Random Walks on Discrete Cylinders and Random Interlacements

Details Random Walks on Discrete Cylinders and Random Interlacements Random Walks on Discrete Cylinders and Random Interlacements

2008-05-29

arxiv.org/abs/0805.4516v1

Probab. Relat. Fields, 145, 143-174, 2009.

Mathematics

Probability

27 pages, accepted for publication in Probability Theory and Related Fields

Scientific paper

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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