Decomposition of Levy trees along their diameter
Abstract
We study the diameter of L\'evy trees that are random compact metric spaces obtained as the scaling limits of Galton-Watson trees. L\'evy 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 L\'evy trees and we prove that it is realized by a unique pair of points. We prove that the law of L\'evy trees conditioned to have a fixed diameter r ∈ (0, ∞) is obtained by glueing at their respective roots two independent size-biased L\'evy 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 L\'evy trees according to their diameter, we characterize the joint law of the height and the diameter of stable L\'evy 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.