Thresholds On Hyperbolic, Hierarchical, and Tree Lattices
Visualization of a triangular hyperbolic lattice {3,7} projected on the Poincaré disk
Depiction of the non-planar Hanoi network HN-NP
| Lattice | z | Site Percolation Threshold | Bond Percolation Threshold | ||
|---|---|---|---|---|---|
| Lower | Upper | ||||
| {4,5} hyperbolic | 5 | 5 | 0.27 | 0.52 | |
| {7,3} hyperbolic | 3 | 3 | 0.72 | 0.53 | |
| {3,7} hyperbolic | 7 | 7 | 0.20 | 0.37 | |
| {∞,3} Cayley tree | 3 | 3 | 1/2 | 1/2 | 1 |
| Enhanced binary tree (EBT) | 0.304(1) | 0.48, 0.564(1) | |||
| Enhanced binary tree dual | 0.436(1) | 0.696(1) | |||
| Non-Planar Hanoi Network (HN-NP) | 0.319445 | 0.381996 | |||
| Cayley tree with grandparents | 8 | 0.158656326 | |||
Note: {m,n} is the Shläfli symbol, signifying a hyperbolic lattice in which n regular m-gons meet at every vertex
Cayley tree (Bethe latttice) with coordination number z: pc= 1 / (z - 1)
Cayley tree with a distribution of z with mean, mean-square pc= (site or bond threshold)
Read more about this topic: Percolation Threshold
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