The cut-tree of large recursive trees

Abstract

Imagine a graph which is progressively destroyed by cutting its edges one after the other in a uniform random order. The so-called cut-tree records key steps of this destruction process. It can be viewed as a random metric space equipped with a natural probability mass. In this work, we show that the cut-tree of a random recursive tree of size n, rescaled by the factor n−1 lnn, converges in probability as n → ∞ in the sense of GromovHausdorff-Prokhorov, to the unit interval endowed with the usual distance and Lebesgue measure. This enables us to explain and extend some recent results of Kuba and Panholzer [15] on multiple isolation of nodes in random recursive trees.