Complex Outliers of Hermitian Random Matrices

In this paper, we study the asymptotic behavior of the outliers of the sum a Hermitian random matrix and a finite rank matrix which is not necessarily Hermitian. We observe several possible convergence rates and outliers locating around their limits at the vertices of regular polygons as in Benaych-Georges and Rochet (Probab Theory Relat Fields, 2015), as well as possible correlations between outliers at macroscopic distance as in Knowles and Yin (Ann Probab 42(5):1980–2031, 2014) and Benaych-Georges and Rochet (2015). We also observe that a single spike can generate several outliers in the spectrum of the deformed model, as already noticed in Benaych-Georges and Nadakuditi (Adv Math 227(1):494–521, 2011) and Belinschi et al. (Outliers in the spectrum of large deformed unitarily invariant models 2012, arXiv:1207.5443v1). In the particular case where the perturbation matrix is Hermitian, our results complete the work of Benaych-Georges et al. (Electron J Probab 16(60):1621–1662, 2011), as we consider fluctuations of outliers lying in “holes” of the limit support, which happen to exhibit surprising correlations.