Three-Colorings of the Square Lattice: A Hard Squares Model

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Abstract
The evaluation of the partition function of the 2‐dimensional ice model is equivalent to counting the number of ways of coloring the faces of the square lattice with three colors so that no two adjacent faces are colored alike. In this paper we solve a generalized problem in which activities are associated with the colors. If one of the colors is regarded as a particle and the others as forming a background, then the model is reminiscent of the hard‐square lattice gas. It is found to undergo a phase transition with infinite compressibility at the density ρ = 1/3.