## 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 language | English |
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Pages (from-to) | 13-65 |

Number of pages | 53 |

Journal | Journal of Statistical Physics |

Volume | 150 |

Issue number | 1 |

DOIs | |

Publication status | Published - 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