Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues

TL;DR

This paper establishes a new central limit theorem for the fluctuations of interlacing eigenvalues of Wigner matrices and their minors, revealing a smaller fluctuation scale and connecting random matrix theory with Young diagram asymptotics.

Contribution

It proves a CLT for the difference of eigenvalue statistics of Wigner matrices and their minors, highlighting the fluctuation behavior of Kerov's rectangular Young diagrams.

Findings

Fluctuation of eigenvalue differences is smaller than individual fluctuations.

Identifies the fluctuation of Kerov's rectangular Young diagrams around their limit shape.

Shows convergence of spectral distribution fluctuations to a Gaussian free field in derivative sense.

Abstract

We prove a new CLT for the difference of linear eigenvalue statistics of a Wigner random matrix $H$ and its minor $\hat H$ 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 $\hat H$. In particular our theorem identifies the fluctuation of Kerov's rectangular Young diagrams, defined by the interlacing eigenvalues of $H$ and $\hat H$, around their asymptotic shape, the Vershik-Kerov-Logan-Shepp curve. This result demonstrates yet another aspect of the close connection between random matrix theory and Young diagrams equipped with the Plancherel measure known from representation theory. For the latter a CLT has been obtained in [18] 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.