On the universality of distribution of ranked cluster masses at critical percolation

TL;DR

This paper investigates the universal distribution of scaled cluster masses at the critical percolation threshold, revealing a Gaussian universal function for large ranks across different lattice sizes.

Contribution

It demonstrates the universal behavior of the scaled cluster mass distribution at criticality and identifies a Gaussian form for the universal scaling function.

Findings

Universal distribution of scaled cluster masses across lattice sizes

Gaussian form of the universal scaling function for large ranks

Scaling behavior characterized by specific exponents y and ζ

Abstract

The distribution of masses of clusters smaller than the infinite cluster is evaluated at the percolation threshold. The clusters are ranked according to their masses and the distribution $P(M/L^D,r)$ of the scaled masses M for any rank r shows a universal behaviour for different lattice sizes L (D is the fractal dimension). For different ranks however, there is a universal distribution function only in the large rank limit, i.e., $P(M/L^D,r)r^{-y\zeta } \sim g(Mr^y/L^D)$ (y and $\zeta$ are defined in the text), where the universal scaling function g is found to be Gaussian in nature.