The site percolation threshold of the square lattice

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.

Problem

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

p_c = \inf\bigl\{\, p \in [0,1] : \mathbb{P}_p(\text{there is an infinite connected cluster of open sites}) > 0 \,\bigr\}.

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.

Background

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.

What is known

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.
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].
According to the source, neither a closed form nor a conjectured exact expression is known.

What would count as progress

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.
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.

References

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
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
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
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
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

Source

Imported from HorizonMath site_percolation_square, 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.