k-cut model for the Brownian continuum random tree

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.