Random walks on groups and KMS states

2020/06/25 by Johannes Christensen, Klaus Thomsen, Christensen, Johannes +1
Mathematics · #FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR) #math.OA #math.PR

paper · pdf · doi:10.48550/arxiv.2006.14433

arxiv created 2020/06/25 · arxiv updated 2020/06/26

Abstract

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) \rtimes_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.

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