License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.AofA.2018.34
URN: urn:nbn:de:0030-drops-89276
URL: https://drops.dagstuhl.de/opus/volltexte/2018/8927/
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Stufler, Benedikt

Local Limits of Large Galton-Watson Trees Rerooted at a Random Vertex

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Abstract

We prove limit theorems describing the asymptotic behaviour of a typical vertex in random simply generated trees as their sizes tends to infinity. In the standard case of a critical Galton-Watson tree conditioned to be large, the limit is the invariant random sin-tree constructed by Aldous (1991). Our main contribution lies in the condensation regime where vertices of macroscopic degree appear. Here we describe in complete generality the asymptotic local behaviour from a random vertex up to its first ancestor with "large" degree. Beyond this distinguished ancestor, different behaviours may occur, depending on the branching weights. In a subregime of complete condensation, we obtain convergence toward a novel limit tree, that describes the asymptotic shape of the vicinity of the full path from a random vertex to the root vertex. This includes the important case where the offspring distribution follows a power law up to a factor that varies slowly at infinity.

BibTeX - Entry

@InProceedings{stufler:LIPIcs:2018:8927,
  author =	{Benedikt Stufler},
  title =	{{Local Limits of Large Galton-Watson Trees Rerooted at a Random Vertex}},
  booktitle =	{29th International Conference on Probabilistic,  Combinatorial and Asymptotic Methods for the Analysis of Algorithms  (AofA 2018)},
  pages =	{34:1--34:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-078-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{110},
  editor =	{James Allen Fill and Mark Daniel Ward},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8927},
  URN =		{urn:nbn:de:0030-drops-89276},
  doi =		{10.4230/LIPIcs.AofA.2018.34},
  annote =	{Keywords: Galton-Watson trees, local weak limits}
}

Keywords: Galton-Watson trees, local weak limits
Collection: 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)
Issue Date: 2018
Date of publication: 18.06.2018


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