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

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 $p\in (0,1)$ to remain and with probability $(1-p)$ to be removed. As $n\to \infty ,$ we show that the proportion of the remaining clusters of size $k$ is of order $k^{-1-\frac{1}{p}}$, 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.

24 pages, 4 figures