Core size of a random partition for the Plancherel measure

TL;DR

This paper demonstrates that the size of the e-core of a partition under the Poissonised Plancherel measure converges to a sum of independent Gamma distributions, revealing new probabilistic structure in partition cores.

Contribution

It establishes the asymptotic distribution of the e-core size under the Poissonised Plancherel measure, extending known results from uniform measures and utilizing determinantal point process techniques.

Findings

e-core size converges to a sum of Gamma distributions

descent set forms a determinantal point process

central limit theorem applies to the process

Abstract

We prove that the size of the e-core of a partition taken under the Poissonised Plancherel measure converges in distribution to, as the Poisson parameter goes to infinity and after a suitable renormalisation, a sum of e-1 mutually independent Gamma distributions with explicit parameters. Such a result already exists for the uniform measure on the set of partitions of n as n goes to infinity, the parameters of the Gamma distributions being all equal. We rely on the fact that the descent set of a partition is a determinantal point process under the Poissonised Plancherel measure and on a central limit theorem for such processes.