Localization and delocalization for heavy tailed band matrices

Florent Benaych-Georges (1) , Sandrine Péché (2)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Mots clés: Random matrices; band matrices; heavy tailed random variables

Domaines: Probabilités [math.PR]

Florent Benaych-Georges, Sandrine Péché. Localization and delocalization for heavy tailed band matrices. 2012. ⟨hal-00746679v2⟩