Beta random matrix ensembles

Abstract

In applications of random matrix theory to physics, time reversal symmetry implies one of three exponents β = 1; 2 or 4 for the repulsion sβ between eigenvalues at spacing s, s → 0. However, in the corresponding eigenvalue probability density functions (p.d.f.'s), β is a natural positive real variable. The general β p.d.f.'s have alternative physical interpretations in terms of certain classical log-potential Coulomb systems and quantum many body systems. Each of these topics is reviewed, along with the problem of computing correlation functions for general β. There are also random matrix constructions which give rise to these general β p.d.f.'s. An inductive approach to the latter topic usi..