Absolutely continuous spectrum for a random potential on a tree with strong transverse correlations and large weighted loops

TL;DR

This paper proves the presence of absolutely continuous spectrum at low disorder for specific random Schrödinger operators on trees with strong correlations and large loops, contrasting with localization in fully correlated cases.

Contribution

It introduces two models of random Schrödinger operators on trees with strong correlations and weighted loops, demonstrating absolutely continuous spectrum at small disorder.

Findings

Absolutely continuous spectrum exists at small disorder in the models.

Strong correlations can lead to delocalization contrary to localization.

Weighted loops influence spectral properties significantly.

Abstract

We consider random Schr\"odinger operators on tree graphs and prove absolutely continuous spectrum at small disorder for two models. The first model is the usual binary tree with certain strongly correlated random potentials. These potentials are of interest since for complete correlation they exhibit localization at all disorders. In the second model we change the tree graph by adding all possible edges to the graph inside each sphere, with weights proportional to the number of points in the sphere.