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.