On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees

TL;DR

This paper investigates the non-Gaussian fluctuations of the giant cluster in supercritical percolation on random recursive trees, providing an explanation based on the growth phases of the trees, with potential applications to other tree classes.

Contribution

It offers a new analysis of the non-Gaussian fluctuations of the giant cluster, focusing on the effects of percolation during different growth phases of recursive trees.

Findings

Giant cluster size converges to e^{-c} with non-Gaussian fluctuations.

Analysis of percolation effects during different growth phases.

Potential applicability to other classes of trees.

Abstract

We consider a Bernoulli bond percolation on a random recursive tree of size $n\gg 1$, with supercritical parameter $p_n=1-c/\ln n$ for some $c>0$ fixed. It is known that with high probability, there exists then a unique giant cluster of size $G_n\sim \e^{-c}$, and it follows from a recent result of Schweinsberg \cite{Sch} that $G_n$ has non-gaussian fluctuations. We provide an explanation of this by analyzing the effect of percolation on different phases of the growth of recursive trees. This alternative approach may be useful for studying percolation on other classes of trees, such as for instance regular trees.