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The Poisson boundary of lamplighter random walks on trees
Authors:Anders Karlsson  Wolfgang Woess
Institution:1.Department of Mathematics,Royal Institute of Technology,Stockholm,Sweden;2.Institut für Mathematik C,Technische Universit?t Graz,Graz,Austria
Abstract:Let $${\mathbb{T}_q}$$ be the homogeneous tree with degree q + 1 ≥ 3 and $${\mathcal{F}}$$ a finitely generated group whose Cayley graph is $${\mathbb{T}_q}$$ . The associated lamplighter group is the wreath product $${\mathcal{L} \wr \mathcal{F}}$$ , where $${{\mathcal{L}}}$$ is a finite group. For a large class of random walks on this group, we 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. We also prove that the Dirichlet problem at infinity is solvable for continuous functions on the active part of the boundary, if the lamplighter “operates at bounded range”. Supported by ESF program RDSES and by Austrian Science Fund (FWF) P15577.
Keywords:Random walk  Wreath product  Tree  Poisson boundary  Dirichlet problem
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