Exact Series-Expansion Study of the Monomer-Dimer Problem

The monomer-dimer problem is studied for several two- and three-dimensional lattices (coordination number $q$ by deriving between 8 and 16 coefficients of the exact series expansions in powers of the activity $z$ and density $\rho$ of dimers. Analysis of the series by the ratio and Padé-approximant techniques shows that while there is no phase transition, close packing is a singular point so that as $\rho$ approaches $\frac{1}{q}$ (the close-packing density), $z\approx \frac{A}{{(1-q\rho )}^{\gamma }}$. Our results enable us to conjecture that $\gamma =2\mathrm{}$ for the close-packed lattices (irrespective of dimensionality), but for the loose-packed lattices $\gamma =1.75\mathrm{}$ in two dimensions and 1.95 in three dimensions. Estimates of the amplitude $A$ are given.