Percolation Threshold

The percolation threshold of a model in percolation theory is the probability that "marks the arrival" (Grimmett 1999) of an infinite connected component (i.e., of a percolation) within the 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. A separate critical-value question asks whether a cluster of infinite size already occurs at . For nearest-neighbor Bernoulli bond percolation on , an AI-generated Lean 4 development claims for every (Anthropic 2026). No independent human verification had been reported as of Sep. 9, 2026.