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 Gaussian distributed random variables with $0 \leq \alpha < \frac{1}{7}$, 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 \cite{schenker, peled, et. al.}, and the strong Wegner and Minami estimates \cite{peled, et. al.}. 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.