THE ANDERSON-MODEL AS A MATRIX-MODEL

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 matric es which appear are approximately of the free type well known to physi cists and mathematicians, and their asymptotic eigenvalue distribution is therefore simply Wigner's law. However in d = 3 the natural random matrices that appear have non-trivial constraints of a geometrical or igin. It would be interesting to develop a general theory of these con strained random matrices, which presumably play an interesting role fo r many non-integrable problems related to diffusion. We present a firs t step in this direction, namely a rigorous bound on the tail of the e igenvalue distribution of such objects based on large deviation and gr aphical estimates. This bound allows to prove regularity and decay pro perties of the averaged Green's functions and the density of states fo r a three dimensional model with a thin conducting band and an energy close to the border of the band, for sufficiently small coupling const ant.