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

Abstract

We consider a discrete time simple symmetric random walk among Bernoulli obstacles on ${\mathbb{Z}}^d$, $d\ge 2$, 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 $d\ge 2$. 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 $d\ge 3$ remains open. We complete the picture by showing that for any $d\ge 2$, 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 $a>0$.

46 pages, 3 figures, minor corrections, to appear in Probability Theory and Related Fields