The Poisson boundary of wreath products

In this paper, we give a complete description of the Poisson boundary of wreath products {A\wr B\coloneqq \bigoplus_{B} A\rtimes 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.