Recursive construction of continuum random trees

TL;DR

This paper presents a recursive method for constructing continuum random trees (CRTs) from independent random structures, proving their existence and applying the method to various tree growth processes and genealogical models.

Contribution

It introduces a general recursive construction technique for CRTs, including new models for genealogies of growth fragmentations and applications to embedding problems.

Findings

Established existence of CRTs via recursive distribution equations.

Provided new constructions for self-similar CRTs and genealogical trees.

Solved an open problem on binary embedding of stable line-breaking CRTs.

Abstract

We introduce a general recursive method to construct continuum random trees (CRTs) from independent copies of a random string of beads, that is, any random interval equipped with a random discrete probability measure, and from related structures. We prove the existence of these CRTs as a new application of the fixpoint method for recursive distribution equations formalised in high generality by Aldous and Bandyopadhyay. We apply this recursive method to show the convergence to CRTs of various tree growth processes. We note alternative constructions of existing self-similar CRTs in the sense of Haas, Miermont and Stephenson, and we give for the first time constructions of random compact R-trees that describe the genealogies of Bertoin's self-similar growth fragmentations. In forthcoming work, we develop further applications to embedding problems for CRTs, providing a binary embedding of the stable line-breaking construction that solves an open problem of Goldschmidt and Haas.