# Singular Spectrum and Recent Results on Hierarchical Operators

## TL;DR

This paper employs trace class scattering theory to analyze the spectral properties of hierarchical operators, demonstrating the absence of absolutely continuous spectrum and exploring localization phenomena in both deterministic and random hierarchical models.

## Contribution

It introduces new methods to exclude absolutely continuous spectrum in hierarchical operators and surveys recent localization and spectral statistics results in hierarchical Anderson models.

## Key findings

- Absence of absolutely continuous spectrum in large classes of hierarchical operators
- Localization effects in deterministic hierarchical structures
- Spectral statistics of ultrametric random matrix ensembles

## Abstract

We use trace class scattering theory to exclude the possibility of absolutely continuous spectrum in a large class of self-adjoint operators with an underlying hierarchical structure and provide applications to certain random hierarchical operators and matrices. We proceed to contrast the localizing effect of the hierarchical structure in the deterministic setting with previous results and conjectures in the random setting. Furthermore, we survey stronger localization statements truly exploiting the disorder for the hierarchical Anderson model and report recent results concerning the spectral statistics of the ultrametric random matrix ensemble.