New high-density series data for the mean number, percolation probability and 'susceptibility' of finite clusters are presented for bond and site percolation on four standard three-dimensional lattices. A Pade approximant analysis of both high- and low-density series makes particular use of the rather precise unbiased estimates of the percolation threshold, p c , obtained recently by Heermann and Stauffer (1981) for the simple cubic lattice. For this lattice the authors obtain the biased estimate gamma =1.73+or-0.03 for the site problem and a similar estimate but with larger uncertainties for the bond problem. Such a value is significantly larger than earlier series estimates. Assuming gamma to be universal they obtain precise, although biased, estimates of p c for both bond and site percolation on all four lattices. Using the bond estimates of p c they find an overall biased estimate of beta =0.454+or-0.008 for bond percolation on all three-dimensional lattices. (The corresponding site problem requires further study.) Scaling estimates of other critical exponents are alpha =-0.64+or-0.05, delta =4.81+or-0.14, Delta =2.18+or-0.04, nu =0.88+or-0.02 and nu =0.03+or-0.03.
Cluster statistics are obtained by computer simulation for percolation processes on
d-dimensional lattices with
d=2 through 7. For all
d,
ns, the number of
s-site clusters per site, is found to satisfy reasonably well the scaling hypothesis first proposed by Stauffer. The scaling functions are analyzed for dimensional dependence, and it is found that as
d increases they approach very rapidly the exactly known result for the Bethe lattice corresponding to $d=\ensuremath{\infty}$. Corrections to scaling are also studied, and at the upper critical dimension
dc=6, a deviation from scaling consistent with a logarithmic correction is obtained. Some universal quantities such as the ratio of the amplitudes
C+ and ${C}_{\ensuremath{-}}$ of the "susceptibility" (second moment of
ns) below and above the percolation threshold
pc are also found to approach the Bethe lattice limit very quickly. In addition, our data suggest that $\frac{{C}_{+}}{{C}_{\ensuremath{-}}}$ already assumes the limiting value of unity for
d=6. This is consistent with the exact relation for the asymmetric decay of
ns proved by Kunz and Souillard if we adopt the hypothesis that the asymmetry of
ns about
pc only enters the corrections to the leading scaling term for
d> 6.
The behavior of the magnetic susceptibility of a two-dimensional Ising model with nonmagnetic impurities is investigated numerically. A new method for determining the critical amplitudes and critical temperature is developed. The results of a numerical investigation of the ratio of the critical amplitudes of the magnetic susceptibility are presented. It is shown that the ratio of the critical amplitudes is universal right up to impurity concentrations q ≤ 0.25 (the percolation point of a square lattice is q c = 0.407254). The behavior of the effective critical exponent γ( q ) of the magnetic susceptibility is discussed. Apparently, a transition from Ising-type universal behavior to percolation behavior should occur in a quite narrow concentration range near the percolation point of the lattice.
Monte Carlo (MC) simulations and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the ferromagnetic 4-state Potts model on the square lattice are analyzed in a vicinity of the critical point in order to estimate universal combinations of critical amplitudes. The quality of the fits is improved using predictions of the renormalization group (RG) approach and of conformal invariance, and restricting the data within an appropriate temperature window. The RG predictions on the cancelation of the logarithmic corrections in the universal amplitude ratios are tested. A direct calculation of the effective ratio of the energy amplitudes using duality relations explicitly demonstrates this cancelation of logarithms, thus supporting the predictions of RG. We emphasize the role of corrections and of background terms on the determination of the amplitudes. The ratios of the critical amplitudes of the susceptibilities obtained in our analysis differ significantly from those predicted theoretically and supported by earlier SE and MC analysis. This disagreement might signal that the two-kink approximation used in the analytical estimates is not sufficient to describe with fair accuracy the amplitudes of the 4-state model.
Monte Carlo (MC) and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the two-dimensional 4-state Potts model in the vicinity of the critical point are analysed. The role of logarithmic corrections is discussed and an approach is proposed in order to account numerically for these corrections in the determination of critical amplitudes. Accurate estimates of universal amplitude ratios A + / A - , Γ + /Γ - , Γ T /Γ - and R C ± are given, which arouse new questions with respect to previous works.