Low density expansion for Lyapunov exponents

TL;DR

This paper develops a perturbation theory for Lyapunov exponents in weakly disordered quasi-one-dimensional media with sparse, strong impurities, revealing linear growth and anomalies at rational quasi-momenta.

Contribution

It introduces a low-density expansion method for Lyapunov exponents in Anderson models with strong impurities, extending understanding of disorder effects.

Findings

Lyapunov exponent grows linearly with impurity density.

Anomalies appear at rational quasi-momenta in lowest order.

Perturbation theory captures effects of sparse, strong impurities.

Abstract

In some quasi-one-dimensional weakly disordered media, impurities are large and rare rather than small and dense. For an Anderson model with a low density of strong impurities, a perturbation theory in the impurity density is developed for the Lyapunov exponent and the density of states. The Lyapunov exponent grows linearly with the density. Anomalies of the Kappus-Wegner type appear for all rational quasi-momenta even in lowest order perturbation theory.