Eur. Phys. J. B (2014) 87: 179
https://doi.org/10.1140/epjb/e2014-40996-4

Regular Article

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

¹ Environmental Science and Engineering Research Center, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, P.R. China
² Department of Environmental Engineering, Jinan University, Guangzhou 510632, P.R. China

^a e-mail: lichaolin@hitsz.edu.cn

Received: 11 November 2013
Received in final form: 20 May 2014
Published online: 7 August 2014

Abstract

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 log s + 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^y_t, 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.