Outlier Eigenvalues for Deformed I.I.D. Random Matrices

We consider a square random matrix of size N of the form A + Y where A is deterministic and Y has i.i.d. entries with variance 1/N. Under mild assumptions, as N grows the empirical distribution of the eigenvalues of A + Y converges weakly to a limit probability measure β on the complex plane. This work is devoted to the study of the outlier eigenvalues, i.e., eigenvalues in the complement of the support of β. Even in the simplest cases, a variety of interesting phenomena can occur. As in earlier works, we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of Y or the Jordan decomposition of A. We also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a Gaussian analytic function. © 2016 Wiley Periodicals, Inc.