Poly-logarithmic localization for random walks among random obstacles

Jian Ding, Changji Xu

Published 2017-03-20Version 1

Place an obstacle with probability $1-\mathsf{p}$ independently at each vertex of $\mathbb Z^d$, and run a simple random walk until hitting one of the obstacles. For $d\geq 2$ and $\mathsf{p}$ strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite cluster of non-obstacles; we show that for environments with probability tending to 1 as $n\to \infty$, conditioning on survival after $n$ steps the simple random walk is localized in a region of size poly-logarithmic in $n$ with probability tending to 1 as $n\to \infty$.