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 ${K_2(n)}$ in the classical dimer model on a triangular lattice, with the horizontal and vertical weights ${w_h=w_v=1}$ and the diagonal weight ${w_d=t > 0}$, between two monomers at vertices q and r that are 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 ${n\to\infty, K_2(n)=K_2(\infty)\left[1-\frac{e^{-n/\xi}}{n}\,\Big(C_1+C_2(-1)^n+ \mathcal{O}(n^{-1})\Big) \right]}$, 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 ${n\to\infty, K_2(n)=K_2(\infty)\Bigg[1-\frac{e^{-n/\xi}}{n}\, \Big(C_1\cos(\omega n+\varphi_1)+C_2(-1)^n\cos(\omega n+\varphi_2)+ C_3+C_4(-1)^n + \mathcal{O}(n^{-1})\Big)\Bigg]}$, 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.