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2010 Spectral Asymptotics for Stable Trees
David Croydon, Benjamin Hambly
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Electron. J. Probab. 15: 1772-1801 (2010). DOI: 10.1214/EJP.v15-819


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


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David Croydon. Benjamin Hambly. "Spectral Asymptotics for Stable Trees." Electron. J. Probab. 15 1772 - 1801, 2010.


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

zbMATH: 1236.60082
MathSciNet: MR2738338
Digital Object Identifier: 10.1214/EJP.v15-819

Primary: 35P20
Secondary: 28A80 , 58G25 , 60J35 , 60J80

Keywords: heat kernel , self-similar decomposition , Spectral asymptotics , stable tree


Vol.15 • 2010
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