Cover levels and random interlacements
type: Item
instance of: scholarly article
title: Cover levels and random interlacements
zbMATH Open document ID: 1271.60057
DOI: 10.1214/11-AAP770
author: David Belius
published in: The Annals of Applied Probability
publication date: 2012-05-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1103.2072
full work available at URL: https://projecteuclid.org/euclid.aoap/1333372006
reviewed by: Achim Klenke
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 82C41
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 6034163
zbMATH Keywords: cover level
zbMATH Keywords: random interlacements
zbMATH Keywords: cover time
zbMATH Keywords: uncovered sets
zbMATH Keywords: Gumbel distributional limit
MaRDI profile type: Publication
arXiv ID: 1103.2072
cites work: On the disconnection of a discrete cylinder by a random walk
cites work: Random walks on highly symmetric graphs
cites work: Q4343010
cites work: Q3101597
cites work: Q4218912
cites work: Covering problems for Markov chains
cites work: Random Walks in Multidimensional Spaces, Especially on Periodic Lattices
cites work: Q3881651
cites work: Limiting crossing probabilities of random fields
cites work: Q3771297
cites work: Percolation for the vacant set of random interlacements
cites work: On the domination of a random walk on a discrete cylinder by random interlacements
cites work: Random walks on discrete cylinders and random interlacements
cites work: Upper bound on the disconnection time of discrete cylinders and random interlacements
cites work: Vacant set of random interlacements and percolation
cites work: Interlacement percolation on transient weighted graphs
cites work: THREE TRIPLE INTEGRALS
cites work: Random walk on a discrete torus and random interlacements
OpenAlex ID: W3098219325
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review text: The random interlacements model was introduced by \textit{A.-S. Sznitman} [Ann. Math. (2) 171, No. 3, 2039--2087 (2010; Zbl 1202.60160)]. It is a certain Poisson point process indexed by time \(u\) and taking values in the set of doubly infinite random walk paths (modulo time shift) on the \(d\)-dimensional integer lattice (\(d \geq 3\)). Denote by \(I^u\) the union of paths of this process up to time \(u\). For any finite subset \(A\) of the integer lattice, denote by \(M(A)\) the cover time of this process, that is, the smallest \(u\) such that \(A\subset I^u\). Finally, let \(g\) denote the Green function of a random walk on the \(d\)-dimensional integer lattice.NEWLINENEWLINEThe author shows that \(M(A)/g(0)\log|A|\) converges to \(1\) as the cardinality of \(A\) increases to infinity. More precisely, he shows that, for all finite \(A\), NEWLINE\[NEWLINE\sup_{z\in\mathbb{R}}\left|\operatorname{P}\left[\frac{M(A)}{g(0)}-\log(|A|)\,\leq z\right]-\exp\big(-e^{-z}\big)\right|\leq c|A|^{-c_1}NEWLINE\]NEWLINE for some \(c_1>0\) which is given more or less explicitly. Presumably, \(c\) denotes a constant that is independent of \(A\). In particular, \(M(A)/g(0)-\log(|A|)\) converges to the Gumbel distribution as \(|A|\to\infty\).
description: scientific article; zbMATH DE number 6034163