Positive Lyapunov Exponents and a Large Deviation Theorem for Continuum Anderson Models, Briefly

TL;DR

This paper proves the positivity of Lyapunov exponents for 1D continuum Anderson models using classical inverse spectral theory, simplifying previous proofs and establishing a uniform Large Deviation Theorem.

Contribution

It introduces a simpler proof for Lyapunov exponent positivity that applies to a broader class of random models and confirms a uniform Large Deviation Theorem.

Findings

Positivity of Lyapunov exponents for 1D continuum Anderson models.

A simplified proof method leveraging inverse spectral theory.

Establishment of a uniform Large Deviation Theorem.

Abstract

In this short note, we prove positivity of the Lyapunov exponent for 1D continuum Anderson models by leveraging some classical tools from inverse spectral theory. The argument is much simpler than the existing proof due to Damanik--Sims--Stolz, and it covers a wider variety of random models. Along the way we note that a Large Deviation Theorem holds uniformly on compacts.