Outliers in the spectrum of large deformed unitarily invariant models

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.

44 pages, LaTeX, one figure. Several changes form the previous version, including a correction in Lemma 3.1 and the addition of several remarks and of a numerical example, kindly provided by Charles Bordenave. Same title as arXiv:1207.5443, which it greatly expands