Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density.
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 mxn two-dimensional rectangular lattice strips with fixed dimer density rho . For any dimer density 0<rho<1 , we find a logarithmic correction term in the finite-size correction of the free energy per lattice site. The coefficient of the logarithmic correction term is exactly -12 . This logarithmic correction term is explained by the newly developed asymptotic theory of Pemantle and Wilson. The sequence of the free energy of lattice strips with cylinder boundary condition converges so fast that very accurate free energy f{2}(rho) for large lattices can be obtained. For example, for a half-filled lattice, f{2}(12)=0.633195588930 , while f{2}(14)=0.4413453753046 and f{2}(34)=0.64039026 . For rho<0.65 , f{2}(rho) is accurate at least to ten decimal digits. The function f{2}(rho) reaches the maximum value f{2}(rho{*})=0.662798972834 at rho{*}=0.6381231 , 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.
Kong
Yong
Y
Department of Mathematics, National University of Singapore, Singapore 117543. matky@nus.edu.sg