Self-similar Markov trees and scaling limits

TL;DR

This paper introduces self-similar Markov trees, a broad class of random real trees with self-similarity and Markov properties, unifying many known models and establishing invariance principles for related Galton–Watson trees.

Contribution

It defines self-similar Markov trees, explores their properties, and proves invariance principles linking them to Galton–Watson trees with integer types.

Findings

Encompasses Brownian CRT, stable Levy trees, and fragmentation trees.

Provides explicit spinal decompositions using length and harmonic measures.

Establishes invariance principles for Galton–Watson trees with various combinatorial classes.

Abstract

Self-similar Markov trees constitute a remarkable family of random compact real trees carrying a decoration function that is positive on the skeleton. As the terminology suggests, they are self-similar objects that further satisfy a Markov branching property. They are built from the combination of the recursive construction of real trees by gluing line segments with the seminal observation of Lamperti, which relates positive self-similar Markov processes and Levy processes via a time change. They carry natural length and harmonic measures, which can be used to perform explicit spinal decompositions. Self-similar Markov trees encompass a large variety of random real trees that have been studied over the last decades, such as the Brownian CRT, stable Levy trees, fragmentation trees, and growth-fragmentation trees. We establish general invariance principles for Galton--Watson trees with integer types and illustrate them with many combinatorial classes of random trees that have been studied in the literature.