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 Teor. Veroyatnost. i Primenen., 2018, Volume 63, Issue 1, Pages 117–144 (Mi tvp5136)

Poisson statistics of eigenvalues in the hierarchical Dyson model

A. Bendikova, A. Bravermanb, J. Pikec

a Institute of Mathematics, University of Wroclaw, Wroclaw, Poland
b School of Operation Research and Industrial Engineering, Cornell University, Ithaca, NY, USA
c Department of Mathematics, Cornell University, Ithaca, NY, USA

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 $\mathcal{B}$ of all balls $B\subset X$, 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 $\{\varepsilon(B)\}_{B\in \mathcal{B}}$ of i.i.d. random variables, we define the perturbed function $\mathcal{C}(B)=C(B)(1+\varepsilon(B))$ and the perturbed hierarchical Laplacian $\mathcal{L}=L_{\mathcal{C}}$. All outcomes of the perturbed operator $\mathcal{L}$ are hierarchical Laplacians. In particular they all have purely point spectrum. We study the empirical point process $M$ defined in terms of $\mathcal{L}$-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 $\mathfrak{D}^{\alpha }$, the $p$-adic fractional derivative of order $\alpha > 0$.

Keywords: Poisson approximation, hierarchical Laplacian, ultrametric measure space, field of $p$-adic numbers, fractional derivative, point spectrum, integrated density of states, Stein's method.

 Funding Agency Grant Number National Science Centre (Narodowe Centrum Nauki) DEC 2015/17/B/ST1/00062 National Science Foundation DMS-0739164 The first author was supported by National Science Centre, Poland (grant DEC 2015/17/B/ST1/00062). The third author was supported in part by NSF grant DMS-0739164.

DOI: https://doi.org/10.4213/tvp5136

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English version:
Theory of Probability and its Applications, 2018, 63:1, 94–116

Bibliographic databases:

Document Type: Article
Accepted:30.06.2016
Language: English

Citation: A. Bendikov, A. Braverman, J. Pike, “Poisson statistics of eigenvalues in the hierarchical Dyson model”, Teor. Veroyatnost. i Primenen., 63:1 (2018), 117–144; Theory Probab. Appl., 63:1 (2018), 94–116

Citation in format AMSBIB
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