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

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.

arXiv comment21 pages, 2 figures. Corrected some typos in previous version

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