The Annals of Probability

The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations

Persi Diaconis, Eddy Mayer-Wolf, Ofer Zeitouni, and Martin P. W. Zerner

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We consider a Markov chain 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 (if the sampled parts are distinct) or splitting the part uniformly (if the same part was sampled twice). We prove a conjecture of Vershik stating that the Poisson--Dirichlet law with parameter $\theta=1$ is the unique invariant distribution for this Markov chain. Our proof uses a combination of probabilistic, combinatoric and representation-theoretic arguments.

Article information

Ann. Probab., Volume 32, Number 1B (2004), 915-938.

First available in Project Euclid: 11 March 2004

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60G55: Point processes

Partitions coagulation fragmentation invariant measures Poisson--Dirichlet


Diaconis, Persi; Mayer-Wolf, Eddy; Zeitouni, Ofer; Zerner, Martin P. W. The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations. Ann. Probab. 32 (2004), no. 1B, 915--938. doi:10.1214/aop/1079021468.

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