Yule processes with rare mutation and their applications to percolation on b-ary trees

TL;DR

This paper analyzes the size fluctuations of giant clusters in supercritical percolation on large b-ary trees, revealing non-Gaussian behavior and extending previous results to scale-free trees using branching process techniques.

Contribution

It introduces a novel approach using branching processes with rare mutations to study giant cluster fluctuations in percolation on b-ary and scale-free trees.

Findings

Giant cluster size exhibits non-Gaussian fluctuations.

Results extend to scale-free trees.

Method based on analysis of sub-populations with ancestral types.

Abstract

We consider supercritical Bernoulli bond percolation on a large $b$-ary tree, in the sense that with high probability, there exists a giant cluster. We show that the size of the giant cluster has non-gaussian fluctuations, which extends a result due to Schweinsberg in the case of random recursive trees. Using ideas in the recent work of Bertoin and Uribe Bravo, the approach developed in this work relies on the analysis of the sub-population with ancestral type in a system of branching processes with rare mutations, which may be of independent interest. This also allows us to establish the analogous result for scale-free trees.