Critical probability and scaling functions of bond percolation on two-dimensional random lattices

We locate the critical probability of bond percolation on two-dimensional random lattices as p(c) = 0.3329(6). Because of the symmetry with respect to permutation of the two axes for random lattices, we expect that for an aspect ratio of unity and sufficiently large lattices, the probability of horizontal spanning equals the probability of vertical spanning. This is confirmed by our Monte Carlo simulations. We show that the ideas of universal scaling functions and nonuniversal metric factors can be extended to random lattices by studying the existence probability E-p and the percolation probability P on finite square, planar triangular, and random lattices with periodic boundary conditions using a histogram Monte Carlo method. Our results also indicate that the metric factors may be the same between random lattices and planar triangular lattices provided that the aspect ratios are 1 and root(3/2).