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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. 17721801. ISSN 10836489

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Official URL: http://www.math.washington.edu/~ejpecp/viewarticle...
Abstract
We calculate the mean and almostsure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on alphastable trees, which lead in turn to shorttime heat kernel asymptotics for these random structures. In particular, the conclusions we obtain demonstrate that the spectral dimension of an alphastable tree is almostsurely equal to 2 alpha/(2 alpha1), 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 alpha1). To prove our results, we adapt a selfsimilar 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 alphastable trees.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of 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:  10836489 
Official Date:  14 November 2010 
Volume:  Vol.15 
Number:  No.57 
Number of Pages:  30 
Page Range:  pp. 17721801 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Open Access 
References:  [1] D. Aldous, The continuum random tree. III, Ann. Probab. 21 (1993), no. 1, 248–289. 
URI:  http://wrap.warwick.ac.uk/id/eprint/4863 
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