A note on the Poisson boundary of lamplighter random walks

The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups $Γ$ endowed with a rich boundary. The starting point is the Strip Criterion of identification of the Poisson boundary for random walks on discrete groups due to Kaimanovich. A geometrical method for constructing the strip as a subset of the lamplighter group starting with a smaller strip in the base group $Γ$ is developed. Then, this method is applied to several classes of base groups $Γ$: groups with infinitely many ends, hyperbolic groups in the sense of Gromov, and Euclidean lattices. We show that under suitable hypothesis the Poisson boundary for a class of random walks on lamplighter groups is the space of infinite limit configurations.

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