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

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.

5 pages, 4 fiures

Cited by in corpus (6)

• Large-deviations for spatial diffusion of cold atoms
• Exact derivation of a finite-size-scaling law and corrections to scaling in the geometric Galton-Watson process
• Phase transition, scaling of moments, and order-parameter distributions in Brownian particles and branching processes with finite-size effects
• Dynamically accelerated cover times
• From Boltzmann to Zipf through Shannon and Jaynes
• Correspondence between noisy sample space reducing process and records in correlated random events