The fluctuations of the giant cluster for percolation on random split trees

TL;DR

This paper investigates the size fluctuations of the giant cluster in Bernoulli bond percolation on split trees, revealing stable Cauchy law behavior and generalizing previous results for specific tree types.

Contribution

It introduces a unified approach to analyze giant cluster fluctuations on split trees, extending known results to a broader class of random trees.

Findings

Existence of a unique giant cluster in large split trees.

Fluctuations of the giant cluster size follow a stable Cauchy distribution.

Method applicable to other trees with logarithmic height.

Abstract

A split tree of cardinality $n$ is constructed by distributing $n$ "balls" in a subset of vertices of an infinite tree which encompasses many types of random trees such as $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees, fringe-balanced trees, digital search trees and random simplex trees. In this work, we study Bernoulli bond percolation on arbitrary split trees of large but finite cardinality $n$. We show for appropriate percolation regimes that depend on the cardinality $n$ of the split tree that there exists a unique giant cluster, the fluctuations of the size of the giant cluster as $n \rightarrow \infty$ are described by an infinitely divisible distribution that belongs to the class of stable Cauchy laws. This work generalizes the results for the random $m$-ary recursive trees in Berzunza (2015). Our approach is based on a remarkable decomposition of the size of the giant percolation cluster as a sum of essentially independent random variables which may be useful for studying percolation on other trees with logarithmic height; for instance in this work we study also the case of regular trees.