Communications in Mathematical Physics

, Volume 356, Issue 2, pp 397–425

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

Article

## 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.

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## Authors and Affiliations

• Estelle Basor
• 1
• Pavel Bleher
• 2
1. 1.American Institute of MathematicsSan JoseUSA
2. 2.Department of Mathematical SciencesIndiana University-Purdue University IndianapolisIndianapolisUSA