  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.
