Effect of anisotropy on finite-size scaling in percolation theory

We investigate the effects of anisotropy on finite-size scaling of site percolation in two dimensions. We consider a lattice of size n(x) x n(y). We define an aspect ratio omega=n(x)/n(y) and consider the mean connected fraction P (averaged over the realizations) as a function of the site occupancy probability (p), the system size (n(x)), and this aspect ratio. It is clear that there is an easy direction for percolation, which is in the short direction (i.e., y if omega>1) and a difficult direction which is along the long axis. We define an apparent percolation threshold in each direction as the value of p when 50% of realizations connect in that direction. We show that standard finite-size scaling applies if we use this apparent threshold. We also find a finite-size scaling for the fluctuations about this mean connected fraction.