Outliers in the spectrum of large deformed unitarily invariant models

TL;DR

This paper analyzes the asymptotic eigenvalue behavior of large deformed unitarily invariant matrices, identifying conditions under which certain eigenvalues (outliers) deviate from the bulk spectrum using free probability theory.

Contribution

It extends previous work by characterizing outliers in large unitarily invariant models with finite-rank spikes through free probability and subordination functions.

Findings

Only specific spikes generate outliers outside the spectral support.

Outliers are described via free additive convolution and subordination functions.

Finite rank perturbations are generalized to broader deformed models.

Abstract

We investigate the asymptotic behavior of the eigenvalues of the sum A+U*BU, where A and B are deterministic N by N Hermitian matrices having respective limiting compactly supported distributions \mu, \nu, and U is a random N by N unitary matrix distributed according to Haar measure. We assume that A has a fixed number of fixed eigenvalues (spikes) outside the support of \mu, whereas the distances between the other eigenvalues of A and the support of \mu, and between the eigenvalues of B and the support of \nu, uniformly go to zero as N goes to infinity. We establish that only a particular subset of the spikes will generate some eigenvalues of A+U*BU outside the support of the limiting spectral measure, called outliers. This phenomenon is fully described in terms of free probability involving the subordination function related to the free additive convolution of \mu\ and \nu. Only finite rank perturbations had been considered up to now.