Self-similar Markov trees and scaling limits

TL;DR

This work constructs and analyzes self-similar Markov trees (ssMt), a broad class of decorated real trees built by recursive gluing of line segments whose decorations evolve as positive self-similar Markov processes via Lamperti-type time changes. The authors develop a rigorous topological framework for decorated trees, link ssMt to general branching processes with real types, and introduce a cumulant-based description via a characteristic quadruplet (σ^2, a, Λ; α). They establish finite weighted-length and harmonic measures, prove a first Cramér condition for the harmonic mass, and derive spine decompositions that isolate a distinguished spine with conditionally independent ssMt hanging from it. The invariance principles show that multi-type Galton–Watson trees converge to ssMt under appropriate scaling, with concrete illustrations including Brownian CRT, stable trees, fragmentation trees, and Brownian growth-fragmentation trees, and connections to random planar maps and Liouville quantum gravity. The framework yields deep geometric and probabilistic tools—Hausdorff dimensions, spine decompositions, and intrinsic martingales—to study scaling limits and geometry in random trees and related planar structures.

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.

Figures (50)

• Figure 1: Illustration of a self-similar Markov tree (embedded in the plane $\mathbb{R}^{2}$) where its decoration function is represented in the third (vertical) dimension.
• Figure 2: Illustration of the Markov property (left) and the self-similar property (right). The color of the root of a tree is meant for its decoration. Left: Conditionally on the black structure of the tree up to height $h$ and on the decoration (colors) of the root vertices, the dangling subtrees (in gray) are independent.
• Figure 3: A simulation of a Brownian CRT. The tree is embedded (non-isometrically in $\mathbb{R}^{2}$) and the decoration function representing the $\upmu$-mass above each point is depicted in the vertical coordinate.
• Figure 4: A simulation of a Brownian CRT normalized by the height. The tree is embedded non-isometrically in $\mathbb{R}^{2}$; the decoration function represents the height of fringe subtrees and is depicted in the vertical coordinate.
• Figure 5: The decorated random tree $\mathcal{T}_{{\mathrm{e}}}$ associated with a half-planar Brownian excursion. The tree is embedded non-isometrically in $\mathbb{R}^{2}$; the decoration function represents the horizontal $X$ displacement in fringe subtrees and is depicted in the vertical coordinate.

Theorems & Definitions (206)

• Definition 1.1
• Lemma 1.2
• proof
• Lemma 1.3
• proof
• Lemma 1.4
• Theorem 1.5
• proof
• Remark 1.6
• proof : Proof of Lemma \ref{['L:newL3']}