Calogero-Sutherland techniques in the physics of disorderd wires

TL;DR

This paper explores the relationship between random matrix theory and Calogero-Sutherland models to better understand the physics of disordered wires, providing new insights into their mathematical description.

Contribution

It establishes a connection between random matrix ensembles and Calogero-Sutherland models, enabling the derivation of properties of disordered wires using symmetric space functions.

Findings

Different random matrix ensembles correspond to specific CS models.

Properties of the DMPK equation are derived from symmetric space analysis.

Results align with known behaviors of disordered wires.

Abstract

We discuss the connection between the random matrix approach to disordered wires and the Calogero-Sutherland models. We show that different choices of random matrix ensembles correspond to different classes of CS models. In particular, the standard transfer matrix ensembles correspond to CS model with sinh-type interaction, constructed according to the $C_n$ root lattice pattern. By exploiting this relation, and by using some known properties of the zonal spherical functions on symmetric spaces we can obtain several properties of the Dorokhov-Mello-Pereyra-Kumar equation, which describes the evolution of an ensemble of quasi one-dimensional disordered wires of increasing length $L$. These results are in complete agreement with all known properties of disordered wires. (To appear in the Proceedings of the Conference: Recent Developments in Statistical Mechanics and Quantum Field Theory (Trieste, 1995))