Functional CLT for non-Hermitian random matrices

TL;DR

This paper establishes a central limit theorem for the fluctuations of analytic functions of large non-Hermitian random matrices, revealing Gaussian behavior and providing a new variance formula involving matrix and function norms.

Contribution

It introduces a functional CLT for non-Hermitian matrices, detailing Gaussian fluctuations and a novel variance expression for the traceless component.

Findings

Gaussian fluctuations of Trf(X)A for large matrices

New variance formula involving Frobenius norm and boundary L2-norm

Decomposition into tracial and traceless fluctuation modes

Abstract

For large dimensional non-Hermitian random matrices $X$ with real or complex independent, identically distributed, centered entries, we consider the fluctuations of $f(X)$ as a matrix where $f$ is an analytic function around the spectrum of $X$. We prove that for a generic bounded square matrix $A$, the quantity $\mathrm{Tr}f(X)A$ exhibits Gaussian fluctuations as the matrix size grows to infinity, which consists of two independent modes corresponding to the tracial and traceless parts of $A$. We find a new formula for the variance of the traceless part that involves the Frobenius norm of $A$ and the $L^{2}$-norm of $f$ on the boundary of the limiting spectrum.