The Poisson boundary of lamplighter random walks on trees
type: Item
instance of: scholarly article
title: The Poisson boundary of lamplighter random walks on trees
zbMATH Open document ID: 1125.60082
author: Anders Karlsson
author: Wolfgang Woess
published in: Geometriae Dedicata
publication date: 2007-08-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0606046
review text: Let \(T=T_q\) be the homogeneous tree with degree \(q+1\geq 3\) and let \(F\) be a finitely generated group which acts simply transitively on \(T\) by graph automorphisms. It is possible to view \(T\) as a Cayley graph \(F\) being a free product of two elements and infinite cyclic groups. The associated lamplighter group is the wreath product \(L\wr F\), where \(L\) is a finite group. For a large class of random walks on this group the authors prove almost sure convergence to a natural geometric boundary. If the probability law governing the random walk has finite first moment, then the probability space formed by this geometric boundary together with the limit distribution of the random walk is proved to be maximal, that is, the Poisson boundary. Moreover, they also prove that the Dirichlet problem at infinity is solvable for continuous functions on the active part of the boundary, if the lamplighter operates on bounded range
Mathematics Subject Classification ID: 60J50
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 31C20
zbMATH DE Number: 5180050
zbMATH Keywords: random walk
zbMATH Keywords: wreath product
zbMATH Keywords: tree
zbMATH Keywords: Poisson boundary
zbMATH Keywords: Dirichlet problem
reviewed by: N. G. Gamkrelidze
MaRDI profile type: Publication
OpenAlex ID: W1963659692
arXiv ID: math/0606046
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DOI: 10.1007/S10711-006-9104-X
description: scientific article