We study random walks on the lampshuffler group \(\textrm{FSym}(H)\rtimes H\) , where H is a finitely generated group and \(\textrm{FSym}(H)\) 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 \((\textrm{FSym}(\mathbb {Z})\rtimes \mathbb {Z},\mu )\) .
