The Anderson Model as a Matrix Model
Abstract
In this paper we describe a strategy to study the Anderson model of an electron in a random potential at weak coupling by a renormalization group analysis. There is an interesting technical analogy between this problem and the theory of random matrices. In d = 2 the random matrices which appear are approximately of the free type well known to physicists and mathematicians, and their asymptotic eigenvalue distribution is therefore simply Wigner's law. However in d = 3 the natural random matrices that appear have nontrivial constraints of a geometrical origin. It would be interesting to develop a general theory of these constrained random matrices, which presumably play an interesting role for many nonintegrable problems related to diffusion. We present a first step in this direction, namely a rigorous bound on the tail of the eigenvalue distribution of such objects based on large deviation and graphical estimates. This bound allows to prove regularity and decay properties of the averaged Green's functions and the density of states for a three dimensional model with thin conducting band and an energy close to the border of the band, for sufficiently small coupling constant.
 Publication:

Nuclear Physics B Proceedings Supplements
 Pub Date:
 September 1997
 DOI:
 10.1016/S09205632(97)004209
 arXiv:
 arXiv:condmat/9611236
 Bibcode:
 1997NuPhS..58..149M
 Keywords:

 Condensed Matter
 EPrint:
 23 pages, LateX, ps file available at http://cpth.polytechnique.fr/cpth/rivass/articles.html