Eigenvalue Statistics and Localization for Random Band Matrices with Fixed Width and Wegner Orbital Model

We discuss two models from the study of disordered quantum systems. The first is the Random Band Matrix with a fixed band width and Gaussian or more general disorder. The second is the Wegner $n$-orbital model. We establish that the point process constructed from the eigenvalues of finite size matrices converge to a Poisson Point Process in the limit as the matrix size goes to infinity. The proof is based on the method of Minami for the Anderson tight-binding model. As a first step, we expand upon the localization results by Schenker and Peled-Schenker-Shamis-Sodin to account for complex energies. We use the fractional moment method of Aizenman-Molchanov to derive these bounds. In addition, we establish convergence and smoothness of the density of states functions by modifying estimates of Dolai-Krishna-Mallick to allow for unbounded random variables. From there we follow the Daley and Vere-Jones criteria for establishing the convergence of the eigenvalue point process to the Poisson Point process. The analysis is first presented for the band matrix with adjustments for the orbital model following after. Other properties of these models such as ergodicity and Lyapunov exponents are discussed