A Monte Carlo renormalisation group approach using a scaling transformation in real space is applied to the critical properties for the site percolation problem on the simple cubic lattice. The authors find a sequence of estimates for the critical concentration p c and the eigenvalue lambda at various values of the rescaling length b. Extrapolation of these sequences to the limit b to infinity yields the site percolation threshold p c =0.3115+0.0004/-0.0003 and the connectedness length exponent nu =0.88+or-0.04.
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 obtain the critical parameters for the site-percolation problem on the square lattice to a high degree of accuracy (comparable to that of series expansions) by using a Monte Carlo position-space renormalization-group procedure directly on the site-occupation probability. Our method involves calculating recursion relations using progressively larger lattice rescalings,
b. We find smooth sequences for the value of the critical percolation concentration
pc(b) and for the scaling powers
yp(b) and
yh(b). Extrapolating these sequences to the limit $b\ensuremath{\rightarrow}\ensuremath{\infty}$ leads to quite accurate numerical predictions. Further, by considering other weight functions or "rules" which also embody the essential connectivity feature of percolation, we find that the numerical results in the infinite-cell limit are in fact "rule independent." However, the actual fashion in which this limit is approached does depend upon the rule chosen. A connection between extrapolation of our renormalization-group results and finite-size scaling is made. Furthermore, the usual finite-size scaling arguments lead to independent estimates of
pc and
yp. Combining both the large-cell approach and the finite-size scaling results, we obtain ${y}_{p}=0.7385\ifmmode\pm\else\textpm\fi{}0.0080$ and ${y}_{h}=1.898\ifmmode\pm\else\textpm\fi{}0.003$. Thus we find ${\ensuremath{\alpha}}_{p}=\ensuremath{-}0.708\ifmmode\pm\else\textpm\fi{}0.030$, ${\ensuremath{\beta}}_{p}=0.138(+0.006,\ensuremath{-}0.005)$, ${\ensuremath{\gamma}}_{p}=2.432\ifmmode\pm\else\textpm\fi{}0.035$, ${\ensuremath{\delta}}_{p}=18.6\ifmmode\pm\else\textpm\fi{}0.6$, ${\ensuremath{\nu}}_{p}=1.354\ifmmode\pm\else\textpm\fi{}0.015$, and $2\ensuremath{-}{\ensuremath{\eta}}_{p}=1.796\ifmmode\pm\else\textpm\fi{}0.006$. The site-percolation threshold is found for the square lattice at ${p}_{c}=0.5931\ifmmode\pm\else\textpm\fi{}0.0006$. We note that our calculated value of ${\ensuremath{\nu}}_{p}$ is in considerably better agreement with the proposal of Klein et al. that ${\ensuremath{\nu}}_{p}=\frac{\mathrm{ln}\sqrt{3}}{\mathrm{ln}(\frac{3}{2})}\ensuremath{\cong}1.3548$, than with den Nijs' recent conjecture, which predicts ${\ensuremath{\nu}}_{p}=\frac{4}{3}$. However, our results cannot entirely rule out the latter possibility.
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).