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

TL;DR

This paper rigorously analyzes site-percolation on random recursive trees, revealing the size distribution of clusters follows a Yule-Simon distribution and identifying the scaling behavior of the largest cluster.

Contribution

It provides the first rigorous proof of the cluster size distribution and largest cluster scaling in site-percolation on random recursive trees, connecting to branching processes.

Findings

Cluster size distribution follows a Yule-Simon distribution.

Largest cluster size scales as n^p.

Proved the asymptotic proportion of clusters of size k.

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.