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 β-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.2010 Mathematics Subject Classification. 60C05, 05E05 (secondary:60B20).

Gaussian fluctuations of Jack-deformed random Young diagrams

“…where we use a shorthand notation S k = S k (λ), R k = R k (λ). In fact, the above-cited papers [Bia98,DFŚ10] concerned only the special isotropic case α = 1, however the passage to the anisotropic case α = 1 does not create any difficulties, see the work of Lassalle [Las09] (who used a different normalization constants) as well as of the first-named author and Féray [DF16] (whose normalization we use).…”

“…We will regard ∆ n as a Schwartz distribution on the real line R or, more precisely, as a random vector from this space. The following result is a generalization of Kerov's CLT [Ker93a, IO02] which concerned Plancherel measure in the special case α = 1 as well as a generalization of its extension by the first-named author and Féray [DF16] for the generic fixed value of α > 0. Indeed, Example 1.2 shows that the assumptions of the following theorem are fulfilled for χ n := χ reg , thus CLT holds for Jack-Plancherel measure in a wider generality, when α = α(n) may vary with n.…”

Rigidity and Edge Universality of Discrete β‐Ensembles

“…In general, it is called the Jack deformation of Plancherel measure, because of its connection to Jack polynomials. We refer to [21,39,53,74,75,86] for recent researches on the Jack measure. We prove that the distribution of the lengths of the first few rows in Young diagrams under the Jack deformation of Plancherel measure, after proper scaling, converges to the Tracy-Widom β distribution.…”

Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q,t)-Gelfand-Tsetlin Graph

“…We remark that Macdonald symmetric group characters have not been considered yet, to the author's best knowledge. However, there have been many articles studying the structural theory and asymptotics on Jack symmetric group characters, notably several recent works by Maciej Do lȩga, Valentin Féray and PiotrŚniady, see, e.g., [16,17,18,46].…”