We study a two-color loop model known as the $C^{(1)}_2$ loop model. Wedefine a free-fermionic regime for this model, and show that under thisassumption it can be transformed into a double dimer model. We then compute itsfree energy on periodic planar graphs. We also study the star-triangle relationor Yang-Baxter equations of this model, and show that after a properparametrization they can be summed up into a single relation known as Kashaev'srelation. This is enough to identify the solution of Kashaev's relation as thepartition function of a $C^{(1)}_2$ loop model with some boundary conditions,thus solving an open question of Kenyon and Pemantle about the combinatorics ofKashaev's relation.
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机译：我们研究了一种称为$ C ^ {（（1）} _ 2 $循环模型）的双色循环模型。我们为该模型定义了一个自由铁离子体系，并表明在这种假设下它可以转化为双二聚体模型。然后，我们在周期平面图上计算其自由能。我们还研究了该模型的星形三角关系器Yang-Baxter方程，并表明在适当的参数化之后，它们可以总结为一个称为Kashaev关系的单个关系。这足以将Kashaev关系的解决方案识别为具有一定边界条件的$ C ^ {（1）} _ 2 $循环模型的分区函数，从而解决了Kenyon和Pemantle关于Kashaev关系的组合问题的公开问题。
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