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

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We consider the dimer model on the rectangular 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 () 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 () and square () geometries 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 () for aspect ratio to the coefficients of the free energy for square geometries, multiplied by , that is , is also equal to 1/2 in the limit of . With these results, one can find the asymptotic behavior of the from that of the , whose asymptotic behavior is derived explicitly here. We find that the corner contribution to the free energy for the dimer model on rectangular lattices with free boundary conditions is equal to zero and explain that result in the framework of conformal field theory, with two consistent values of the central charge, namely, c = −2 for the construction of a conformal field theory using a mapping of spanning trees and c = 1 for the height function description. We also derive a simple exact expression for the free energy of open strips of arbitrary width.