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The existence of a giant cluster for percolation on large Crump–Mode–Jagers trees

Part of: Markov processes Special processes Graph theory

Published online by Cambridge University Press:  29 April 2020

G. Berzunza
G. Berzunza*
Affiliation:
Department of Mathematics, Uppsala University
*
*Postal address: Lägerhyddsvägen 1, Hus 1, 6 och 7, Box 480, 751 06 Uppsala, Sweden. Email address: gabriel.berzunza-ojeda@math.uu.se
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Abstract

In this paper we consider random trees associated with the genealogy of Crump–Mode–Jagers processes and perform Bernoulli bond-percolation whose parameter depends on the size of the tree. Our purpose is to show the existence of a giant percolation cluster for appropriate regimes as the size grows. We stress that the family trees of Crump–Mode–Jagers processes include random recursive trees, preferential attachment trees, binary search trees for which this question has been answered by Bertoin [7], as well as (more general) m-ary search trees, fragmentation trees, and median-of-( $2\ell+1$ ) binary search trees, to name a few, where to our knowledge percolation has not yet been studied.

Keywords

Random treepercolationgiant componentCrump–Mode–Jagers processes

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 05C05: Trees
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 1 , March 2020 , pp. 266 - 290
Copyright
© Applied Probability Trust 2020

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