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March 2010 Random walks on discrete cylinders with large bases and random interlacements
David Windisch
Ann. Probab. 38(2): 841-895 (March 2010). DOI: 10.1214/09-AOP497

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 GN×ℤ, where GN 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 |GN|2 in a neighborhood of a point with ℤ-component of order |GN| 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 GN, 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.

Citation

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David Windisch. "Random walks on discrete cylinders with large bases and random interlacements." Ann. Probab. 38 (2) 841 - 895, March 2010. https://doi.org/10.1214/09-AOP497

Information

Published: March 2010
First available in Project Euclid: 9 March 2010

zbMATH: 1191.60062
MathSciNet: MR2642893
Digital Object Identifier: 10.1214/09-AOP497

Subjects:
Primary: 60G50, 60K35, 82C41

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 2 • March 2010
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