Central Limit Theorem for Linear Eigenvalue Statistics of Non‐Hermitian Random Matrices

• Published in: Communications on pure and applied mathematics Vol. 76; no. 5; pp. 946 - 1034
• Main Authors: Cipolloni, Giorgio, Erdős, László, Schröder, Dominik
• Format: Journal Article
• Language: English
• Published: Melbourne John Wiley & Sons Australia, Ltd 01.05.2023; John Wiley and Sons, Limited
• Subjects: Applications of mathematics; Central limit theorem; Eigenvalues; Mathematical analysis
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.22028

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+ϵ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. 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 [32]. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.