Localization and delocalization for heavy tailed band matrices

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 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.
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  • HAL Id : hal-00746679, version 5
  • ARXIV : 1210.7677

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Florent Benaych-Georges, Sandrine Péché. Localization and delocalization for heavy tailed band matrices. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institute Henri Poincaré, 2014, 50 (4), pp.doi:10.1214/13-AIHP562. ⟨hal-00746679v5⟩

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