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 $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that
Wn for site and bond percolation on sc lattices have universal finite-size scaling functions.
In this paper, we briefly review our recent results in the calculation of universal scaling functions for site and bond percolation on finite square, plane triangular, and honeycomb lattices.We find that, by choosing an aspect ratio for each lattice and a very small number of nonuniversal metric factors, all scaled data of the existance probability E p and the percolation probability P fallon the same universal scaling functions.We also find that free and periodic boundary conditions share the same nonuniversal metric factors.When the aspect ratio of each lattice is reduced by the same factor, the nonouniversal metric factors remain the same.The probabilities for the "appearance of n, n = 1, 2, 3, ..., percolating clusters for bond and site percolations on various planar lattices also have universal scaling functions.The implications of such results on numerical and experimental studies of critical phenomena are discussed.
The universality of the crossing probability ${\ensuremath{\pi}}_{\mathrm{hs}}$ of a system to percolate only in the horizontal direction was investigated numerically by a cluster Monte Carlo algorithm for the q-state Potts model for
q=2,3,4 and for percolation
q=1. We check the percolation through Fortuin-Kasteleyn clusters near the critical point on the square lattice by using representation of the Potts model as the correlated site-bond percolation model. It was shown that probability of a system to percolate only in the horizontal direction ${\ensuremath{\pi}}_{\mathrm{hs}}$ has the universal form ${\ensuremath{\pi}}_{\mathrm{hs}}=A(q)Q(z)$ for
q=1,2,3,4 as a function of the scaling variable $z={{b(q)L}^{1/\ensuremath{\nu}(q)}[p\ensuremath{-}{p}_{c}(q,L)]{}}^{\ensuremath{\zeta}(q)}.$ Here, $p=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta})$ is the probability of a bond to be closed,
A(q) is the nonuniversal crossing amplitude,
b(q) is the nonuniversal metric factor, $\ensuremath{\nu}(q)$ is the correlation length index, and $\ensuremath{\zeta}(q)$ is the additional scaling index. The universal function $Q(x)\ensuremath{\simeq}\mathrm{exp}(\ensuremath{-}|z|).$ The nonuniversal scaling factors were found numerically.