Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Localization and delocalization for heavy tailed band matrices

Florent Benaych-Georges and Sandrine Péché

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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 for full matrices by Soshnikov in (Electron. Comm. Probab. 9 (2004) 82–91, In Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensembles (2006) 351–364) when $\alpha<2$ and by Auffinger et al. in (Ann. Inst. H. Poincarè Probab. Statist. 45 (2005) 589–610) 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.

Résumé

On considère des matrices aléatoires à structure bande dont la bande a pour largeur $N^{\mu}$ et dont les coefficients sont indépendants à queue de distribution en $x^{-\alpha}$. On s’intéresse aux plus grandes valeurs propres et aux vecteurs propres associés et prouve la transition de phase suivante. D’une part, quand $\alpha<2(1+\mu^{-1})$, les plus grandes valeurs propres ont pour ordre $N^{(1+\mu)/\alpha}$, sont asymptotiquement distribuées selon un processus de Poisson et les vecteurs propres associés sont essentiellement portés par deux coordonnées (ce phénomène a déjà été remarqué pour des matrices pleines par Soshnikov dans (Electron. Comm. Probab. 9 (2004) 82–91, In Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensembles (2006) 351–364) quand $\alpha<2$, et par Auffinger et al. dans (Ann. Inst. H. Poincarè Probab. Statist. 45 (2005) 589–610) quand $\alpha<4$). D’autre part, quand $\alpha>2(1+\mu^{-1})$, les plus grandes valeurs propres ont pour ordre $N^{\mu/2}$ et la plupart des vecteurs propres de la matrice sont délocalisés, i.e. approximativement uniformément distribués sur leurs $N$ coordonnées.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 4 (2014), 1385-1403.

Dates
First available in Project Euclid: 17 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1413555505

Digital Object Identifier
doi:10.1214/13-AIHP562

Mathematical Reviews number (MathSciNet)
MR3269999

Zentralblatt MATH identifier
1307.15054

Subjects
Primary: 15A52 60F05: Central limit and other weak theorems

Keywords
Random matrices Band matrices Heavy tailed random variables

Citation

Benaych-Georges, Florent; Péché, Sandrine. Localization and delocalization for heavy tailed band matrices. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 4, 1385--1403. doi:10.1214/13-AIHP562. https://projecteuclid.org/euclid.aihp/1413555505


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