Cutting edges at random in large recursive trees

TL;DR

This paper explores the process of randomly cutting edges in large recursive trees, analyzing the steps to disconnect key vertices, and connecting the process to percolation theory and cluster sizes.

Contribution

It provides new probabilistic insights into the destruction process of recursive trees and links it to Bernoulli bond percolation, with a focus on the cut-tree and component size tree frameworks.

Findings

Analysis of steps to isolate vertices in large RRTs

Probabilistic explanations via cut-tree and component size tree

Connection established between destruction process and supercritical percolation regimes

Abstract

We comment on old and new results related to the destruction of a random recursive tree (RRT), in which its edges are cut one after the other in a uniform random order. In particular, we study the number of steps needed to isolate or disconnect certain distinguished vertices when the size of the tree tends to infinity. New probabilistic explanations are given in terms of the so-called cut-tree and the tree of component sizes, which both encode different aspects of the destruction process. Finally, we establish the connection to Bernoulli bond percolation on large RRT's and present recent results on the cluster sizes in the supercritical regime.