An Elementary Proof of Hawkes's Conjecture on Galton-Watson Trees

TL;DR

This paper offers an elementary proof of Hawkes's conjecture on the Hausdorff gauge function for the boundary of supercritical Galton-Watson trees, relaxing previous assumptions using size-biased trees.

Contribution

It provides a simpler proof of Hawkes's conjecture under weaker conditions by employing size-biased Galton-Watson trees.

Findings

Elementary proof of Hawkes's conjecture achieved

Weaker assumptions than previous proofs used

Enhanced understanding of fractal properties of Galton-Watson tree boundaries

Abstract

In 1981, J. Hawkes conjectured the exact form of the Hausdorff gauge function for the boundary of supercritical Galton-Watson trees under a certain assumption on the tail at the infinity of the total mass of the branching measure. Hawkes's conjecture has been proved by T. Watanabe in 2007 as well as other other precise results on fractal properties of the boundary of Galton-Watson trees. The goal of this paper is to provide an elementary proof of Hawkes's conjecture under a less restrictive assumption than in T. Watanabe's paper, by use of size-biased Galton-Watson trees introduced by Lyons, Pemantle and Peres in 1995.