On the internal distance in the interlacement set

TL;DR

This paper establishes a shape theorem and large deviation estimates for the internal graph distance within the interlacement set of a random interlacement model on high-dimensional integer lattices, with applications to random walk ranges.

Contribution

It introduces a shape theorem and large deviation bounds for the internal distance in the interlacement set, advancing understanding of its geometric structure and random walk behavior.

Findings

Shape theorem for internal distance in interlacement set

Large deviation estimates for internal distance of distant points

Application to internal distance in random walk range

Abstract

We prove a shape theorem for the internal (graph) distance on the interlacement set $\mathcal{I}^u$ of the random interlacement model on $\mathbb Z^d$, $d\ge 3$. We provide large deviation estimates for the internal distance of distant points in this set, and use these estimates to study the internal distance on the range of a simple random walk on a discrete torus.