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

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

2012-01-19

arxiv.org/abs/1201.4081v1

Mathematics

Probability

25 pages, 3 figures

Scientific paper

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 CRT in the weak sense induced by the Gromov-Prokhorov topology. This yields a multi-dimensional extension of a limit theorem due to Janson \cite{Janson} for the number of random cuts needed to isolate the root in Galton-Watson trees conditioned by their sizes, and generalizes also a recent result \cite{Be} obtained in the special case of Cayley trees.

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