The Poisson boundary of lamplighter random walks on trees

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”.