Abstract
We show that the Anderson model has a transition from localization to delocalization at exactly two-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 two-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.
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Communicated by Jean Bellissard.
The research of C.S. has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA grant agreement number 291734.
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Sadel, C. Anderson Transition at Two-Dimensional Growth Rate on Antitrees and Spectral Theory for Operators with One Propagating Channel. Ann. Henri Poincaré 17, 1631–1675 (2016). https://doi.org/10.1007/s00023-015-0456-3
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DOI: https://doi.org/10.1007/s00023-015-0456-3