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Localization and delocalization for heavy tailed band matrices. (English. French summary) Zbl 1307.15054

The authors consider some random band matrices with band-width \(N^\mu\) whose entries are independent random variables with distribution tail in \(x^{-\alpha}\). They 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 A. Soshnikov [Electron. Commun. Probab. 9, 82–91 (2004; Zbl 1060.60013); Lect. Notes Phys. 690, 351–364 (2006; Zbl 1169.15302)], when \(\alpha <2\) and by A. Auffinger et al. [Ann. Inst. Henri Poincaré, Probab. Stat. 45, No. 3, 589–610 (2009; Zbl 1177.15037)], 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.

MSC:

15B52 Random matrices (algebraic aspects)
60F05 Central limit and other weak theorems
60B20 Random matrices (probabilistic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
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