Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices

Alain Comtet 12, Christophe Texier 13, Yves Tourigny 4

Journal of Physics A: Mathematical and Theoretical 46 (2013) 254003

The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the localisation length. The Lyapunov exponent also features prominently in the theory of products of random matrices pioneered by Furstenberg. After a brief historical survey, we describe some recent work that exploits the close connections between these topics. We review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case. Finally, we point out some limitations of the Lyapunov exponent as a means of studying localisation properties.

  • 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 : Laboratoire de Physique des Solides (LPS)
    CNRS : UMR8502 – Université Paris XI - Paris Sud
  • 4 : School of Mathematics [Bristol]
    University of Bristol