Scaling and universality in the spanning probability for percolation

We discuss the spanning percolation probability function for three different spanning rules, in general dimensions, with both free and periodic boundary conditions. Our discussion relies on the renormalization group theory, which indicates that, apart from a few scale factors, the scaling functions for the spanning probability are determined by the fixed point and therefore are universal for every system with the same dimensionality, spanning rule, aspect ratio, and boundary conditions. For square and rectangular systems, we combine this theory with simple relations between the spanning rules and with duality arguments, and find strong relations among different derivatives of the spanning function with respect to the scaling variables, thus yielding several universal amplitude ratios and allowing a systematic study of the corrections to scaling, both singular and analytic, in the system size. The theoretical predictions are numerically confirmed with excellent accuracy.