Eigenvectors of random matrices

We’ll discuss the phenomenon of eigenvector delocalization within a somewhat large class of random matrices. If you don’t care about random matrices, it turns out that the story is not all that different for Schrödinger operators, in some sense. Namely, I’ll talk about the role of the resolvent in studying the problem of eigenvector delocalization for random matrices and the necessary multi-scale analysis for random matrices; turns out, multi-scale analysis of the resolvent is the key ingredient in Bourgain–Kenig '05 to establish a delocalization result for Schrödinger operators. I’ll end the talk by discussing the stability of eigenvector delocalization under a particular matrix flow, which itself consists of a magical duality and standard parabolic theory.