1. Central Limit Theorem for Linear Eigenvalue Statistics of <scp>Non‐Hermitian</scp> Random Matrices
- Author
-
Dominik Schröder, László Erdős, and Giorgio Cipolloni
- Subjects
Independent and identically distributed random variables ,Applied Mathematics ,General Mathematics ,Gaussian ,Hermitian matrix ,symbols.namesake ,Matrix (mathematics) ,Distribution (mathematics) ,Statistics ,symbols ,Random matrix ,Eigenvalues and eigenvectors ,Central limit theorem ,Mathematics - 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, Virag 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, Erdős, Schroder 2019].
- Published
- 2021