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 eigenvalues of one-dimensional, $$(2N+1) \times (2N+1)$$ random band matrices independent, identically distributed, real variables and width growing as $$N^\alpha $$ , for $$0< \alpha < \frac{1}{2}$$ . consider limit points ,E}^N [I]$$ $$I \subset \mathbb {R}$$ $$E \in (-2,2)$$ For Gaussian distributed $$0 \le \frac{1}{7}$$ we prove that this family has nontrivial almost every these are Poisson positive intensities. The proof is based on an analysis characteristic functions quantities related to intensities, N tends towards infinity, employs known localization bounds (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019, Schenker Commun Math Phys 290:1065–1097, 2009), strong Wegner Minami estimates 2019). Our more general result applies having absolutely continuous distributions bounded densities. Under hypothesis hold any according distributions.