The free-fermionic $C^{(1)}_2$ loop model, double dimers and Kashaev's recurrence

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.