The Poisson boundary of wreath products

Abstract

In this paper, we give a complete description of the Poisson boundary of wreath products $A\wr B:={⨁}_{B}A⋊B$ of countable groups $A$ and $B$, for probability measures $\mu$ with finite entropy where lamp configurations stabilize almost surely. If, in addition, the projection of $\mu$ to $B$ is Liouville, we prove that the Poisson boundary of $(A\wr B,\mu )$ is equal to the space of limit lamp configurations, endowed with the corresponding hitting measure. In particular, this answers an open question asked by Kaimanovich (2001) and Lyons–Peres (2021) for $B={\mathbb{Z}}^d$, $d\ge 3$, and measures $\mu$ with a finite first moment.