Percolation thresholds, critical exponents, and scaling functions on planar random lattices and their duals.

Abstract

The bond-percolation process is studied on periodic planar random lattices and their duals. The thresholds and critical exponents of the percolation transition are determined. The scaling functions of the percolating probability, the existence probability of the appearance of percolating clusters, and the mean cluster size are also calculated. The simulation result of the percolation threshold is p(c)=0.3333+/-0.0001 for planar random lattices, and 0.6670+/-0.0001 for the duals of planar random lattices. We conjecture that the exact value of p(c) is 1/3 for a planar random lattice and 2/3 for the dual of a planar random lattice. By taking possible errors into account, the results of our critical exponents agree with the values given by the universality hypothesis. By properly adjusting the metric factors on random lattices and their duals, we demonstrate explicitly that the idea of a universal scaling function with nonuniversal metric factors in the finite-size scaling theory can be extended to random lattices and their duals for the existence probability, the percolating probability, and the mean cluster size.