Central limit theorems for the spectra of classes of random fractals

TL;DR

This paper establishes central limit theorems for the spectral asymptotics of certain random fractals, including fractal boundaries and the continuum random tree, using branching process techniques.

Contribution

It introduces new central limit theorems for spectral fluctuations of random fractals generated by Dirichlet distributions, extending previous results with weaker conditions.

Findings

Almost sure second order spectral asymptotics for fractal boundaries

Central limit theorem for spectral fluctuations of the continuum random tree

Conditions for the existence of a CLT in random fractal spectra

Abstract

We discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal, the continuum random tree. In the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit. We will show examples from a class of random fractals generated from Dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem. The Brownian continuum random tree can also be viewed as a random fractal generated by a Dirichlet distribution. The first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this, though the positivity of the variance arising in the central limit theorem is left open. In both cases these fractals can be described through a general Crump-Mode-Jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics. Our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known.