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arXiv · 1201.4081

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

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

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Jean Bertoin, Grégory Miermont. 2012-01-19. The cut-tree of large Galton-Watson trees and the Brownian CRT. https://doi.org/10.1214/12-aap877

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