Random walks on discrete cylinders with large bases and random interlacements

TL;DR

This paper studies the microscopic structure of random walks on large cylindrical graphs and links it to random interlacements, revealing the limiting distribution of unvisited points and local neighborhoods in various graph models.

Contribution

It establishes a connection between the local behavior of random walks on large graphs and the theory of random interlacements, extending previous results to new graph structures.

Findings

Unvisited set converges to random interlacement model.

Local neighborhoods follow the law of random interlacements.

Results apply to various graph types like boxes, Sierpinski graphs, and trees.

Abstract

Following the recent work of Sznitman (arXiv:0805.4516), we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form G_N x Z, where G_N is a large finite connected weighted graph, and relate it to the model of random interlacements on infinite transient weighted graphs. Under suitable assumptions, the set of points not visited by the random walk until a time of order |G_N|^2 in a neighborhood of a point with Z-component of order |G_N| converges in distribution to the law of the vacant set of a random interlacement on a certain limit model describing the structure of the graph in the neighborhood of the point. The level of the random interlacement depends on the local time of a Brownian motion. The result also describes the limit behavior of the joint distribution of the local pictures in the neighborhood of several distant points with possibly different limit models. As examples of G_N, we treat the d-dimensional box of side length N, the Sierpinski graph of depth N and the d-ary tree of depth N, where d >= 2.