Poisson-Dirichlet asymptotics in condensing particle systems

TL;DR

This paper investigates the asymptotic behavior of measures on random partitions from condensing particle systems, revealing Poisson-Dirichlet statistics in the condensed phase under general conditions.

Contribution

It provides general conditions on stationary weights that lead to Poisson-Dirichlet limits and characterizes the macroscopic distribution using size-biased sampling and ensemble equivalence.

Findings

Poisson-Dirichlet distribution describes the condensed phase

Conditions on stationary weights ensure Poisson-Dirichlet asymptotics

Concentration results for the macroscopic phase

Abstract

We study measures on random partitions, arising from condensing stochastic particle systems with stationary product distributions. We provide fairly general conditions on the stationary weights, which lead to Poisson-Dirichlet statistics of the condensed phase in the thermodynamic limit. The Poisson-Dirichlet distribution is known to be the unique reversible measure of split-merge dynamics for random partitions, which we use to characterize the limit law. We also establish concentration results for the macroscopic phase, using size-biased sampling techniques and the equivalence of ensembles to characterize the bulk distribution of the system.