# The cut-tree of large recursive trees

Bertoin, Jean (2015). The cut-tree of large recursive trees. Annales de l'Institut Henri Poincaré (B) Probabilities et Statistiques, 51(2):478-488.

## 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 $n^{-1}\ln n$ ln $n$, converges in probability as $n\to\infty$ in the sense of Gromov-Hausdorff-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 (Multiple isolation of nodes in recursive trees (2013) Preprint) on multiple isolation of nodes in large random 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 $n^{-1}\ln n$ ln $n$, converges in probability as $n\to\infty$ in the sense of Gromov-Hausdorff-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 (Multiple isolation of nodes in recursive trees (2013) Preprint) on multiple isolation of nodes in large random recursive trees.

## Citations

1 citation in Web of Science®
1 citation in Scopus®

## Altmetrics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 1 May 2015 14 Jan 2016 12:53 05 Apr 2016 19:19 Elsevier 0246-0203 https://doi.org/10.1214/13-AIHP597