Percolation thresholds on elongated lattices

We investigate the percolation thresholds of both random and invasion percolation in two and three dimensions on elongated lattices; lattices with a geometry of L^(d−1) × nL in d dimensions, where n denotes the aspect ratio of the lattice. Scaling laws for the threshold and spanning cluster density for random percolation are derived and simulation confirms the behaviour. A direct relationship between thresholds obtained for random percolation and invasion percolation is given and verified numerically