The site percolation threshold of the square lattice

Find a closed form for the site percolation threshold of the square lattice, p_c ≈ 0.5927460508; bond percolation has p_c = 1/2.

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1	# The site percolation threshold of the square lattice
2
3	Open each site of the square grid independently with probability $p$. Above a critical value $p_c \approx 0.5927$ an infinite cluster of open sites appears. For the bond version of the model $p_c = 1/2$ exactly, but no exact expression is known for the site threshold. The task is to find one.
4
5	## Problem
6
7	Declare each vertex of $\mathbb{Z}^2$ open with probability $p$ and closed otherwise, independently of all other vertices. Two open vertices are connected when they are nearest neighbours, at distance 1. Let
8
9	$$
10	p_c = \inf\bigl\{\, p \in [0,1] : \mathbb{P}_p(\text{there is an infinite connected cluster of open sites}) > 0 \,\bigr\}.
11	$$
12
13	Find a closed-form expression for $p_c = 0.59274605079210\ldots$: a finite combination of algebraic numbers, standard constants such as $\pi$, $e$, $\gamma$, Catalan's constant $G$ and $\zeta(n)$, and special functions at algebraic arguments.
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15	## Background
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17	Broadbent and Hammersley introduced percolation in 1957 to study how the random structure of a medium affects the spread of a fluid through it [1]. For bond percolation on the square lattice, where edges rather than sites are open, a self-duality argument gives $p_c = 1/2$; Kesten proved this in 1980 [2]. That argument does not apply to site percolation, and the square-lattice site threshold has become a standard benchmark for high-precision numerical methods. Rigorous bounds on it have improved only slowly.
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19	## What is known
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21	- Numerical value: $p_c = 0.59274605079210(2)$ (Jacobsen, 2015), from an eigenvalue formulation of graph-polynomial methods in periodic Temperley–Lieb algebras [3]. HorizonMath uses this 14-digit value as its reference.
22	- Rigorous bounds are far weaker: van den Berg and Ermakov (1996) proved a lower bound [4], and Wierman and Oberly (2024) a new upper bound [5].
23	- According to the source, neither a closed form nor a conjectured exact expression is known.
24
25	## What would count as progress
26
27	- A closed-form expression that matches the reference value to 13 significant digits, all but the last available one. HorizonMath's evaluator works at 100-digit precision and rejects numerical shortcuts such as quadrature, root-finding, infinite series and fitted constants; its default test asks for 20 matching digits, which no answer can reach against this 14-digit reference.
28	- Thirteen digits can be matched by chance, so a convincing candidate needs either a derivation or confirmation against new digits. An independent computation of $p_c$ beyond 14 digits is useful progress in its own right.
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30	## References
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32	1. S. R. Broadbent and J. M. Hammersley, "Percolation processes. I. Crystals and mazes", Math. Proc. Cambridge Philos. Soc. 53 (1957), 629–641. [doi:10.1017/S0305004100032680](https://doi.org/10.1017/S0305004100032680)
33	2. H. Kesten, "The critical probability of bond percolation on the square lattice equals 1/2", Comm. Math. Phys. 74 (1980), 41–59. [doi:10.1007/BF01197577](https://doi.org/10.1007/BF01197577)
34	3. J. L. Jacobsen, "Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras", J. Phys. A 48 (2015), 454003. [doi:10.1088/1751-8113/48/45/454003](https://doi.org/10.1088/1751-8113/48/45/454003)
35	4. J. van den Berg and A. Ermakov, "A new lower bound for the critical probability of site percolation on the square lattice", Random Structures Algorithms 8 (1996), 199–212. [doi link](https://doi.org/10.1002/%28SICI%291098-2418%28199605%298%3A3%3C199%3A%3AAID-RSA4%3E3.0.CO%3B2-T)
36	5. J. C. Wierman and S. P. Oberly, "A new upper bound for the site percolation threshold of the square lattice", in Combinatorics, Graph Theory and Computing, Springer Proc. Math. Stat., 2024, 129–138. [doi:10.1007/978-3-031-62166-6_9](https://doi.org/10.1007/978-3-031-62166-6_9)
37
38	## Source
39
40	Imported from [HorizonMath site_percolation_square](https://github.com/ewang26/HorizonMath/blob/4ef0b61a8a6d4c5ded661fab7c2e896af84abe6e/data/problems_full.json#L781-L791), curated by Erik Y. Wang, Sumeet R. Motwani et al. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.