Exact solution of the classical dimer model on a triangular lattice: Monomer-monomer correlations

We obtain an asymptotic formula, as $n\to \mathrm{\infty }$, for the monomer-monomer correlation function $K_2(x,y)$ in the classical dimer model on a triangular lattice, with the horizontal and vertical weights $w_h=w_v=1$ and the diagonal weight $w_d=t>0$, where $x$ and $y$ are sites $n$ spaces apart in adjacent rows. We find that $t_c=\frac{1}{2}$ is a critical value of $t$. We prove that in the subcritical case, $0<t<\frac{1}{2}$, as $n\to \mathrm{\infty }$, $K_2(x,y)=K_2(\infty)\left[1-\frac{e^{-n/\xi}}{n},\Big(C_1+C_2(-1)^n+\mathcal O(n^{-1})\Big)\right]$,withexplicitformulaefor$K_2(\infty)$,$\xi$,$C_1$, and $C_2$. In the supercritical case, $\frac{1}{2}<t<1$, we prove that as $n\to \mathrm{\infty }$, $K_2(x,y)=K_2(\infty)\Bigg[1- \frac{e^{-n/\xi}}{n}, \Big(C_1\cos(\omega n+\varphi_1)+C_2(-1)^n\cos(\omega n+\varphi_2)+ C_3+C_4(-1)^n$ $+\mathcal{O}(n^{-1})\left)\right]$, with explicit formulae for $K_2(\mathrm{\infty })$, $\xi$, $\omega$, and $C_1$, $C_2$, $C_3$, $C_4$, ${\phi }_1$, ${\phi }_2$. 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.