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Phase transition in the exit boundary problem for random walks on groups

Authors
  • Vershik, A. M.1
  • Malyutin, A. V.1
  • 1 St. Petersburg State University, Institute for Information Transmission Problems of RAS, St. Petersburg Department of Steklov Institute of Mathematics, St. Petersburg, Russia , St. Petersburg (Russia)
Type
Published Article
Journal
Functional Analysis and Its Applications
Publisher
Springer US
Publication Date
Apr 01, 2015
Volume
49
Issue
2
Pages
86–96
Identifiers
DOI: 10.1007/s10688-015-0090-3
Source
Springer Nature
Keywords
License
Yellow

Abstract

We describe the full exit boundary of random walks on homogeneous trees, in particular, on free groups. This model exhibits a phase transition; namely, the family of Markov measures under study loses ergodicity as a parameter of the random walk changes. The problem under consideration is a special case of the problem of describing the invariant (central) measures on branching graphs, which covers a number of problems in combinatorics, representation theory, and probability and was fully stated in a series of recent papers by the first author [1]–[3]. On the other hand, in the context of the theory of Markov processes, close problems were discussed as early as 1960s by E. B. Dynkin.

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