Electronic Communications in Probability

An elementary proof of Hawkes's conjecture on Galton-Watson trees.

Thomas Duquesne

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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 infinity of the total mass of the branching measure. Hawkes's conjecture has been proved by T. Watanabe in 2007 as well as 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.

Article information

Electron. Commun. Probab., Volume 14 (2009), paper no. 15, 151-164.

Accepted: 19 April 2009
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 28A78: Hausdorff and packing measures

Galton-Watson tree exact Hausdorff measure boundary branching measure size-biased tree

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Duquesne, Thomas. An elementary proof of Hawkes's conjecture on Galton-Watson trees. Electron. Commun. Probab. 14 (2009), paper no. 15, 151--164. doi:10.1214/ECP.v14-1454. https://projecteuclid.org/euclid.ecp/1465234724

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