Scaling of cluster heterogeneity in percolation transitions

TL;DR

This paper uncovers a new critical scaling law for cluster heterogeneity in percolation transitions across various dimensions, revealing how it diverges near the critical point and introducing a novel finite-size-scaling exponent.

Contribution

It analytically derives a new scaling law for cluster heterogeneity and introduces a novel finite-size-scaling exponent in percolation theory.

Findings

H diverges as |p - p_c|^{-1/σ} near criticality

Finite-size-scaling governed by a new exponent ν_H = (1+d_f/d)ν

Numerical simulations confirm the analytical results

Abstract

We investigate a critical scaling law for the cluster heterogeneity $H$ in site and bond percolations in $d$-dimensional lattices with $d=2,...,6$. The cluster heterogeneity is defined as the number of distinct cluster sizes. As an occupation probability $p$ increases, the cluster size distribution evolves from a monodisperse distribution to a polydisperse one in the subcritical phase, and back to a monodisperse one in the supercritical phase. We show analytically that $H$ diverges algebraically approaching the percolation critical point $p_c$ as $H\sim |p-p_c|^{-1/\sigma}$ with the critical exponent $\sigma$ associated with the characteristic cluster size. Interestingly, its finite-size-scaling behavior is governed by a new exponent $\nu_H = (1+d_f/d)\nu$ where $d_f$ is the fractal dimension of the critical percolating cluster and $\nu$ is the correlation length exponent. The corresponding scaling variable defines a singular path to the critical point. All results are confirmed by numerical simulations.