Gaussian fluctuations of Young diagrams and structure constants of Jack characters

Abstract

In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first and second-order asymptotics of the bulk of a random Young diagram under this distribution, which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the first order asymptotics) and Kerov (for the second order asymptotics). This gives more evidence of the connection with Gaussian $\beta$-ensemble, already suggested by some work of Matsumoto. Our main tool is a polynomiality result for the structure constant of some quantities that we call Jack characters, recently introduced by Lassalle. We believe that this result is also interested in itself and we give several other applications of it.

71 pages. Minor modifications from version 1. An extended abstract of this work, with significantly fewer results and a different title, is available as arXiv:1201.1806

References in corpus (5)

• Jack polynomials and free cumulants
• Asymptotics of characters of symmetric groups, genus expansion and free probability
• The uses of random partitions
• Structure coefficients of the Hecke algebra of $(S_{2n},B_n)$
• Corrigendum to "Generators of the Hecke algebra of (S_{2n},B_n)"

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• On the matchings-Jack and hypermap-Jack conjectures for labelled matchings and star hypermaps
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• Polynomiality of shifted Plancherel averages and content evaluations
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• Quantum Mechanics of Plancherel Growth
• Planar algebras for the Young graph and the Khovanov Heisenberg category
• Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures
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