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, μ) with any ergodic ($\left( {G,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mu } } \right)$ )-space is still ergodic, which in particular yields a new proof of weak mixing for the double Poisson boundary of (G, μ) when μ is symmetric. Finally, we characterize the failure of weak-mixing for an ergodic (G, μ)-space as the existence of a non-trivial measure-preserving isometric factor.