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

TL;DR

This paper proves that the cut-tree of large critical Galton-Watson trees converges to a Brownian CRT after rescaling, extending previous results on random cuts and tree structures.

Contribution

It establishes the convergence of the cut-tree of conditioned Galton-Watson trees to a Brownian CRT, generalizing prior specific case results.

Findings

Convergence of scaled cut-trees to Brownian CRT

Extension of Janson's limit theorem to multi-dimensional case

Generalization of results from Cayley trees to Galton-Watson trees

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 Poincar\'{e} Probab. Stat. (2012) 48 909-921] obtained in the special case of Cayley trees.