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

Gabriel Berzunza Ojeda

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arXiv · 1806.10686

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

Gabriel Berzunza Ojeda

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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, 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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Gabriel Berzunza Ojeda. 2018-06-27. The existence of a giant cluster for percolation on large Crump-Mode-Jagers trees. https://arxiv.org/abs/1806.10686

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