Size distribution of clusters in site-percolation on random recursive tree

2024/08/22 by Gu, Chenlin, Yuan, Linglong
#60J27 #60J80 #60J85 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2408.12515

Abstract

We prove rigorously several results about the site-percolation on random recursive trees, observed in the previous work by Kalay and Ben-Naim [J. Phys. A48(2015), no.4, 0405001, 15 pp.]. For a random recursive tree of size n, let every site have probability p ∈ (0,1) to remain and with probability (1-p) to be removed. As n→∞, we show that the proportion of the remaining clusters of size k is of order k^-1-(1)/(p), resulting in a Yule-Simon distribution; the largest cluster size is of order n^p, and admits a non-trivial scaling limit. The proofs are based on the embedding of this model in the multi-type branching processes, and a coupling with the bond-percolation on random recursive trees.

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