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

Abstract : 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 :
Type de document :
Pré-publication, Document de travail
2011
Domaine :

https://hal.archives-ouvertes.fr/hal-00536164
Contributeur : Catherine Donati-Martin <>
Soumis le : lundi 19 septembre 2011 - 10:15:00
Dernière modification le : mardi 11 octobre 2016 - 14:05:04

### Fichier

EJP1309.pdf
Fichiers produits par l'(les) auteur(s)

### Identifiants

• HAL Id : hal-00536164, version 2

### Citation

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>

Consultations de
la notice

## 217

Téléchargements du document