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

TL;DR

This paper investigates the emergence of a giant cluster in percolation processes on large random trees derived from Crump-Mode-Jagers processes, extending understanding to various tree models.

Contribution

It demonstrates the existence of a giant percolation cluster in large Crump-Mode-Jagers trees under certain conditions, a case not previously studied.

Findings

Giant cluster exists in large Crump-Mode-Jagers trees under specific regimes.

Includes analysis of various tree models like recursive, preferential attachment, and search trees.

Extends percolation theory to new classes of random trees.

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.