## Abstract

We study the local eigenvalue statistics ξω,EN associated with the eigenvalues of one-dimensional, (2 N+ 1) × (2 N+ 1) random band matrices with independent, identically distributed, real random variables and band width growing as N^{α}, for 0<α<12. We consider the limit points associated with the random variables ξω,EN[I], for I⊂ R, and E∈ (- 2 , 2). For random band matrices with Gaussian distributed random variables and for 0≤α<17, we prove that this family of random variables has nontrivial limit points for almost every E∈ (- 2 , 2) , and that these limit points are Poisson distributed with positive intensities. The proof is based on an analysis of the characteristic functions of the random variables ξω,EN[I] and associated quantities related to the intensities, as N tends towards infinity, and employs known localization bounds of (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019, Schenker in Commun Math Phys 290:1065–1097, 2009), and the strong Wegner and Minami estimates (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019). Our more general result applies to random band matrices with random variables having absolutely continuous distributions with bounded densities. Under the hypothesis that the localization bounds hold for 0<α<12, we prove that any nontrivial limit points of the random variables ξω,EN[I] are distributed according to Poisson distributions.

Original language | English |
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Article number | 26 |

Journal | Journal of Statistical Physics |

Volume | 187 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2022 |

### Bibliographical note

Funding Information:PDH is supported in part by a Simons Foundation Collaboration Grant for Mathematicians No. 843327.

Publisher Copyright:

© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

## Keywords

- Eigenvalue statistics
- Localization
- Random band matrices

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics