Localization and delocalization for heavy tailed band matrices
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1385-1403.

On considère des matrices aléatoires à structure bande dont la bande a pour largeur N μ et dont les coefficients sont indépendants à queue de distribution en x -α . On s’intéresse aux plus grandes valeurs propres et aux vecteurs propres associés et prouve la transition de phase suivante. D’une part, quand α<2(1+μ -1 ), les plus grandes valeurs propres ont pour ordre N (1+μ)/α , 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 α<2, et par Auffinger et al. dans (Ann. Inst. H. Poincarè Probab. Statist. 45 (2005) 589-610) quand α<4). D’autre part, quand α>2(1+μ -1 ), les plus grandes valeurs propres ont pour ordre N μ/2 et la plupart des vecteurs propres de la matrice sont délocalisés, i.e. approximativement uniformément distribués sur leurs N coordonnées.

We consider some random band matrices with band-width N μ whose entries are independent random variables with distribution tail in x -α . We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when α<2(1+μ -1 ), the largest eigenvalues have order N (1+μ)/α , 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 α<2 and by Auffinger et al. in (Ann. Inst. H. Poincarè Probab. Statist. 45 (2005) 589-610) when α<4). On the other hand, when α>2(1+μ -1 ), the largest eigenvalues have order N μ/2 and most eigenvectors of the matrix are delocalized, i.e. approximately uniformly distributed on their N coordinates.

DOI : 10.1214/13-AIHP562
Classification : 15A52, 60F05
Mots clés : random matrices, band matrices, heavy tailed random variables
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Benaych-Georges, Florent; Péché, Sandrine. Localization and delocalization for heavy tailed band matrices. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1385-1403. doi : 10.1214/13-AIHP562. http://www.numdam.org/articles/10.1214/13-AIHP562/

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