The poisson-dirichlet law is the unique invariant distribution for uniform split-merge transformations

Abstract

We consider a Markov chain on the space of (countable) partitions of the interval, 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 θ = 1 is the unique invariant distribution for this Markov chain. Our proof uses a combination of probabilistic, combinatoric and representation-theoretic arguments.