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

We obtain an asymptotic formula, as n -> infinity, 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) = 1/2 is a critical value of t. We prove that in the subcritical case, 0 < t < 1/2, as n -> infinity, K-2 (n) = K-2 (infinity) [1 - e(-n/xi)/n (C-1 + C-2(-1)(n) + O(n(-1)))], with explicit formulae for K-2(infinity), xi, C-1, and C-2. In the supercritical case, 1/2 < t < 1, we prove that as n -> infinity, K-2 (n) = K-2(infinity)[1 - 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) + O(n(-1))))], with explicit formulae for K-2(infinity), 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.