{"main_file_link":[{"url":"http://arxiv.org/abs/1501.04287","open_access":"1"}],"publist_id":"5558","intvolume":" 17","day":"01","date_updated":"2021-01-12T06:51:58Z","language":[{"iso":"eng"}],"year":"2016","ec_funded":1,"doi":"10.1007/s00023-015-0456-3","publication":"Annales Henri Poincare","page":"1631 - 1675","oa_version":"Preprint","oa":1,"abstract":[{"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 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. ","lang":"eng"}],"publisher":"Birkhäuser","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","publication_status":"published","quality_controlled":"1","volume":17,"status":"public","_id":"1608","type":"journal_article","department":[{"_id":"LaEr"}],"issue":"7","date_created":"2018-12-11T11:53:00Z","scopus_import":1,"title":"Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel","project":[{"name":"International IST Postdoc Fellowship Programme","grant_number":"291734","call_identifier":"FP7","_id":"25681D80-B435-11E9-9278-68D0E5697425"}],"date_published":"2016-07-01T00:00:00Z","month":"07","author":[{"full_name":"Sadel, Christian","last_name":"Sadel","first_name":"Christian","id":"4760E9F8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-8255-3968"}],"citation":{"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.","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","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.","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.","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","short":"C. Sadel, Annales Henri Poincare 17 (2016) 1631–1675."}}