Random walks on groups and KMS states

Johannes Christensen; Klaus Thomsen

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arXiv · 2006.14433

Random walks on groups and KMS states

Johannes Christensen·Klaus Thomsen

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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.

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Johannes Christensen, Klaus Thomsen. 2020-06-25. Random walks on groups and KMS states. https://arxiv.org/abs/2006.14433

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