The Poisson boundary of a random walk on a group is a probability space that encodes the possible asymptotic trajectories that a sample path might take. Given a group $G$ and a probability measure $\mu$, a natural problem is to provide an explicit model of the corresponding Poisson boundary in terms of the geometric properties of $G$.

In this talk, I will discuss the identification problem for random walks on wreath products, with the main example being $\Z/2\Z\wr \Z^d$, $d\ge 3$. I will explain joint work with Joshua Frisch that holds for step distribution that have finite entropy and satisfy a stabilization condition, without any moment assumptions. I will also motivate how infinitely supported probability measures (with an infinite first moment) are relevant within the theory of Poisson boundaries.