The cut-tree of large recursive trees
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, p. 478-488

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 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.

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 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.

DOI : https://doi.org/10.1214/13-AIHP597
Classification:  60D05,  60F15
Keywords: random recursive tree, destruction of graphs, Gromov–Hausdorff–Prokhorov convergence, multiple isolation of nodes
@article{AIHPB_2015__51_2_478_0,
     author = {Bertoin, Jean},
     title = {The cut-tree of large recursive trees},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {2},
     year = {2015},
     pages = {478-488},
     doi = {10.1214/13-AIHP597},
     zbl = {1351.60010},
     mrnumber = {3335011},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_2_478_0}
}
Bertoin, Jean. The cut-tree of large recursive trees. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, pp. 478-488. doi : 10.1214/13-AIHP597. http://www.numdam.org/item/AIHPB_2015__51_2_478_0/

[1] L. Addario-Berry, N. Broutin and C. Holmgren. Cutting down trees with a Markov chainsaw. Ann. Appl. Probab. 24 (2014) 2297–2339. | MR 3262504 | Zbl 06371851

[2] J. Bertoin. Fires on trees. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 909–921. | Numdam | MR 3052398 | Zbl 1263.60083

[3] J. Bertoin. Sizes of the largest clusters for supercritical percolation on random recursive trees. Random Structures Algorithms 44 (2014) 29–44. | MR 3143589 | Zbl 1280.05117

[4] J. Bertoin and G. Miermont. The cut-tree of large Galton–Watson trees and the Brownian CRT. Ann. Appl. Probab. 23 (2013) 1469–1493. | MR 3098439 | Zbl 1279.60035

[5] M. Drmota, A. Iksanov, M. Möhle and U. Rösler. A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Structures Algorithms 34 (2009) 319–336. | MR 2504401 | Zbl 1187.05068

[6] K. B. Erickson. Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151 (1970) 263–291. | MR 268976 | Zbl 0212.51601

[7] A. Greven, P. Pfaffelhuber and A. Winter. Convergence in distribution of random metric measure spacesProbab. Theory Related Fields 145 (2009) 285–322. | MR 2520129 | Zbl 1215.05161

[8] M. Gromov. Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics 152. Birkhäuser, Boston, MA, 1999. | MR 1699320 | Zbl 0953.53002

[9] B. Haas and G. Miermont. Scaling limits of Markov branching trees, with applications to Galton–Watson and random unordered trees. Ann. Probab. 40 (2012) 2589–2666. | MR 3050512 | Zbl 1259.60033

[10] C. Holmgren. Random records and cuttings in binary search trees. Combin. Probab. Comput. 19 (2010) 391–424. | MR 2607374 | Zbl 1215.05162

[11] C. Holmgren. A weakly 1-stable distribution for the number of random records and cuttings in split trees. Adv. in Appl. Probab. 43 (2011) 151–177. | MR 2761152 | Zbl 1213.05037

[12] A. Iksanov and M. Möhle. A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree. Electron. Commun. Probab. 12 (2007) 28–35. | MR 2407414 | Zbl 1133.60012

[13] S. Janson. Random records and cuttings in complete binary trees. In Mathematics and Computer Science III. Trends Math. 241–253. Birkhäuser, Basel, 2004. | MR 2090513 | Zbl 1063.60018

[14] S. Janson. Random cutting and records in deterministic and random trees. Random Structures Algorithms 29 (2006) 139–179. | MR 2245498 | Zbl 1120.05083

[15] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Probability and its Applications (New York). Springer, New York, 2002. | MR 1876169 | Zbl 0892.60001

[16] M. Kuba and A. Panholzer. Multiple isolation of nodes in recursive trees. Preprint, 2013. Available at http://arxiv.org/abs/1305.2880. | MR 3259471 | Zbl 1300.05285

[17] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of LlogL citeria for mean behaviour of branching processes. Ann. Probab. 23 (1995) 1125–1138. | MR 1349164 | Zbl 0840.60077

[18] A. Meir and J. W. Moon. Cutting down random trees. J. Aust. Math. Soc. 11 (1970) 313–324. | MR 284370 | Zbl 0196.27602

[19] A. Meir and J. W. Moon. Cutting down recursive trees. Math. Biosci. 21 (1974) 173–181. | Zbl 0288.05102

[20] A. Panholzer. Destruction of recursive trees. In Mathematics and Computer Science III. Trends Math. 267–280. Birkhäuser, Basel, 2004. | MR 2090518 | Zbl 1060.05022

[21] A. Panholzer. Cutting down very simple trees. Quaest. Math. 29 (2006) 211–227. | MR 2233368 | Zbl 1120.05020