Self-similarity and spectral asymptotics for the continuum random tree

TL;DR

This paper explores the self-similar structure of the continuum random tree, establishing its homeomorphism to a fractal with a random metric, and analyzes the spectral properties and heat kernel asymptotics of the associated Dirichlet form.

Contribution

It demonstrates the continuum random tree's homeomorphism to a self-similar fractal and characterizes its spectral asymptotics and heat kernel behavior.

Findings

Determined the asymptotic behavior of the eigenvalue counting function.

Established the homeomorphism to a self-similar fractal with a random metric.

Derived short time asymptotics for the heat kernel.

Abstract

We use the random self-similarity of the continuum random tree to show that it is homeomorphic to a post-critically finite self-similar fractal equipped with a random self-similar metric. As an application we determine 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 the continuum random tree. We also obtain short time asymptotics for the trace of the heat semigroup and the annealed on-diagonal heat kernel associated with this Dirichlet form.