Random Walks on countable groups

TL;DR

This paper provides new proofs and characterizations related to random walks on countable groups, including the Liouville property, Poisson boundary ergodicity, and weak mixing, enhancing understanding of their probabilistic and dynamical properties.

Contribution

It offers novel proofs of key properties of symmetric random walks and Poisson boundaries, and characterizes weak-mixing failures via measure-preserving isometric factors.

Findings

Equivalence between Liouville property and vanishing drift proved anew.

Product of Poisson boundary with ergodic space remains ergodic.

Weak-mixing failure characterized by measure-preserving isometric factors.

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.