RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor
Subscription
Guidelines for authors
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Teor. Veroyatnost. i Primenen.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

Full text: PDF file (670 kB)
First page: PDF file
References: PDF file   HTML file

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

Bibliographic databases:

Document Type: Article
Received: 18.12.2015
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
\Bibitem{BenBraPik18}
\by A.~Bendikov, A.~Braverman, J.~Pike
\paper Poisson statistics of eigenvalues in the hierarchical Dyson model
\jour Teor. Veroyatnost. i Primenen.
\yr 2018
\vol 63
\issue 1
\pages 117--144
\mathnet{http://mi.mathnet.ru/tvp5136}
\crossref{https://doi.org/10.4213/tvp5136}
\elib{http://elibrary.ru/item.asp?id=32428154}
\transl
\jour Theory Probab. Appl.
\yr 2018
\vol 63
\issue 1
\pages 94--116
\crossref{https://doi.org/10.1137/S0040585X97T988939}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000448195400006}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85064685692}


Linking options:
  • http://mi.mathnet.ru/eng/tvp5136
  • https://doi.org/10.4213/tvp5136
  • http://mi.mathnet.ru/eng/tvp/v63/i1/p117

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Теория вероятностей и ее применения Theory of Probability and its Applications
    Number of views:
    This page:104
    References:9
    First page:12

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019