ar X iv : m at h - ph / 0 40 40 57 v 1 2 5 A pr 2 00 4 THE SUPERSYMMETRY METHOD OF RANDOM MATRIX THEORY

A prominent theme of modern condensed matter physics is electronic transport – in particular the electrical conductivity – of disordered metallic systems at very low temperatures. From the Landau theory of weakly interacting Fermi liquids one expects the essential aspects of the situation to be captured by the single-electron approximation. Mathematical models that have been proposed and studied in this context include random Schrödinger operators and band random matrices. If the physical system has infinite size, two distinct possibilities exist: the quantum single-electron motion may either be bounded or unbounded. In the former case the disordered electron system is an insulator, in the latter case a metal with finite conductivity (if the electron motion is not critical but diffusive). Metallic behavior is expected for weakly disordered systems in three dimensions; insulating behavior sets in when the disorder is increased or the space dimension reduced. The main theoretical tool used in the physics literature on the subject is the supersymmetry method pioneered by Wegner and Efetov (1979-1983). Over the past twenty years, physicists have applied the method in many instances, and a rather complete picture of weakly disordered metals has emerged. Several excellent reviews of these developments are available in print. From the perspective of mathematics, however, the method has not always been described correctly, and what is sorely lacking at present is an exposition of how to implement the method rigorously. [Unfortunately, the correct exposition by Schäfer and Wegner (1980) was largely ignored or forgotten by later authors.] In this encyclopedia article an attempt will be made to help remedy the situation, by giving a careful review of the Wegner-Efetov supersymmetry method for the case of Hermitian band random matrices. 2. Gaussian Ensembles