Random graphs as models of quantum disorder

A disordered quantum system is mathematically described by a large Hermitian random matrix. One of the most remarkable phenomena expected to occur in such systems is a localization-delocalization transition for the eigenvectors. Originally proposed in the 1950s to model conduction in semiconductors with random impurities, the phenomenon is now recognized as a general feature of wave transport in disordered media, and is one of the most influential ideas in modern condensed matter physics. A simple and natural model of such a system is given by the adjacency matrix of a random graph. I report on recent progress in analysing the phase diagram for the Erdös-Renyi model of random graphs. In particular, I explain the emergence of fully localized and fully delocalized phases, which are separated by a mobility edge. I also explain how to obtain optimal delocalization bounds using a new Bernoulli flow method. Based on joint work with Johannes Alt, Raphael Ducatez, and Joscha Henheik.