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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


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.''

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.

© 1996-9 Eric W. Weisstein