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 , rescaled by the factor , converges in probability as 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 , et renormalisé par un facteur , converge en probabilité quand 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.
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}, pages = {478--488}, publisher = {Gauthier-Villars}, volume = {51}, number = {2}, year = {2015}, doi = {10.1214/13-AIHP597}, mrnumber = {3335011}, zbl = {1351.60010}, language = {en}, url = {http://www.numdam.org/articles/10.1214/13-AIHP597/} }
TY - JOUR AU - Bertoin, Jean TI - The cut-tree of large recursive trees JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 478 EP - 488 VL - 51 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/13-AIHP597/ DO - 10.1214/13-AIHP597 LA - en ID - AIHPB_2015__51_2_478_0 ER -
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/articles/10.1214/13-AIHP597/
[1] Cutting down trees with a Markov chainsaw. Ann. Appl. Probab. 24 (2014) 2297–2339. | DOI | MR | Zbl
, and .[2] Fires on trees. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 909–921. | DOI | Numdam | MR | Zbl
.[3] Sizes of the largest clusters for supercritical percolation on random recursive trees. Random Structures Algorithms 44 (2014) 29–44. | DOI | MR | Zbl
.[4] The cut-tree of large Galton–Watson trees and the Brownian CRT. Ann. Appl. Probab. 23 (2013) 1469–1493. | DOI | MR | Zbl
and .[5] 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. | DOI | MR | Zbl
, , and .[6] Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151 (1970) 263–291. | DOI | MR | Zbl
.[7] Convergence in distribution of random metric measure spacesProbab. Theory Related Fields 145 (2009) 285–322. | DOI | MR | Zbl
, and .[8] Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics 152. Birkhäuser, Boston, MA, 1999. | MR | Zbl
.[9] Scaling limits of Markov branching trees, with applications to Galton–Watson and random unordered trees. Ann. Probab. 40 (2012) 2589–2666. | DOI | MR | Zbl
and .[10] Random records and cuttings in binary search trees. Combin. Probab. Comput. 19 (2010) 391–424. | DOI | MR | Zbl
.[11] A weakly 1-stable distribution for the number of random records and cuttings in split trees. Adv. in Appl. Probab. 43 (2011) 151–177. | DOI | MR | Zbl
.[12] 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. | DOI | MR | Zbl
and .[13] Random records and cuttings in complete binary trees. In Mathematics and Computer Science III. Trends Math. 241–253. Birkhäuser, Basel, 2004. | MR | Zbl
.[14] Random cutting and records in deterministic and random trees. Random Structures Algorithms 29 (2006) 139–179. | DOI | MR | Zbl
.[15] Foundations of Modern Probability, 2nd edition. Probability and its Applications (New York). Springer, New York, 2002. | DOI | MR | Zbl
.[16] Multiple isolation of nodes in recursive trees. Preprint, 2013. Available at http://arxiv.org/abs/1305.2880. | MR | Zbl
and .[17] Conceptual proofs of citeria for mean behaviour of branching processes. Ann. Probab. 23 (1995) 1125–1138. | DOI | MR | Zbl
, and .[18] Cutting down random trees. J. Aust. Math. Soc. 11 (1970) 313–324. | DOI | MR | Zbl
and .[19] Cutting down recursive trees. Math. Biosci. 21 (1974) 173–181. | DOI | Zbl
and .[20] Destruction of recursive trees. In Mathematics and Computer Science III. Trends Math. 267–280. Birkhäuser, Basel, 2004. | MR | Zbl
.[21] Cutting down very simple trees. Quaest. Math. 29 (2006) 211–227. | DOI | MR | Zbl
.Cited by Sources: