Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations

Abstract

The site percolation threshold for the random Voronoi network is determined numerically, with the result $p_c=0.71410\pm 0.00002$, using Monte Carlo simulation on periodic systems of up to $40000$ sites. The result is very close to the recent theoretical estimate $p_c\approx 0.7151$ of Neher et al. For the bond threshold on the Voronoi network, we find $p_c=0.666931\pm 0.000005$ implying that, for its dual, the Delaunay triangulation $p_c=0.333069\pm 0.000005$. These results rule out the conjecture by Hsu and Huang that the bond thresholds are 2/3 and 1/3, respectively, but support the conjecture of Wierman that, for fully triangulated lattices other than the regular triangular lattice, the bond threshold is less than $2\text{sin}\pi /18\approx 0.3473$.