Localization and delocalization for heavy tailed band matrices

Benaych-Georges, Florent; Péché, Sandrine

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 $N^{\mu }$ 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.

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.

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