Poisson statistics of eigenvalues in the hierarchical dyson model

A. Bendikov, A. Braverman, J. Pike

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Let (X, d) be a locally compact separable ultrametric space. Given a measure m on X and a function C defined on the set B of all balls B ⊂ X, we consider the hierarchical Laplacian L = L C . The operator L acts in L 2 (X, m), is essentially self-adjoint, and has a purely point spectrum. Choosing a family {ε(B)} B ∈B of i.i.d. random variables, we define the perturbed function C(B) =C(B)(1 + ε(B)) and the perturbed hierarchical Laplacian L = L C . All outcomes of the perturbed operator L are hierarchical Laplacians. In particular they all have purely point spectrum. We study the empirical point process M defined in terms of L-eigenvalues. Under some natural assumptions, M can be approximated by a Poisson point process. Using a result of Arratia, Goldstein, and Gordon based on the Chen–Stein method, we provide total variation convergence rates for the Poisson approximation. We apply our theory to random perturbations of the operator D α ,thep-adic fractional derivative of order α>0.

Original languageEnglish (US)
Pages (from-to)94-116
Number of pages23
JournalTheory of Probability and its Applications
Volume63
Issue number1
DOIs
StatePublished - 2018

Keywords

  • Field of p-adic numbers
  • Fractional derivative
  • Hierarchical Laplacian
  • Integrated density of states
  • Point spectrum
  • Poisson approximation
  • Stein’s method
  • Ultrametric measure space

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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