Percolation Threshold

 

In the field of percolation theory, the term percolation threshold is used to denote the probability which "marks the arrival" (Grimmett 1999) of an infinite connected component (i.e., of a percolation) within a particular model. The percolation threshold is commonly denoted  and is sometimes called the critical phenomenon of the model.

Special attention is paid to probabilities  both below and above the percolation threshold; a percolation model for which  is called a subcritical percolation while a model satisfying  is called a supercritical percolation. Because of this distinction, the value  is also sometimes called the phase transition of the model as it marks the exact point of transition between the subcritical phase  and the supercritical phase . Note that by definition, subcritical percolation models are necessarily devoid of infinite connected components, whereas supercritical models always contain at least one such component.

A great deal of literature has been devoted to the identification of the percolation threshold within a number of models and, indeed, nearly all literature which highlights a specific percolation model does so in order to study and present information related to that model's percolation threshold.

This concept is particularly well-studied in the case of discrete percolation theory on certain classes of "well-behaved" point lattices. In this context, the percolation threshold is the fraction of lattice points that must be filled to create a continuous path of nearest neighbors from one side to another.

The following table is taken from Stauffer and Aharony (1992, p. 17). Entries indicated with an asterisk (*) have known exact solutions.

lattice	(site percolation)	(bond percolation)
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*

Exactly known values include

			(1)
			(2)
			(3)
			(4)

Determining an exact expression for other percolation thresholds, including of the square site percolation, remains an open problem.

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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. R. "Percolation Cluster Density Constants." §5.18 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 371-378, 2003.

Grimmett, G. Percolation, 2nd ed. Berlin: Springer-Verlag, 1999.

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.

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