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The packing of two species of polygons on the square lattice. (English) Zbl 1206.82018
Summary: We decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons. We end up with a 24-vertex model which is known in the literature as the fully packed double loop model (FPL2). In the particular case in which the fugacities of the polygons are the same, the model admits an exact solution. The solution is obtained using coordinate Bethe ansatz and provides a closed expression for the free energy. In particular, we find the free energy of the four-colouring model and the double Hamiltonian walk and recover the known entropy of the Ice model. When both fugacities are set equal to 2 the model undergoes an infinite-order phase transition.

82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
82B26Phase transitions (general)
82B41Random walks, random surfaces, lattice animals, etc. (statistical mechanics)
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