Random band matrices and the extended states conjecture

Random matrices were introduced in the 80s to model disordered quantum systems on large graphs (typically lattices). They provide a means of interpolating between random Schrodinger operators and mean-field models such as Wigner matrices. On the one-dimensional lattice it is conjectured that as one increases the band width a sharp transition occurs from the localized to the delocalized regime. In parallel, the local spectral statistics undergo a transition from Poisson to random matrix statistics.

I give an overview of recent progress in understanding the eigenvector and eigenvalue distribution of random band matrices. I mainly focus on the derivation of delocalization bounds on the eigenvectors. I outline two approaches: one based on perturbative renormalization and the other on the averaging of fluctuations among resolvent entries.