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Symmetric Functions 2001: Surveys of Developments and Perspectives pp 93–151Cite as

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

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Part of the book series: NATO Science Series ((NAII,volume 74))

Abstract

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 EquationSource$$ \mathfrak{S}_n $$ 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 It is largely based on Kerov’s unpublished work notes, 1999

In memory of Sergei Kerov (1946–2000)

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Author information

Authors and Affiliations

  1. Moscow State University, Russia

    Vladimir Ivanov

  2. Dobrushin Mathematics Laboratory, Russia

    Grigori Olshanski

Authors
  1. Vladimir Ivanov
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  2. Grigori Olshanski
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, Michigan, USA

    Sergey Fomin

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Ivanov, V., Olshanski, G. (2002). Kerov’s Central Limit Theorem for the Plancherel Measure on Young Diagrams. In: Fomin, S. (eds) Symmetric Functions 2001: Surveys of Developments and Perspectives. NATO Science Series, vol 74. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0524-1_3

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  • DOI: https://doi.org/10.1007/978-94-010-0524-1_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-0774-3

  • Online ISBN: 978-94-010-0524-1

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