On stochastic spatial patterns and neuronal polarity

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 \(x = D(p - p_c )L^{y_t }\) , 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 .