Absolutely continuous spectrum for random operators on trees of finite cone type

TL;DR

This paper investigates the spectral properties of random operators on trees with finitely many cone types, demonstrating the stability of the absolutely continuous spectrum under small random perturbations.

Contribution

It establishes the stability of the absolutely continuous spectrum for a broad class of trees with finitely many cone types under small random perturbations.

Findings

Large parts of the absolutely continuous spectrum remain stable under small disorder.

The trees can be constructed via substitution rules, allowing for a wide class of structures.

The study covers perturbations of Laplace-type operators by random potentials or hopping terms.

Abstract

We study the spectrum of random operators on a large class of trees. These trees have finitely many cone types and they can be constructed by a substitution rule. The random operators are perturbations of Laplace type operators either by random potentials or by random hopping terms, i.e., perturbations of the off-diagonal elements. We prove stability of arbitrary large parts of the absolutely continuous spectrum for sufficiently small but extensive disorder.