---
_id: '1608'
abstract:
- lang: eng
text: '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
rΔdr+λ on ℤd, where r is an orthogonal radial projection, Δd the discrete
adjacency operator (Laplacian) on ℤd and λ a random potential. '
author:
- first_name: Christian
full_name: Sadel, Christian
id: 4760E9F8-F248-11E8-B48F-1D18A9856A87
last_name: Sadel
orcid: 0000-0001-8255-3968
citation:
ama: Sadel C. Anderson transition at 2 dimensional growth rate on antitrees and
spectral theory for operators with one propagating channel. Annales Henri Poincare.
2016;17(7):1631-1675. doi:10.1007/s00023-015-0456-3
apa: Sadel, C. (2016). Anderson transition at 2 dimensional growth rate on antitrees
and spectral theory for operators with one propagating channel. Annales Henri
Poincare. Birkhäuser. https://doi.org/10.1007/s00023-015-0456-3
chicago: Sadel, Christian. “Anderson Transition at 2 Dimensional Growth Rate on
Antitrees and Spectral Theory for Operators with One Propagating Channel.” Annales
Henri Poincare. Birkhäuser, 2016. https://doi.org/10.1007/s00023-015-0456-3.
ieee: C. Sadel, “Anderson transition at 2 dimensional growth rate on antitrees and
spectral theory for operators with one propagating channel,” Annales Henri
Poincare, vol. 17, no. 7. Birkhäuser, pp. 1631–1675, 2016.
ista: Sadel C. 2016. Anderson transition at 2 dimensional growth rate on antitrees
and spectral theory for operators with one propagating channel. Annales Henri
Poincare. 17(7), 1631–1675.
mla: Sadel, Christian. “Anderson Transition at 2 Dimensional Growth Rate on Antitrees
and Spectral Theory for Operators with One Propagating Channel.” Annales Henri
Poincare, vol. 17, no. 7, Birkhäuser, 2016, pp. 1631–75, doi:10.1007/s00023-015-0456-3.
short: C. Sadel, Annales Henri Poincare 17 (2016) 1631–1675.
date_created: 2018-12-11T11:53:00Z
date_published: 2016-07-01T00:00:00Z
date_updated: 2021-01-12T06:51:58Z
day: '01'
department:
- _id: LaEr
doi: 10.1007/s00023-015-0456-3
ec_funded: 1
intvolume: ' 17'
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1501.04287
month: '07'
oa: 1
oa_version: Preprint
page: 1631 - 1675
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
publication: Annales Henri Poincare
publication_status: published
publisher: Birkhäuser
publist_id: '5558'
quality_controlled: '1'
scopus_import: 1
status: public
title: Anderson transition at 2 dimensional growth rate on antitrees and spectral
theory for operators with one propagating channel
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 17
year: '2016'
...