The Lyapunov exponent of products of random $2\times2$ matrices close to the identity

TL;DR

This paper analyzes the Lyapunov exponent for products of random 2x2 matrices near the identity, deriving a scaling form expressed through special functions, unifying various known results in disordered systems.

Contribution

It introduces a continuum regime analysis using Iwasawa decomposition, deriving a universal scaling form for the Lyapunov exponent involving hypergeometric and other special functions.

Findings

Scaling form of Lyapunov exponent in the continuum regime

Expression of the scaling function via hypergeometric function

Recovery of known results from disordered systems

Abstract

We study products of arbitrary random real $2 \times 2$ matrices that are close to the identity matrix. Using the Iwasawa decomposition of $\text{SL}(2,{\mathbb R})$, we identify a continuum regime where the mean values and the covariances of the three Iwasawa parameters are simultaneously small. In this regime, the Lyapunov exponent of the product is shown to assume a scaling form. In the general case, the corresponding scaling function is expressed in terms of Gauss' hypergeometric function. A number of particular cases are also considered, where the scaling function of the Lyapunov exponent involves other special functions (Airy, Bessel, Whittaker, elliptic). The general solution thus obtained allows us, among other things, to recover in a unified framework many results known previously from exactly solvable models of one-dimensional disordered systems.