Anderson Localization in dimension two for singular noise

Mathematical Physics and Analysis Working Seminar
Tuesday, February 21, 2023 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
Omar Hurtado – UC Irvine
Burak Hatinoglu

We will discuss the work of Ding-Smart (2019) which showed Anderson localization at the bottom of the spectrum for random discrete Schroedinger operators with arbitrary bounded noise, i.e. without any suppositionĀ of regularity of the distribution. In this talk, we will discuss at a high level the basic idea behind a multi-scale analysis, as well as the usual ingredients involved in one: resolvent decay at large scales and the Wegner-type estimate.

We will then discuss the obstacles posed by singular distributions, and the various methods used to overcome these obstacles in various regimes, discussing briefly the transfer matrix method used for d=1 by Carmona-Klein-Martinelli (1987) before examining the unique continuation principles used by Bourgain-Kenig (2005) and the Ding-Smart work which are used in d=2 in the continuum and discrete cases respectively, highlighting the unique challenges arising in the discrete case.