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On the Local Eigenvalue Statistics for Random Band Matrices in the Localization Regime


We study the local eigenvalue statistics \(\xi _{\omega ,E}^N\) associated with the eigenvalues of one-dimensional, \((2N+1) \times (2N+1)\) random band matrices with independent, identically distributed, real random variables and band width growing as \(N^\alpha \), for \(0< \alpha < \frac{1}{2}\). We consider the limit points associated with the random variables \(\xi _{\omega ,E}^N [I]\), for \(I \subset \mathbb {R}\), and \(E \in (-2,2)\). For random band matrices with Gaussian distributed random variables and for \(0 \le \alpha < \frac{1}{7}\), we prove that this family of random variables has nontrivial limit points for almost every \(E \in (-2,2)\), 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 \(\xi _{\omega ,E}^N [I]\) 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 \(0< \alpha < \frac{1}{2}\), we prove that any nontrivial limit points of the random variables \(\xi _{\omega ,E}^N [I]\) are distributed according to Poisson distributions.

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Correspondence to Peter D. Hislop.

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Communicated by Eric A. Carlen.

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

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Hislop, P.D., Krishna, M. On the Local Eigenvalue Statistics for Random Band Matrices in the Localization Regime. J Stat Phys 187, 26 (2022).

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  • Random band matrices
  • Eigenvalue statistics
  • Localization