Mathematical Physics
[Submitted on 18 Jan 2015 (v1), last revised 15 Jun 2015 (this version, v2)]
Title:Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel
View PDFAbstract: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.
Submission history
From: Christian Sadel [view email][v1] Sun, 18 Jan 2015 12:16:15 UTC (51 KB)
[v2] Mon, 15 Jun 2015 12:56:12 UTC (53 KB)
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