Largest and second largest cluster statistics at the percolation threshold of hypercubic lattices

We investigate the scale invariance of the average ratio between the masses of the largest and second largest clusters at percolation. We employ a finite size scaling method to estimate percolation thresholds based on the simulations of relatively small lattices, and report on estimates for p(c) in hypercubic lattices with d=2-7, in full agreement with the best literature estimates. Also, we find the critical mass ratio to be strongly dependent on the boundary conditions, decreasing with the lattice dimension. Further, we compute several relevant mass distribution functions associated with the two largest clusters, which approach to limiting distributions for d>6. Finally, we discuss the main relevant features of the mass distributions in light of the relative role played by the spanning and nonspanning clusters.