The monomer-dimer constant of the square lattice

Place non-overlapping dominoes (dimers) on the edges of an $m \times n$ grid and leave the other sites empty (monomers). The number of such arrangements grows like $e^{h\,mn}$, with $h \approx 0.6627989727$. Kasteleyn, Temperley and Fisher solved the case of complete domino coverings in 1961, but no closed form is known for $h$.

Problem

Let $\Lambda_{m,n}$ be the $m \times n$ rectangle in $\mathbb{Z}^2$ with free boundary. A configuration is a matching: a set of pairwise disjoint edges (dimers). Every uncovered vertex is a monomer. With monomer weight $z$ and dimer weight 1, the partition function is

$$Z_{m,n}(z) = \sum_{\text{matchings } M} z^{\#\text{monomers}(M)}.$$

At $z = 1$ every matching counts once. The monomer-dimer constant is

$$h = \lim_{m,n \to \infty} \frac{1}{mn} \log Z_{m,n}(1) = 0.6627989727\ldots,$$

and the limit does not depend on the boundary conditions. Find a closed-form expression for $h$: 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. The defining limit, transfer-matrix eigenvalues and numerical approximations do not count.

Background

Without monomers ($z = 0$, $m$ and $n$ even) the model is exactly solvable. Kasteleyn, and Temperley and Fisher, showed in 1961 that the entropy per site of domino tilings is $G/\pi$ [1, 2]. With monomers present, the problem on two-dimensional lattices is #P-complete [4], and $h$, the entropy per site at unit activity, is known only numerically.

What is known

• Butera and Pernici (2012) quote $h = 0.6627989727(1)$ as the most precise determination, citing Baxter (1968) and Friedland and Peled (2005) [3, 5]. They also quote $0.66279897190 \le h \le 0.662798972844913$ as the tightest bounds then known [6].
• Kong (2006) obtained $h = 0.662798972834$ from exact enumerations on strips and claimed 11 correct digits [4]. HorizonMath uses this 12-digit value as its reference.
• According to the source, no closed form is known for any $z > 0$, including $z = 1$.

What would count as progress

• A closed-form expression that matches Kong's estimate to 11 significant digits, the accuracy claimed for it, and respects the bounds above. 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 12-digit reference.
• Eleven digits is a weak filter, so new and more precise values of $h$ are valuable in their own right, for example from larger transfer-matrix or series computations.
• A proof that an expression equals $h$, for example through an exact solution of the model at $z = 1$.

References

1. P. W. Kasteleyn, "The statistics of dimers on a lattice", Physica 27 (1961), 1209–1225. doi:10.1016/0031-8914(61)90063-5
2. H. N. V. Temperley and M. E. Fisher, "Dimer problem in statistical mechanics — an exact result", Phil. Mag. 6 (1961), 1061–1063. doi:10.1080/14786436108243366
3. R. J. Baxter, "Dimers on a rectangular lattice", J. Math. Phys. 9 (1968), 650–654. doi:10.1063/1.1664623
4. Y. Kong, "Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density", Phys. Rev. E 74 (2006), 061102. doi:10.1103/PhysRevE.74.061102, arXiv:cond-mat/0610690
5. S. Friedland and U. N. Peled, "Theory of computation of multidimensional entropy with an application to the monomer–dimer problem", Adv. Appl. Math. 34 (2005), 486–522. doi:10.1016/j.aam.2004.08.005
6. P. Butera and M. Pernici, "Yang–Lee edge singularities from extended activity expansions of the dimer density for bipartite lattices of dimensionality 2 ≤ d ≤ 7", Phys. Rev. E 86 (2012), 011104. doi:10.1103/PhysRevE.86.011104

Source

Imported from HorizonMath monomer_dimer_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.