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 nₓ×ny. We define an aspect ratio ω=nₓ∕ny and consider the mean connected fraction P (averaged over the realizations) as a function of the site occupancy probability $(p)$, the system size (nₓ), 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 ω>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.D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor and Francis, London, 1992).