How thin does random interlacement or constrained random walk have to be so that a random walk can see through it? (Work in Progress)

Abstract

The random interlacement $\mathscr{I}^u$ at level $u$ has been introduced by Sznitman, as a Poissonian collection of independent simple random walk trajectories on $\ZZ^d$, $d\geq 3$, with intensity $u>0$. Since then, several works investigated the properties of the random interlacement intersected with large sets of~$\ZZ^d$. In this paper, we study the asymptotic behavior of the capacity of $\mathscr{I}^u\cap D_n$, where $D_n$ is the blow up of a compact set $D$. We determine the correct window $(u_n)_{n\geq 0}$ of the intensity parameter for which the capacity $\mathrm{Cap}(\mathscr{I}^{u_n}\cap D_n)$ 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 $\mathscr{I}^{u_n}\cap D_n$.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 $t_n = u_n n^d$.