The existence of a giant cluster for percolation on large Crump–Mode–Jagers trees. (March 2020)

• Record Type: Journal Article
• Authors: Berzunza, G.
• Is Part Of: Advances in applied probability. Volume 52:Number 1(2020)
• Page Start: 266
• Page End: 290
• Publication Date: 2020-03
• Subjects: Random tree, -- percolation, -- giant component, -- Crump–Mode–Jagers processes; 60J80, -- 60K35, -- 05C05; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2019.61
• Languages: English
• ISSNs: 0001-8678

Abstract

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.