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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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1463434881

Digital Object Identifier
doi:10.1214/EJP.v7-107

Mathematical Reviews number (MathSciNet)
MR1902841

Zentralblatt MATH identifier
1007.60100

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

Keywords
Partitions coagulation fragmentation invariant measures Poisson-Dirichlet

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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