  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 $W^{-2+\delta },$ where $\delta\to 0$ when $W\to \infty.$ The proof
uses the supersymmetric approach and extends results by Disertori, Pinson and
Spencer from three to two dimensions.
