Fluctuation based proof of the stability of ac spectra of random operators on tree graphs

TL;DR

This paper reviews recent research demonstrating the stability of absolutely continuous spectra in various random operators on tree graphs, highlighting the robustness of spectral properties under disorder.

Contribution

It provides a comprehensive summary of recent advances in understanding the stability of ac spectra for different classes of random operators on trees.

Findings

Absolutely continuous spectra are stable under certain random perturbations.

Various models like Schrödinger and quantum graph operators exhibit spectral stability.

Disorder effects on spectral properties are systematically analyzed.

Abstract

We summarize recent works on the stability under disorder of the absolutely continuous spectra of random operators on tree graphs. The cases covered include: Schr\"odinger operators with random potential, quantum graph operators for trees with randomized edge lengths, and radial quasi-periodic operators perturbed by random potentials.