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Pré-Publication, Document De Travail Année : 2023

The Poisson boundary of wreath products

Résumé

We give a complete description of the Poisson boundary of wreath products $A\wr B= \bigoplus_{B} A\rtimes B$ 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, and Lyons-Peres, for $B=\mathbb{Z}^d$, $d\ge 3$, and measures $\mu$ with a finite first moment.

Dates et versions

hal-04313048 , version 1 (28-11-2023)

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Joshua Frisch, Eduardo Silva. The Poisson boundary of wreath products. 2023. ⟨hal-04313048⟩
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