The Anderson Model as a Matrix Model

TL;DR

This paper applies a renormalization group approach to the Anderson model, revealing connections to random matrix theory and providing bounds on eigenvalue distributions to analyze electron behavior in disordered systems.

Contribution

It introduces a novel renormalization group analysis for the Anderson model and develops bounds for constrained random matrices relevant to three-dimensional diffusion problems.

Findings

Eigenvalue distribution bounds for constrained random matrices

Regularity and decay properties of Green's functions

Density of states near band edges in 3D models

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 non-trivial 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 non-integrable 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 a thin conducting band and an energy close to the border of the band, for sufficiently small coupling constant.