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

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.