Monomer-dimer model on fractal lattices

Abstract

Within the grand canonical ensemble formalism, the entropy of the monomer-dimer model is obtained on several fractal lattices. Entropies as functions of monomer density are concave and asymmetric around the maxima which are located at densities lower than half coverage. Details of the lattice structure are almost irrelevant for very high monomer densities, whereas in the low monomer limit values of entropies are predominantly determined by the coordination number. In the zero monomer limit entropies of the close-packed dimer model are retrieved. Possible criticality of the model in the close-packed dimer limit is examined, but it is found that all thermodynamic quantities are analytic functions, which is in contrast with the square lattice where the model proved to be critical at close-packing.