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Central limit theorems for the spectra of classes of random fractals

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Charmoy, Philippe H. A., Croydon, David A. and Hambly, Ben M. (2017) Central limit theorems for the spectra of classes of random fractals. Transactions of the American Mathematical Society, 369 . 8967-9013 . doi:10.1090/tran/7147

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Official URL: https://doi.org/10.1090/tran/7147

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Abstract

We discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal, the continuum random tree. In the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit. We will show examples from a class of random fractals generated from Dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem. The Brownian continuum random tree can also be viewed as a random fractal generated by a Dirichlet distribution. The first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this, though the positivity of the variance arising in the central limit theorem is left open. In both cases these fractals can be described through a general Crump-Mode-Jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics. Our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known.

Item Type: Journal Article
Alternative Title:
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Statistics
Library of Congress Subject Headings (LCSH): Fractals., Spectral sequences (Mathematics), Random walks (Mathematics)
Journal or Publication Title: Transactions of the American Mathematical Society
Publisher: American Mathematical Society
ISSN: 0002-9947
Official Date: May 2017
Dates:
DateEvent
May 2017Available
7 December 2016Accepted
Volume: 369
Page Range: 8967-9013
DOI: 10.1090/tran/7147
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Funder: Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung [Swiss National Science Foundation] (SNSF), Berrow Foundation Scholarships
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