Symmetric Functions 2001: Surveys of Developments and Perspectives

Volume 74 of the series NATO Science Series pp 93-151

Kerov’s Central Limit Theorem for the Plancherel Measure on Young Diagrams

  • Vladimir IvanovAffiliated withMoscow State University
  • , Grigori OlshanskiAffiliated withDobrushin Mathematics Laboratory

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Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure M n. That is, the weight M n(λ) of a diagram λ equals dim2 λ/n!, where dim λ denotes the dimension of the irreducible representation of the symmetric group indexed by λ. As n → ∞, the boundary of the (appropriately rescaled) random shape λ concentrates near a curve Ω (Logan-Shepp 1977, Vershik-Kerov 1977). In 1993, Kerov announced a remarkable theorem describing Gaussian fluctuations around the limit shape Ω. Here we propose a reconstruction of his proof. It is largely based on Kerov’s unpublished work notes, 1999

Key words

random Young diagrams Plancherel measure central limit theorem generalized Gaussian processes