Thresholds On 2D Random and Quasi-lattices
Left to right: (a) Voronoi diagram (solid lines) and its dual, the Delaunay triangulation (dotted lines), for a Poisson distribution of points, (b) Delaunay triangulation only, (c) Voronoi diagram (black lines) and the covering or line graph (dotted red lines), (d) the Relative Neighborhood Graph (black lines) superimposed on the Delaunay triangulation (black plus grey lines) for the same set of 128 uniformly distributed random points.
| Lattice | z | Site Percolation Threshold | Bond Percolation Threshold | |
|---|---|---|---|---|
| Relative neighborhood graph | 2.5576 | 0.796(2) | 0.771(2) | |
| Voronoi tessellation | 3 | 0.71410(2), 0.7151* | 0.68, 0.666931(5), 0.6670(1) | |
| Voronoi covering | 4 | 0.666931(2) | 0.53618(2) | |
| Penrose rhomb dual | 4 | 0.6381(3) | 0.5233(2) | |
| Penrose rhomb | 4 | 0.5837(3), 0.58391(1) | 0.4770(2) | |
| Delaunay triangulation | 6 | 1/2 | 0.333069(2) |
*Theoretical estimate
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