Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues Journal Article

Author(s): Erdős, László; Schröder, Dominik
Article Title: Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues
Affiliation IST Austria
Abstract: We prove a new central limit theorem (CLT) for the difference of linear eigenvalue statistics of a Wigner random matrix H and its minor Hˆ and find that the fluctuation is much smaller than the fluctuations of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of H and Hˆ . In particular, our theorem identifies the fluctuation of Kerov’s rectangular Young diagrams, defined by the interlacing eigenvalues of H and Hˆ , around their asymptotic shape, the Vershik–Kerov–Logan–Shepp curve. Young diagrams equipped with the Plancherel measure follow the same limiting shape. For this, algebraically motivated, ensemble a CLT has been obtained in Ivanov and Olshanski [20] which is structurally similar to our result but the variance is different, indicating that the analogy between the two models has its limitations. Moreover, our theorem shows that Borodin’s result [7] on the convergence of the spectral distribution of Wigner matrices to a Gaussian free field also holds in derivative sense.
Keywords: Vershik-Kerov-Logan-Shepp curve; CLT; Young diagrams
Journal Title: International Mathematics Research Notices
ISSN: 1687-0247
Publisher: Oxford University Press  
Date Published: 2017-02-06
DOI: 10.1093/imrn/rnw330
Open access: yes (repository)
IST Austria Authors
  1. László Erdős
    102 Erdős