Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel

TL;DR

This paper investigates the Anderson transition from localization to delocalization on antitrees with 2-dimensional growth, extending spectral theory for operators with one propagating channel, revealing a critical transition at exactly 2D growth.

Contribution

It introduces a spectral analysis framework for operators with one propagating channel and applies it to demonstrate the Anderson transition at 2D growth on antitrees.

Findings

Transition from localization to delocalization at 2D growth

Existence of singular continuous spectrum at 2D growth with small disorder

Change from singular to absolutely continuous spectrum at 3D growth

Abstract

We show that the Anderson model has a transition from localization to delocalization at exactly 2 dimensional growth rate on antitrees with normalized edge weights which are certain discrete graphs. The kinetic part has a one-dimensional structure allowing a description through transfer matrices which involve some Schur complement. For such operators we introduce the notion of having one propagating channel and extend theorems from the theory of one-dimensional Jacobi operators that relate the behavior of transfer matrices with the spectrum. These theorems are then applied to the considered model. In essence, in a certain energy region the kinetic part averages the random potentials along shells and the transfer matrices behave similar as for a one-dimensional operator with random potential of decaying variance. At $d$ dimensional growth for $d>2$ this effective decay is strong enough to obtain absolutely continuous spectrum, whereas for some uniform $d$ dimensional growth with $d<2$ one has pure point spectrum in this energy region. At exactly uniform $2$ dimensional growth also some singular continuous spectrum appears, at least at small disorder. As a corollary we also obtain a change from singular spectrum ($d\leq 2$) to absolutely continuous spectrum ($d\geq 3)$ for random operators of the type $\mathcal{P}_r \Delta_d \mathcal{P}_r+\lambda \mathcal{V}$ on $\mathbb{Z}^d$, where $\mathcal{P}_r$ is an orthogonal radial projection, $\Delta_d$ the discrete adjacency operator (Laplacian) on $\mathbb{Z}^d$ and $\lambda \mathcal{V}$ a random potential.