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Jack deformations of Plancherel measures and traceless Gaussian random matrices. (English) Zbl 1159.60009

Summary: We study random partitions \(\lambda=(\lambda_1,\lambda_2,\dots,\lambda_d)\) 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 \(\lambda_1,\dots, \lambda_d\) 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 \(\lambda\).

MSC:

60C05 Combinatorial probability
05E10 Combinatorial aspects of representation theory
60F05 Central limit and other weak theorems
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