The cut-tree of large recursive trees

Bertoin, Jean

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 478-488.

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 $n^{-1}lnn$, 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.

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 $n\to \infty$ 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.

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