Thresholds On 2D Bowtie and Martini Lattices
To the left, center, and right are: the martini lattice, the martini-A lattice, the martini-B lattice. Below: the martini covering lattice, same as the 2x2, 1x1 subnet for kagome-type lattices (removed).
Some other examples of generalized bow-tie lattices (a-d) and the duals of the lattices (e-h)
| Lattice | z | Site Percolation Threshold | Bond Percolation Threshold | |
|---|---|---|---|---|
| martini (3/4)(3,92)+(1/4)(93) | 3 | 3 | 0.764826..., 1 +p4 - 3p3=0 | 0.707107... = 1/√2 |
| bow-tie (c) | 3,4 | 3 1/7 | 0.672929..., 1-2p3-2p4-2p5-7p6+18p7+11p8-35p9+21p10-4p11=0 | |
| bow-tie (d) | 3,4 | 3⅓ | 0.625457..., 1-2p2-3p3+4p4-p5=0 | |
| martini-A (2/3)(3,72)+(1/3)(3,73) | 3,4 | 3⅓ | 1/√2 | 0.625457..., 1-2p2-3p3+4p4-p5=0 |
| bow-tie dual lattice (e) | 3,4 | 3⅔ | 0.595482..., 1-pcbond (bow-tie (a)) | |
| bow-tie (b) | 3,4,6 | 3⅔ | 0.533213..., 1-p- 2p3 -4p4-4p5+156+ 13p7-36p8+19p9+ p10 + p11=0 | |
| martini covering (1/2)(33,9)+(1/2)(3,9,3,9) | 4 | 4 | 0.707107... = 1/√2 | 0.57086651(33) |
| martini-B (1/2)(3,5,3,52)+(1/2)(3,52) | 3, 5 | 4 | 0.618034... = 2/(1 +√5)..., 1- p2-p=0 | 1/2 |
| bow-tie dual lattice (f) | 3,4,8 | 4 2/5 | 0.466787..., 1-pcbond (bow-tie (b)) | |
| bow-tie (a) (1/2)(32,4,32,4)+(1/2)(3,4,3) | 4,6 | 5 | 0.5472(2) | 0.404518..., 1 - p - 6p2 +6p3-p5=0 |
| bow-tie dual lattice (h) | 3,6,8 | 5 | 0.374543..., 1-pcbond(bow-tie (d)) | |
| bow-tie dual lattice (g) | 3,6,10 | 5½ | 0.327071..., 1-pcbond(bow-tie (c)) |
Read more about this topic: Percolation Threshold
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