Dynamics of a rank-one multiplicative perturbation of a unitary matrix

TL;DR

This paper studies how the spectrum of a unitary matrix changes under a rank-one multiplicative perturbation, revealing deterministic spectral bounds and the separation of outliers from typical eigenvalues over time.

Contribution

It introduces a dynamical framework for analyzing spectral evolution under rank-one multiplicative perturbations of unitary matrices, with high-probability spectral bounds.

Findings

Deterministic domains bound the spectrum with high probability.

Outliers are separated from typical eigenvalues at all sub-critical times.

Results apply broadly to various unitary random matrix models.

Abstract

We provide a dynamical study of a model of multiplicative perturbation of a unitary matrix introduced by Fyodorov. In particular, we identify a flow of deterministic domains that bound the spectrum with high probability, separating the outlier from the typical eigenvalues at all sub-critical timescales. These results are obtained under generic assumptions on $U$ that hold for a variety of unitary random matrix models.