Universal finite-size scaling functions for percolation on three-dimensional lattices

Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability Eₚ, the percolation probability $P,$ and the probability Wₙ for the appearance of n percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions $L.$ Using the HMCSM, we calculate the existence probability Eₚ and the percolation probability P for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of Eₚ and P in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that Eₚ and P have universal finite-size scaling functions. This implies that the critical Eₚ is a universal quantity, which is 0.265±0.005 for free boundary conditions and 0.924±0.005 for periodic boundary conditions. We also find that Wₙ for site and bond percolation on sc lattices have universal finite-size scaling functions.