Percolation on trees as a Brownian excursion: from Gaussian to Kolmogorov-Smirnov to Exponential statistics

TL;DR

This paper derives the distribution of percolation cluster sizes on trees across different phases using a novel mapping to Brownian excursions, revealing Gaussian, Kolmogorov-Smirnov, and exponential statistics.

Contribution

It introduces a continuum tree to Brownian excursion mapping to analytically compute cluster size distributions in percolation phases.

Findings

Gaussian distribution in subcritical phase

Kolmogorov-Smirnov distribution at criticality

Exponential distribution in supercritical phase

Abstract

We calculate the distribution of the size of the percolating cluster on a tree in the subcritical, critical and supercritical phase. We do this by exploiting a mapping between continuum trees and Brownian excursions, and arrive at a diffusion equation with suitable boundary conditions. The exact solution to this equation can be conveniently represented as a characteristic function, from which the following distributions are clearly visible: Gaussian (subcritical), Kolmogorov-Smirnov (critical) and exponential (supercritical). In this way we provide an intuitive explanation for the result reported in R. Botet and M. Ploszajczak, Phys. Rev. Lett 95, 185702 (2005) for critical percolation.