  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.
