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Scaling limit of trees with vertices of fixed degrees and heights

Arthur Blanc-Renaudie, Emmanuel Kammerer

Abstract

We consider large uniform random trees where we fix for each vertex its degree and height. We prove, under natural conditions of convergence for the profile, that those trees properly renormalized converge. To this end, we study the paths from random vertices to the root using coalescent processes. As an application, we obtain scaling limits of Bienaymé-Galton-Watson trees in varying environment.

Scaling limit of trees with vertices of fixed degrees and heights

Abstract

We consider large uniform random trees where we fix for each vertex its degree and height. We prove, under natural conditions of convergence for the profile, that those trees properly renormalized converge. To this end, we study the paths from random vertices to the root using coalescent processes. As an application, we obtain scaling limits of Bienaymé-Galton-Watson trees in varying environment.
Paper Structure (9 sections, 4 theorems, 7 equations, 2 figures)

This paper contains 9 sections, 4 theorems, 7 equations, 2 figures.

Table of Contents

  1. Acknowledgements.
  2. Introduction
  3. Model
  4. Main results
  5. Connections with previous works and applications
  6. Plan of the paper:
  7. Definition of the limit
  8. Description of the distance matrix between uniform vertices in the limit tree
  9. Leaf-tightness

Key Result

Theorem 1.2

Assume that for some $\nu, \rho,\Theta$ the conditions hyp:naissance, hyp:coalescence, hyp:atomes, hyp:splitatomes, hyp:tightGP hold. Then there exists a random measured metric space $\mathcal{T}(\nu,\rho,\Theta)$ such that the following weak convergence holds for the Gromov--Prokhorov topology (see where we recall that $\mathcal{T}^n/n$ is the metric space obtained by dividing the shortest path d

Figures (2)

  • Figure 1: Simulation of a uniform random tree of height $100$ whose profile scales towards $t \mapsto \sin( \pi t)$ with vertices of degrees $0$ and $4$ with some additional vertices with large degrees. The profile is drawn on the right. The low vertices are green while the high vertices are red.
  • Figure 2: An example of tree $\mathcal{T}$ such that $d_{0,1}=4$, $d_{1,1}=3$, $d_{1,2}=1$, $d_{1,3}=1$, $d_{2,1}=2$, $d_{2,2}=1$, $d_{3,1}=2$ and all the other degrees are null. The vertices of $\mathcal{T}$ are represented in red, the root as a red dot, and the other vertices as black dots. Here, $D_0=4, D_1=5, D_2=3, D_3=2, D_4=0$.

Theorems & Definitions (8)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof