Jack deformations of Plancherel measures and traceless Gaussian random matrices

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 $\lambda$.

18 pages

References in corpus (1)

• The uses of random partitions