Spectral asymptotics for stable trees

TL;DR

This paper derives the spectral asymptotics and heat kernel behavior of $oldsymbol{ ext{α}}$-stable trees, revealing their spectral dimension and asymptotic eigenvalue distribution using self-similar fractal and spinal decomposition techniques.

Contribution

It provides the first detailed spectral asymptotics for $oldsymbol{ ext{α}}$-stable trees, including spectral dimension and eigenvalue counting function behavior.

Findings

Spectral dimension of an $ ext{α}$-stable tree is almost surely $2 ext{α}/(2 ext{α}-1)$.

The second term in the eigenvalue counting asymptotics has an exponent no greater than $1/(2 ext{α}-1)$.

Results match spectral properties of related discrete models.

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.