Representation theory and products of random matrices in $\text{SL}(2,{\mathbb R})$

TL;DR

This paper investigates the statistical properties of products of random matrices in SL(2,R), deriving explicit formulas for growth rates and variance using group representation theory, with applications to disordered systems.

Contribution

It applies group representation theory to analyze the generalized Lyapunov exponent for products of random matrices in SL(2,R), providing explicit formulas for growth and variance.

Findings

Derived explicit formulas for the generalized Lyapunov exponent.

Connected transfer operators to second-order difference/differential operators.

Applied representation theory to models of disordered systems.

Abstract

The statistical behaviour of a product of independent, identically distributed random matrices in $\text{SL}(2,{\mathbb R})$ is encoded in the generalised Lyapunov exponent $\Lambda$; this is a function whose value at the complex number $2 \ell$ is the logarithm of the largest eigenvalue of the transfer operator obtained when one averages, over $g \in \text{SL}(2,{\mathbb R})$, a certain representation $T_\ell (g)$ associated with the product. We study some products that arise from models of one-dimensional disordered systems. These models have the property that the inverse of the transfer operator takes the form of a second-order difference or differential operator. We show how the ideas expounded by N. Ja. Vilenkin in his book [Special Functions and the Theory of Group Representations, American Mathematical Society, 1968.] can be used to study the generalised Lyapunov exponent. In particular, we derive explicit formulae for the almost-sure growth and for the variance of the corresponding products.