Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence of trees with a given degree sequence and of their associated laminations

Gabriel Berzunza Ojeda, Cecilia Holmgren, Paul Thévenin

TL;DR

The paper analyzes scaling limits for uniform rooted plane trees with prescribed degree sequences (TGDS), proving convergence to an Inhomogeneous CRT $igl( obreak ext{T}_{ heta}, r_{ heta}, ho_{ heta}, obreak obreak obreak obreak obreak obreak ext{-} obreak )igr$ after renormalisation, with the limiting object characterized by a parameter sequence $ heta$. The approach hinges on a modified Lukasiewicz path and its excursion-height representation, linking discrete encodings to the exploration process of the Inhomogeneous CRT. The authors develop a comprehensive lamination framework that codes the tree fragmentation via non-crossing chords in the unit disk, proving an equivalence between planar Gromov-weak convergence of discrete trees and convergence of their lamination-valued processes, both in discrete and continuum settings. In a specific regime, they obtain a Gromov–Hausdorff–Prohorov convergence to the Inhomogeneous CRT, and they show that the fragmentation and the associated mass processes converge to Bertoin’s fragmentation of the Inhomogeneous CRT, with the mass dynamics encoded by the lamination faces. Overall, the work unifies discrete tree scaling limits, lamination theory, and fragmentation processes, providing a robust bridge from TGDS to continuum fracturing structures with practical implications for the study of random planar maps and related combinatorial models.

Abstract

In this paper, we study uniform rooted plane trees with given degree sequence. We show, under some natural hypotheses on the degree sequence, that these trees converge toward the so-called Inhomogeneous Continuum Random Tree after renormalisation. Our proof relies on the convergence of a modification of the well-known Lukasiewicz path. We also give a unified treatment of the limit, as the number of vertices tends to infinity, of the fragmentation process derived by cutting-down the edges of a tree with a given degree sequence, including its geometric representation by a lamination-valued process. The latter is a collection of nested laminations that are compact subsets of the unit disk made of non-crossing chords. In particular, we prove an equivalence between Gromov-weak convergence of discrete trees and the convergence of their associated lamination-valued processes.

Convergence of trees with a given degree sequence and of their associated laminations

TL;DR

The paper analyzes scaling limits for uniform rooted plane trees with prescribed degree sequences (TGDS), proving convergence to an Inhomogeneous CRT after renormalisation, with the limiting object characterized by a parameter sequence . The approach hinges on a modified Lukasiewicz path and its excursion-height representation, linking discrete encodings to the exploration process of the Inhomogeneous CRT. The authors develop a comprehensive lamination framework that codes the tree fragmentation via non-crossing chords in the unit disk, proving an equivalence between planar Gromov-weak convergence of discrete trees and convergence of their lamination-valued processes, both in discrete and continuum settings. In a specific regime, they obtain a Gromov–Hausdorff–Prohorov convergence to the Inhomogeneous CRT, and they show that the fragmentation and the associated mass processes converge to Bertoin’s fragmentation of the Inhomogeneous CRT, with the mass dynamics encoded by the lamination faces. Overall, the work unifies discrete tree scaling limits, lamination theory, and fragmentation processes, providing a robust bridge from TGDS to continuum fracturing structures with practical implications for the study of random planar maps and related combinatorial models.

Abstract

In this paper, we study uniform rooted plane trees with given degree sequence. We show, under some natural hypotheses on the degree sequence, that these trees converge toward the so-called Inhomogeneous Continuum Random Tree after renormalisation. Our proof relies on the convergence of a modification of the well-known Lukasiewicz path. We also give a unified treatment of the limit, as the number of vertices tends to infinity, of the fragmentation process derived by cutting-down the edges of a tree with a given degree sequence, including its geometric representation by a lamination-valued process. The latter is a collection of nested laminations that are compact subsets of the unit disk made of non-crossing chords. In particular, we prove an equivalence between Gromov-weak convergence of discrete trees and the convergence of their associated lamination-valued processes.
Paper Structure (27 sections, 25 theorems, 134 equations, 3 figures)

This paper contains 27 sections, 25 theorems, 134 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Scaling limits of trees
  3. Fragmentations and laminations
  4. Organization of the paper
  5. Notation.
  6. Plane trees and their encoding paths
  7. $\L$ukasiewicz path, reverse-$\L$ukasiewicz path and Prim path.
  8. Height process.
  9. Contour function.
  10. Convergence of the trees with given degree sequences
  11. The exploration process of the Inhomogeneous CRT
  12. The modified $\L$ukasiewicz path
  13. Gromov-Hausdorff-Prohorov convergence of a specific model of TGDS
  14. Lamination-valued processes
  15. The set of laminations
  16. ...and 12 more sections

Key Result

Theorem 1

For $n \in \mathbb{N}$, let $\mathbf{s}_{n} = (N_{i}^{n}, i \geq 0)$ be a degree sequence. Let $(d^{n}(i), 1 \leq i \leq V_{n})$ denote the associated child sequence, obtained by writing $N_{0}^{n}$ zeros, $N_{1}^{n}$ ones, etc., and ordering the resulting sequence decreasingly. Assume that there ex Then, for all $q \geq 1$, we have that where $\mathcal{T}_{\theta}$ is an Inhomogeneous CRT with p

Figures (3)

  • Figure 1: Simulation of an Inhomogeneous CRT $\mathcal{T}_\theta$ and its associated lamination $\mathbb{L}(\mathcal{T}_\theta)$, for $\theta = (1/\sqrt{7},2/\sqrt{7}, 1/\sqrt{7}, 1/\sqrt{7}, 0, \ldots)$.
  • Figure 2: A tree $\tau$, its contour function $C_{\tau}$, and the associated lamination $\mathbb{L}(\tau)$.
  • Figure 3: A tree $\tau$ with $4$ marked vertices, and the reduced tree $\tau^{(4)}$ with its edge lengths.

Theorems & Definitions (60)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof : Proof of Theorem \ref{['Theo6']}
  • Theorem 5
  • proof : Proof of Theorem \ref{['Theo2']}
  • ...and 50 more