自由费米子$C^{(1)}_2$圈模型、双二聚体和Kashaev回归
The free-fermionic $C^{(1)}_2$ loop model, double dimers and Kashaev's recurrence
摘要：
我们研究了一种称为$C^{(1)}_2$循环模型的双色循环模型。我们为这个模型定义了一个自由的费米子制度，并表明在这个假设下，它可以转化为一个双二聚体模型。然后我们计算它在周期可平面图上的自由能。我们还研究了该模型的星三角关系或杨-巴克斯特方程，并表明，经过适当的参数化，它们可以归结为一个单一的关系称为Kashaev的关系。这足以将Kashaev关系的解识别为具有某些边界条件的$C^{(1)}_2$回路模型的分拆函数，从而解决了Kenyon和Pemantle关于Kashaev关系的组合学的公开问题。
英文摘要：
We study a two-color loop model known as the $C^{(1)}_2$ loop model. We define a free-fermionic regime for this model, and show that under this assumption it can be transformed into a double dimer model. We then compute its free energy on periodic planar graphs. We also study the star-triangle relation or Yang-Baxter equations of this model, and show that after a proper parametrization they can be summed up into a single relation known as Kashaev's relation. This is enough to identify the solution of Kashaev's relation as the partition function of a $C^{(1)}_2$ loop model with some boundary conditions, thus solving an open question of Kenyon and Pemantle about the combinatorics of Kashaev's relation.
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