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.
"The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations." Ann. Probab. 32 (1B) 915 - 938, January 2004. https://doi.org/10.1214/aop/1079021468