Random walks on groups and KMS states

A classical construction associates to a transient random walk on a discrete group Γ a compact Γ -space ∂ M Γ known as the Martin boundary. The resulting crossed product C ∗ -algebra C ( ∂ M Γ ) ⋊ r Γ 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 Γ 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.