Positivity of Lyapunov exponents for a continuous matrix-valued Anderson model

TL;DR

This paper proves that for a continuous matrix-valued Anderson model, the two leading Lyapunov exponents are positive and distinct across most energies above 2, implying the absence of absolutely continuous spectrum in that range.

Contribution

It extends previous results by establishing positivity and distinctness of Lyapunov exponents for a broader class of distributions, including Bernoulli, using advanced Lie group methods.

Findings

Positivity of Lyapunov exponents for energies in (2, +∞)

Distinctness of the two leading Lyapunov exponents

Absence of absolutely continuous spectrum in (2, +∞)

Abstract

We study a continuous matrix-valued Anderson-type model. Both leading Lyapunov exponents of this model are proved to be positive and distinct for all ernergies in $(2,+\infty)$ except those in a discrete set, which leads to absence of absolutely continuous spectrum in $(2,+\infty)$. This result is an improvement of a previous result with Stolz. The methods, based upon a result by Breuillard and Gelander on dense subgroups in semisimple Lie groups, and a criterion by Goldsheid and Margulis, allow for singular Bernoulli distributions.