Record process on the Continuum Random Tree

TL;DR

This paper studies a continuous pruning process on the Brownian continuum random tree, introducing a new random variable related to the number of cuts needed to isolate the root, and shows its convergence properties.

Contribution

It constructs a new random variable for the pruning process on the continuum tree and proves its convergence as the number of leaves increases.

Findings

The variable $ heta$ matches Janson's limit distribution for cuts in Galton-Watson trees.

The number of cuts to isolate the root converges almost surely as the subtree size grows.

The pruning process on the continuum tree is rigorously characterized.

Abstract

By considering a continuous pruning procedure on Aldous's Brownian tree, we construct a random variable $\Theta$ which is distributed, conditionally given the tree, according to the probability law introduced by Janson as the limit distribution of the number of cuts needed to isolate the root in a critical Galton-Watson tree. We also prove that this random variable can be obtained as the a.s. limit of the number of cuts needed to cut down the subtree of the continuum tree spanned by $n$ leaves.