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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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:1 May 2015
Deposited On:14 Jan 2016 12:53
Last Modified:05 Apr 2016 19:19
Publisher:Elsevier
ISSN:0246-0203
Publisher DOI:https://doi.org/10.1214/13-AIHP597

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