Growing Self-Similar Markov Trees

TL;DR

This paper explores how self-similar Markov trees, which generalize Brownian and stable continuum random trees, can be manipulated through cutting and growth processes, revealing new dynamics and couplings in their structure.

Contribution

It introduces a framework for understanding self-similar Markov trees and characterizes their laws, providing new insights into their growth, fragmentation, and coupling properties.

Findings

Coupling of trees into nested subtrees for specific cases.

Derivation of simple dynamics for Brownian and stable CRTs.

Connection to leaf-growth algorithms and their scaling limits.

Abstract

Can we obtain a Brownian CRT of mass $1/2$ from a CRT of mass $1$ by cutting certain branches? In this paper, we will answer that question in the much more general setting of self-similar Markov trees. Self-similar Markov trees (ssMt) are random decorated trees that encode the genealogy of a system of particles carrying positive labels, and where particles undergo splitting and growth depending on their labels in a self-similar fashion. Introduced and developed in the recent monograph (Bertoin-Curien-Riera, 2024), they provide a broad generalization of Brownian and stable continuum random trees and arise naturally in various models of random geometry such as the Brownian sphere/disk. The law of a ssMt is characterized by its quadruplet $(\mathrm{a}, \sigma^2, \boldsymbol{\Lambda}; \alpha)$, which specifies the features of the underlying growth-fragmentation mechanism, together with the initial decoration $x>0$. In this work, we focus on special cases of ssMt in which the trees started from different initial values $x>0$ can be coupled into a continuous, increasing family of nested subtrees. In the case of the Brownian and stable continuum random trees, this yields surprisingly simple novel dynamics corresponding to the scaling limit of the leaf-growth algorithms of Luczak-Winkler and Caraceni-Stauffer.