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

2018/06/27 by Ojeda, Gabriel Berzunza
#05C05 #60J80 #60K35 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1806.10686

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, as well as (more general) m-ary search trees, fragmentation trees, median-of-(2l+1) binary search trees, to name a few, where up to our knowledge percolation has not been studied yet.

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