Rotating random trees with Skorokhod's $M_1$ topology

TL;DR

This paper extends the coding of measured $ ext{R}$-trees to c ext{a}dl ext{a}g excursion functions using parametric representations, studying their topological properties and applications to stable Lévy processes and tree rotations.

Contribution

It introduces a new continuous coding framework for c ext{a}dl ext{a}g excursions and analyzes the effects of tree rotations under different offspring distributions.

Findings

The new coding is continuous with respect to Gromov-Hausdorff-Prokhorov and Skorokhod's $M_1$ topologies.

Rotation acts as a dilation on large uniform and Gaussian-atracted trees.

For $ ext{α}$-stable offspring distributions, the rotated trees converge to a new $ ext{R}$-tree $ ext{T}_{x^{( ext{α})}}$.

Abstract

We extend the classical coding of measured $\mathbb R$-trees by continuous excursion-type functions to c\`adl\`ag 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 c\`adl\`ag functions. As a first application, we study the $\mathbb R$-trees $\mathcal T_{x^{(\alpha)}}$ encoded by excursions of spectrally positive $\alpha$-stable L\'evy processes for $\alpha \in (1,2]$. 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 Bienaym\'e 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 $\alpha$-stable law with $\alpha \in (1,2)$, and instead we prove that the scaling limit of the rotated trees is $\mathcal T_{x^{(\alpha)}}$.