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 (\hat{\beta} = \beta / \nu) and (\hat{\gamma} = \gamma / \nu) remain the same as in the ordinary model, which is a manifestation of weak universality.
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.
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 investigate the site-percolation problem on the square and simple-cubic lattices by means of a Monte Carlo algorithm that in fact simulates systems with size ${L}^{d\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$, where
L specifies the linear system size. This algorithm can be regarded either as an extension of the Hoshen-Kopelman method or as a special case of the transfer-matrix Monte Carlo technique. Various quantities, such as the magnetic correlation function, are sampled in the finite directions of the above geometry. Simulations are arranged such that both bulk and surface quantities can be sampled. On the square lattice, we locate the percolation threshold at
pc=0.5927465(4), and determine two universal quantities as
Qgbc=0.93034(1) and
Qgsc=0.79372(3), which are associated with bulk and surface correlations, respectively. These values agree well with the exact values ${2}^{\ensuremath{-}5∕48}$ and ${2}^{\ensuremath{-}1∕3}$, respectively, which follow from conformal invariance. On the simple-cubic lattice, we locate the percolation threshold at
pc=0.3116077(4). We further determine the bulk thermal and magnetic exponents as
yt=1.1437(6) and
yh=2.5219(2), respectively, and the surface magnetic exponent at the ordinary phase transition as
yhs(o)=1.0248(3).
Using finite-size scaling methods we measure the thermal and magnetic exponents of the site percolation in four dimensions, obtaining a value for the anomalous dimension very different from the results found in the literature. We also obtain the leading corrections-toscaling exponent and, with great accuracy, the critical density. Key words: Lattice. Monte Carlo. Percolation. Critical exponents. Finite size scaling. ffl-expansion. PACS: 12.40.Ee;64.60.A;75.40.Mg;75.40.Cx hector@lattice.fis.ucm.es y laf@lattice.fis.ucm.es z victor@lattice.fis.ucm.es x sudupe@lattice.fis.ucm.es -- parisi@roma1.infn.it k ruiz@chimera.roma1.infn.it 1 Introduction From the point of view of its definition, the simplest statistical system is perhaps the percolation. In the case of the site percolation, we fill the sites of a given lattice with probability p. Then we construct the clusters as sets of contiguous filled sites. The critical properties of the system can be described in terms of the...
Using the Average Occupation Probability in site percolation, the critical exponent β for a lattice (L × L) is determined. The results for different lattice sizes show that as the lattice becomes larger, the calculated β agrees better with that obtained from 2D scaling theory. There appears a common range of p – pc which can be used for any L to compute for β. We estimate that a minimum lattice size of 200 × 200 is necessary for the procedure to yield an acceptable value of the critical exponent.
We consider the cluster-size statistics of two types of site-bond-correlated percolation. In model A (B) [see Phys. Rev. B 40, 10 986 (1989)], the activation of a bond between two first-neighbor occupied sites is also dependent on the occupancy of both (either) of their own nearest neighbors, along the line joining them. The critical concentration for models A and B, obtained from simulations on square lattices of sizes up to $2000\ifmmode\times\else\texttimes\fi{}2000$, is found to be $0.7405\ifmmode\pm\else\textpm\fi{}0.0005$ and $0.6145\ifmmode\pm\else\textpm\fi{}0.0005$, respectively. The resulting critical exponents indicate that both models are in the same universality class as ordinary independent (site or bond) percolation.