Journal of Statistical Mechanics: Theory and Experiment
Volume 2008
January 2008
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Richard A Neher et al J. Stat. Mech. (2008) P01011 doi:10.1088/1742-5468/2008/01/P01011
Topological estimation of percolation thresholds
Richard A Neher1,3, Klaus Mecke2 and Herbert Wagner1
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| Figure 1. (a) Black clusters partition the white vertices into an aggregate of complementary clusters. For the square lattice, white vertices are connected by lattice bonds and by diagonal bonds across the faces of the lattice. (b) The black and white clusters of size 1 and 2 on the square lattice. The white perimeter sites of a black cluster are connected (and vice versa). |
| Figure 2. The MEC of the square lattice (left) and of the triangular lattice (right). Note that p0 is slightly above pc for the square lattice. The triangular lattice is self-matching and pc = p0 = 1/2. |
Figure 3. The percolation threshold pc is slightly below the zero crossing
p0 of the MEC whenever  . This order is reversed, if  , as is apparent in the panel on bond percolation. The solution
p* of equation (17) provides a very accurate estimate of
pc. The deviation |pc–p*| exceeds 0.01 only for very open lattices with high percolation thresholds. Lattices are in the order of decreasing percolation threshold. For vertex configurations, numerical values of
pc and p* for the 2-uniform lattices; see table 4 in the supplementary material. Numerical estimates of
pcsite for the Archimedean lattices are taken from [14]; values for
pcbond are from [15, 16]. |
Figure 4. (a) A lattice face is decorated, when all diagonal connections across the faces have been added to the lattice graph, as illustrated here for three faces of the hexagonal lattice. We
consider randomly decorated lattices where each face is decorated with probability
pdec. (b) The site percolation threshold pc(pdec) decreases smoothly as the degree of decoration is varied from
pdec = 0 to
1. The zero crossing p0(pdec) of
χ(p, pdec) provides a
tight upper to pc(pdec) if  and vice versa. The solution of equation (17), p*(pdec), lies within 0.005 of pc(pdec) with the largest deviations at full or no decoration. |
Figure 5. Site percolation on the square lattice: the solution  of equation (16) approaches pc with increasing s0. |
| Figure 6. The graphs of the MEC of the simple cubic (sc) and body-centered cubic (bcc) lattice. The bcc lattice has the same black and white connectivities and is hence invariant to the substitution p = 1–p. |
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