Complex outliers of Hermitian random matrices

TL;DR

This paper investigates the asymptotic behavior and correlations of outliers in Hermitian random matrices combined with finite rank perturbations, revealing new convergence rates and spectral phenomena.

Contribution

It extends previous studies by analyzing outlier fluctuations, correlations, and multiple outliers generated by single spikes in deformed Hermitian matrices.

Findings

Outliers can converge at various rates and cluster around polygon vertices.

Single spikes can produce multiple outliers in the spectrum.

Surprising correlations are observed among outliers in the spectral holes.

Abstract

In this paper, we study the asymptotic behavior of the outliers of the sum a Hermitian random matrix and a finite rank matrix which is not necessarily Hermitian. We observe several possible convergence rates and outliers locating around their limits at the vertices of regular polygons as in a previous work by Benaych-Georges and Rochet, as well as possible correlations between outliers at macroscopic distance as in works by Knowles, Yin, Benaych-Georges and Rochet. We also observe that a single spike can generate several outliers in the spectrum of the deformed model, as already noticed in several previous works. In the particular case where the perturbation matrix is Hermitian, our results complete a previous work of Benaych-Georges, Guionnet and Ma\"ida, as we consider fluctuations of outliers lying in "holes" of the limit support, which happen to exhibit surprising correlations.