Stable trees as mixings of inhomogeneous continuum random trees

TL;DR

This paper proves that all stable Lévy trees can be represented as mixtures of inhomogeneous continuum random trees, using a novel method to recover tree distances from graphical spanning trees.

Contribution

It provides a rigorous proof of the claim that stable Lévy trees are mixtures of inhomogeneous continuum random trees, introducing a new distance recovery procedure.

Findings

Stable Lévy trees are mixtures of inhomogeneous continuum random trees.

A new method for recovering tree distances from graphical spanning trees.

The proof applies to stable trees and inhomogeneous continuum random trees.

Abstract

It has been claimed in Aldous, Miermont and Pitman [PTRF, 2004] that all L\'evy trees are mixings of inhomogeneous continuum random trees. We give a rigorous proof of this claim in the case of a stable branching mechanism, relying on a new procedure for recovering the tree distance from the graphical spanning trees that works simultaneously for stable trees and inhomogeneous continuum random trees.