Exact finite-size corrections in the dimer model on a planar square lattice

We consider the dimer model on the rectangular \(2M \times 2N\) lattice with free boundary conditions. We derive exact expressions for the coefficients in the asymptotic expansion of the free energy in terms of the elliptic theta functions (\(\theta_2, \theta_3, \theta_4\)) and the elliptic integral of second kind (\(E\)), up to 22nd order. Surprisingly we find that ratio of the coefficients \(f_p\) in the free energy expansion for strip (\(f_p^\mathrm{strip}\)) and square (\(f_p^\mathrm{sq}\)) geometries \(r_p={f_p^\mathrm{strip}}/{f_p^\mathrm{sq}}\) in the limit of large \(p\) tends to \(1/2\). Furthermore, we predict that the ratio of the coefficients \(f_p\) in the free energy expansion for rectangular (\(f_p(\rho)\)) for aspect ratio \(\rho > 1\) to the coefficients of the free energy for square geometries, multiplied by \(\rho^{-p-1}\), that is \(r_p=\rho^{-p-1} {f_p(\rho)}/{f_p^\mathrm{sq}}\), is also equal to \(1/2\) in the limit of \(p \to \infty\). We find that the corner contribution to the free energy for the dimer model on rectangular \(2M \times 2N\) lattices with free boundary conditions is equal to zero and explain that result in the framework of conformal field theory, in which the central charge of the considering model is \(c=-2\). We also derive a simple exact expression for the free energy of open strips of arbitrary width.