The exact packing measure of L\'evy trees

TL;DR

This paper establishes that Le9vy trees, as scaling limits of Galton-Watson trees, possess an exact packing measure, with an explicit gauge function showing the measure aligns with the mass measure.

Contribution

It provides the first explicit computation of the packing gauge function for Le9vy trees and proves the packing measure equals the mass measure up to a constant.

Findings

Le9vy trees have an exact packing measure.

The packing gauge function is explicitly computed.

Packing measure coincides with the mass measure up to a constant.

Abstract

We study fine properties of L\'evy trees that are random compact metric spaces introduced by Le Gall and Le Jan in 1998 as the genealogy of continuous state branching processes. L\'evy trees are the scaling limits of Galton-Watson trees and they generalize Aldous's continuum random tree which corresponds to the Brownian case. In this paper we prove that L\'evy trees have always an exact packing measure: We explicitely compute the packing gauge function and we prove that the corresponding packing measure coincides with the mass measure up to a multiplicative constant.