Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps

We show that large critical multi-type Galton–Watson trees, when conditioned to be large, converge locally in distribution to an infinite tree which is analogous to Kesten’s infinite monotype Galton–Watson tree. This is proven when we condition on the number of vertices of one fixed type, and with a...

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Bibliographic Details
Published inJournal of theoretical probability Vol. 31; no. 1; pp. 159 - 205
Main Author Stephenson, Robin
Format Journal Article
LanguageEnglish
Published New York Springer US 01.03.2018
Springer Nature B.V
Subjects
Conditioning
Convergence
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
Statistics
Trees
05C63
Random maps
60B10
Galton–Watson processes
05C81
05C80
Local convergence
Random trees
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ISSN0894-9840
1572-9230
DOI10.1007/s10959-016-0707-3

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Summary:We show that large critical multi-type Galton–Watson trees, when conditioned to be large, converge locally in distribution to an infinite tree which is analogous to Kesten’s infinite monotype Galton–Watson tree. This is proven when we condition on the number of vertices of one fixed type, and with an extra technical assumption if we count at least two types. We then apply these results to study local limits of random planar maps, showing that large critical Boltzmann-distributed random maps converge in distribution to an infinite map.
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ISSN:0894-9840
1572-9230
DOI:10.1007/s10959-016-0707-3