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Physica A Statistical Mechanics and its Applications

Critical properties of random-site percolation in two and three dimensions: A Monte Carlo study

Martin J. Corsten, Naeem Jan, R. J. Jerrard

Published Apr 1, 1989• 15 citations
Publisher
DOI
10.1016/0378-4371(89)90020-4

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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.
We study the site-bond-correlated percolation model in two and three dimensions. In this model, two first-neighbor sites are considered to be joined by an active bond if both are present and if their first neighbors, along the line joining them, are also present. Using a Monte Carlo renormalization-group approach, we calculate percolation thresholds and critical exponents for both square and simple cubic lattices. Our results indicate that the correlated model is in the same universality class as usual percolation, in two and three dimensions; the critical probabilities for the square and simple cubic lattices are found to be 0.7402\ifmmode\pm\else\textpm\fi{}0.0002 and 0.5660\ifmmode\pm\else\textpm\fi{}0.0004, respectively.