Absolutely continuous spectrum implies ballistic transport for quantum particles in a random potential on tree graphs

TL;DR

This paper demonstrates that the presence of absolutely continuous spectrum in a quantum particle on a random tree graph guarantees ballistic transport, linking spectral properties to dynamical behavior.

Contribution

It establishes a non-perturbative connection between absolutely continuous spectrum and ballistic transport for quantum particles on tree graphs.

Findings

Absolutely continuous spectrum implies ballistic transport.

Rare fluctuation-enabled resonances are key to spectral properties.

The result is proven non-perturbatively.

Abstract

We discuss the dynamical implications of the recent proof that for a quantum particle in a random potential on a regular tree graph absolutely continuous spectrum occurs non-perturbatively through rare fluctuation-enabled resonances. The main result is spelled in the title.