The hard-square entropy constant

How many ways are there to place tokens on an $n \times n$ grid so that no two tokens are side by side? The count grows like $\kappa^{n^2}$ with $\kappa \approx 1.5030$. Its triangular-lattice cousin, the hard-hexagon model, was solved exactly by Baxter, but $\kappa$ itself has no known closed form.

Problem

Let $F(m,n)$ be the number of $m \times n$ matrices with entries in $\{0,1\}$ in which no two 1s are horizontally or vertically adjacent. Equivalently, $F(m,n)$ counts the independent sets of the $m \times n$ grid graph. The hard-square entropy constant is

$$\kappa = \lim_{n \to \infty} F(n,n)^{1/n^2} = 1.50304808247533226432\ldots$$

The double limit $\lim_{m,n\to\infty} F(m,n)^{1/(mn)}$ exists and has the same value [1]. Find a closed-form expression for $\kappa$: a finite combination of algebraic numbers, standard constants and special functions such as $\Gamma$, hypergeometric functions, elliptic integrals and polylogarithms. Transfer-matrix eigenvalues, truncated series and infinite products without a closed evaluation do not count.

Background

In statistical mechanics this is the hard-core lattice gas on $\mathbb{Z}^2$: particles on lattice sites, with nearest neighbours excluded. At unit activity every allowed configuration has weight 1, so $\kappa$ is the partition function per site and $\log \kappa \approx 0.40749$ is the entropy per site [3]. In symbolic dynamics, $\log\kappa$ is the topological entropy of the hard-square shift [4]. Baxter solved the hard-hexagon model, the same exclusion rule on the triangular lattice, exactly in 1980 [2]. No exact solution of the square-lattice model is known.

What is known

• $\kappa = 1.5030480824753322643220663294755536893857810\ldots$. Baxter (1999) computed the partition function per site at unit activity to 43 decimal places with corner transfer matrices [3]. These digits are listed in OEIS A085850, and HorizonMath uses them as its reference value.
• Calkin and Wilf (1998) proved that the double limit exists and that $1.503047782\ldots \le \kappa \le 1.5035148\ldots$ [1].
• Pavlov (2012) proved that differences of strip entropies converge to $\log\kappa$ at least exponentially fast. Hence $\log\kappa$ can be computed to within $1/n$ in time polynomial in $n$. He also noted that no closed form is known [4].

What would count as progress

• A closed-form expression that matches $\kappa$ to 20 significant digits, HorizonMath's acceptance threshold. Its evaluator checks this at 100-digit precision and rejects numerical shortcuts such as quadrature, root-finding, infinite series and fitted constants. A candidate that also reproduces all 43 published decimals would be compelling evidence.
• A proof that some expression equals $\kappa$, for instance through an exact solution of the hard-square model at unit activity.

References

1. N. J. Calkin and H. S. Wilf, "The number of independent sets in a grid graph", SIAM J. Discrete Math. 11 (1998), 54–60. doi:10.1137/S089548019528993X
2. R. J. Baxter, "Hard hexagons: exact solution", J. Phys. A 13 (1980), L61–L70. doi:10.1088/0305-4470/13/3/007
3. R. J. Baxter, "Planar lattice gases with nearest-neighbour exclusion", Ann. Comb. 3 (1999), 191–203. doi:10.1007/BF01608783, arXiv:cond-mat/9811264
4. R. Pavlov, "Approximating the hard square entropy constant with probabilistic methods", Ann. Probab. 40 (2012), 2362–2399. doi:10.1214/11-AOP681

Source

Imported from HorizonMath hard_square_entropy, 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.