Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The cut-tree of large recursive trees

Jean Bertoin

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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 $n^{-1}\ln n$, 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.

Résumé

Imaginons la destruction progressive d’un graphe auquel on retire ses arêtes une à une dans un ordre aléatoire uniforme. Le “cut-tree” permet de coder les étapes essentielles du processus de destruction; il peut être vu comme un espace métrique aléatoire muni d’une mesure de probabilité naturelle. Dans cet article, nous montrons que le cut-tree d’un arbre récursif aléatoire de taille $n$, et renormalisé par un facteur $n^{-1}\ln n$, converge en probabilité quand $n\to\infty$ au sens de Gromov–Hausdorff–Prokhorov, vers l’intervale unité muni de la distance usuelle et de la mesure de Lebesgue. Ceci nous permet d’expliquer et d’étendre des résultats récents de Kuba and Panholzer (Multiple isolation of nodes in recursive trees (2013) Preprint) sur l’isolation multiple de sommets dans un grand arbre récursif aléatoire.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 2 (2015), 478-488.

Dates
First available in Project Euclid: 10 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1428672677

Digital Object Identifier
doi:10.1214/13-AIHP597

Mathematical Reviews number (MathSciNet)
MR3335011

Zentralblatt MATH identifier
1351.60010

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F15: Strong theorems

Keywords
Random recursive tree Destruction of graphs Gromov–Hausdorff–Prokhorov convergence Multiple isolation of nodes

Citation

Bertoin, Jean. The cut-tree of large recursive trees. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 2, 478--488. doi:10.1214/13-AIHP597. https://projecteuclid.org/euclid.aihp/1428672677


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