The free energies of six-vertex models and the $n$-equivalence relation

The free energies of six-vertex models on a general domain $D$ with various boundary conditions are investigated with the use of the $n$-equivalence relation, which help classify the thermodynamic limit properties. It is derived that the free energy of the six-vertex model on the rectangle is unique in the limit $(height,width)→(∞,∞)$⁠. It is derived that the free energies of the model on the domain $D$ are classified through the densities of left∕down arrows on the boundary. Specifically, the free energy is identical to that obtained by Lieb [Phys. Rev. Lett. 18, 1046 (1967); 19, 108 (1967); Phys. Rev. 162, 162 (1967)] and Sutherland [Phys. Rev. Lett 19, 103 (1967)] with the cyclic boundary condition when the densities are both equal to $12$⁠. This fact explains several results already obtained through the transfer matrix calculation. The relation to the domino tiling (or dimer, or matching) problems is also noted.