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Convergence of the pruning processes of stable Galton-Watson trees

Gabriel Berzunza Ojeda, Anita Winter

TL;DR

The work proves that pruning processes of critical Galton-Watson trees with offspring distributions in the domain of attraction of an $α$-stable law converge, after suitable rescaling, to the pruning process of the $α$-stable Lévy tree in the leaf-sampling weak vague topology. It unifies discrete bond-percolation and site-percolation pruning with their continuous analogues by embedding both into bi-measure $\mathbb{R}$-trees and employing the LWV-topology, alongside a Skorokhod $\mathrm{J}_1$-type convergence in the path space. The key contributions include Theorem newTheo establishing the convergence of the full pruning dynamics for ske, bra, and mix pruning, and the technical development of continuity results for functionals in the Skorokhod topology, as well as a novel coding-based approach to comparing bi-measure trees. The results provide a rigorous scaling limit linking dynamic percolation on random trees to its Lévy-tree counterpart, with implications for understanding the interplay between discrete random trees and their continuous limits in pruning-type dynamics.

Abstract

Since the work of Aldous and Pitman (1998), several authors have studied the pruning processes of Galton-Watson trees and their continuous analogue Lévy trees. Löhr, Voisin and Winter (2015) introduced the space of bi-measure $\mathbb{R}$-trees equipped with the so-called leaf sampling weak vague topology which allows them to unify the discrete and the continuous picture by considering them as instances of the same Feller-continuous Markov process with different initial conditions. Moreover, the authors show that these so-called pruning processes converge in the Skorokhod space of càdlàg paths with values in the space of bi-measure $\mathbb{R}$-trees, whenever the initial bi-measure $\mathbb{R}$-trees converge. In this paper we provide an application to the above principle by verifying that a sequence of suitably rescaled critical conditioned Galton-Watson trees whose offspring distributions lie in the domain of attraction of a stable law of index $α\in (1,2]$ converge to the $α$-stable Lévy-tree in the leaf-sampling weak vague topology.

Convergence of the pruning processes of stable Galton-Watson trees

TL;DR

The work proves that pruning processes of critical Galton-Watson trees with offspring distributions in the domain of attraction of an -stable law converge, after suitable rescaling, to the pruning process of the -stable Lévy tree in the leaf-sampling weak vague topology. It unifies discrete bond-percolation and site-percolation pruning with their continuous analogues by embedding both into bi-measure -trees and employing the LWV-topology, alongside a Skorokhod -type convergence in the path space. The key contributions include Theorem newTheo establishing the convergence of the full pruning dynamics for ske, bra, and mix pruning, and the technical development of continuity results for functionals in the Skorokhod topology, as well as a novel coding-based approach to comparing bi-measure trees. The results provide a rigorous scaling limit linking dynamic percolation on random trees to its Lévy-tree counterpart, with implications for understanding the interplay between discrete random trees and their continuous limits in pruning-type dynamics.

Abstract

Since the work of Aldous and Pitman (1998), several authors have studied the pruning processes of Galton-Watson trees and their continuous analogue Lévy trees. Löhr, Voisin and Winter (2015) introduced the space of bi-measure -trees equipped with the so-called leaf sampling weak vague topology which allows them to unify the discrete and the continuous picture by considering them as instances of the same Feller-continuous Markov process with different initial conditions. Moreover, the authors show that these so-called pruning processes converge in the Skorokhod space of càdlàg paths with values in the space of bi-measure -trees, whenever the initial bi-measure -trees converge. In this paper we provide an application to the above principle by verifying that a sequence of suitably rescaled critical conditioned Galton-Watson trees whose offspring distributions lie in the domain of attraction of a stable law of index converge to the -stable Lévy-tree in the leaf-sampling weak vague topology.
Paper Structure (17 sections, 11 theorems, 84 equations, 2 figures)

This paper contains 17 sections, 11 theorems, 84 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Bi-measure $\mathbb{R}$-trees and the LWV-topology
  3. The pruning process
  4. Main results
  5. The Gromov-weak topology
  6. Galton-Watson trees and random walks
  7. Definitions
  8. Coding planar trees by discrete paths
  9. The height process.
  10. The contour process.
  11. The $\alpha$-stable Lévy tree
  12. Further properties
  13. Proof of Theorem \ref{['NewTheo']}
  14. The Brownian case, $\alpha =2$.
  15. The stable case, $\alpha \in (1,2)$.
  16. ...and 2 more sections

Key Result

Theorem 1.4

For $N \in \mathbb{N}$ and $\ast \in \{ {\rm ske}, {\rm bra}, {\rm mix}\}$, let $\mathbf{t}_{N}^{\ast} = (\mathbf{t}_{N}, a_{N} \cdot r_{N}^{{\rm gr}}, \rho_{N}, \mu_{N}^{\rm nod}, \nu_{N}^{\ast})$ be the bi-measure $\mathbb{R}$-tree associated with an $\alpha$-stable ${\rm GW}$-tree. Similarly, let

Figures (2)

  • Figure 1: From left to right, a plane tree with vertices labeled in lexicographical and reverse lexicographical order.
  • Figure 2: From left to right, the paths $W^{\uparrow}$ and $W^{\downarrow}$ of the labeled plane trees in Figure \ref{['Fig2']}.

Theorems & Definitions (24)

  • Definition 1.1: Bi-measure $\mathbb{R}$-trees LohrVoisinWinter2015
  • Definition 1.2: LWV-topology
  • Definition 1.3: THE pruning process LohrVoisinWinter2015
  • Theorem 1.4
  • Corollary 1.5
  • proof
  • Definition 2.1
  • Theorem 3.1: Al1993, Du2003, Kortchemski2013
  • Lemma 3.2: Dieu2015
  • Theorem 3.3
  • ...and 14 more