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Rotating random trees with Skorokhod's $M_1$ topology

Antoine Aurillard

Abstract

We extend the classical coding of measured $\mathbb R$-trees by continuous excursion-type functions to càdlàg excursion-type functions through the notion of parametric representations. The main feature of this extension is its continuity properties with respect to the Gromov-Hausdorff-Prokhorov topology for $\mathbb R$-trees and Skorokhod's $M_1$ topology for càdlàg functions. As a first application, we study the $\mathbb R$-trees $\mathcal T_{x^{(α)}}$ encoded by excursions of spectrally positive $α$-stable Lévy processes for $α\in (1,2]$. In a second time, we use this setting to study the large-scale effects of a well-known bijection between plane trees and binary trees, the so-called rotation. Marckert has proved that the rotation acts as a dilation on large uniform trees, and we show that this remains true when the rotation is applied to large critical Bienaymé trees with offspring distribution attracted to a Gaussian distribution. However, this does not hold anymore when the offspring distribution falls in the domain of attraction of an $α$-stable law with $α\in (1,2)$, and instead we prove that the scaling limit of the rotated trees is $\mathcal T_{x^{(α)}}$.

Rotating random trees with Skorokhod's $M_1$ topology

Abstract

We extend the classical coding of measured -trees by continuous excursion-type functions to càdlàg excursion-type functions through the notion of parametric representations. The main feature of this extension is its continuity properties with respect to the Gromov-Hausdorff-Prokhorov topology for -trees and Skorokhod's topology for càdlàg functions. As a first application, we study the -trees encoded by excursions of spectrally positive -stable Lévy processes for . In a second time, we use this setting to study the large-scale effects of a well-known bijection between plane trees and binary trees, the so-called rotation. Marckert has proved that the rotation acts as a dilation on large uniform trees, and we show that this remains true when the rotation is applied to large critical Bienaymé trees with offspring distribution attracted to a Gaussian distribution. However, this does not hold anymore when the offspring distribution falls in the domain of attraction of an -stable law with , and instead we prove that the scaling limit of the rotated trees is .

Paper Structure

This paper contains 49 sections, 28 theorems, 108 equations, 20 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Skorokhod's $M_1$ topology and (measured) $\mathds R$-trees.
  3. Large-scale effects of the rotation on Bienaymé trees.
  4. Main ideas of the proof.
  5. Properties of $\mathscr T_{\mathbbm x}$ and links with transformations related to the rotation.
  6. Acknowledgement.
  7. Outline.
  8. Basics on finite plane trees
  9. Ulam-Harris-Neveu formalism.
  10. Large critical Bienaymé trees.
  11. Encoding processes.
  12. $\mathds R$-trees and contour functions
  13. Standard setting for scaling limits of trees
  14. The Gromov-Hausdorff and Gromov-Hausdorff-Prokhorov distances.
  15. $\mathds R$-trees.
  16. ...and 34 more sections

Key Result

Proposition 1.1

For all $x,y\in D_0([0,1],\mathds R_+)$,

Figures (20)

  • Figure 1: Recursive definition of $\mathop{\mathrm{Rot}}\nolimits$
  • Figure 2: Realisations of $\mathop{\mathrm{Rot}}\nolimits \mathcal{T}_{5000}$ for several values of $\alpha$: on the left $\alpha=1.01$, in the middle $\alpha=1.5$, on the right $\alpha=1.8$.
  • Figure 3: A plane tree (as a subtree of $\mathcal{U}$). Its lexicographical enumeration is $u_0=\varnothing,u_1=1,u_2=11,u_3=12, u_4=2, u_5=3,u_6=31$.
  • Figure 4: Height, contour and Łukasiewicz processes of the tree depicted in Figure \ref{['fig:lexicoEnumeration']}. Each of them fully characterizes this tree.
  • Figure 5: The completed graph of $x=\mathbbm 1_{[1/2,1]}$ on the left, and one of its parametric representations on the right.
  • ...and 15 more figures

Theorems & Definitions (70)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Definition 3.1
  • Remark
  • Remark
  • Lemma 3.2: Legall2005RandomTreesApplications
  • ...and 60 more