ar X iv : m at h - ph / 0 40 40 58 v 1 2 5 A pr 2 00 4 SYMMETRY CLASSES IN RANDOM MATRIX THEORY

A classification of random-matrix ensembles by symmetries was first established by Dyson, in an influential 1962 paper with the title “the threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics”. Dyson’s threefold way has since become fundamental to various areas of theoretical physics, including the statistical theory of complex manybody systems, mesoscopic physics, disordered electron systems, and the field of quantum chaos. Over the last decade, a number of random-matrix ensembles beyond Dyson’s classification have come to the fore in physics and mathematics. On the physics side these emerged from work on the low-energy Dirac spectrum of quantum chromodynamics, and from the mesoscopic physics of low-energy quasi-particles in disordered superconductors. In the mathematical research area of number theory, the study of statistical correlations in the values of Riemann zeta and similar functions has prompted some of the same generalizations. In this article, Dyson’s fundamental result will be reviewed from a modern perspective, and the recent extension of Dyson’s threefold way will be motivated and described. In particular, it will be explained why symmetry classes are associated with large families of symmetric spaces.