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An elementary proof of Hawkes’s conjecture on Galton-Watson trees. (English) Zbl 1189.60155

Summary: In 1981, J. Hawkes [J. Lond. Math. Soc., II. Ser. 24, 373–384 (1981; Zbl 0468.60081)] 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 [Ann. Probab. 35, No. 3, 1007–1038 (2007; Zbl 1127.60083)] 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 R. Lyons, R. Pemantle and Y. Peres in [Ann. Probab. 23, No. 3, 1125–1138 (1995; Zbl 0840.60077)].

MSC:

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