Numerical analysis of percolation cluster size distribution in two-dimensional and three-dimensional lattices

To investigate the statistical behavior in the sizes of finite clusters for percolation, cluster size distribution n(s)(p) for site and bond percolations at different lattices and dimensions was simulated using a modified algorithm. An equation to approximate the finite cluster size distribution n(s)(p) was obtained and expressed as: log(n(s)(p)) = as - b logs + c. Based on the analysis of simulation data, we found that the equation is valid for p from 0 to 1 on site and for the bond percolation of two-dimensional (2D) and three-dimensional (3D) lattices. Furthermore, the relationship between the coefficients of the equation and the occupied ratio p was studied using the finite-size scaling method. When p was scaled to x = D(p - p(c))L-yt, p < p(c), and D was a nonuniversal metric factor. a was found to be related only to p, and the a-x curves of different lattices were nearly overlapped; b was related to the dimensions and p, and the scaled data of the b of all lattices with the same dimension tended to fall on the same curves. Unlike a and b, c apparently had a quadratic relation with x in 2D lattices and linear relation with x in 3D lattices. The results of this paper could significantly reduce the amount of tasks required to obtain numerical data of on the cluster size distribution for p from 0 to p(c).