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

We study the local eigenvalue statistics ξω,EN associated with the eigenvalues of one-dimensional, (2 N+ 1) × (2 N+ 1) random band matrices with independent, identically distributed, real random variables and band width growing as N, for 0<α<12. We consider the limit points associated with the random variables ξω,EN[I], for I⊂ R, and E∈ (- 2 , 2). For random band matrices with Gaussian distributed random variables and for 0≤α<17, we prove that this family of random variables has nontrivial limit points for almost every E∈ (- 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 ξω,EN[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<α<12, we prove that any nontrivial limit points of the random variables ξω,EN[I] are distributed according to Poisson distributions.