$k$-cut model for the Brownian Continuum Random Tree

TL;DR

This paper studies the $k$-cut model on random trees, showing that the scaled number of cuts needed to isolate the root converges to a limit related to the Brownian CRT, with a new construction using fragmentation processes.

Contribution

It provides a direct construction of the limit distribution for the $k$-cut model on Galton--Watson trees, extending previous convergence results.

Findings

The scaled number of cuts converges to a limit distribution.

A new construction of the limit variable using Aldous-Pitman fragmentation.

Connection established between $k$-cut model and Brownian CRT.

Abstract

To model the destruction of a resilient network, Cai, Holmgren, Devroye and Skerman introduced the $k$-cut model on a random tree, as an extension to the classic problem of cutting down random trees. Berzunza, Cai and Holmgren later proved that the total number of cuts in the $k$-cut model to isolate the root of a Galton--Watson tree with a finite-variance offspring law and conditioned to have $n$ nodes, when divided by $n^{1-1/2k}$, converges in distribution to some random variable defined on the Brownian CRT. We provide here a direct construction of the limit random variable, relying upon the Aldous-Pitman fragmentation process and a deterministic time change.