Electronic Journal of Probability

Spectral Asymptotics for Stable Trees

David Croydon and Benjamin Hambly

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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

Article information

Electron. J. Probab. Volume 15 (2010), paper no. 57, 1772-1801.

Accepted: 14 November 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Secondary: 28A80: Fractals [See also 37Fxx] 58G25 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

stable tree self-similar decomposition spectral asymptotics heat kernel

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Croydon, David; Hambly, Benjamin. Spectral Asymptotics for Stable Trees. Electron. J. Probab. 15 (2010), paper no. 57, 1772--1801. doi:10.1214/EJP.v15-819. http://projecteuclid.org/euclid.ejp/1464819842.

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