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Core size of a random partition for the Plancherel measure

Salim Rostam

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.

Core size of a random partition for the Plancherel measure

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.

Paper Structure

This paper contains 35 sections, 35 theorems, 232 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Setting
  4. Partitions
  5. Partitions
  6. Young diagrams
  7. Russian convention
  8. Descent set
  9. Plancherel measure
  10. Plancherel measure
  11. Correlation function
  12. Discrete Bessel kernel
  13. Cores
  14. Covariance calculations
  15. Central limit theorem
  16. ...and 20 more sections

Key Result

Theorem A

Under the Poissonised Plancherel measure $\mathrm{pl}_{t}$, as $t \to +\infty$ we have: for all $i,j \in \mathbb{Z}/e\mathbb{Z}$. In particular, we have $\mathrm{Var}_{t}x_{i}(\lambda) \sim \frac{4\sqrt t}{\pi e^2}\cot\frac{\pi}{2e}$.

Figures (7)

  • Figure 1: Universal limit shape for the partitions under the Plancherel measure (here with $n = 700$).
  • Figure 2: The hook $h_{(a,b)}$ and the rim hook $r_{(a,b)}$ for $(a,b) = (1,2)$ and $\lambda = (5,5,5,4,2)$.
  • Figure 3: The $8$-core of $\lambda = (5,5,5,4,2)$ is $\overline{\lambda} = (3,2).$
  • Figure 4: Russian convention for the Young diagram of $\lambda = (4,3,2,2)$
  • Figure 5: Descent set $\mathcal{D}(\lambda) = \{3,1,-1,-2,-5,-6,-7,\dots\}$ for $\lambda= (4,3,2,2)$
  • ...and 2 more figures

Theorems & Definitions (74)

  • Theorem A
  • Theorem B
  • Theorem C
  • Example 2.1
  • Definition 2.7
  • Definition 2.9
  • Definition 2.11
  • Lemma 2.13
  • proof
  • Definition 2.14: boojohansson
  • ...and 64 more