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 $n\to \infty$, $K_2(x,y)=K_2(\infty )\left[1-\frac{e^{-n/\xi }}{n}(C_1+C_2(-1{)}^n+\mathcal{O}(n^{-1}))\right]$, 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 $n\to \infty$, $K_2(x,y)=K_2(\infty )[1-\frac{e^{-n/\xi }}{n}(C_1\cos (\omega n+{\phi }_1)+C_2(-1{)}^n\cos (\omega n+{\phi }_2)+C_3+C_4(-1{)}^n$ $+\mathcal{O}(n^{-1})\left)\right]$, with explicit formulae for $K_2(\infty )$, $\xi$, $\omega$, 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.

References in corpus (2)

• On Ising and dimer models in two and three dimensions
• Asymptotics of block Toeplitz determinants and the classical dimer model

Cited by in corpus (3)

• Site monotonicity and uniform positivity for interacting random walks and the spin O(N) model with arbitrary N
• Modified Szego-Widom Asymptotics for Block Toeplitz Matrices with Zero Modes
• Exactly Solvable Topological Phase Transition in a Quantum Dimer Model