Free Convolution with a Semicircular Distribution and Eigenvalues of Spiked Deformations of Wigner Matrices

Abstract

We investigate the asymptotic spectrum of spiked perturbations of Wigner matrices. The entries of the Wigner matrix have a distribution which is symmetric and satisfies a Poincaré inequality. The spectral measure of the deterministic Hermitian perturbation matrix converges to some probability measure with compact support. We also assume that this perturbation matrix has a fixed number of fixed eigenvalues (spikes) outside the support of its limiting spectral measure whereas the distance between the other eigenvalues and this support uniformly goes to zero as the dimension goes to infinity. We establish that only a particular subset of the spikes will generate some eigenvalues of the deformed model, which will converge to some limiting points outside the support of the limiting spectral measure. This phenomenon can be fully described in terms of free probability involving the subordination function related to the free additive convolution by a semicircular distribution. Note that only finite rank perturbations had been considered up to now (even in the deformed GUE case).