Central Limit Theorem for Linear Eigenvalue Statistics of non-Hermitian Random Matrices

TL;DR

This paper proves that linear eigenvalue statistics of large non-Hermitian random matrices are asymptotically Gaussian for a broad class of test functions, extending previous results and identifying the variance dependence on the fourth cumulant.

Contribution

It establishes a central limit theorem for eigenvalue statistics of non-Hermitian matrices with minimal smoothness assumptions and introduces novel analytical tools for the proof.

Findings

Eigenvalue linear statistics are asymptotically Gaussian for test functions with $2+psilon$ derivatives.

The limiting variance explicitly depends on the fourth cumulant of the matrix entries.

New methods include a local law for resolvent products and coupling of Dyson Brownian motions.

Abstract

We consider large non-Hermitian random matrices $X$ with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having $2+\epsilon$ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [Rider, Silverstein 2006], or the distribution of the matrix elements needed to be Gaussian [Rider, Vir\'ag 2007], or at least match the Gaussian up to the first four moments [Tao, Vu 2016; Kopel 2015]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of $X$ with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian Motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices $X$ that are presented in the companion paper [Cipolloni, Erd\H{o}s, Schr\"oder 2019].