## Duke Mathematical Journal

### 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, 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 $\beta$-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

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

Dates
Revised: 27 May 2015
First available in Project Euclid: 4 February 2016

https://projecteuclid.org/euclid.dmj/1454594831

Digital Object Identifier
doi:10.1215/00127094-3449566

Mathematical Reviews number (MathSciNet)
MR3498866

Zentralblatt MATH identifier
1338.60017

#### Citation

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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