Jack Deformations of Plancherel Measures and Traceless Gaussian Random Matrices

We study random partitions λ = (λ1, λ2, . . . , λd) of n whose length is not bigger than a fixed number d. Suppose a random partition λ is distributed according to the Jack measure, which is a deformation of the Plancherel measure with a positive parameter α > 0. We prove that for all α > 0, in the limit as n → ∞, the joint distribution of scaled λ1, . . . , λd converges to the joint distribution of some random variables from a traceless Gaussian β-ensemble with β = 2/α. We also give a short proof of Regev’s asymptotic theorem for the sum of β-powers of f, the number of standard tableaux of shape λ. MSC-class: primary 60C05 ; secondary 05E10