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 onedimensional 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 onedimensional 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 onedimensional 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 rΔdr+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:

14240661

Publisher:

Birkhäuser

Date Published:

20160701

Start Page: 
1631

End Page:

1675

URL: 

DOI: 
10.1007/s0002301504563

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) 