Random walk on a discrete torus and random interlacements

TL;DR

This paper studies how the local environment of a simple random walk on a high-dimensional discrete torus resembles the random interlacement model as the size of the torus grows, establishing a convergence result for local pictures.

Contribution

It demonstrates that the local neighborhoods of a random walk on a large torus converge in distribution to independent random interlacements, linking the walk's local behavior to a well-studied stochastic model.

Findings

Local pictures converge to random interlacements as N grows

Joint distribution of neighborhoods becomes independent

Results hold for dimensions d >= 3

Abstract

We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus (Z/NZ)^d, d >= 3, until u N^d time steps, u > 0, and the model of random interlacements recently introduced by Sznitman. In particular, we show that for large N, the joint distribution of the local pictures in the neighborhoods of finitely many distant points left by the walk up to time u N^d converges to independent copies of the random interlacement at level u.