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Random walks on groups and KMS states

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A classical construction associates to a transient random walk on a discrete group Γ a compact Γ -space ∂MΓ known as the Martin boundary. The resulting crossed product C-algebra C(∂MΓ) ⋊ rΓ 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.

Original languageEnglish
JournalMonatshefte fur Mathematik
Volume196
Issue1
Pages (from-to)15-37
Number of pages23
ISSN0026-9255
DOIs
Publication statusPublished - Sept 2021

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.

    Research areas

  • KMS states, Martin boundary, Random walks

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