Jack deformations of Plancherel measures and traceless Gaussian random matrices

Details Jack deformations of Plancherel measures and traceless Gaussian random matrices Jack deformations of Plancherel measures and traceless Gaussian random matrices

2008-10-31

arxiv.org/abs/0810.5619v2

The Electronic Journal of Combinatorics 15 (2008), R149

Mathematics

Combinatorics

18 pages

Scientific paper

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 $\lambda$.

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