Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues

• Published in: International mathematics research notices Vol. 2018; no. 10; pp. 3255 - 3298
• Main Authors: Erdős, László, Schröder, Dominik
• Format: Journal Article
• Language: English
• Published: Oxford University Press 18-05-2018
• ISSN: 1073-7928; 1687-0247
• DOI: 10.1093/imrn/rnw330

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 $\widehat H$ 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 $\widehat H$. In particular, our theorem identifies the fluctuation of Kerov’s rectangular Young diagrams, defined by the interlacing eigenvalues of $H$ and $\widehat H$, 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.