# 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.

## Key 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.