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This article is cited in 5 scientific papers (total in 5 papers)
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.
Received: 21.08.2014 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. Math., 79:5 (2015), 859–893
Linking options:
https://www.mathnet.ru/eng/im8294https://doi.org/10.1070/IM2015v079n05ABEH002764 https://www.mathnet.ru/eng/im/v79/i5/p3
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