Impurity models and products of random matrices

TL;DR

This paper introduces the theory of one-dimensional disordered systems and products of 2x2 random matrices, highlighting impurity models with localized interactions and illustrating Furstenberg's theorem for matrix products.

Contribution

It connects impurity models with random matrix products and provides elementary mathematical insights into their spectral properties and exponential growth conditions.

Findings

Spectral theory of impurity models explained

Furstenberg's theorem illustrated with examples

Conditions for exponential growth of matrix products established

Abstract

This is an introduction to the theory of one-dimensional disordered systems and products of random matrices, confined to the 2 by 2 case. The notion of impurity model--- that is, a system in which the interactions are highly localised--- links the two themes and enables their study by elementary mathematical tools. After discussing the spectral theory of some impurity models, we state and illustrate Furstenberg's theorem, which gives sufficient conditions for the exponential growth of a product of independent, identically-distributed matrices.