Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel Journal Article


Author(s): Sadel, Christian
Article Title: Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel
Affiliation IST Austria
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≤2) to absolutely continuous spectrum (d≥3) for random operators of the type rΔdr+λ on ℤd, where r is an orthogonal radial projection, Δd the discrete adjacency operator (Laplacian) on ℤd and λ a random potential.
Journal Title: Annales Henri Poincare
Volume: 17
Issue 7
ISSN: 1424-0661
Publisher: Birkhäuser  
Date Published: 2016-07-01
Start Page: 1631
End Page: 1675
URL:
DOI: 10.1007/s00023-015-0456-3
Notes: 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.
Open access: yes (repository)