Anderson transition at 2 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.