Random walks on groups and KMS states

TL;DR

This paper explores the structure of KMS states on crossed product C*-algebras associated with random walks on groups, providing a complete description in specific cases and illustrating the complexity in general.

Contribution

It offers a comprehensive analysis of KMS states for Martin boundary flows, especially for trivial Poisson boundaries and hyperbolic groups, and constructs examples of more complex structures.

Findings

Complete description of KMS states for trivial Poisson boundary

Analysis of KMS states for hyperbolic groups

Examples showing complex KMS state structures

Abstract

A classical construction associates to a transient random walk on a discrete group $\Gamma$ a compact $\Gamma$-space $\partial_M \Gamma$ known as the Martin boundary. The resulting crossed product $C^*$-algebra $C(\partial_M \Gamma) \rtimes_r \Gamma$ comes equipped with a one-parameter group of automorphisms given by the Martin kernels that define the Martin boundary. In this paper we study the KMS states for this flow and obtain a complete description when the Poisson boundary of the random walk is trivial and when $\Gamma$ is a torsion free non-elementary hyperbolic group. We also construct examples to show that the structure of the KMS states can be more complicated beyond these cases.