Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices

Understand

We investigate the asymptotic behavior of the eigenvalues of spiked perturbations of Wigner matrices when the dimension goes to infinity.

• The entries of the Hermitian Wigner matrix have a distribution which is symmetric and satisfies a Poincar\'e inequality.
• The perturbation matrix is a deterministic Hermitian matrix whose spectral measure converges to some probability measure with compact support.
• We 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.