@article{EJP819,
author = {David Croydon and Benjamin Hambly},
title = {Spectral Asymptotics for Stable Trees},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {15},
year = {2010},
keywords = {stable tree; self-similar decomposition; spectral asymptotics; heat kernel},
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},
pages = {no. 57, 1772-1801},
issn = {1083-6489},
doi = {10.1214/EJP.v15-819},
url = {http://ejp.ejpecp.org/article/view/819}}