Self-similar Markov trees and scaling limits

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.