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

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.