Universal Gaussian fluctuations of non-Hermitian matrix ensembles: from weak convergence to almost sure CLTs

TL;DR

This paper applies a universality principle for Gaussian Wiener chaos to prove multi-dimensional CLTs for spectral moments of i.i.d. random matrices, providing bounds and an almost sure CLT for functions of traces.

Contribution

It introduces a novel application of the Gaussian Wiener chaos universality principle to random matrix spectral moments, including bounds and an almost sure CLT.

Findings

Proved multi-dimensional CLTs for spectral moments of i.i.d. matrices.

Established bounds for smooth functions of spectral traces.

Derived an almost sure central limit theorem involving logarithmic means.

Abstract

In the paper [25], written in collaboration with Gesine Reinert, we proved a universality principle for the Gaussian Wiener chaos. In the present work, we aim at providing an original example of application of this principle in the framework of random matrix theory. More specifically, by combining the result in [25] with some combinatorial estimates, we are able to prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. Our approach has the advantage of yielding, without extra effort, bounds over classes of smooth (i.e., thrice differentiable) functions, and it allows to deal directly with discrete distributions. As a further application of our estimates, we provide a new "almost sure central limit theorem", involving logarithmic means of functions of vectors of traces.