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November 2020 Hausdorff dimension of the uniform measure of Galton–Watson trees without the XlogX condition
Elie Aïdékon
Ann. Inst. H. Poincaré Probab. Statist. 56(4): 2301-2306 (November 2020). DOI: 10.1214/19-AIHP1031

Abstract

We consider a Galton–Watson tree with offspring distribution $\nu $ of finite mean. The uniform measure on the boundary of the tree is obtained by putting mass $1$ on each vertex of the $n$-th generation and taking the limit $n\to \infty $. In the case $E[\nu \log (\nu )]<\infty $, this measure has been well studied, and it is known that the Hausdorff dimension of the measure is equal to $\log (m)$ (J. Lond. Math. Soc. (2) 24 (1981) 373–384; Ergodic Theory Dynam. Systems 15 (1995) 593–619). When $E[\nu \log (\nu )]=\infty $, we show that the dimension drops to $0$. This answers a question of Lyons, Pemantle and Peres (In Classical and Modern Branching Processes. Proceedings of the IMA Workshop (1997) 223–237 Springer).

Nous considérons un arbre de Galton–Watson dont le nombre d’enfants $\nu $ a une moyenne finie. La mesure uniforme sur la frontière de l’arbre s’obtient en chargeant chaque sommet de la $n$-ième génération avec une masse $1$, puis en prenant la limite $n\to \infty $. Dans le cas $E[\nu \log (\nu )]<\infty $, cette mesure est bien étudiée, et l’on sait que la dimension de Hausdorff de la mesure est égale à $\log (m)$ (J. Lond. Math. Soc. (2) 24 (1981) 373–384; Ergodic Theory Dynam. Systems 15 (1995) 593–619). Lorsque $E[\nu \log (\nu )]=\infty $, nous montrons que la dimension est $0$. Cela répond à une question posée par Lyons, Pemantle et Peres (In Classical and Modern Branching Processes. Proceedings of the IMA Workshop (1997) 223–237 Springer).

Citation

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Elie Aïdékon. "Hausdorff dimension of the uniform measure of Galton–Watson trees without the XlogX condition." Ann. Inst. H. Poincaré Probab. Statist. 56 (4) 2301 - 2306, November 2020. https://doi.org/10.1214/19-AIHP1031

Information

Received: 15 June 2010; Revised: 12 September 2017; Accepted: 8 October 2019; Published: November 2020
First available in Project Euclid: 21 October 2020

MathSciNet: MR4164837
Digital Object Identifier: 10.1214/19-AIHP1031

Subjects:
Primary: 28A78, 60J80

Rights: Copyright © 2020 Institut Henri Poincaré

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Vol.56 • No. 4 • November 2020
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