We simulate systems up to 4*1012 sites on a square lattice at the percolation threshold, pc=0.592746. We confirm τ=187/91, and report that the correction to scaling exponent, Δ1=0.65±0.05. The mass of the largest and second largest cluster scales with the fractal dimensionality, df=91/48. These exponents, apart from the correction-to-scaling exponent, Δ1, are expected from exact calculations and previous substantial numerical work.
Random-walk simulations on square-lattice percolation clusters are performed for (a) all clusters at criticality; (b) the largest cluster at nominal criticality; and (c) the largest cluster at exact criticality for each realization (new method). The short-time correction-to-scaling exponent is obtained: \ensuremath{\omega}=-0.48 (for N
<103 Monte Carlo steps). The scaling exponent appears to be in better agreement with the recent Aharony-Stauffer conjecture for the spectral dimension (
ds=1.309) than with the Alexander-Orbach-Rammal-Toulouse conjecture (
ds=1.333).
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.
We study the ratio of the number of sites in the largest and second largest clusters in random percolation. Using the scaling hypothesis that the ratio <M1>/<M2> of the mean cluster sizes M1 and M2 scales as f ((p - pc) L1/ν), we employ finite-size scaling analysis to find that <M1>/<M2> is nonuniversal with respect to the boundary conditions imposed. The mean <M1/M2> of the ratios behaves similarly although with a distinct critical value reflecting the relevance of mass fluctuations at the percolation threshold. These zero exponent ratios also allow for reliable estimates of the critical parameters at percolation from relatively small lattices.
We propose a site random-cluster model by introducing an additional cluster weight in the partition function of the traditional site percolation. To simulate the model on a square lattice, we combine the color-assignation and the Swendsen-Wang methods to design a highly efficient cluster algorithm with a small critical slowing-down phenomenon. To verify whether or not it is consistent with the bond random-cluster model, we measure several quantities, such as the wrapping probability Re, the percolating cluster density P∞, and the magnetic susceptibility per site χp, as well as two exponents, such as the thermal exponent yt and the fractal dimension yh of the percolating cluster. We find that for different exponents of cluster weight q=1.5, 2, 2.5, 3, 3.5, and 4, the numerical estimation of the exponents yt and yh are consistent with the theoretical values. The universalities of the site random-cluster model and the bond random-cluster model are completely identical. For larger values of q, we find obvious signatures of the first-order percolation transition by the histograms and the hysteresis loops of percolating cluster density and the energy per site. Our results are helpful for the understanding of the percolation of traditional statistical models.
We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to
L=512, and estimate the percolation thresholds to be
pc(bond)=0.24881182(10) and
pc(site)=0.3116077(2). By performing extensive simulations at these estimated critical points, we then estimate the critical exponents $1/\ensuremath{\nu}=1.1410(15)$, $\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)$, the leading correction exponent ${y}_{i}=\ensuremath{-}1.2(2)$, and the shortest-path exponent
dmin=1.3756(3). Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent $\ensuremath{\approx}$$\ensuremath{-}2$, rather than ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.
A site-percolation model with two different sizes of particles is newly introduced on a square lattice. To estimate the fractal dimension and critical exponents with high accuracy, a finite-size scaling analysis is performed with a Monte Carlo simulation. The obtained fractal dimension coincides with that of the ordinary model, while the estimated exponents β, γ, and ν are slightly different from those of the ordinary model. However, the normalized exponents
β^=β/ν and
γ^=γ/ν remain the same as in the ordinary model, which is a manifestation of weak universality.