Density of States for Random Band Matrices in Two Dimensions

We consider a two-dimensional random band matrix ensemble, in the limit of infinite volume and fixed but large band width W . For this model, we rigorously prove smoothness of the averaged density of states. We also prove that the resulting expression coincides with Wigner’s semicircle law with a precision W - 2 + δ , where δ → 0 when W → ∞ . The proof uses the supersymmetric approach and extends results by Disertori et al. (Commun Math Phys 232(1):83–124, 2002 ) from three to two dimensions.