Percolation Threshold

The critical fraction of lattice points which must be filled to create a continuous path of nearest neighbors from one side to another. The following table is from Stauffer and Aharony (1992, p. 17).

Lattice	Site	Bond
Cubic (Body-Centered)	0.246	0.1803
Cubic (Face-Centered)	0.198	0.119
Cubic (Simple)	0.3116	0.2488
Diamond	0.43	0.388
Honeycomb	0.6962	0.65271
4-Hypercubic	0.197	0.1601
5-Hypercubic	0.141	0.1182
6-Hypercubic	0.107	0.0942
7-Hypercubic	0.089	0.0787
Square	0.592746	0.50000
Triangular	0.50000	0.34729

The square bond value is $1/2$ exactly, as is the triangular site. $p_c=2\sin(\pi/18)$ for the triangular bond and $p_c=1-2\sin(\pi/18)$ for the honeycomb bond. An exact answer for the square site percolation threshold is not known.

See also Percolation Theory

References

Essam, J. W.; Gaunt, D. S.; and Guttmann, A. J. ``Percolation Theory at the Critical Dimension.'' J. Phys. A 11, 1983-1990, 1978.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/rndprc/rndprc.html

Kesten, H. Percolation Theory for Mathematicians. Boston, MA: Birkhäuser, 1982.

Stauffer, D. and Aharony, A. Introduction to Percolation Theory, 2nd ed. London: Taylor & Francis, 1992.