Interlacement limit of a stopped random walk trace on a torus

TL;DR

This paper studies the behavior of a stopped random walk on a large torus, showing that its trace converges to an interlacement process, extending previous results in the field.

Contribution

It establishes the convergence of the stopped random walk trace on a torus to an interlacement process at a specific level, generalizing earlier work to higher dimensions.

Findings

Convergence of the trace to an interlacement process for large tori.

Identification of the interlacement level as proportional to the exit time of Brownian motion.

Extension of previous results to higher dimensions and different scaling regimes.

Abstract

We consider a simple random walk on $\mathbb{Z}^d$ started at the origin and stopped on its first exit time from $(-L,L)^d \cap \mathbb{Z}^d$. Write $L$ in the form $L = m N$ with $m = m(N)$ and $N$ an integer going to infinity in such a way that $L^2 \sim A N^d$ for some real constant $A > 0$. Our main result is that for $d \ge 3$, the projection of the stopped trajectory to the $N$-torus locally converges, away from the origin, to an interlacement process at level $A d \sigma_1$, where $\sigma_1$ is the exit time of a Brownian motion from the unit cube $(-1,1)^d$ that is independent of the interlacement process. The above problem is a variation on results of Windisch (2008) and Sznitman (2009).