The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations
Diaconis, Persi ; Mayer-Wolf, Eddy ; Zeitouni, Ofer ; Zerner, Martin P. W.
Ann. Probab., Tome 32 (2004) no. 1A, p. 915-938 / Harvested from Project Euclid
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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.
Publié le : 2004-01-14
Classification:  Partitions,  coagulation,  fragmentation,  invariant measures,  Poisson--Dirichlet,  60K35,  60J27,  60G55 
@article{1079021468,
     author = {Diaconis, Persi and Mayer-Wolf, Eddy and Zeitouni, Ofer and Zerner, Martin P. W.},
     title = {The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations},
     journal = {Ann. Probab.},
     volume = {32},
     number = {1A},
     year = {2004},
     pages = { 915-938},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1079021468}
}

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., Tome 32 (2004) no. 1A, pp.  915-938. http://gdmltest.u-ga.fr/item/1079021468/