Absolutely continuous spectrum for the Anderson model on a product of a tree with a finite graph

TL;DR

This paper proves that the Anderson model on a product of a regular tree and a finite graph with large loops has an almost sure absolutely continuous spectrum at low disorder levels, advancing understanding of spectral properties in complex structures.

Contribution

It demonstrates the existence of absolutely continuous spectrum for the Anderson model on a tree-product graph with large loops at low disorder, a novel result for such complex structures.

Findings

Absolutely continuous spectrum exists at low disorder

Spectrum is almost surely absolutely continuous

Applicable to graphs with unbounded loops

Abstract

We prove the almost sure existence of absolutely continuous spectrum at low disorder for the Anderson model on the simplest example of a product of a regular tree with a finite graph. This graph contains loops of unbounded size.