THE CUT-TREE OF LARGE GALTON-WATSON TREES AND THE BROWNIAN CRT

• 成果类型: Article
• 署名作者: Bertoin, Jean; Miermont, Gregory
• 署名单位: University of Zurich; Universite Paris Saclay
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/12-AAP877
• 发表日期: 2013
• 页码: 1469-1493
• 关键词: coalescent

摘要

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 -> infinity to a Brownian continuum random tree (CRT) in the weak sense induced by the Gromov-Prolchorov topology. This yields a multi-dimensional extension of a limit theorem due to Janson [Random Structures Algorithms 29 (2006) 139-1791 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 Poincare Probab. Stat. (2012) 48 909-921] obtained in the special case of Cayley trees.