Random Walks on countable groups

Abstract

We begin by giving a new proof of the equivalence between the Liouville property and vanishing of the drift for symmetric random walks with finite first moments on finitely generated groups; a result which was first established by Kaimanovich-Vershik and Karlsson-Ledrappier. We then proceed to prove that the product of the Poisson boundary of any countable measured group $(G,\mu)$ with any ergodic $(G,\check{\mu})$-space is still ergodic, which in particular yields a new proof of weak mixing for the double Poisson boundary of $(G,\mu)$ when $\mu$ is symmetric. Finally, we characterize the failure of weak-mixing for an ergodic $(G,\mu)$-space as the existence of a non-trivial measure-preserving isometric factor.