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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.
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https://hal.archives-ouvertes.fr/hal-01574018
Contributor : Paul Melotti <>
Submitted on : Friday, November 9, 2018 - 10:38:49 AM
Last modification on : Friday, March 27, 2020 - 3:01:01 AM
Document(s) archivé(s) le : Sunday, February 10, 2019 - 1:19:01 PM

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

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

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