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.

arXiv comment71 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

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• Asymptotic Formulas for Macdonald Polynomials and the Boundary of the $(q,t)$-Gelfand-Tsetlin Graph
• Gaussian fluctuations of Jack-deformed random Young diagrams
• Energy-dependent correlations in the $S$-matrix of chaotic systems
• $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral
• Cumulants of Jack symmetric functions and $b$-conjecture
• Structure coefficients for Jack characters: approximate factorization property
• Asymptotics of Jack characters
• Time delay statistics for finite number of channels in all symmetry classes
• A general framework for the polynomiality property of the structure coefficients of double-class algebras
• A Frobenius formula for the structure coefficients of double-class algebras of Gelfand pairs
• A recurrence formula for Jack connection coefficients
• A Spin Analogue of Kerov Polynomials
• The top-degree part in the Matchings-Jack Conjecture
• Statistical properties of eigenvalues of Laplace-Beltrami operators
• Macdonald cumulants, $G$-inversion polynomials and $G$-parking functions
• Bijection between oriented maps and weighted non-oriented maps
• On the matchings-Jack and hypermap-Jack conjectures for labelled matchings and star hypermaps
• Normalized characters of symmetric groups and Boolean cumulants via Khovanov's Heisenberg category
• Polynomiality of shifted Plancherel averages and content evaluations
• Polynomial properties of Jack connection coefficients and generalization of a result by Dénes
• 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
• Algebras with two multiplications and their cumulants