We obtain an asymptotic formula, as n→∞\documentclass[12pt]{minimal}
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\begin{document}$${n\to\infty}$$\end{document}, for the monomer–monomer correlation function K2(n)\documentclass[12pt]{minimal}
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\begin{document}$${K_2(n)}$$\end{document} in the classical dimer model on a triangular lattice, with the horizontal and vertical weights wh=wv=1\documentclass[12pt]{minimal}
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\begin{document}$${w_h=w_v=1}$$\end{document} and the diagonal weight wd=t>0\documentclass[12pt]{minimal}
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\begin{document}$${w_d=t > 0}$$\end{document}, between two monomers at vertices q and r that are n spaces apart in adjacent rows. We find that tc=12\documentclass[12pt]{minimal}
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\begin{document}$${t_c=\frac{1}{2}}$$\end{document} is a critical value of t. We prove that in the subcritical case, 0<t<12\documentclass[12pt]{minimal}
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\begin{document}$${0 < t < \frac{1}{2}}$$\end{document}, as n→∞,K2(n)=K2(∞)1-e-n/ξn(C1+C2(-1)n+O(n-1))\documentclass[12pt]{minimal}
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\begin{document}$${n\to\infty, K_2(n)=K_2(\infty)\left[1-\frac{e^{-n/\xi}}{n}\,\Big(C_1+C_2(-1)^n+ \mathcal{O}(n^{-1})\Big) \right]}$$\end{document}, with explicit formulae for K2(∞),ξ,C1\documentclass[12pt]{minimal}
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\begin{document}$${K_2(\infty), \xi, C_1}$$\end{document}, and C2\documentclass[12pt]{minimal}
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\begin{document}$${C_2}$$\end{document}. In the supercritical case, 12<t<1\documentclass[12pt]{minimal}
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\begin{document}$${\frac{1}{2} < t < 1}$$\end{document}, we prove that as n→∞,K2(n)=K2(∞)[1-e-n/ξn(C1cos(ωn+φ1)+C2(-1)ncos(ωn+φ2)+C3+C4(-1)n+O(n-1))]\documentclass[12pt]{minimal}
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\begin{document}$${n\to\infty, K_2(n)=K_2(\infty)\Bigg[1-\frac{e^{-n/\xi}}{n}\, \Big(C_1\cos(\omega n+\varphi_1)+C_2(-1)^n\cos(\omega n+\varphi_2)+ C_3+C_4(-1)^n + \mathcal{O}(n^{-1})\Big)\Bigg]}$$\end{document}, with explicit formulae for K2(∞),ξ,ω\documentclass[12pt]{minimal}
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\begin{document}$${K_2(\infty), \xi,\omega}$$\end{document}, and C1,C2,C3,C4,φ1,φ2\documentclass[12pt]{minimal}
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\begin{document}$${C_1, C_2, C_3, C_4, \varphi_1, \varphi_2}$$\end{document}. The proof is based on an extension of the Borodin–Okounkov–Case–Geronimo formula to block Toeplitz determinants and on an asymptotic analysis of the Fredholm determinants in hand.
