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

Abstract

We obtain an asymptotic formula, as n→∞, for the monomer–monomer correlation function K2(n) in the classical dimer model on a triangular lattice, with the horizontal and vertical weights wh=wv=1 and the diagonal weight wd=t>0, between two monomers at vertices q and r that are n spaces apart in adjacent rows. We find that tc=12 is a critical value of t. We prove that in the subcritical case, 0<t<12, as n→∞,K2(n)=K2(∞)[1−e−n/ξn(C1+C2(−1)n+O(n−1))], with explicit formulae for K2(∞),ξ,C1, and C2. In the supercritical case, 12<t<1, we prove that as n→∞,K2(n)=K2(∞)[1−e−n/ξn(C1cos(ωn+φ1)+C2(−1)ncos(ωn+φ2)+C3+C4(−1)n+O(n−1))], with explicit formulae for K2(∞),ξ,ω, and C1,C2,C3,C4,φ1,φ2. 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.