Core size of a random partition for the Plancherel measure

Taille du cœur d'une partition tirée selon la mesure de Plancherel

Salim Rostam (1, 2)

1 UR - Université de Rennes
2 IRMAR - Institut de Recherche Mathématique de Rennes

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 +∞ 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 → +∞, 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.

Mots clés: plancherel measure; partition; core; determinantal point process; central limit theorem

Domaines: Probabilités [math.PR]; Combinatoire [math.CO]; Théorie des représentations [math.RT]

Salim Rostam. Core size of a random partition for the Plancherel measure. 2021. ⟨hal-03426306v1⟩