Percolation thresholds on three-dimensional lattices with three nearest neighbors

We present a study of site and bond percolation on periodic lattices with three nearest neighbors per site. Essentially all previous studies of percolation in 3D have considered coordination numbers of 4 or higher, but 3-coordinated lattices have attracted recent interest for their unusual symmetries and relevance to self-assembled materials. We have considered four lattices, with different symmetries, different underlying Bravais lattices, and different degrees of longer-range connections. As expected, we find that the site and bond percolation thresholds in all of the 3-connected lattices studied here are significantly higher than in the diamond lattice (a 4-connected lattice). Thresholds for different lattices are similar to within a few per cent, despite the differences between the lattices at scales beyond nearest and next-nearest neighbors. They also confirm an approximate analytical result for the relationship between coordination number and percolation threshold, one that had previously only been compared with simulation results for coordination numbers of 4 or higher.