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Communications in Mathematical Physics

, Volume 356, Issue 2, pp 397–425 | Cite as

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

  • Estelle Basor
  • Pavel BleherEmail author
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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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.American Institute of MathematicsSan JoseUSA
  2. 2.Department of Mathematical SciencesIndiana University-Purdue University IndianapolisIndianapolisUSA

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