Site percolation threshold for honeycomb and square lattices

Abstract

A new method of estimating the critical percolation threshold is proposed, based on Stauffer's cluster number scaling hypothesis and the universality with respect to lattice structure of the corresponding Stauffer scaling function. This method is illustrated by obtaining estimates of the site percolation threshold for the honeycomb lattice, p_c=0.6962+or-0.0006, and for the square lattice, p_c=0.5923+or-0.0007. The error bars or 'confidence limits' of our estimates are substantially smaller than previous series estimates, and-for the honeycomb lattice-would have to be multiplied by the factor 18 to include the value, p_c=2^-12/, recently proposed by Kondor (1980) to be exact. An additional result is R=4.95+or-0.15, where R=lim_{s to infinity} R_s, and R_s is the ratio of the cluster number scaling function at its maximum to its value at p_c for clusters of size s.