Lyapunov Exponents for Unitary Anderson Models

TL;DR

This paper analyzes a unitary Anderson model, characterizing when the Lyapunov exponent is positive or zero across the spectrum, including cases with Bernoulli distributions, and extends results to a unitary random dimer model.

Contribution

It provides a complete characterization of the Lyapunov exponent's behavior for a unitary Anderson model and its extension to a unitary random dimer model.

Findings

Positivity and vanishing of Lyapunov exponent characterized across the spectrum.

Existence of critical spectral values with zero Lyapunov exponent for Bernoulli distributions.

Results extended to a unitary version of the random dimer model.

Abstract

We study a unitary version of the one-dimensional Anderson model, given by a five diagonal deterministic unitary operator multiplicatively perturbed by a random phase matrix. We fully characterize positivity and vanishing of the Lyapunov exponent for this model throughout the spectrum and for arbitrary distributions of the random phases. This includes Bernoulli distributions, where in certain cases a finite number of critical spectral values, with vanishing Lyapunov exponent, exists. We establish similar results for a unitary version of the random dimer model.