Absence of mobility edge for the Anderson random potential on tree graphs at weak disorder

TL;DR

This paper demonstrates that for the Anderson model on tree graphs with bounded random potentials, a transition to Anderson localization does not occur at weak disorder, challenging traditional expectations.

Contribution

The authors show that the previously established criterion implies no localization transition at weak disorder for bounded potentials on tree graphs.

Findings

No spectral transition to localization at weak disorder

Localization only occurs when disorder is sufficiently strong

Challenges conventional understanding of Anderson localization on trees

Abstract

Our recently established criterion for the formation of extended states on tree graphs in the presence of disorder is shown to have the surprising implication that for bounded random potentials, as in the Anderson model, there is no transition to a spectral regime of Anderson localization, in the form usually envisioned, unless the disorder is strong enough.