The Localization Transition in the Ultrametric Ensemble

TL;DR

This paper investigates a hierarchical random matrix model, demonstrating a phase transition between localized and delocalized eigenstates through eigenfunction localization and eigenvalue statistics analysis.

Contribution

It establishes the full localization regime for the ultrametric ensemble, identifying a phase transition in eigenstate behavior and eigenvalue distribution.

Findings

Eigenfunction localization in the model

Poisson eigenvalue statistics in the localized regime

Existence of a phase transition between localized and delocalized phases

Abstract

We study the hierarchical analogue of power-law random band matrices, a symmetric ensemble of random matrices with independent entries whose variances decay exponentially in the metric induced by the tree topology on $\mathbb{N}$. We map out the entirety of the localization regime by proving the localization of eigenfunctions and Poisson statistics of the suitably scaled eigenvalues. Our results complement existing works on complete delocalization and random matrix universality, thereby proving the existence of a phase transition in this model.