The Poisson boundary of lamplighter random walks on trees
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
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