The Lyapunov Exponent of Products of Random 2x2 Matrices Close to the Identity

Alain Comtet, Jean-Marc Luck*, Christophe Texier, Yves Tourigny

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

20 Citations (Scopus)

Abstract

We study products of arbitrary random real 2x2 matrices that are close to the identity matrix. Using the Iwasawa decomposition of SL(2,a"e), 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.

Original languageEnglish
Pages (from-to)13-65
Number of pages53
JournalJournal of Statistical Physics
Volume150
Issue number1
DOIs
Publication statusPublished - Jan 2013

Keywords

  • GREENS-FUNCTIONS
  • WEAK DISORDER EXPANSION
  • Anderson localization
  • Iwasawa decomposition
  • Quantum mechanics
  • ELECTRONIC STATES
  • BROWNIAN-MOTION
  • FIELD ISING CHAINS
  • DIMENSIONAL RANDOM-SYSTEMS
  • LOCALIZED STATES
  • Random matrices
  • DIFFUSION
  • MODEL
  • Disordered one-dimensional systems
  • REFLECTION
  • Lyapunov exponent

Cite this