Free convolution with a semicircular distribution and eigenvalues of spiked deformations of Wigner matrices

Mireille Capitaine (1) , Catherine Donati-Martin (2) , Delphine Féral (3) , Maxime Fevrier (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
3 IMB - Institut de Mathématiques de Bordeaux

We investigate the asymptotic behavior of the eigenvalues of spiked perturbations of Wigner matrices dened by $M_N = W_N + A_N$, where $W_N$ is a rescaled Wigner Hermitian matrix of size N whose entries have a distri- bution $\mu$ which is symmetric and satises a Poincare inequality and $A_N$ is a deterministic Hermitian matrix of size N whose spectral measure converges to some probability measure with compact support. We assume that $A_N$ has a fixed number of fixed eigenvalues (spikes) outside the support of $\mu$ whereas the distance between the other eigenvalues and the support of $\mu$ uniformly goes to zero as N goes to innity. We establish that only a particular subset of the spikes will generate some eigenvalues of $M_N$ 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 of $\mu$ by a semicircular distribution. Note that only finite rank perturbations had been considered up to now (even in the deformed GUE case).

Keywords: Random matrices; Free convolution

Domains: Probability [math.PR]

Mireille Capitaine, Catherine Donati-Martin, Delphine Féral, Maxime Fevrier. Free convolution with a semicircular distribution and eigenvalues of spiked deformations of Wigner matrices. 2011. ⟨hal-00536164v2⟩