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Teoriya Veroyatnostei i ee Primeneniya, 1999, Volume 44, Issue 1, Pages 55–73
DOI: https://doi.org/10.4213/tvp597
(Mi tvp597)
 

This article is cited in 16 scientific papers (total in 16 papers)

Stationary random partitions of positive integers

N. V. Tsilevich

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
Abstract: This paper gives a description of stationary random partitions of positive integers (equivalently, stationary coherent sequences of random permutations) under the action of the infinite symmetric group. Equivalently, all stationary coherent sequences of random permutations are described. This result gives a new characterization of the Poisson–Dirichlet distribution PD(1) with the unit parameter, which turns out to be the unique invariant distribution for a family of Markovian operators on the infinite-dimensional simplex. This result also provides a new characterization of the Haar measure on the projective limit of finite symmetric groups.
Keywords: random partitions, random permutations, stationary distribution, Markovian operator, Poisson–Dirichlet distribution.
Received: 15.09.1998
English version:
Theory of Probability and its Applications, 2000, Volume 44, Issue 1, Pages 60–74
DOI: https://doi.org/10.1137/S0040585X97977331
Bibliographic databases:
Language: Russian
Citation: N. V. Tsilevich, “Stationary random partitions of positive integers”, Teor. Veroyatnost. i Primenen., 44:1 (1999), 55–73; Theory Probab. Appl., 44:1 (2000), 60–74
Citation in format AMSBIB
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\pages 55--73
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\zmath{https://zbmath.org/?q=an:0960.60012}
\transl
\jour Theory Probab. Appl.
\yr 2000
\vol 44
\issue 1
\pages 60--74
\crossref{https://doi.org/10.1137/S0040585X97977331}
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Linking options:
  • https://www.mathnet.ru/eng/tvp597
  • https://doi.org/10.4213/tvp597
  • https://www.mathnet.ru/eng/tvp/v44/i1/p55
  • This publication is cited in the following 16 articles:
    1. Nils Caci, Peter Mühlbacher, Daniel Ueltschi, Stefan Wessel, “Poisson-Dirichlet distributions and weakly first-order spin-nematic phase transitions”, Phys. Rev. B, 107:2 (2023)  crossref
    2. Sam Olesker-Taylor, “Cutoff for rewiring dynamics on perfect matchings”, Ann. Appl. Probab., 33:1 (2023)  crossref
    3. Benassi C., Ueltschi D., “Loop Correlations in Random Wire Models”, Commun. Math. Phys., 374:2 (2020), 525–547  crossref  isi
    4. Bjoernberg J.E., Kotowski M., Lees B., Milos P., “The Interchange Process With Reversals on the Complete Graph”, Electron. J. Probab., 24 (2019), 108  crossref  isi
    5. Kammoun M.S., “Monotonous Subsequences and the Descent Process of Invariant Random Permutations”, Electron. J. Probab., 23 (2018), 118  crossref  mathscinet  zmath  isi
    6. Joseph Najnudel, Ashkan Nikeghbali, Lecture Notes in Mathematics, 2123, Séminaire de Probabilités XLVI, 2014, 481  crossref
    7. Grosskinsky S., Lovisolo A.A., Ueltschi D., “Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths”, J. Stat. Phys., 146:6 (2012), 1105–1121  crossref  mathscinet  zmath  adsnasa  isi  scopus
    8. Goldschmidt Ch., Ueltschi D., Windridge P., “Quantum Heisenberg Models and their Probabilistic Representations”, Entropy and the Quantum II, Contemporary Mathematics, 552, eds. Sims R., Ueltschi D., Amer Mathematical Soc, 2011, 177–224  crossref  mathscinet  zmath  isi
    9. Bertoin J., “Two-parameter Poisson-Dirichlet measures and reversible exchangeable fragmentation–coalescence processes”, Combin. Probab. Comput., 17:3 (2008), 329–337  crossref  mathscinet  zmath  isi  elib  scopus
    10. A. M. Vershik, “Does There Exist a Lebesgue Measure in the Infinite-Dimensional Space?”, Proc. Steklov Inst. Math., 259 (2007), 248–272  mathnet  crossref  mathscinet  zmath  elib  elib
    11. Gamburd A., “Poisson-Dirichlet distribution for random Belyi surfaces”, Ann. Probab., 34:5 (2006), 1827–1848  crossref  mathscinet  zmath  isi  elib  scopus
    12. Brooks R., Makover E., “Random construction of Riemann surfaces”, J. Differential Geom., 68:1 (2004), 121–157  crossref  mathscinet  zmath  isi  scopus
    13. Diaconis P., Mayer-Wolf E., Zeitouni O., Zerner M.P.W., “The Poisson–Dirichlet law is the unique invariant distribution for uniform split–merge transformations”, Ann. Probab., 32:1B (2004), 915–938  crossref  mathscinet  zmath  isi
    14. Brooks R., “A statistical model of Riemann surfaces”, Complex analysis and dynamical systems, Contemp. Math., 364, Amer. Math. Soc., Providence, RI, 2004, 15–25  crossref  mathscinet  zmath  isi
    15. Pitman J., “Poisson–Dirichlet and GEM invariant distributions for split-and-merge transformations of an interval partition”, Combin. Probab. Comput., 11:5 (2002), 501–514  crossref  mathscinet  zmath  isi  scopus
    16. Eddy Mayer-Wolf, Ofer Zeitouni, Martin Zerner, “Asymptotics of Certain Coagulation-Fragmentation Processes and Invariant Poisson-Dirichlet Measures”, Electron. J. Probab., 7:none (2002)  crossref
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