The monomer-dimer constant of the square lattice

Find a closed form for the entropy per site of monomer-dimer configurations on the square lattice, h ≈ 0.6627989727.

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	1	+# The monomer-dimer constant of the square lattice
	2	+
	3	+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 $h$.
	4	+
	5	+## Problem
	6	+
	7	+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
	8	+
	9	+$$
	10	+Z_{m,n}(z) = \sum_{\text{matchings } M} z^{\#\text{monomers}(M)}.
	11	+$$
	12	+
	13	+At $z = 1$ every matching counts once. The monomer-dimer constant is
	14	+
	15	+$$
	16	+h = \lim_{m,n \to \infty} \frac{1}{mn} \log Z_{m,n}(1) = 0.6627989727\ldots,
	17	+$$
	18	+
	19	+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.
	20	+
	21	+## Background
	22	+
	23	+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.
	24	+
	25	+## What is known
	26	+
	27	+- 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].
	28	+- 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.
	29	+- According to the source, no closed form is known for any $z > 0$, including $z = 1$.
	30	+
	31	+## What would count as progress
	32	+
	33	+- 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.
	34	+- 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.
	35	+- A proof that an expression equals $h$, for example through an exact solution of the model at $z = 1$.
	36	+
	37	+## References
	38	+
	39	+1. P. W. Kasteleyn, "The statistics of dimers on a lattice", Physica 27 (1961), 1209–1225. [doi:10.1016/0031-8914(61)90063-5](https://doi.org/10.1016/0031-8914(61)90063-5)
	40	+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](https://doi.org/10.1080/14786436108243366)
	41	+3. R. J. Baxter, "Dimers on a rectangular lattice", J. Math. Phys. 9 (1968), 650–654. [doi:10.1063/1.1664623](https://doi.org/10.1063/1.1664623)
	42	+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](https://doi.org/10.1103/PhysRevE.74.061102), [arXiv:cond-mat/0610690](https://arxiv.org/abs/cond-mat/0610690)
	43	+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](https://doi.org/10.1016/j.aam.2004.08.005)
	44	+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](https://doi.org/10.1103/PhysRevE.86.011104)
	45	+
	46	+## Source
	47	+
	48	+Imported from [HorizonMath monomer_dimer_entropy](https://github.com/ewang26/HorizonMath/blob/4ef0b61a8a6d4c5ded661fab7c2e896af84abe6e/data/problems_full.json#L693-L703), 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.