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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...
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: |
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| 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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