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Solidification estimates for random walks on supercritical percolation clusters

Alberto Chiarini, Zhizhou Liu, Maximilian Nitzschner

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.

Solidification estimates for random walks on supercritical percolation clusters

Abstract

We consider the simple random walk on the infinite cluster of a general class of percolation models on , , 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 . These controls substantially generalize previous results obtained in arXiv:1706.07229 for Brownian motion in and in arXiv:2012.05230 for random walks on equipped with uniformly elliptic edge weights to a manifestly non-elliptic framework.

Paper Structure

This paper contains 11 sections, 17 theorems, 161 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and model assumptions on percolation clusters
  3. Notation
  4. Model assumptions and examples
  5. Random walks on the infinite percolation cluster
  6. Uniform large-scale behavior over very large regions
  7. Solidification of porous interfaces on percolation clusters
  8. Local density functions
  9. Resonance set
  10. Proof of Theorem \ref{['thm:solidification']}
  11. Proof of Proposition \ref{['prop:Quantitative-control-volume']}

Key Result

Proposition 3.1

For $\alpha \in (0,1)$, there exist positive constants $\kappa_{\mathrm{d}i}(\alpha)$, $i=1,2$, such that for $R\geq 1$, (with $\Delta_{\mathrm{S}}$ defined in eq:funcS). Moreover, the function $\alpha \mapsto \kappa_{\mathrm{d}2}(\alpha)$ is increasing on $(0,1)$.

Figures (2)

  • Figure 1: Schematic illustration of a possible choice of $U_0$ with boundary $S = \partial_{\mathcal{S}_\infty} U_0$ and $\Sigma$ in $\mathcal{S}^\omega_{U_0,\epsilon,\chi}$. In the picture, $U_0$ appears as the intersection between $\mathcal{S}_\infty$ and the shaded area, and $S$ consists of the squares at the boundary of the shaded area.
  • Figure 2: Schematic illustration of the role of $r_{\alpha,R}$, for $R \geq R_{\min}$. All boxes with center inside $B(0,r_{\alpha,R}) \cap \mathcal{S}_\infty$ and radius greater or equal to $R$ are well-behaved, but degeneracies (depicted here for the relative volume) occur outside of $B(0,r_{\alpha,R})$.

Theorems & Definitions (35)

  • Remark 2.1
  • Remark 2.2: Examples
  • Proposition 3.1: $d\geq 2$, \ref{['A']}
  • Proposition 3.2: $d\geq 2$, \ref{['A']}
  • proof
  • Lemma 3.3: $d\geq 2$, \ref{['A']}
  • proof
  • Proposition 3.4: $d\geq 2$, \ref{['A']}
  • Theorem 4.1: $d\geq 3$, \ref{['A']}, \ref{['eq:ass-bounded']}, \ref{['eq:a_N-bound']}
  • Corollary 4.2: $d\geq 3$, \ref{['A']}, \ref{['eq:ass-bounded']}, \ref{['eq:a_N-bound']}
  • ...and 25 more