Random walks on groups and KMS states

A classical construction associates to a transient random walk on a discrete group $\mathrm{\Gamma }$ a compact $\mathrm{\Gamma }$-space ${\mathrm{\partial }}_M\mathrm{\Gamma }$ known as the Martin boundary. The resulting crossed product $C^{\ast }$-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 $\mathrm{\Gamma }$ 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.