## Electronic Communications in Probability

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

Thomas Duquesne

#### 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 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465234724

Digital Object Identifier
doi:10.1214/ECP.v14-1454

Mathematical Reviews number (MathSciNet)
MR2497323

Zentralblatt MATH identifier
1189.60155

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

#### Citation

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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