Outliers in the spectrum of large deformed unitarily invariant models

TL;DR

This paper characterizes outliers in the spectrum of large deformed unitarily invariant models, revealing that a single spike can generate multiple outliers, with free subordination functions being central to the analysis.

Contribution

It introduces a comprehensive spectral outlier characterization for large deformed unitarily invariant models, including non-trivial spectral measures and multiple outliers from a single spike.

Findings

A single spike can produce multiple outliers.

Free subordination functions are crucial in spectral analysis.

The models include non-trivial spectral measures and outliers.

Abstract

In this paper we characterize the possible outliers in the spectrum of large deformed unitarily invariant additive and multiplicative models, as well as the eigenvectors corresponding to them. We allow both the non-deformed unitarily invariant model and the perturbation matrix to have non-trivial limiting spectral measures and spiked outliers in their spectrum. We uncover a remarkable new phenomenon: a single spike can generate asymptotically several outliers in the spectrum of the deformed model. The free subordination functions play a key role in this analysis.