The Poisson boundary of lampshuffler groups

We study random walks on the lampshuffler group $\mathrm{F}\mathrm{S}\mathrm{y}\mathrm{m}(H)⋊H$, where $H$ is a finitely generated group and $\mathrm{F}\mathrm{S}\mathrm{y}\mathrm{m}(H)$ 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 $(\mathrm{F}\mathrm{S}\mathrm{y}\mathrm{m}(\mathbb{Z})⋊\mathbb{Z},\mu )$.