Anderson Transition at Two-Dimensional Growth Rate on Antitrees and Spectral Theory for Operators with One Propagating Channel

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.