Universal scaling functions for site and bond percolations on planar lattices

Universality and scaling are two important concepts in the theory of critical phenomena. It is generally believed that site and bond percolations on lattices of the same dimensions have the same set of critical exponents, but they have different scaling functions. In this paper, we briefly review our recent Monte Carlo results about universal scaling functions for site and bond percolation on planar lattices. We find that, by choosing an aspect ratio for each lattice and a very small number of non-universal metric factors, all scaled data of the existence probability Ep and the percolation probability P for site and bond percolations on square, plane triangular, and honeycomb lattices may fall on the same universal scaling functions. We also find that free and periodic boundary conditions share the same non-universal metric factors. When the aspect ratio of each lattice is reduced by the same factor, the non-universal metric factors remain the same. The implications of such results are discussed.