Statistical Mechanics of Dimers on a Plane Lattice. II. Dimer Correlations and Monomers

In part I of this paper, exact expressions were obtained for the partition function and thermodynamic properties of an $m\ifmmode\times\else\texttimes\fi{}n$ plane square lattice filled with $\frac{1}{2}\mathrm{mn}$ rigid dimers each occupying two adjacent lattice sites. In this part the correlation properties of the model are studied with the aid of a general perturbation theory for Pfaffians. Closed formulas are derived for the changes in the probability of a dimer occupying a given bond that are induced by the proximity of an edge or a corner of the lattic (singlet correlations) and, in the center of the lattice by the fixed position of another dimer (pair correlations). We show how to calculate the number of configurations of a dimer lattice containing a pair of monomers (or holes) a fixed distance apart. The explicit result when the separation vector is ($p$, 0) or ($p$, 1) involves a Toeplitz determinant $|{a}_{i\ensuremath{-}j+1}|$ ($i, j=1, 2, \ensuremath{\cdots}p$) defined by $\ensuremath{\Sigma}\stackrel{\ensuremath{\infty}}{n=\ensuremath{-}\ensuremath{\infty}}{a}_{n}{e}^{\mathrm{in}\ensuremath{\theta}}=\mathrm{sgn}{cos\ensuremath{\theta}}\mathrm{exp}[\ensuremath{-}iinvcot(\ensuremath{\tau}cos\ensuremath{\theta})],$ where $\ensuremath{\tau}=\frac{x}{y}$ and $x$ and $y$ are the activities of $x$ and $y$ dimers. A similar result holds along the diagonals ($p, p\ifmmode\pm\else\textpm\fi{}1$). The relative number of configurations decays to zero with radial separation $r$ as $\frac{B}{{r}^{\frac{1}{2}}}$.