# Localization and delocalization for heavy tailed band matrices

Abstract : We consider some random band matrices with band-width $N^\mu$ whose entries are independent random variables with distribution tail in $x^{-\alpha}$. We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when $\alpha<2(1+\mu^{-1})$, the largest eigenvalues have order $N^{(1+\mu)/\alpha}$, are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked by Soshnikov for full matrices with heavy tailed entries,i.e. when $\alpha<2$, and by Auffinger, Ben Arous and Péché when $\alpha<4$). On the other hand, when $\alpha>2(1+\mu^{-1})$, the largest eigenvalues have order $N^{\mu/2}$ and most eigenvectors of the matrix are delocalized, i.e. approximately uniformly distributed on their $N$ coordinates.
Keywords :
Type de document :
Article dans une revue
Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institute Henri Poincaré, 2014, 50 (4), pp.doi:10.1214/13-AIHP562
Domaine :

https://hal.archives-ouvertes.fr/hal-00746679
Contributeur : Florent Benaych-Georges <>
Soumis le : mercredi 24 juin 2015 - 13:23:44
Dernière modification le : jeudi 27 avril 2017 - 09:46:06
Document(s) archivé(s) le : mardi 15 septembre 2015 - 22:51:18

### Fichiers

Benaych-Peche-Rev10.pdf
Fichiers produits par l'(les) auteur(s)

### Identifiants

• HAL Id : hal-00746679, version 5
• ARXIV : 1210.7677

### Citation

Florent Benaych-Georges, Sandrine Péché. Localization and delocalization for heavy tailed band matrices. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institute Henri Poincaré, 2014, 50 (4), pp.doi:10.1214/13-AIHP562. <hal-00746679v5>

Consultations de
la notice

## 146

Téléchargements du document