Fluctuations for analytic test functions in the Single Ring Theorem

TL;DR

This paper investigates the fluctuations of analytic functions of non-Hermitian random matrices under the Single Ring Theorem, establishing convergence and central limit theorems for spectral statistics and applications to outlier detection.

Contribution

It proves convergence in distribution of traces involving analytic functions of matrices and derives CLTs for spectral statistics and matrix entries, extending understanding of eigenvalue fluctuations.

Findings

Convergence in distribution of Tr(f(A)M) for analytic f on the support ring.

Central limit theorems for linear spectral statistics of A.

Application to locating outliers in matrix perturbations.

Abstract

We consider a non-Hermitian random matrix $A$ whose distribution is invariant under the left and right actions of the unitary group. The so-called Single Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni, states that the empirical eigenvalue distribution of $A$ converges to a limit measure supported by a ring $S$. In this text, we establish the convergence in distribution of random variables of the type $Tr (f(A)M)$ where $f$ is analytic on $S$ and the Frobenius norm of $M$ has order $\sqrt{N}$. As corollaries, we obtain central limit theorems for linear spectral statistics of $A$ (for analytic test functions) and for finite rank projections of $f(A)$ (like matrix entries). As an application, we locate outliers in multiplicative perturbations of $A$.