Abstract:  Hermitian random matrix models are known to exhibit phase transitions  regarding both their local eigenvalue statistics and in the eigenvectors’ localisation properties. The poster child of such is the Rosenzweig-Porter model, which is based on the interpolation between 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 local GOE statistics. In this talk, I will explain the main ideas behind the emergence of this phase.  I will also address the motivation of these questions and consequences for the ultra-metric ensemble.