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Funktsional. Anal. i Prilozhen.:

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Funktsional. Anal. i Prilozhen., 2015, Volume 49, Issue 2, Pages 7–20 (Mi faa3195)  

This article is cited in 7 scientific papers (total in 8 papers)

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

A. M. Vershikabc, A. V. Malyutinb

a Saint Petersburg State University
b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow

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. 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.

Keywords: phase transition, Markov chain, Martin boundary, Poisson–Furstenberg boundary, Laplace operator, free group, homogeneous tree, Bratteli diagram, intrinsic metric, pascalization, central measure, de Finetti's theorem, dynamic Cayley graph, tail filtration

Funding Agency Grant Number
Russian Science Foundation 14-11-00581
Supported by RSF grant 14-11-00581.


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English version:
Functional Analysis and Its Applications, 2015, 49:2, 86–96

Bibliographic databases:

UDC: 517.9
Received: 31.03.2015

Citation: A. M. Vershik, A. V. Malyutin, “Phase transition in the exit boundary problem for random walks on groups”, Funktsional. Anal. i Prilozhen., 49:2 (2015), 7–20; Funct. Anal. Appl., 49:2 (2015), 86–96

Citation in format AMSBIB
\by A.~M.~Vershik, A.~V.~Malyutin
\paper Phase transition in the exit boundary problem for random walks on groups
\jour Funktsional. Anal. i Prilozhen.
\yr 2015
\vol 49
\issue 2
\pages 7--20
\jour Funct. Anal. Appl.
\yr 2015
\vol 49
\issue 2
\pages 86--96

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    This publication is cited in the following articles:
    1. È. B. Vinberg, S. E. Kuznetsov, “Evgenii (Eugene) Borisovich Dynkin (obituary)”, Russian Math. Surveys, 71:2 (2016), 345–371  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. A. M. Vershik, “Asymptotic theory of path spaces of graded graphs and its applications”, Jpn. J. Math., 11:2 (2016), 151–218  crossref  mathscinet  zmath  isi  scopus
    3. A. M. Vershik, “The theory of filtrations of subalgebras, standardness, and independence”, Russian Math. Surveys, 72:2 (2017), 257–333  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. S. K. Nechaev, M. V. Tamm, O. V. Valba, “Path counting on simple graphs: from escape to localization”, J. Stat. Mech. Theory Exp., 2017, 053301, 17 pp.  crossref  mathscinet  isi  scopus
    5. A. M. Vershik, A. V. Malyutin, “Infinite geodesics in the discrete Heisenberg group”, J. Math. Sci. (N. Y.), 232:2 (2018), 121–128  mathnet  crossref
    6. A. M. Vershik, A. V. Malyutin, “The Absolute of Finitely Generated Groups: II. The Laplacian and Degenerate Parts”, Funct. Anal. Appl., 52:3 (2018), 163–177  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. A. M. Vershik, A. V. Malyutin, “The absolute of finitely generated groups: I. Commutative (semi)groups”, Eur. J. Math., 4:4 (2018), 1476–1490  crossref  mathscinet  isi  scopus
    8. A. M. Vershik, A. V. Malyutin, “Asymptotic behavior of the number of geodesics in the discrete Heisenberg group”, J. Math. Sci. (N. Y.), 240:5 (2019), 525–534  mathnet  crossref
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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