Growing Self-Similar Markov Trees

TL;DR

The paper develops a general theory of growing self-similar Markov trees (ssMt), showing that when a growth condition on the splitting mechanism is satisfied, one can construct continuous, increasing couplings of trees across starting masses. It reduces the problem to a flow of decoration-reproduction processes driven by SDEs with jumps and exploits a generator framework to verify growth via a divergence-PDE, enabling canonical couplings for Brownian CRTs and β-stable trees, among others. Key contributions include the growing theorem, explicit constructions for Brownian CRTs with locally largest and size-biased bifurcators, and a unified generator-based approach that applies to a wide class of fragmentation trees, tying together mass and height decorations and linking to leaf-growth scaling limits. The results provide new, intrinsic dynamics for scaling limits of random trees and offer pathways to analyze monotonicity, continuity, and Markov properties in a broad continuum of self-similar random geometries.

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 $(\mathrm{a}, σ^2, \boldsymbolΛ; α)$, 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.

Figures (12)

• Figure 1: Illustration of the increasing family of ssMt in the case of the Brownian CRT: the underlying Brownian CRT of mass $1$ is displayed as the base tree, whereas its subtrees of mass $0.3 , 0.6$ and $1$ are depicted as hypographs.
• Figure 2: Illustration of the existence of the growing functions $G$ preserving the splitting intensity and obeying a monotonicity condition. The individual correspond to the particles with heavier lines.
• Figure 3: Illustration of Markov property of the construction. We get $T_x$ from $T_{x'}$ by gluing on $p_{x'}(i)$ independent ssMt with starting decorations $w_{x' \to x}(i)$. The points $p_{x'}(i)$ and the weights $w_{x' \to x}(i)$ depend on $\mathtt{T}_{x'}$ and on its characteristics, i.e. on the bifurcator chosen to construct it.
• Figure 4: Plot of the function $\mathtt{f}_x$ for different values of $x$. Although explicit, this family of functions is not particularly obvious to guess! Notice in particular that if $x<1$ then $\mathtt{f}_x(u)$ is closer to $1/2$ compared to $u$.
• Figure 5: Illustration of the leaf-growth algorithm producing a sequence of increasing uniform binary trees which converge in the scaling limit towards the locally largest growing coupling of Brownian CRT's.

Theorems & Definitions (53)

• Theorem 1: Growing self-similar Markov trees, informal
• Definition 2.1: $G$-Growing
• Lemma 1: Upper bound on $\alpha$
• proof
• Lemma 2: Critical self-similarity
• proof
• Remark
• Theorem 2: Baby version of Theorem \ref{['thm:main']} for a single branch
• proof : Proof of Theorem \ref{['thm:deco-repro-grow']}
• Proposition 2.1