Outlier eigenvalue fluctuations of perturbed iid matrices

Abstract: 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 paper is concerned with fluctuations of these outlier eigenvalues of iid matrices $X_n$ under bounded rank and bounded operator norm perturbations $A_n$, namely with $\lambda(\frac{X_n}{\sqrt{n}}+A_n)-\lambda(A_n)$. The perturbations we consider are allowed to be of arbitrary Jordan type and have (left and right) eigenvectors satisfying a mild condition. We obtain the joint convergence of the (normalized) asymptotic fluctuations of the outlier eigenvalues in this setting with a unified approach.