Random Walks on Discrete Cylinders and Random Interlacements

TL;DR

This paper investigates the relationship between the local behavior of simple random walks on large discrete cylinders and the model of random interlacements, revealing that the unvisited set near certain points converges to the vacant set of interlacements influenced by Brownian local time.

Contribution

It establishes a connection between the local trace of random walks on large cylinders and random interlacements, including the limiting distribution of unvisited sites near specific points.

Findings

Unvisited sites near certain points resemble the vacant set of random interlacements.

The local picture converges to a distribution influenced by Brownian local time.

Results extend to multiple points with joint local pictures.

Abstract

We explore some of the connections between the local picture left by the trace of simple random walk on a discrete cylinder with base a d-dimensional torus, d at least 2, of side-length N running for times of order N^{2d} and the model of random interlacements recently introduced in arXiv:0704.2560. In particular we show that when the base becomes large, in the neighborhood of a point of the cylinder with a vertical component of order N^d, the complement of the set of points visited by the walk up to times of order N^{2d}, is close in distribution to the law of the vacant set of random interlacements at a level which is determined by an independent Brownian local time. The limit of the local pictures in the neighborhood of finitely many points is also derived.