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 Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 5, Pages 3–38 (Mi izv8294)

On a class of random perturbations of the hierarchical Laplacian

A. D. Bendikova, A. A. Grigor'yanb, S. A. Molchanovc, G. P. Samorodnitskyd

a Institute of Mathematics, Wrocław University
b Bielefeld University, Department of Mathematics
c Department of Mathematics, University of North Carolina Charlotte
d School of Operations Research and Information Engineering, Cornell University

Abstract: Let $(X,d)$ be a locally compact separable ultrametric space. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set $B$ of all balls of positive measure of $X$, we consider the hierarchical Laplacian $L=L_{C}$. The operator $L$ acts on $L^{2}(X,m)$. It is essentially self-adjoint and has a pure point spectrum. By choosing a family $\{\varepsilon (B)\}$ of independent identically distributed random variables, we define the perturbed function $C(B,\omega)$ and the perturbed hierarchical Laplacian $L^{\omega }=L_{C(\omega)}$. We study the arithmetic means $\bar{\lambda }(\omega)$ of the eigenvalues of $L^{\omega }$. Under some mild assumptions the normalized arithmetic means $( \bar{\lambda }-\mathbb{E}\bar{\lambda })/\sigma [\bar{\lambda }]$ converge to $N(0,1)$ in distribution. We also give examples when the normal convergence fails. We prove the existence of an integrated density of states. Introducing an empirical point process $N^{\omega }$ for the eigenvalues of $L^{\omega }$ and assuming that the density of states exists and is continuous, we prove that the finite-dimensional distributions of $N^{\omega }$ converge to those of the Poisson point process. As an example we consider random perturbations of the Vladimirov operator acting on $L^{2}(X,m)$, where $X=\mathbb{Q}_{p}$ is the ring of $p$-adic numbers and $m$ is the Haar measure.

Keywords: ultrametric measure space, field of $p$-adic numbers, hierarchical Laplacian, fractional derivative, Vladimirov Laplacian, point spectrum, integrated density of states, Bernoulli convolutions, Erdős problem, point process, Poisson convergence.

 Funding Agency Grant Number National Science Centre (Narodowe Centrum Nauki) NCN DEC-2012/05/B/ST 1/00613 Deutsche Forschungsgemeinschaft SFB 701 National Science Foundation The first author was supported by the Polish National Centre of Sciences (grant DEC-2012/05/B/ST 1/00613). The second author was supported by the German Research Council (grant SFB 701). The third and fourth authors were supported by the NSF (USA).

DOI: https://doi.org/10.4213/im8294

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English version:
Izvestiya: Mathematics, 2015, 79:5, 859–893

Bibliographic databases:

Document Type: Article
UDC: 517.983+517.1+519.2
MSC: 05C05, 47S10, 60J25, 81Q10
Revised: 01.12.2014

Citation: A. D. Bendikov, A. A. Grigor'yan, S. A. Molchanov, G. P. Samorodnitsky, “On a class of random perturbations of the hierarchical Laplacian”, Izv. RAN. Ser. Mat., 79:5 (2015), 3–38; Izv. Math., 79:5 (2015), 859–893

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv8294
• https://doi.org/10.4213/im8294
• http://mi.mathnet.ru/eng/izv/v79/i5/p3

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