We study the local eigenvalue statistics ξω,EN\documentclass[12pt]{minimal}
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\begin{document}$$\xi _{\omega ,E}^N$$\end{document} associated with the eigenvalues of one-dimensional, (2N+1)×(2N+1)\documentclass[12pt]{minimal}
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\begin{document}$$(2N+1) \times (2N+1)$$\end{document} random band matrices with independent, identically distributed, real random variables and band width growing as Nα\documentclass[12pt]{minimal}
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\begin{document}$$N^\alpha $$\end{document}, for 0<α<12\documentclass[12pt]{minimal}
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\begin{document}$$0< \alpha < \frac{1}{2}$$\end{document}. We consider the limit points associated with the random variables ξω,EN[I]\documentclass[12pt]{minimal}
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\begin{document}$$\xi _{\omega ,E}^N [I]$$\end{document}, for I⊂R\documentclass[12pt]{minimal}
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\begin{document}$$I \subset \mathbb {R}$$\end{document}, and E∈(-2,2)\documentclass[12pt]{minimal}
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\begin{document}$$E \in (-2,2)$$\end{document}. For random band matrices with Gaussian distributed random variables and for 0≤α<17\documentclass[12pt]{minimal}
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\begin{document}$$0 \le \alpha < \frac{1}{7}$$\end{document}, we prove that this family of random variables has nontrivial limit points for almost every E∈(-2,2)\documentclass[12pt]{minimal}
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\begin{document}$$E \in (-2,2)$$\end{document}, 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]\documentclass[12pt]{minimal}
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\begin{document}$$\xi _{\omega ,E}^N [I]$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$0< \alpha < \frac{1}{2}$$\end{document}, we prove that any nontrivial limit points of the random variables ξω,EN[I]\documentclass[12pt]{minimal}
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\begin{document}$$\xi _{\omega ,E}^N [I]$$\end{document} are distributed according to Poisson distributions.
