Distribution of the random walk conditioned on survival among quenched Bernoulli obstacles

TL;DR

This paper studies the behavior of a random walk in a randomly obstacle-filled environment, proving localization within obstacle-free regions and analyzing the walk's distribution conditioned on survival.

Contribution

It demonstrates that the localized region is obstacle-free and derives the limiting distribution of the walk conditioned on survival, using obstacle modifications and eigenvalue estimates.

Findings

The walk localizes in obstacle-free regions of volume proportional to log n.

The conditioned walk's distribution converges to a limiting distribution.

Obstacle modifications influence the probability and eigenvalues of configurations.

Abstract

Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$ and consider a simple symmetric random walk that is killed upon hitting one of the obstacles. For $d \geq 2$ and $p$ strictly above the critical threshold for site percolation, we condition on the environment such that the origin is contained in an infinite connected component free of obstacles. It has previously been shown that with high probability, the random walk conditioned on survival up to time $n$ will be localized in a ball of volume asymptotically $d\log_{1/p}n$. In this work, we prove that this ball is free of obstacles, and we derive the limiting one-time distributions of the random walk conditioned on survival. Our proof is based on obstacle modifications and estimates on how such modifications affect the probability of the obstacle configurations as well as their associated Dirichlet eigenvalues, which is of independent interest.