Finite size scaling for percolation on elongated lattices in two and three dimensions

We derive scaling laws for the percolation properties of an elongated lattice, i.e., those with dimensions of L^(d-1)×nL in d dimensions, where n denotes the aspect ratio of the lattice. Based on statistical arguments it is shown that, in the direction of the extension, the percolation threshold scales approximately as ln n^(1/a) in both two and three dimensions. Extensive Monte Carlo simulations of the site percolation model confirm this scaling behavior. It is further shown that the density of the incipient infinite cluster at the percolation threshold scales differently in two and three dimensions