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.

经典构造与离散组\（\ Gamma \）紧凑\（\ Gamma \） -空间\（\ partial _M \ Gamma \）上的瞬态随机游动相关联，这就是马丁边界。得到的交叉积\(C^*\) -algebra \(C(\partial _M \Gamma ) \rtimes _r \Gamma \)配备了由定义 Martin 边界的 Martin 核给出的单参数自同构组. 在本文中，我们研究了该流的 KMS 状态，并在随机游走的泊松边界微不足道时以及当\(\Gamma \)是无扭非初等双曲群。我们还构建了示例以表明 KMS 状态的结构可以在这些情况之外更加复杂。