On a class of random perturbations of the hierarchical Laplacian

Let be a locally compact separable ultrametric space. Given a measure on and a function defined on the set of all balls of positive measure of , we consider the hierarchical Laplacian . The operator acts on . It is essentially self-adjoint and has a pure point spectrum. By choosing a family of independent identically distributed random variables, we define the perturbed function and the perturbed hierarchical Laplacian . We study the arithmetic means of the eigenvalues of . Under some mild assumptions the normalized arithmetic means converge to in distribution. We also give examples when the normal convergence fails. We prove the existence of an integrated density of states. Introducing an empirical point process for the eigenvalues of and assuming that the density of states exists and is continuous, we prove that the finite-dimensional distributions of converge to those of the Poisson point process. As an example we consider random perturbations of the Vladimirov operator acting on , where is the ring of -adic numbers and is the Haar measure.