Beyond universality in random matrix theory

TL;DR

This paper investigates finite random matrices with non-Gaussian entries, focusing on the $1/N$ expansion of local eigenvalue statistics to understand the influence of kurtosis on the spectrum.

Contribution

It introduces a detailed $1/N$ expansion analysis for local eigenvalue statistics, highlighting the dependence on the fourth moment of matrix entries.

Findings

Eigenvalue statistics depend on the kurtosis of entries.

Provides insights into the smallest singular value behavior.

Uses complex Gaussian divisible ensembles for analysis.

Abstract

In order to have a better understanding of finite random matrices with non-Gaussian entries, we study the $1/N$ expansion of local eigenvalue statistics in both the bulk and at the hard edge of the spectrum of random matrices. This gives valuable information about the smallest singular value not seen in universality laws. In particular, we show the dependence on the fourth moment (or the kurtosis) of the entries. This work makes use of the so-called complex Gaussian divisible ensembles for both Wigner and sample covariance matrices.