It is known that in various random matrix models, large perturbations create outlier eigenvalues which lie, asymptotically, in the complement of the support of the limiting spectral density. This thesis studies fluctuations of these outlier eigenvalues of iid matrices @math under bounded rank and bounded operator norm perturbations @math , namely the fluctuations @math . The perturbations @math that we consider belong to a large class, where we allow for arbitrary Jordan types and almost minimal assumptions on the left and right eigenvectors. We obtain the joint convergence of the normalized asymptotic fluctuations of the outlier eigenvalues in this setting with a unified approach.
We consider a square random matrix of size N of the form A + Y where A is deterministic and Y has iid 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 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 . 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.