Localization for transversally periodic random potentials on binary trees

TL;DR

This paper studies a random Schrödinger operator on a binary tree with a combined radially symmetric and transversally periodic potential, revealing localization and delocalization transitions influenced by the coupling strength.

Contribution

Introduces a novel dynamical systems approach with hyperbolic space estimates to analyze localization in a complex tree model with two Anderson transitions.

Findings

Proves localization for small and large coupling constants.

Identifies two Anderson transitions as the coupling varies.

Provides a new proof of one-dimensional Anderson localization.

Abstract

We consider a random Schr\"odinger operator on the binary tree with a random potential which is the sum of a random radially symmetric potential, $Q_r$, and a random transversally periodic potential, $\kappa Q_t$, with coupling constant $\kappa$. Using a new one-dimensional dynamical systems approach combined with Jensen's inequality in hyperbolic space (our key estimate) we obtain a fractional moment estimate proving localization for small and large $\kappa$. Together with a previous result we therefore obtain a model with two Anderson transitions, from localization to delocalization and back to localization, when increasing $\kappa$. As a by-product we also have a partially new proof of one-dimensional Anderson localization at any disorder.