Critical behaviors and universality classes of percolation phase transitions on two-dimensional square lattice

TL;DR

This study investigates the critical behaviors and universality classes of site and bond percolation on a 2D square lattice, revealing that the universality class depends on the percolation rule and differs between site and bond percolation under the product rule.

Contribution

It introduces finite-size scaling forms for giant cluster size and size jump, and determines critical exponents, highlighting differences in universality classes between site and bond percolation with different rules.

Findings

Random site and bond percolation share the same universality class.

Critical exponents for site percolation under the product rule differ from other cases.

Universality class varies between site and bond percolation when using the product rule.

Abstract

We have investigated both site and bond percolation on two dimensional lattice under the random rule and the product rule respectively. With the random rule, sites or bonds are added randomly into the lattice. From two candidates picked randomly, the site or bond with the smaller size product of two connected clusters is added when the product rule is taken. Not only the size of the largest cluster but also its size jump are studied to characterize the universality class of percolation. The finite-size scaling forms of giant cluster size and size jump are proposed and used to determine the critical exponents of percolation from Monte Carlo data. It is found that the critical exponents of both size and size jump in random site percolation are equal to that in random bond percolation. With the random rule, site and bond percolation belong to the same universality class. We obtain the critical exponents of the site percolation under the product rule, which are different from that of both random percolation and the bond percolation under the product rule. The universality class of site percolation differs different from that of bond percolation when the product rule is used.