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

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.
Type de document :
Pré-publication, Document de travail
34 pages. 2017
Domaine :

https://hal.archives-ouvertes.fr/hal-01574018
Contributeur : Paul Melotti <>
Soumis le : vendredi 11 août 2017 - 13:29:10
Dernière modification le : mercredi 21 mars 2018 - 18:56:49

### Identifiants

• HAL Id : hal-01574018, version 1
• ARXIV : 1708.03239

### Citation

Paul Melotti. The free-fermionic $C^{(1)}_2$ loop model, double dimers and Kashaev's recurrence. 34 pages. 2017. 〈hal-01574018〉

### Métriques

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