Regenerative tree growth: structural results and convergence

TL;DR

This paper introduces regenerative tree growth processes, providing a structural representation via dislocation measures and establishing conditions for their scaling limits, extending previous models to include non-exchangeable trees.

Contribution

It develops a unified framework for regenerative tree growth, including non-exchangeable models, and characterizes their growth rules through dislocation measures, enabling new scaling limit results.

Findings

Representation of growth rules via dislocation measures

Necessary and sufficient conditions for scaling limits

Extension of previous models to non-exchangeable trees

Abstract

We introduce regenerative tree growth processes as consistent families of random trees with n labelled leaves, n>=1, with a regenerative property at branch points. This framework includes growth processes for exchangeably labelled Markov branching trees, as well as non-exchangeable models such as the alpha-theta model, the alpha-gamma model and all restricted exchangeable models previously studied. Our main structural result is a representation of the growth rule by a sigma-finite dislocation measure kappa on the set of partitions of the natural numbers extending Bertoin's notion of exchangeable dislocation measures from the setting of homogeneous fragmentations. We use this representation to establish necessary and sufficient conditions on the growth rule under which we can apply results by Haas and Miermont for unlabelled and not necessarily consistent trees to establish self-similar random trees and residual mass processes as scaling limits. While previous studies exploited some form of exchangeability, our scaling limit results here only require a regularity condition on the convergence of asymptotic frequencies under kappa, in addition to a regular variation condition.