The Annals of Applied Probability

The cut-tree of large Galton–Watson trees and the Brownian CRT

Jean Bertoin and Grégory Miermont

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Abstract

Consider the edge-deletion process in which the edges of some finite tree $T$ are removed one after the other in the uniform random order. Roughly speaking, the cut-tree then describes the genealogy of connected components appearing in this edge-deletion process. Our main result shows that after a proper rescaling, the cut-tree of a critical Galton–Watson tree with finite variance and conditioned to have size $n$, converges as $n\to\infty$ to a Brownian continuum random tree (CRT) in the weak sense induced by the Gromov–Prokhorov topology. This yields a multi-dimensional extension of a limit theorem due to Janson [Random Structures Algorithms 29 (2006) 139–179] for the number of random cuts needed to isolate the root in Galton–Watson trees conditioned by their sizes, and also generalizes a recent result [Ann. Inst. Henri Poincaré Probab. Stat. (2012) 48 909–921] obtained in the special case of Cayley trees.

Article information

Source
Ann. Appl. Probab. Volume 23, Number 4 (2013), 1469-1493.

Dates
First available in Project Euclid: 21 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1371834035

Digital Object Identifier
doi:10.1214/12-AAP877

Mathematical Reviews number (MathSciNet)
MR3098439

Zentralblatt MATH identifier
1279.60035

Subjects
Primary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Galton–Watson tree cut-tree Brownian continuum random tree

Citation

Bertoin, Jean; Miermont, Grégory. The cut-tree of large Galton–Watson trees and the Brownian CRT. Ann. Appl. Probab. 23 (2013), no. 4, 1469--1493. doi:10.1214/12-AAP877. https://projecteuclid.org/euclid.aoap/1371834035


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