Percolation on random recursive trees

TL;DR

This paper investigates the asymptotic behavior of percolation clusters on large random recursive trees, revealing their convergence to a continuous-state branching process and analyzing cluster size distributions.

Contribution

It establishes the convergence of cluster size trees to a continuous-state branching process and links this to the emergence of tree components in a destruction process.

Findings

Convergence of cluster trees to a continuous-state branching process

Asymptotic sizes of largest and second-largest clusters determined

Connection between percolation clusters and tree destruction process established

Abstract

We study Bernoulli bond percolation on a random recursive tree of size $n$ with percolation parameter $p(n)$ converging to $1$ as $n$ tends to infinity. The sizes of the percolation clusters are naturally stored in a tree. We prove convergence in distribution of this tree to the genealogical tree of a continuous-state branching process in discrete time. As a corollary we obtain the asymptotic sizes of the largest and next largest percolation clusters, extending thereby a recent work of Bertoin (2014) which deals with cluster sizes in the supercritical regime. In a second part, we show that the same limit tree appears in the study of the tree components which emerge from a continuous-time destruction of a random recursive tree. We comment on the connection to our first result on Bernoulli bond percolation.