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

Abstract

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.

LaTeX, 23 pages, 3 pdf figures ; review for a special issue on "Lyapunov analysis" ; v2 : typo corrected in eq.(3) & minor changes

References in corpus (7)

• Anderson Transitions
• The Lyapunov exponent of products of random $2\times 2$ matrices close to the identity
• Products of random matrices and generalised quantum point scatterers
• Localization for one-dimensional random potentials with large local fluctuations
• Breaking supersymmetry in a one-dimensional random Hamiltonian
• Statistics of quantum transmission in one dimension with broad disorder
• On the statistics of superlocalized states in self-affine disordered potentials

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