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 probability 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 point of the model as it marks the exact point of transition between the subcritical phase and the supercritical phase . For Bernoulli percolation on an infinite point lattice, subcritical models almost surely have no infinite connected components, whereas supercritical models almost surely contain at least one such connected 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. For site percolation on a point lattice, the percolation threshold is the critical occupation probability for the existence of an infinite connected component.

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 that of site percolation on the square point lattice, 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), where is the probability that the origin belongs to an infinite cluster. No independent human verification had been reported as of Sep. 9, 2026.

Bou-Rabee (2026) reported that nearest-neighbor Bernoulli site percolation on satisfies for every , so there is no infinite open cluster at criticality. ChatGPT supplied the mathematical argument, with Claude and GLM supporting formalization. As of Oct. 8, 2026, the Lean development had a successful public build and its statement had been compared with the lattice question, but an independent rebuild and independent mathematical review had not been reported (VibeMathed 2026).