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Spectral asymptotics for stable trees

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Croydon, David A. and Hambly, Ben M. (2010) Spectral asymptotics for stable trees. Electronic Journal of Probability, Vol.15 (No.57). pp. 1772-1801. ISSN 1083-6489.

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Official URL: http://www.math.washington.edu/~ejpecp/viewarticle...

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Abstract

We calculate the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on alpha-stable trees, which lead in turn to short-time heat kernel asymptotics for these random structures. In particular, the conclusions we obtain demonstrate that the spectral dimension of an alpha-stable tree is almost-surely equal to 2 alpha/(2 alpha-1), matching that of certain related discrete models. We also show that the exponent for the second term in the asymptotic expansion of the eigenvalue counting function is no greater than 1/(2 alpha-1). To prove our results, we adapt a self-similar fractal argument previously applied to the continuum random tree, replacing the decomposition of the continuum tree at the branch point of three suitably chosen vertices with a recently developed spinal decomposition for alpha-stable trees.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Statistics
Library of Congress Subject Headings (LCSH): Decomposition (Mathematics), Eigenvalues, Dirichlet forms
Journal or Publication Title: Electronic Journal of Probability
Publisher: University of Washington. Dept. of Mathematics
ISSN: 1083-6489
Official Date: 14 November 2010
Dates:
DateEvent
14 November 2010Published
Volume: Vol.15
Number: No.57
Number of Pages: 30
Page Range: pp. 1772-1801
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Open Access (Creative Commons)
Date of first compliant deposit: 3 December 2015
Date of first compliant Open Access: 3 December 2015

Data sourced from Thomson Reuters' Web of Knowledge

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