Supercritical percolation on large scale-free random trees

TL;DR

This paper studies supercritical bond percolation on large scale-free trees, analyzing the sizes of large clusters and extending previous results using branching process techniques.

Contribution

It introduces a weak limit theorem for the sizes of the second-largest clusters in supercritical scale-free trees, extending prior work on recursive trees.

Findings

Weak limit theorem for cluster sizes

Extension of results to scale-free trees

Analysis of branching processes with neutral mutations

Abstract

We consider Bernoulli bond percolation on a large scale-free tree in the supercritical regime, meaning informally that there exists a giant cluster with high probability. We obtain a weak limit theorem for the sizes of the next largest clusters, extending a recent result for large random recursive trees. The approach relies on the analysis of the asymptotic behavior of branching processes subject to rare neutral mutations, which may be of independent interest.