A binary embedding of the stable line-breaking construction

TL;DR

This paper constructs a binary continuum random tree embedding the stable tree, solving an open problem, and expresses it through recursive methods and Ford CRT replacements, advancing the understanding of complex random tree structures.

Contribution

It introduces a novel binary embedding of the stable tree into a CRT using recursive construction and bead distributions, addressing an open problem in the field.

Findings

Embedded stable tree into a binary CRT

Expressed the CRT via recursive construction and Ford CRTs

Solved an open problem in stable tree embeddings

Abstract

We embed Duquesne and Le Gall's stable tree into a binary compact continuum random tree (CRT) in a way that solves an open problem posed by Goldschmidt and Haas. This CRT can be obtained by applying a recursive construction method of compact CRTs as presented in earlier work to a specific distribution of a random string of beads, i.e. a random interval equipped with a random discrete measure. We also express this CRT as a tree built by replacing all branch points of a stable tree by rescaled i.i.d. copies of a Ford CRT. Some of these developments are carried out in a space of infinity-marked metric spaces generalising Miermont's notion of a k-marked metric space.