Asymptotics of the monomer-dimer model on two-dimensional semi-infinite lattices

Abstract

By using the asymptotic theory of Pemantle and Wilson [R. Pemantle and M. C. Wilson, J. Comb. Theory, Ser. AJCBTA70097-316510.1006/jcta.2001.3201 97, 129 (2002)], asymptotic expansions of the free energy of the monomer-dimer model on two-dimensional semi-infinite ×n lattices in terms of dimer density are obtained for small values of n, at both high- and low-dimer-density limits. In the high-dimer-density limit, the theoretical results confirm the dependence of the free energy on the parity of n, a result obtained previously by computational methods by Y. Kong [Y. Kong, Phys. Rev. EPLEEE81063-651X10.1103/ PhysRevE.74.061102 74, 061102 (2006); Phys. Rev. EPLEEE81063-651X10.1103/ PhysRevE.73.016106 73, 016106 (2006);Phys. Rev. EPLEEE81063-651X10.1103/ PhysRevE.74.011102 74, 011102 (2006)]. In the low-dimer-density limit, the free energy on a cylinder ×n lattice strip has exactly the same first n terms in the series expansion as that of an infinite × lattice. © 2007 The American Physical Society.