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 N. When the decay rate of the off-diagonal variances is sufficiently slow, we prove that the spectral measures are uniformly θ-Hölder continuous for all θ∈ (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.