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

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.

arXiv comment63 pages, 1 figure, 1 table. A couple of additions and updates

References in corpus (9)

• Single parameter scaling in one-dimensional localization revisited
• Band-center anomaly of the conductance distribution in one-dimensional Anderson localization
• DIFFUSION IN ONE DIMENSIONAL RANDOM MEDIUM AND HYPERBOLIC BROWNIAN MOTION
• Products of random matrices and generalised quantum point scatterers
• Supersymmetric quantum mechanics with Levy disorder in one dimension
• Perturbative test of single parameter scaling for 1D random media
• Ergodic properties of a model for turbulent dispersion of inertial particles
• Random Lie group actions on compact manifolds: A perturbative analysis
• Breaking supersymmetry in a one-dimensional random Hamiltonian

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