Decomposition of Lévy trees along their diameter

Duquesne, Thomas and Wang, Minmin (2017) Decomposition of Lévy trees along their diameter. Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, 53 (2). pp. 539-593. ISSN 0246-0203

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Abstract

We study the diameter of Lévy trees that are random compact metric spaces obtained as the scaling limits of Galton–Watson trees. Lévy trees have been introduced by Le Gall & Le Jan (Ann. Probab. 26 (1998) 213–252) and they generalise Aldous’ Continuum Random Tree (1991) that corresponds to the Brownian case. We first characterize the law of the diameter of Lévy trees and we prove that it is realized by a unique pair of points. We prove that the law of Lévy trees conditioned to have a fixed diameter r ∈ (0, ∞) is obtained by glueing at their respective roots two independent size-biased Lévy trees conditioned to have height r/2 and then by uniformly re-rooting the resulting tree; we also describe by a Poisson point measure the law of the subtrees that are grafted on the diameter. As an application of this decomposition of Lévy trees according to their diameter, we characterize the joint law of the height and the diameter of stable Lévy trees conditioned by their total mass; we also provide asymptotic expansions of the law of the height and of the diameter of such normalised stable trees, which generalises the identity due to Szekeres (In Combinatorial Mathematics, X (Adelaide, 1982) (1983) 392–397 Springer) in the Brownian case.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Probability and Statistics Research Group
Depositing User: Minmin Wang
Date Deposited: 11 Feb 2019 15:06
Last Modified: 21 Oct 2019 14:30
URI: http://sro.sussex.ac.uk/id/eprint/81894

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