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

**Eddy Mayer-Wolf**

*(Technion)*

**Ofer Zeitouni**

*(Technion)*

**Martin P.W. Zerner**

*(Stanford University)*

#### 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.

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Pages: 1-25

Publication Date: February 14, 2002

DOI: 10.1214/EJP.v7-107

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