Renormalization group methods and applications: First results for the weakly coupled Anderson model

We first recall the renormalization group approach to the study of weakly interacting Fermions where the singularity of the free propagator lies on a sphere i.e. the Fermi surface [1][2]. The main results are in two dimensions because, in this case, the vertices of the interaction are approximatively factorized [3]. It has allowed to prove the existence of a Fermi liquid in two dimensions [4][5]. This suggests a new point of view on the two dimensional Anderson model of an electron in a random potential at small coupling lambda (where there are almost no rigorous results up to now). We prove that there exists kappa: > 0 such that for epsilon = lambda(2+kappa), the density of states [GRAPHICS] is analytic in E in a band of width lambda(2) which is the expected optimal width [8]. A non perturbative stability argument should complete this work by taking the epsilon --> 0 limit.