The Poisson boundary of lampshuffler groups

TL;DR

This paper investigates the Poisson boundary of lampshuffler groups, showing that for certain groups, the permutation component stabilizes and fully characterizes the boundary, especially when the base group is the integers.

Contribution

It characterizes the Poisson boundary of lampshuffler groups for finitely generated base groups, especially $bZ$, revealing the stabilization of the permutation component.

Findings

Permutation coordinate stabilizes pointwise for transient walks

Poisson boundary described explicitly for $H=\mathbb{Z}$

Main result links stabilization to boundary characterization

Abstract

We study random walks on the lampshuffler group $\mathrm{FSym}(H)\rtimes H$, where $H$ is a finitely generated group and $\mathrm{FSym}(H)$ is the group of finitary permutations of $H$. We show that for any step distribution $\mu$ with a finite first moment that induces a transient random walk on $H$, the permutation coordinate of the random walk almost surely stabilizes pointwise. Our main result states that for $H=\mathbb{Z}$, the above convergence completely describes the Poisson boundary of the random walk $(\mathrm{FSym}(\mathbb{Z})\rtimes \mathbb{Z},\mu)$.