Symmetry classes in random matrix theory

TL;DR

This paper reviews Dyson's classification of matrix ensembles, explores its extension to disordered fermion problems, and explains the connection between symmetry classes and symmetric spaces from a modern perspective.

Contribution

It provides a modern review of Dyson's symmetry classes and extends the classification to disordered fermion systems, highlighting their geometric connections.

Findings

Symmetry classes correspond to large families of symmetric spaces.

The extension of Dyson's classification to disordered fermions is motivated and described.

The paper clarifies the geometric interpretation of symmetry classes.

Abstract

Dyson's (1962) classification of matrix ensembles is reviewed from a modern perspective, and its recent extension to disordered fermion problems is motivated and described. It is explained in particular why symmetry classes are associated with large families of symmetric spaces.