A functional CLT for partial traces of random matrices

TL;DR

This paper establishes a functional central limit theorem for partial traces of functions of random matrices, revealing how fluctuations of individual elements relate to eigenvalue statistics in large matrix limits.

Contribution

It introduces a new functional CLT that connects partial traces of matrix functions with eigenvalue fluctuations for invariant ensembles.

Findings

The limit process interpolates between element fluctuations and eigenvalue statistics.

The CLT applies to orthogonal and unitary invariant ensembles.

The result extends understanding of fluctuations in random matrix theory.

Abstract

In this paper we show a functional central limit theorem for the sum of the first $\lfloor t n \rfloor$ diagonal elements of $f(Z)$ as a function in $t$, for $Z$ a random real symmetric or complex Hermitian $n\times n$ matrix. The result holds for orthogonal or unitarily invariant distributions of $Z$, in the cases when the linear eigenvalue statistic $\operatorname{tr} f(Z)$ satisfies a CLT. The limit process interpolates between the fluctuations of individual matrix elements as $f(Z)_{1,1}$ and of the linear eigenvalue statistic. It can also be seen as a functional CLT for processes of randomly weighted measures.