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


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.)