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.

8 pages, no figures. Substantial overlap with the (longer) paper "Five remarks about random walks on groups", http://arxiv.org/abs/1406.0763

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