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 $E_p,$ the percolation probability $P,$ and the probability $W_n$ 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 $L.\mathrm{}$ Using the HMCSM, we calculate the existence probability $E_p$ 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 $E_p$ 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 $E_p$ and $P$ have universal finite-size scaling functions. This implies that the critical $E_p$ is a universal quantity, which is $0.265\pm 0.005$ for free boundary conditions and $0.924\pm 0.005$ for periodic boundary conditions. We also find that $W_n$ for site and bond percolation on sc lattices have universal finite-size scaling functions.