The cut-tree of large galton-watson trees and the brownian crt

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 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. DOI: https://doi.org/10.1214/12-AAP877 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-85277 Published Version Originally published at: Bertoin, Jean; Miermont, Grégory (2013). The cut-tree of large galton-watson trees and the brownian crt. Annals of Applied Probability, 23(4):1469-1493. DOI: https://doi.org/10.1214/12-AAP877 The Annals of Applied Probability 2013, Vol. 23, No. 4, 1469–1493 DOI: 10.1214/12-AAP877 © Institute of Mathematical Statistics, 2013 THE CUT-TREE OF LARGE GALTON–WATSON TREES AND THE BROWNIAN CRT BY JEAN BERTOIN AND GRÉGORY MIERMONT1 Universität Zürich and Université Paris-Sud 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 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.