Monte Carlo study of invasion percolation clusters in two and three dimensions

Abstract

The authors consider the process of growing invasion percolation clusters from a point into an infinite medium. The form considered is the simplest one in which the cluster is permitted to grow into regions it has previously surrounded. It is shown that this process can yield extremely good Monte Carlo estimates of the percolation threshold pc. For the square, triangular and match-square lattices they obtain pc values of 0.5925+or-0.0003, 0.5000+or-0.0003 and 0.4072+or-0.0002, and for the simple cubic lattice 0.31158+or-0.00006. The errors quoted are purely statistical, and represent one standard deviation. Two critical exponents are obtained which they suggest should be identified in terms of the fractal dimension D and gap exponent Delta of ordinary percolation. Based on these identifications they obtain values for 1/D and 1/ Delta of 0.527+or-0.002 and 0.393+or-0.004 in two dimensions and 0.402+or-0.003 and 0.454+or-0.005 in three dimensions. These results are consistent with known exact results and best series and Monte Carlo estimates, suggesting that the form of invasion percolation considered is probably in the same universality class as ordinary percolation.