Abstract
We introduce a large class of random Young diagrams which can be regarded as a natural one-parameter deformation of some classical Young diagram ensembles; a deformation which is related to Jack polynomials and Jack characters. We show that each such a random Young diagram converges asymptotically to some limit shape and that the fluctuations around the limit are asymptotically Gaussian.
Highlights
1.1 Random partitions...An integer partition, called a Young diagram, is a weakly decreasing finite sequence λ = (λ1, . . . , λl ) of positive integers λ1 ≥ · · · ≥ λl > 0
The class of random Young diagrams considered in the current paper as well as the classes from [2,23] are of quite distinct flavors and it is not obvious why they should contain any elements in common, except for the trivial example given by the Jack–Plancherel measure
A side product of the work of Kerov et al is an interesting, natural class of random Young diagrams which fits into the framework which we consider in the current paper, see Sect. 1.16 and the forthcoming paper [7] for more details
Summary
Certain random partitions can be regarded as discrete counterparts of some interesting ensembles of random matrices We shall explore this link on a particular example of random matrices called β-ensembles or β-log gases [8], i.e. the probability distributions on Rn with the density of the form p(x1, . In the special cases β ∈ {1, 2, 4} they describe the joint distribution of the eigenvalues of random matrices with natural symmetries; the investigation of such ensembles for a generic value of β is motivated, among others, by statistical mechanics. In this general case the problem of computing their correlation functions heavily relies on Jack polynomial theory [8, Chapter 13]
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