Random Walks on Discrete Cylinders and Random Interlacements

2008/05/29 by Alain-Sol Sznitman · 1 citation
Mathematics · Physics and Astronomy · #math.PR #math-ph #math.MP #msc:60G50 #msc:60K35 #msc:82C41

paper · pdf

published as Probab. Relat. Fields, 145, 143-174, 2009. · 27 pages, accepted for publication in Probability Theory and Related Fields

arxiv created 2008/05/29 · arxiv updated 2009/12/01

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.

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• Random walks on discrete cylinders with large bases and random interlacements

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