Random walks on discrete cylinders and random interlacements

Abstract

We explore some of the connections between the local picture left by the trace of simple random walk on a cylinder \({(\mathbb {Z} / N\mathbb {Z})^d \times \mathbb {Z}}\) , d ≥ 2, running for times of order N2d and the model of random interlacements recently introduced in Sznitman ( http://www.math.ethz.ch/u/sznitman/preprints). In particular, we show that for large N in the neighborhood of a point of the cylinder with vertical component of order Nd the complement of the set of points visited by the walk up to times of order N2d is close in distribution to the law of the vacant set of random interlacements with a level which is determined by an independent Brownian local time. The limit behavior of the joint distribution of the local pictures in the neighborhood of finitely many points is also derived.