The Phase Transition in the Ultrametric Ensemble and Local Stability of Dyson Brownian Motion

We study the ultrametric random matrix ensemble, whose independent entries have variances decaying exponentially in the metric induced by the tree topology on $\mathbb{N}$, and map out the entire localization regime in terms of eigenfunction localization and Poisson statistics. Our results complement existing works on complete delocalization and random matrix universality, thereby proving the existence of a phase transition in this model. In the simpler case of the Rosenzweig-Porter model, the analysis yields a complete characterization of the transition in the local statistics. The proofs are based on the flow of the resolvents of matrices with a random diagonal component under Dyson Brownian motion, for which we establish submicroscopic stability results for short times. These results go beyond norm-based continuity arguments for Dyson Brownian motion and complement the existing analysis after the local equilibration time.