DECOMPOSITION OF LEVY TREES ALONG THEIR DIAMETER

Thomas Duquesne (1) , Minmin Wang (2)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 and Le Jan (1998) 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 (1983) in the Brownian case.

Mots clés: Lévy trees; height process; diameter; decomposition; asymptotic expansion; stable law

Domaines: Probabilités [math.PR]

Thomas Duquesne, Minmin Wang. DECOMPOSITION OF LEVY TREES ALONG THEIR DIAMETER. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2017, ⟨10.1214/15-AIHP725⟩. ⟨hal-01132231⟩