Random walks on discrete cylinders with large bases and random interlacements
type: Item
instance of: scholarly article
title: Random walks on discrete cylinders with large bases and random interlacements
zbMATH Open document ID: 1191.60062
DOI: 10.1214/09-AOP497
author: David Windisch
published in: The Annals of Probability
publication date: 2010-04-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/0907.1627
review text: \textit{A.-S. Sznitman} [Probab. Theory Relat. Fields 145, No. 1--2, 143--174 (2009; Zbl 1172.60316)] has introduced the model of random interlacements on \(\mathbb{Z}^{d+1}, d>2\), and explored its relation with the microscopic structure left by simple random walk on an infinite discrete cylinder \((\mathbb{Z}/N\mathbb{Z})^d \times Z\) by times of order \(N^{2d}\) . That is generalised to random walk trajectories on a cylinder of the form \(G_N\times \mathbb{Z}\), where \(G_N\) is a large finite connected weighted graph, running to a time of order \(|G_N|^2\), Under suitable assumptions , the set of points not visited by the random walk, in a neighborhood od a point with the \(\mathbb{Z}\) 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 local structure of the graph. As examples of \(G_N\), Sierpinski gasket and \(d\)-ary tree are considered.
reviewed by: Piotr Garbaczewski
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82C41
zbMATH DE Number: 5695678
zbMATH Keywords: random walk
zbMATH Keywords: discrete cylinder
zbMATH Keywords: random interlacements
zbMATH Keywords: percolation
zbMATH Keywords: connected weighted graphs
zbMATH Keywords: local times of a Brownian motion
zbMATH Keywords: excursions
zbMATH Keywords: continuous time
zbMATH Keywords: discrete time
zbMATH Keywords: Sierpinski gasket
zbMATH Keywords: tree
MaRDI profile type: Publication
arXiv ID: 0907.1627
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description: scientific article