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

TL;DR

This paper analyzes how sparse random interlacements need to be for a random walk to effectively 'see through' them, by studying the asymptotic behavior of their capacity intersection with large sets.

Contribution

It determines the critical intensity window for random interlacements where their capacity becomes negligible, revealing when a random walk can perceive the interlacement structure.

Findings

Identifies the threshold intensity for negligible capacity of interlacements.

Provides asymptotic behavior of the capacity of conditioned random walks.

Connects interlacement sparsity with the ability of random walks to detect them.

Abstract

The random interlacements $\mathscr{I}(u)$ at level $u$ has been introduced by Sznitman, as a Poissonian collection of independent simple random walk trajectories on $\mathbb{Z}^d$, $d\geq 3$, 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 $\mathscr{I}(u) \cap D_N$, where $D_N$ is the blow up of a compact set $D$, with typical size $N$. We determine the correct window $(u_N)_{N\geq 1}$ 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 $t_N = u_N N^d$.