Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density

Abstract

The classical monomer-dimer model in two-dimensional lattices has been shown to belong to the " #P -complete" class, which indicates the problem is computationally "intractable." We use exact computational method to investigate the number of ways to arrange dimers on m×n two-dimensional rectangular lattice strips with fixed dimer density ρ. For any dimer density 0 <0.65, f2 (ρ) is accurate at least to ten decimal digits. The function f2 (ρ) reaches the maximum value f2 (ρ*) =0.662 798 972 834 at ρ* =0.638 1231, with 11 correct digits. This is also the monomer-dimer constant for two-dimensional rectangular lattices. The asymptotic expressions of free energy near close packing are investigated for finite and infinite lattice widths. For lattices with finite width, dependence on the parity of the lattice width is found. For infinite lattices, the data support the functional form obtained previously through series expansions. © 2006 The American Physical Society.