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

We investigate the asymptotic behavior of the eigenvalues of spikedperturbations of Wigner matrices when the dimension goes to infinity. Theentries of the Hermitian Wigner matrix have a distribution which is symmetricand satisfies a Poincar\'e inequality. The perturbation matrix is adeterministic Hermitian matrix whose spectral measure converges to someprobability measure with compact support. We assume that this perturbationmatrix has a fixed number of fixed eigenvalues (spikes) outside the support ofits limiting spectral measure whereas the distance between the othereigenvalues and this support uniformly goes to zero as the dimension goes toinfinity. We establish that only a particular subset of the spikes willgenerate some eigenvalues of the deformed model which will converge to somelimiting points outside the support of the limiting spectral measure. Thisphenomenon can be fully described in terms of free probability involving thesubordination function related to the additive free convolution of the limitingspectral measure of the perturbation matrix by a semi-circular distribution.Note that up to now only finite rank perturbations had been considered (even inthe deformed GUE case).