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

A. Comtet ¹, ², J. M. Luck ³, C. Texier ¹, ⁴, Y. Tourigny ⁵

Journal of Statistical Physics 150 (2013) 13-65

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.

• 1 : Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS)
  CNRS : UMR8626 – Université Paris XI – Paris Sud
• 2 : Unite mixte de service de l’institut Henri Poincaré (UMSIHP)
  CNRS : UMS839 – Université Pierre et Marie Curie (UPMC) – Paris VI
• 3 : Institut de Physique Théorique (ex SPhT) (IPHT)
  CNRS : URA2306 – CEA : DSM/IPHT
• 4 : Laboratoire de Physique des Solides (LPS)
  CNRS : UMR8502 – Université Paris XI – Paris Sud
• 5 : Department of Mathematics [Bristol]
  University of Bristol – University Walk