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.

38 pages, 3 figures

References in corpus (1)

• Anomalies and non-log-normal tails in one-dimensional localization with power-law disorder