Non-Lyapunov annealed decay for 1d Anderson eigenfunctions

TL;DR

This paper investigates the decay properties of 1D Anderson eigenfunctions, demonstrating that their expected dynamical decay does not generally match the Lyapunov exponent when disorder is large, contrasting previous results for related models.

Contribution

It provides a counterexample to the assumption that decay rates align with Lyapunov exponents in the 1D Anderson model at high disorder levels.

Findings

Decay in expectation differs from Lyapunov exponent at high disorder

Counterexamples show non-Lyapunov decay behavior

Challenges previous assumptions about eigenfunction decay in disordered systems

Abstract

In [10] Jitomirskaya, Kr\"uger and Liu analysed the dynamical decay in expectation for the super-critical almost-Mathieu operator in function of the coupling parameter , showing that it is equal to the Lyapunov exponent of its transfer matrix cocycle, and asked whether the same is true for the 1d Anderson model. We show that this is essentially never true when the disorder parameter is sufficiently large.