Parametric Furstenberg Theorem on Random Products of $SL(2, \mathbb{R})$ matrices

TL;DR

This paper studies how the Lyapunov exponents of random $SL(2, R)$ matrix products depend on a parameter, showing a universal positive upper exponent and a dense set of parameters with zero lower exponent, and also provides a geometric proof of Anderson localization.

Contribution

It establishes a parametric Furstenberg theorem for $SL(2, R)$ matrices with monotone dependence, revealing detailed Lyapunov exponent behavior and offering a geometric proof of Anderson localization.

Findings

Almost surely positive upper Lyapunov exponent for all parameters.

Dense $G_\delta$ set of parameters with zero lower Lyapunov exponent.

Geometric proof of Anderson localization for 1D random Schrödinger operators.

Abstract

We consider random products of $SL(2, \mathbb{R})$ matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense $G_\delta$ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schr\"odinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice.