Lyapunov exponents of Green's functions for random potentials tending to zero

TL;DR

This paper proves that both quenched and annealed Lyapunov exponents for the Green's function of a random Schrödinger operator scale like the square root of the potential's strength as it tends to zero, with an explicit constant.

Contribution

It provides a probabilistic proof showing the common scaling of quenched and annealed Lyapunov exponents as the potential vanishes, improving previous results.

Findings

Lyapunov exponents scale as c√γ for small γ

The constant c is explicitly computed and identical for both exponents

The approach extends to alternative methods of approaching zero potential

Abstract

We consider quenched and annealed Lyapunov exponents for the Green's function of $-\Delta+\gamma V$, where the potentials $V(x), x\in\Z^d$, are i.i.d. nonnegative random variables and $\gamma>0$ is a scalar. We present a probabilistic proof that both Lyapunov exponents scale like $c\sqrt{\gamma}$ as $\gamma$ tends to 0. Here the constant $c$ is the same for the quenched as for the annealed exponent and is computed explicitly. This improves results obtained previously by Wei-Min Wang. We also consider other ways to send the potential to zero than multiplying it by a small number.