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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 > 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:

Date	Event
14 November 2010	Published

Volume:

Vol.15

Number:

No.57

Number of Pages:

Page Range:

pp. 1772-1801

Status:

Peer Reviewed

Publication Status:

Published

Access rights to Published version:

Open Access

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