Duke Mathematical Journal

Gaussian fluctuations of Young diagrams and structure constants of Jack characters

Maciej Dołęga and Valentin Féray

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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, Logan, and Shepp (for the first-order asymptotics) and Kerov (for the second-order asymptotics). This gives more evidence for the connection with the Gaussian β-ensemble, already suggested by a work of Matsumoto.

Our main tool is a polynomiality result for the structure constants of some quantities that we call Jack characters, recently introduced by Lassalle. We believe that this result is also interesting in itself and we give several other applications of it.

Article information

Duke Math. J., Volume 165, Number 7 (2016), 1193-1282.

Received: 24 September 2014
Revised: 27 May 2015
First available in Project Euclid: 4 February 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 05E05: Symmetric functions and generalizations

random partitions Jack measure bulk fluctuations Jack polynomials polynomial functions on Young diagrams Stein’s method


Dołęga, Maciej; Féray, Valentin. Gaussian fluctuations of Young diagrams and structure constants of Jack characters. Duke Math. J. 165 (2016), no. 7, 1193--1282. doi:10.1215/00127094-3449566. https://projecteuclid.org/euclid.dmj/1454594831

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