The regular tree Anderson model at low disorder

TL;DR

This paper proves that the Anderson model on an infinite regular tree exhibits delocalization at low disorder, extending previous results by covering the entire spectrum and generalizing to disorders with small fourth moments.

Contribution

It extends prior work by establishing delocalization across the full spectrum for the regular tree Anderson model at low disorder, including cases with broader disorder distributions.

Findings

Delocalization occurs at low disorder on regular trees.

Continuity of the Lyapunov exponent as disorder approaches zero.

Results apply to disorders with small fourth moments.

Abstract

We prove delocalization for the Anderson model on an infinite regular tree (or Cayley graph or Bethe lattice) at low disorder. This extends earlier results of Klein and Aizenman--Warzel by filling in the previously missing parts of the spectrum. Our argument generalizes to any disorder with small fourth moment and sufficiently regular density. We prove continuity of the Lyapunov exponent as the disorder vanishes.