The stable trees revisited
Christina Goldschmidt, Liam Hill
TL;DR
This work provides a novel line-breaking representation for the $\alpha$-stable continuum random tree ($1<\alpha<2$) using a measure-changed $(\alpha-1)$-stable subordinator to drive random, jump-weighted attachments. It also delivers a new, self-contained discrete-to-continuum proof of the scaling limit for Bienaymé trees in the α-stable domain, avoiding the height process and linking to Wang’s construction via the common jump structure. The results unify line-breaking methods with discrete random trees, establish compactness and a natural mass measure on the limit, and pave the way for generalizations to broader Lévy trees. Overall, the paper deepens the connection between discrete conditioned trees and their continuous stable-tree limits through a robust line-breaking framework.
Abstract
We introduce a new, relatively simple, line-breaking construction of the $α$-stable tree which realises its random finite-dimensional distributions. This is a direct analogue of Aldous' line-breaking construction of the Brownian continuum random tree, which is based on an inhomogeneous Poisson process. Here, we replace the deterministic rate function from the Brownian setting by a random rate process, given by a certain measure-changed $(α-1)$-stable subordinator. Rather than attaching uniformly, the line-segments now connect to locations chosen with probability proportional to the sizes of the jumps of the rate process. We also give a new proof of an invariance principle originally due to Duquesne, which states that the family tree of a Bienaymé branching process with critical offspring distribution in the domain of attraction of an $α$-stable law (for $α\in (1,2))$, conditioned to have $n$ vertices, converges on rescaling distances appropriately to the $α$-stable tree. Our proof makes use of a discrete line-breaking construction of the branching process tree, which we show converges to our continuous line-breaking construction.