Jack deformations of Plancherel measures and traceless Gaussian random matrices

TL;DR

This paper investigates the asymptotic behavior of random partitions under Jack measures, revealing their convergence to traceless Gaussian beta-ensembles and providing a new proof of Regev's theorem.

Contribution

It establishes the limit distribution of scaled Jack-deformed partitions as traceless Gaussian beta-ensembles for all positive alpha, and offers a concise proof of Regev's asymptotic theorem.

Findings

Convergence of scaled Jack measure partitions to traceless Gaussian beta-ensembles.

Validation of the limit distribution for all positive alpha.

Simplified proof of Regev's asymptotic theorem.

Abstract

We study random partitions $\lambda=(\lambda_1,\lambda_2,...,\lambda_d)$ of $n$ whose length is not bigger than a fixed number $d$. Suppose a random partition $\lambda$ is distributed according to the Jack measure, which is a deformation of the Plancherel measure with a positive parameter $\alpha>0$. We prove that for all $\alpha>0$, in the limit as $n \to \infty$, the joint distribution of scaled $\lambda_1,..., \lambda_d$ converges to the joint distribution of some random variables from a traceless Gaussian $\beta$-ensemble with $\beta=2/\alpha$. We also give a short proof of Regev's asymptotic theorem for the sum of $\beta$-powers of $f^\lambda$, the number of standard tableaux of shape $\lambda$.