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.
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 ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.
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.
Directed percolation in four dimensions is of direct physical relevance to the world with three space and one time dimension. We present a comprehensive analysis of recently extended series for the moments of the cluster-size distribution and for the percolation probability in a ``field'' on the hypercubic lattice. We find a critical threshold,
pc=0.3025\ifmmode\pm\else\textpm\fi{}0.0010, and dominant critical exponents \ensuremath{\gamma}=1.21\ifmmode\pm\else\textpm\fi{}0.05, for the mean cluster size; \ensuremath{\beta}=0.82\ifmmode\pm\else\textpm\fi{}0.03 and 1/\ensuremath{\delta}=0.45\ifmmode\pm\else\textpm\fi{}0.02 for the percolation probability in the thermal and field directions respectively; and a gap exponent of \ensuremath{\Delta}=2.03\ifmmode\pm\else\textpm\fi{}0.06. We find a thermal-correction exponent ${\ensuremath{\Delta}}_{1}$=0.55\ifmmode\pm\else\textpm\fi{}0.15 and a field correction of \ensuremath{\Omega}=0.3\ifmmode\pm\else\textpm\fi{}0.1. We also calculate some universal critical-amplitude ratios.
We present a Monte Carlo study of the bond- and site-directed (oriented) percolation models in (d+1) dimensions on simple-cubic and body-centered-cubic lattices, with 2 ≤ d ≤ 7. A dimensionless ratio is defined, and an analysis of its finite-size scaling produces improved estimates of percolation thresholds. We also report improved estimates for the standard critical exponents. In addition, we study the probability distributions of the number of wet sites and radius of gyration, for 1 ≤ d ≤ 7.
In this work we apply a highly efficient Monte Carlo algorithm recently proposed by Newman and Ziff to treat percolation problems. The site and bond percolations are studied on a number of lattices in two and three dimensions. Quite good results for the wrapping probabilities, correlation length critical exponent, and critical concentration are obtained for the square, simple cubic, hexagonal close packed, and hexagonal lattices by using relatively small systems. We also confirm the universal aspect of the wrapping probabilities regarding site and bond dilution.
The method of high-temperature series expansions is used to study the dependence of T c on the concentration p of interactions in a randomly diluted S= 1 / 2 Heisenberg model. It is found that, for all three cubic lattices, the interval of p over which the Pade approximants to the series converge extends further towards the critical probability for percolation than for the corresponding site diluted model. In this interval the critical curve is linear and extrapolates to intersect the T=0 boundary at a value of p marginally above the critical probability for bond percolation. The anomalous dependence of the susceptibility exponent gamma on p which has recently been reported for the site diluted model is not observed for the model studied here.