets the announced volume bounds.

The difference of our construction of the seed events (A.1)–(A.2) compared to [42] is the dependence on the additional fixed constant $\alpha\in(0,1)$. In the latter reference, $\eta_{1}=\frac{3}{4}\eta(u)$ and $\eta_{2}=\frac{5}{4}\eta(u)$ (see Subsection 3.1 and (35) therein). We have the following analogue of [42, Lemma 3.3].

Lemma A.1**.**

Assume that the sequences $(r_{n})_{n\geq 0}$ and $(\ell_{n})_{n\geq 0}$ satisfy

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Let $K,s\in\mathbb{Z}_{+}$ and $x_{s}\in\mathbb{G}_{s}$. If all the vertices in $\mathbb{G}_{s}\cap Q_{K,s}(x_{s})$ are $s$-good, then $\mathscr{C}_{K,s,L_{0}}(x_{s})$ is uniquely defined and

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Proof.

Without loss of generality, we assume that $x_{s}=0$. Suppose that all the vertices in $\mathbb{G}_{s}\cap Q_{K,s}$ are $s$-good. According to [42, Lemma 3.3] and its proof as well as [42, Corollary 3.4], the cluster $\mathscr{C}_{K,s,L_{0}}$ is uniquely defined and fulfills

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where we used [42, (24)] in the first inequality and abbreviated $\eta_{1}(\alpha)$ by $\eta_{1}$.

On the other hand, abbreviating $\eta_{2}(\alpha)$ by $\eta_{2}$, we have

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where we used [42, (25)] in the first inequality. ∎

Note that $\eta_{1}(\alpha)<\eta(u)<\eta_{2}(\alpha)$ for $\alpha\in(0,1)$. To conclude, it remains to prove that the probabilistic controls used in [42] or [2] remain valid when $(\frac{3}{4}\eta(u),\frac{5}{4}\eta(u))$ is replaced by $(\eta_{1}(\alpha),\eta_{2}(\alpha))$, where $\alpha\in(0,1)$ is fixed.

More precisely, we define $\mathscr{H}_{K,s}^{\alpha}(x_{s})\in\mathscr{F}$ as in [42, Definition 3.7]:

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If $\mathscr{H}_{K,s}^{\alpha}(0)\cap\{0\in\mathscr{S}_{\infty}\}$ occurs, then by definition [math] and $\mathscr{C}_{K,s,L_{0}}$ are connected, so that $\mathscr{C}_{K,s,L_{0}}\subseteq\mathscr{S}_{\infty}\cap Q_{K,s}$. From Lemma A.1, it follows that on $\mathscr{H}_{K,s}^{\alpha}(x_{s})\cap\{0\in\mathscr{S}_{\infty}\}$,

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provided that (A.4) holds. Let $R\geq 1$. We can assume without loss of generality by enlarging $\kappa_{\mathrm{d}1}(\alpha)$ that $R\geq R_{0}(\alpha)$ is sufficiently large. We choose

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and $x_{s}\in\mathbb{G}_{s}$ such that $B(0,R)\subseteq Q_{K,s}(x_{s})$. Note that

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It is therefore possible to choose $R_{0}(\alpha)$ big enough such that

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For instance, a possible choice is

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It remains to show that

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(recall $\mathbb{P}^{u}_{0}[\,\cdot\,]=\mathbb{P}^{u}[\,\cdot\,|\,{0}\in\mathscr{S}_{\infty}]$). To this end, we let $(\ell_{n})_{n\geq 0}$, $(r_{n})_{n\geq 0}$, $(L_{n})_{n\geq 0}$ be defined as in [2, Theorem 6.4], and then apply it to prove the adjusted version of Lemmas 5.2 and 5.4 in [23], with $\frac{3}{4}\eta(u)$ and $\frac{5}{4}\eta(u)$ replaced by $\eta_{1}(\alpha)$ and $\eta_{2}(\alpha)$ respectively. The adjustment is valid since only the relation $\frac{3}{4}\eta(u)<\eta(u)<\frac{5}{4}\eta(u)$ has been used in the proofs to apply the ergodic theorem (see [23, (5.1)]). Hence there exists $C_{1}(\alpha)$ and $C_{2}(\alpha,\ell_{0}(\alpha))$ (the dependence on $u$ is omitted) such that for all $\ell_{0},r_{0}\geq C_{1}(\alpha)$, $L_{0}\geq C_{2}(\alpha,\ell_{0}(\alpha))$, and $n\geq 0$,

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It then follows by the above inequality, the definition of $\mathscr{H}_{K,s}^{\alpha}(x_{s})$, and S1, that

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To conclude (A.13), it remains to show that the right-hand side of the above equation is bounded from above by $C(\alpha)e^{-c(\alpha)(\log R)^{1+\Delta_{\mathrm{S}}}}$. Comparing (A.15) with the inequality below (39) in [42], the claim can be proved following the arguments presented there. This completes the proof of Proposition 3.1. ∎

Acknowledgements While this work was written, AC was associated with INdAM (Istituto Nazionale di Alta Matematica “Francesco Severi”) and the group GNAMPA. MN was partially supported by Hong Kong RGC (Research Grants Council) grants ECS 26301824 and GRF 16303825. The authors wish to thank the referee for the thorough review of the article and for valuable suggestions.