Exact Solution of the Classical Dimer Model on a Triangular Lattice: Monomer–Monomer Correlations

Abstract

We obtain an asymptotic formula, as $n\to\infty$, 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 $n\to\infty$, $K_2(x,y)=K_2(\infty)\left[1-\frac{e^{-n/\xi}}{n}\,\Big(C_1+C_2(-1)^n+\mathcal O(n^{-1})\Big)\right]$, with explicit formulae for $K_2(\infty)$, $\xi$, $C_1$, and $C_2$. In the supercritical case, $\frac{1}{2} < t < 1$, we prove that as $n\to\infty$, $K_2(x,y)=K_2(\infty)\Bigg[1- \frac{e^{-n/\xi}}{n}\, \Big(C_1\cos(\omega n+\varphi_1)+C_2(-1)^n\cos(\omega n+\varphi_2)+ C_3+C_4(-1)^n$ $+\mathcal O(n^{-1})\Big)\Bigg]$, with explicit formulae for $K_2(\infty)$, $\xi$, $\omega$, and $C_1$, $C_2$, $C_3$, $C_4$, $\varphi_1$, $\varphi_2$. 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.