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We prove that the local eigenvalue statistics for $d=1$ random band matriceswith fixed bandwidth and, for example, Gaussian entries, is given by a Poissonpoint process and we identify the intensity of the process. The proof relies onan extension of the localization bounds of Schenker \cite{schenker} and theWegner and Minami estimates. These two estimates are proved using averagingover the diagonal disorder. The new component is a proof of the uniformconvergence and the smoothness of the density of states function. The limitfunction, known to be the semicircle law with a band-width dependent error\cite{bmp,dps,dl,mpk}, is identified as the intensity of the limiting Poissonpoint process. The proof of these results for the density of states relies on anew result that simplifies and extends some of the ideas used by Dolai,Krishna, and Mallick \cite{dkm}. These authors proved regularity properties ofthe density of states for random Schr\"odinger operators (lattice andcontinuum) in the localization regime. The proof presented here applies to therandom Schr\"odinger operators on a class of infinite graphs treated by in\cite{dkm} and extends the results of \cite{dkm} to probability measures withunbounded support. The method also applies to fixed bandwidth RBM for $d=2,3$provided certain localization bounds are known.

The density of states and local eigenvalue statistics for random band matrices of fixed width