Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices

TL;DR

This paper explores the role of Lyapunov exponents in Anderson localisation and products of random matrices, reviewing known solvable models, introducing a new solvable case, and discussing limitations of Lyapunov exponents in localisation studies.

Contribution

It provides a comprehensive review of the connections between Lyapunov exponents, Anderson localisation, and random matrix products, including a new solvable model and analysis of limitations.

Findings

Review of solvable models in disordered quantum mechanics

Introduction of a new solvable case involving random scatterers

Discussion of limitations of Lyapunov exponents in localisation analysis

Abstract

The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the localisation length. The Lyapunov exponent also features prominently in the theory of products of random matrices pioneered by Furstenberg. After a brief historical survey, we describe some recent work that exploits the close connections between these topics. We review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case. Finally, we point out some limitations of the Lyapunov exponent as a means of studying localisation properties.