Poisson approximation related to spectra of hierarchical Laplacians

Abstract

Let [Formula: see text] be a locally compact separable ultrametric space. Given a measure [Formula: see text] on [Formula: see text] and a function [Formula: see text] defined on the set of all non-singleton balls [Formula: see text] of [Formula: see text], we consider the hierarchical Laplacian [Formula: see text]. The operator [Formula: see text] acts in [Formula: see text] is essentially self-adjoint and has a purely point spectrum. Choosing a sequence [Formula: see text] of i.i.d. random variables, we consider the perturbed function [Formula: see text] and the perturbed hierarchical Laplacian [Formula: see text] Under certain conditions, the density of states [Formula: see text] exists and it is a continuous function. We choose a point [Formula: see text] such that [Formula: see text] and build a sequence of point processes defined by the eigenvalues of [Formula: see text] located in the vicinity of [Formula: see text]. We show that this sequence converges in distribution to the homogeneous Poisson point process with intensity [Formula: see text].