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 in Bertoin [Random Structures Algorithms 44 (2014) 29–44] 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.
"Supercritical percolation on large scale-free random trees." Ann. Appl. Probab. 25 (1) 81 - 103, February 2015. https://doi.org/10.1214/13-AAP988