Resonant delocalization on the Bethe strip

TL;DR

This paper extends the concept of resonant delocalization from Bethe lattice operators to matrix-valued random potentials on the Bethe strip, revealing new spectral properties involving loops.

Contribution

It generalizes the resonant delocalization mechanism to matrix-valued potentials on the Bethe strip, a graph with loops, broadening understanding of spectral phenomena.

Findings

Resonant delocalization occurs on the Bethe strip with matrix-valued potentials.

The analysis includes ensembles like the Gaussian Orthogonal Ensemble.

Spectral properties differ from those on the Bethe lattice without loops.

Abstract

Recently, Aizenman and Warzel discovered a mechanism for the appearance of absolutely continuous spectrum for random Schroedinger operators on the Bethe lattice through rare resonances (resonant delocalization). We extend their analysis to operators with matrix-valued random potentials drawn from ensembles such as the Gaussian Orthogonal Ensemble. These operators can be viewed as random operators on the Bethe strip, a graph (lattice) with loops.