The Annals of Probability

Random walks on discrete cylinders with large bases and random interlacements

David Windisch

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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.

Article information

Source
Ann. Probab., Volume 38, Number 2 (2010), 841-895.

Dates
First available in Project Euclid: 9 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1268143534

Digital Object Identifier
doi:10.1214/09-AOP497

Mathematical Reviews number (MathSciNet)
MR2642893

Zentralblatt MATH identifier
1191.60062

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Random walk discrete cylinder random interlacements

Citation

Windisch, David. Random walks on discrete cylinders with large bases and random interlacements. Ann. Probab. 38 (2010), no. 2, 841--895. doi:10.1214/09-AOP497. https://projecteuclid.org/euclid.aop/1268143534


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