Georgia Institute of Technology College of Sciences

We will actually finish our proof of the key technical lemma for the quantitative unique continuation principle of Ding-Smart, reviewing briefly the volumetric bound from the theory of \varepsilon-coverings/nets/packings. From there, we will outline at a high level the strategy for the rest of the proof of the unique continuation principle using this key lemma, before starting the proof in earnest.

We will finish our proof of the key lemma for the probabilistic unique continuation principle used in Ding-Smart. We will also briefly recall enough of the theory of martingales to clarify a use of Azuma's inequality, and the basic definitions of \epsilon-nets and \epsilon-packings required to formulate the basic volumetric bound for these in e.g. the unit sphere, before using these to complete the proof.

Continuing from where we left off, we will go through the proof of the probabilistic unique continuation result in Ding-Smart (2018) for solutions of the eigenequation on large finite boxes in the two-dimensional lattice. We'll briefly discuss the free sites formalism necessary to carry out the multiscale analysis as well, before going through technical lemmas concerning bounds on solutions to our eigenequation on large finite rectangles in the lattice as they propagate from a boundary.

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 supposition 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.