Geometry of the random walk range conditioned on survival among Bernoulli obstacles

TL;DR

This paper studies the shape and boundary size of the range of a random walk conditioned on survival among Bernoulli obstacles, revealing that it contains a large ball and has a boundary of controlled size across dimensions.

Contribution

It completes the understanding of the random walk range's geometric properties conditioned on survival for all dimensions $d\,geq 2$, including the size of the contained ball.

Findings

The range asymptotically contains a large ball of radius close to $ ho_N$.

The boundary of the range is at most of size $ ho_N^{d-1}( ext{log} ho_N)^a$.

Results hold uniformly for all dimensions $d\,geq 2$.

Abstract

We consider a discrete time simple symmetric random walk among Bernoulli obstacles on $\mathbb{Z}^d$, $d\geq 2$, where the walk is killed when it hits an obstacle. It is known that conditioned on survival up to time $N$, the random walk range is asymptotically contained in a ball of radius $\varrho_N=C N^{1/(d+2)}$ for any $d\geq 2$. For $d=2$, it is also known that the range asymptotically contains a ball of radius $(1-\epsilon)\varrho_N$ for any $\epsilon>0$, while the case $d\geq 3$ remains open. We complete the picture by showing that for any $d\geq 2$, the random walk range asymptotically contains a ball of radius $\varrho_N-\varrho_N^\epsilon$ for some $\epsilon \in (0,1)$. Furthermore, we show that its boundary is of size at most $\varrho_N^{d-1}(\log \varrho_N)^a$ for some $a>0$.