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

Abstract

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.