Kerov's central limit theorem for the Plancherel measure on Young diagrams

2003/04/01 by Vladimir Ivanov, Grigori Olshanski · 1 citation
Mathematics · #math.CO #math.PR #math.RT #msc:05E05 #msc:05E10 #msc:20C30 #msc:20C32 #msc:60B10 #msc:60B15

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published as In: S.Fomin, editor. Symmetric Functions 2001: Surveys of Developments and Perspectives (NATO Science Series II. Mathematics, Physics and Chemistry. Vol.74), Kluwer, 2002, pp. 93-151 · AMS-TeX, 49 pages, no figures

arxiv created 2003/04/01 · arxiv updated 2009/11/30

Abstract

Consider random Young diagrams with a fixed number n of boxes, where the probability distribution on diagrams is determined by the Plancherel measure. That is, the weight of a diagram is proportional to the squared dimension of the corresponding irreducible representation of the symmetric group S_n. As n goes to infinity, the boundary of the (suitably scaled) random diagram concentrates near a curve Omega (Logan-Shepp 1977, Vershik-Kerov 1977). In 1993, Kerov announced a central limit theorem describing Gaussian fluctuations of random diagrams around the limit shape Omega. Here we propose a reconstruction of his proof, largely based on Kerov's unpublished work notes (1999). We also discuss a striking similarity between Kerov's result and central limit theorems for random matrices (Diaconis-Shahshahani, Johansson).

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