## 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 language | English (US) |
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Pages (from-to) | 94-116 |

Number of pages | 23 |

Journal | Theory of Probability and its Applications |

Volume | 63 |

Issue number | 1 |

DOIs | |

State | Published - 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