Fluctuations on Plancherel interger partitions around its limit shape

Abstract: For s natural number n, let P_n be the space of all integer partitions \lambda of n. Let P_{pl}(\lambda)=\frac{d_{\lambda}^2}{n!}, where d_{\lambda} stands for the numbers of all standard Young tableanux with shape \lambda. A remarkable result, almost simultaneously obtained by Logan and Shepp, Vershik and Kerov in the seventies, is that there is a limit shape w(x) for suitably scale \lambda under the probability measure P_{pl}. In this talk we will report a Gaussian fluctuation result for \lambda_{[\sqrt{n}x]} around the shape curve w(x). The result complements, in a striking way, the well-known theorem of Kerov on the generalized Gaussian convergence. The proofs are based on the poissonization techniques and the Costin-Lebowitz-Soshnikov central limit theorem for determinantal point processes.

Mathematics

Audience: researchers in the topic

ICCM 2020