How thin does random interlacement have to be so that a random walk can see through it?

Abstract

The random interlacements $\mathcal{I}(u)$ at level $u$ has been introduced by Sznitman, as a Poissonian collection of independent simple random walk trajectories on ${\mathbb{Z}}^d$, $d\ge 3$, with intensity $u>0$. Since then, several works investigated the properties of the random interlacements intersected with large sets of~${\mathbb{Z}}^d$. In this paper, we study the asymptotic behavior of the capacity of $\mathcal{I}(u)\cap D_N$, where $D_N$ is the blow up of a compact set $D$, with typical size $N$. We determine the correct window $(u_N{)}_{N\ge 1}$ of the intensity parameter for which the capacity $\mathrm{cap}(\mathcal{I}(u_N)\cap D_N)$ starts to become negligible compared to $\mathrm{cap}(D_N)$; this roughly means that a random walk starting from far away starts to see through $\mathcal{I}(u_N)\cap D_N$. In the same spirit, we investigate the capacity of the simple random walk conditioned to stay in a large Euclidean ball up to time $t_N$, and find similar asymptotics by taking $t_N=u_N{N}^d$.

22 pages