Transport and localization in random media: theory and applications

Localization-delocalization transitions in random matrix models: a SPDE approach

Simone Warzel

Technical University of Munich
[SLIDES]

Hermitian random matrix models are known to exhibit phase transitions regarding both their local eigenvalue statistics as well as the localisation properties of their eigenvectors. The poster child of such a model is the Rosenzweig--Porter model, i.e. the interpolation of a random diagonal matrix and GOE. Interestingly, this model has recently been shown to exhibit a phase in which the eigenvectors exhibit non-ergodic delocalisation alongside the local GOE statistics. In this talk, I will explain the main ideas behind the emergence of this phase using a SPDE approach. Time permitting, I will also address the motivation for these questions and consequences for the ultra-metric ensemble. (The talk is based on joint works with Per von Soosten.)