Decomposition of Levy trees along their diameter

TL;DR

This paper analyzes the diameter of Lévy trees, providing a detailed decomposition along their diameter, characterizing their law, and exploring implications for stable Lévy trees conditioned on their total mass.

Contribution

It introduces a novel decomposition of Lévy trees along their diameter and characterizes the law of their diameter and height, extending known results to the Lévy case.

Findings

Law of the diameter characterized and shown to be realized by a unique pair of points.

Decomposition of Lévy trees conditioned on diameter via glueing independent trees.

Asymptotic expansions for height and diameter laws of stable Lévy trees.

Abstract

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