Delocalization and continuous spectrum for ultrametric random operators

This paper studies the delocalized regime of an ultrametric random operator whose independent entries have variances decaying in a suitable hierarchical metric on $mathbb{N}$. When the decay-rate of the off-diagonal variances is sufficiently slow, we prove that the spectral measures are uniformly $theta$-H{o}lder continuous for all $theta in (0,1)$. In finite volumes, we prove that the corresponding ultrametric random matrices have completely extended eigenfunctions and that the local eigenvalue statistics converge in the Wigner-Dyson-Mehta universality class.