Large lattice random site percolation

We simulate the two dimensional (2d), simple square and three-dimensional (3d), simple cubic random site percolation systems for L=2000000 (2d) and L=10001 (3d) at the percolation thresholds for these systems. We report excellent agreement with the Fisher exponent, τ, in 2d, with the proposed exact value 187/91 and good agreement with other good high quality simulation results in 3d of 2.186. We have also computed how the first, second, third,…, largest clusters scale with L at the percolation threshold. These clusters all scale with the same fractal dimensionality as the largest cluster.