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Random walks on groups and KMS states

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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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Acknowledgements

The work was supported by the DFF-Research Project 2 ‘Automorphisms and Invariants of Operator Algebras’, No. 7014-00145B.

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  1. Department of Mathematics, Aarhus University, Ny Munkegade, Ny Munkegade, 8000 , Aarhus C, Denmark

    Johannes Christensen & Klaus Thomsen

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  1. Johannes Christensen
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  2. Klaus Thomsen
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Correspondence to Klaus Thomsen.

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Communicated by Adrian Constantin.

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Christensen, J., Thomsen, K. Random walks on groups and KMS states. Monatsh Math 196, 15–37 (2021). https://doi.org/10.1007/s00605-021-01573-1

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