Exceptionally small balls in stable trees

Duquesne, Thomas; Wang, Guanying

doi:10.24033/bsmf.2664

Exceptionally small balls in stable trees
[Boules exceptionnellement petites dans les arbres stables]

Duquesne, Thomas ; Wang, Guanying

Bulletin de la Société Mathématique de France, Tome 142 (2014) no. 2, pp. 223-254

Abstract (VO)
Résumé

The $γ$ -stable trees are random measured compact metric spaces that appear as the scaling limit of Galton-Watson trees whose offspring distribution lies in a $γ$ -stable domain, $γ \in (1, 2]$ . They form a specific class of Lévy trees (introduced by Le Gall and Le Jan in [24]) and the Brownian case $γ = 2$ corresponds to Aldous Continuum Random Tree (CRT). In this paper, we study fine properties of the mass measure, that is the natural measure on $γ$ -stable trees. We first discuss the minimum of the mass measure of balls with radius $r$ and we show that this quantity is of order $r^{\frac{γ}{γ - 1}} {(log 1 / r)}^{- \frac{1}{γ - 1}}$ . We think that no similar result holds true for the maximum of the mass measure of balls with radius $r$ , except in the Brownian case: when $γ = 2$ , we prove that this quantity is of order $r^{2} log 1 / r$ . In addition, we compute the exact constant for the lower local density of the mass measure (and the upper one for the CRT), which continues previous results from [9, 10, 13].

Les arbres $γ$ -stables sont des espaces métriques compacts à mesure aléatoire qui apparaissent en tant que limite de mise à l'échelle d'arbres de Galton-Watson dont les distributions sont situées dans un domaine $γ$ -stable, $γ \in] 1, 2]$ . Ils forment une class spécifique des arbres de Lévy (introduite par Le Gall et Le Jan dans [24]) et le cas brownien $γ = 2$ correspond aux arbres aléatoires du continuum d'Aldous (CRT). Dans cet article nous étudions les propriétés fines de la mesure de masse, qui est la mesure naturelle des arbres $γ$ -stables. Nous discutons d'abord le minimum de la mesure de masse des boules de rayon $r$ et nous montrons que cette quantité est de l'ordre de $r^{\frac{γ}{γ - 1}} {(log 1 / r)}^{- \frac{1}{γ - 1}}$ . Nous pensons qu'aucun résultat similaire n'est vrai pour le maximum des mesures de masse de boule de rayon $r$ , sauf dans le cas brownien : quand $γ = 2$ nous montrons que cette quantité est de l'ordre de $r^{2} log 1 / r$ . D'autre part, nous calculons la constante exacte de la densité local inférieure de la mesure de masse (et la supérieure pour le CRT), à la suite de résultats précédents de [9, 10, 13].

Publié le : 2014-07-21

MR Zbl

DOI : 10.24033/bsmf.2664

Classification : 60G57, 60J80, 28A78
Keywords: Continuum random tree, Lévy trees, stable trees, mass measure, small balls.
Mots-clés : Arbre aléatoire du continuum, arbres de Lévy, arbres stables, mesure de masse, petites boules.

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     author = {Duquesne, Thomas and Wang, Guanying},
     title = {Exceptionally small balls in stable trees},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {223--254},
     year = {2014},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {142},
     number = {2},
     doi = {10.24033/bsmf.2664},
     mrnumber = {3269345},
     zbl = {1327.60168},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2664/}
}

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Duquesne, Thomas; Wang, Guanying. Exceptionally small balls in stable trees. Bulletin de la Société Mathématique de France, Tome 142 (2014) no. 2, pp. 223-254. doi: 10.24033/bsmf.2664

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