Spectral asymptotics for stable trees
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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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|
|Official Date:||14 November 2010|
|Number of Pages:||30|
|Page Range:||pp. 1772-1801|
|Access rights to Published version:||Open Access|
 D. Aldous, The continuum random tree. III, Ann. Probab. 21 (1993), no. 1, 248–289.
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