A note on the Poisson boundary of lamplighter random walks

TL;DR

This paper characterizes the Poisson boundary of lamplighter random walks over various discrete groups using a geometrical approach based on the Strip Criterion, extending understanding of boundary behavior in these stochastic processes.

Contribution

It introduces a geometric method to construct the Poisson boundary for lamplighter groups, applicable to groups with rich boundaries such as hyperbolic and Euclidean groups.

Findings

Poisson boundary identified as the space of infinite limit configurations.

Method applies to groups with infinitely many ends, hyperbolic groups, and Euclidean lattices.

Under certain conditions, the boundary description is complete and explicit.

Abstract

The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups $\Gamma$ 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 $\Gamma$ is developed. Then, this method is applied to several classes of base groups $\Gamma$: 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.