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

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.

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.

DOI : https://doi.org/10.1214/13-AIHP562
Classification:  15A52,  60F05
Keywords: random matrices, band matrices, heavy tailed random variables
@article{AIHPB_2014__50_4_1385_0,
     author = {Benaych-Georges, Florent and P\'ech\'e, Sandrine},
     title = {Localization and delocalization for heavy tailed band matrices},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {4},
     year = {2014},
     pages = {1385-1403},
     doi = {10.1214/13-AIHP562},
     zbl = {06377559},
     mrnumber = {3269999},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_4_1385_0}
}
Benaych-Georges, Florent; Péché, Sandrine. Localization and delocalization for heavy tailed band matrices. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, pp. 1385-1403. doi : 10.1214/13-AIHP562. http://www.numdam.org/item/AIHPB_2014__50_4_1385_0/

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