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 ($d\le 2$) to absolutely continuous spectrum ($d\ge 3)$ 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.

38 pages, 1 figure; Introduction reorganized, Corollary 1.3 added and almost sure essential spectrum now characterized (Proposition 1.4)

References in corpus (3)

• Absolutely continuous spectrum implies ballistic transport for quantum particles in a random potential on tree graphs
• Localization for the Anderson Model on Trees with Finite Dimensions
• Resonances and Partial Delocalization on the Complete Graph

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