Random band matrices

Understand

We survey recent mathematical results about the spectrum of random band matrices.

• We start by exposing the Erd{\H o}s-Schlein-Yau dynamic approach, its application to Wigner matrices, and extension to other mean-field models.
• We then introduce random band matrices and the problem of their Anderson transition.
• We finally describe a method to obtain delocalization and universality in some sparse regimes, highlighting the role of quantum unique ergodicity.