Random matrix theory and symmetric spaces

TL;DR

This review explores the deep connection between random matrix theories and symmetric spaces, revealing how their classifications and properties are intrinsically linked, leading to new insights in disordered systems and quantum transport.

Contribution

It establishes a comprehensive correspondence between random matrix ensembles and symmetric spaces, introducing a new classification based on Cartan's theory and applying it to disordered systems and quantum transport.

Findings

Classification of random matrix ensembles via symmetric spaces

Mapping of Calogero--Sutherland models to symmetric spaces

Applications to disordered systems and quantum transport

Abstract

In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that the integration manifolds of random matrix theories, the eigenvalue distribution, and the Dyson and boundary indices characterizing the ensembles are in strict correspondence with symmetric spaces and the intrinsic characteristics of their restricted root lattices. Several important results can be obtained from this identification. In particular the Cartan classification of triplets of symmetric spaces with positive, zero and negative curvature gives rise to a new classification of random matrix ensembles. The review is organized into two main parts. In Part I the theory of symmetric spaces is reviewed with particular emphasis on the ideas relevant for appreciating the correspondence with random matrix theories. In Part II we discuss various applications of symmetric spaces to random matrix theories and in particular the new classification of disordered systems derived from the classification of symmetric spaces. We also review how the mapping from integrable Calogero--Sutherland models to symmetric spaces can be used in the theory of random matrices, with particular consequences for quantum transport problems. We conclude indicating some interesting new directions of research based on these identifications.