Outliers in the spectrum for products of independent random matrices

TL;DR

This paper studies the eigenvalue distribution of products of independent random matrices with iid entries, focusing on outliers caused by bounded rank perturbations and extending Tao's results to multiple matrices.

Contribution

It extends the understanding of outlier eigenvalues in products of random matrices under various perturbations, generalizing previous single-matrix results.

Findings

Outlier eigenvalues' locations are characterized asymptotically.

The results apply to additive and multiplicative perturbations.

The limiting spectral distribution remains the m-th power of the circular law.

Abstract

For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations.