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

TL;DR

This paper introduces a free-fermionic regime for the $C^{(1)}_2$ loop model, transforms it into a double dimer model, and connects its integrability to Kashaev's relation, solving an open combinatorial problem.

Contribution

It defines a free-fermionic regime for the $C^{(1)}_2$ loop model and links it to Kashaev's relation, providing a new solution to an open problem.

Findings

Transformation into a double dimer model under free-fermionic regime

Explicit computation of free energy on periodic planar graphs

Identification of Kashaev's relation solution with the model's partition function

Abstract

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.