Random walks on discrete cylinders with large bases and random interlacements

Abstract

Following the recent work of Sznitman [Probab. Theory Related Fields 145 (2009) 143–174], we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form G_N×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.