Electronic Journal of Probability

Asymptotics of Certain Coagulation-Fragmentation Processes and Invariant Poisson-Dirichlet Measures

Eddy Mayer-Wolf, Ofer Zeitouni, and Martin Zerner

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We consider Markov chains on the space of (countable) partitions of the interval $[0,1]$, obtained first by size biased sampling twice (allowing repetitions) and then merging the parts with probability $\beta_m$ (if the sampled parts are distinct) or splitting the part with probability $\beta_s$, according to a law $\sigma$ (if the same part was sampled twice). We characterize invariant probability measures for such chains. In particular, if $\sigma$ is the uniform measure, then the Poisson-Dirichlet law is an invariant probability measure, and it is unique within a suitably defined class of "analytic" invariant measures. We also derive transience and recurrence criteria for these chains.

Article information

Electron. J. Probab., Volume 7 (2002), paper no. 8, 25 pp.

Accepted: 14 February 2002
First available in Project Euclid: 16 May 2016

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Partitions coagulation fragmentation invariant measures Poisson-Dirichlet

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Mayer-Wolf, Eddy; Zeitouni, Ofer; Zerner, Martin. Asymptotics of Certain Coagulation-Fragmentation Processes and Invariant Poisson-Dirichlet Measures. Electron. J. Probab. 7 (2002), paper no. 8, 25 pp. doi:10.1214/EJP.v7-107. https://projecteuclid.org/euclid.ejp/1463434881

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  • Aldous, D.J., (1999), Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists, Bernoulli 5, 3-48,
  • Aldous, D.J. and Pitman, J. (1998), The standard additive coalescent, Ann. Probab. 26, 1703-1726,
  • Arratia, R. Barbour, A.D. and Tavare, S. (2001), Logarithmic Combinatorial Structures: A Probabilistic Approach. (Book, preprint, http://www-hto.usc.edu/books/tavare/ABT/index.html.)
  • Bolthausen, E. and Sznitman A.-S. (1998), On Ruelle's probability cascades and an abstract cavity method, Comm. Math. Phys. 197, 247-276,
  • Brooks, R. (1999), private communication.
  • Evans, S.N. and Pitman, J. (1998), Construction of Markovian coalescents, Ann. Inst. Henri Poincaré 34, 339-383.
  • Gnedin, A. and Kerov, S. (2001), A characterization of GEM distributions, Combin. Probab. Comp. 10, 213-217.
  • Jeon, I. (1998), Existence of gelling solutions for coagulation-fragmentation equations, Comm. Math. Phys. 194, 541-567.
  • Kingman, J.F.C. (1975), Random discrete distributions, J. Roy. Statist. Soc. Ser. B 37, 1-22.
  • Kingman, J.F.C. (1993), Poisson Processes, Oxford.
  • Meyn, S.P. and Tweedie, R.L. (1993), Markov Chains and Stochastic Stability, Springer-Verlag, London.
  • Pitman, J. (1996), Random discrete distributions invariant under size-biased permutation, Adv. Appl. Prob. 28, 525-539.
  • Pitman, J., Poisson-Dirichlet and GEM invariant distributions for split and merge transformations of an interval partition, Combin. Prob. Comp., to appear.
  • Pitman, J. and Yor, M. (1997), The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator, Ann. Probab. 25, 855-900.
  • Tsilevich, N.V. (2000), Stationary random partitions of positive integers, Theor. Probab. Appl. 44, 60-74.
  • Tsilevich, N.V. (2001), On the simplest split and merge operator on the infinite-dimensional simplex. (PDMI preprint 03/2001, ftp://ftp.pdmi.ras.ru/pub/publicat/preprint/2001/03-01.ps.gz.)