A line-breaking construction of the stable trees

TL;DR

This paper introduces a new, straightforward line-breaking method to construct the $ ext{alpha}$-stable trees for $ ext{alpha} ext{ in } (1,2]$, generalizing Aldous' construction of the Brownian CRT.

Contribution

It presents a novel inductive line-breaking construction for $ ext{alpha}$-stable trees, extending known methods for the Brownian case to a broader class of stable trees.

Findings

Provides a simple, explicit construction for $ ext{alpha}$-stable trees.

Recovers Aldous' construction for $ ext{alpha} = 2$ (Brownian CRT).

Connects the construction to Markov chain increments.

Abstract

We give a new, simple construction of the $\alpha$-stable tree for $\alpha \in (1,2]$. We obtain it as the closure of an increasing sequence of $\mathbb{R}$-trees inductively built by gluing together line-segments one by one. The lengths of these line-segments are related to the the increments of an increasing $\mathbb{R}_+$-valued Markov chain. For $\alpha = 2$, we recover Aldous' line-breaking construction of the Brownian continuum random tree based on an inhomogeneous Poisson process.