, One can prove in exactly the same way that Kashaev's parametrization of the Ising model [24] is possible whenever the underlying quadrangulation satisfies the assumption of Lemma 7. For instance, the Ising model on any finite isoradial graph admits a Kashaev parametrization

M. Assis, The 16-vertex model and its even and odd 8-vertex subcases on the square lattice, 2017.

Y. Baryshnikov and R. Pemantle, Asymptotics of multivariate sequences, part III: Quadratic points, Advances in Mathematics, vol.228, issue.6, pp.3127-3206, 2011.

V. V. Bazhanov, Trigonometric Solution of Triangle Equations and Classical Lie Algebras, Phys. Lett, vol.159, pp.321-324, 1985.
DOI : 10.1016/0370-2693(85)90259-x

C. Boutillier and B. De-tilière, Height representation of XOR-Ising loops via bipartite dimers, Electron. J. Probab, vol.19, p.33, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00755394

G. D. Carroll and D. Speyer, The cube recurrence, Electr. J. Comb, vol.11, issue.1, 2004.

D. Cimasoni and N. Reshetikhin, Dimers on surface graphs and spin structures. I, Communications in Mathematical Physics, vol.275, issue.1, pp.187-208, 2007.

H. Cohn, R. Kenyon, and J. Propp, A variational principle for domino tilings, Journal of the American Mathematical Society, vol.14, issue.2, pp.297-346, 2001.

J. De-gier, A. Lee, and J. Rasmussen, Discrete holomorphicity and integrability in loop models with open boundaries, J. Stat. Mech, vol.1302, p.2029, 2013.

B. De-tilière, Quadri-tilings of the plane. Probability Theory and Related Fields, vol.137, pp.487-518, 2007.

P. D. Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory. Graduate Texts in Contemporary Physics, 1997.

P. , D. Francesco, and R. Soto-garrido, Arctic curves of the octahedron equation, Journal of Physics A: Mathematical and Theoretical, vol.47, issue.28, p.285204, 2014.
DOI : 10.1088/1751-8113/47/28/285204

URL : https://hal.archives-ouvertes.fr/cea-01002519

C. L. Dodgson, Condensation of determinants, being a new and brief method for computing their arithmetical values, Proceedings of the Royal Society of London, vol.15, pp.150-155, 1866.

J. Dubédat, Exact bosonization of the Ising model, 2011.

C. Fan and F. Y. Wu, Ising model with second-neighbor interaction. I. Some exact results and an approximate solution, Phys. Rev, vol.179, pp.560-569, 1969.

C. Fan and F. Y. Wu, General lattice model of phase transitions, Phys. Rev. B, vol.2, pp.723-733, 1970.
DOI : 10.1103/physrevb.2.723

M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Phys. Rev, vol.124, pp.1664-1672, 1961.
DOI : 10.1103/physrev.124.1664

S. Fomin and A. Zelevinsky, Cluster algebras. I, Foundations. J. Amer. Math. Soc, vol.15, issue.2, pp.497-529, 2002.

S. Fomin and A. Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math, vol.154, issue.1, pp.63-121, 2003.

T. George, Limit shapes for cube groves with periodic conductances, 2017.

C. A. Hurst, New approach to the Ising problem, Journal of Mathematical Physics, vol.7, issue.2, pp.305-310, 1966.
DOI : 10.1063/1.1704933

Y. Ikhlef and J. Cardy, Discretely holomorphic parafermions and integrable loop models, Journal of Physics A: Mathematical and Theoretical, vol.42, issue.10, p.102001, 2009.
DOI : 10.1088/1751-8113/42/10/102001

URL : http://arxiv.org/pdf/0810.5037

J. L. Jacobsen and J. Kondev, Conformal field theory of the Flory model of polymer melting, Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, vol.69, p.66108, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00003755

M. Jimbo, Quantum R matrix for the generalized Toda system, Comm. Math. Phys, vol.102, issue.4, pp.537-547, 1986.
DOI : 10.1007/bf01221646

R. M. Kashaev, On discrete three-dimensional equations associated with the local Yang-Baxter relation, Letters in Mathematical Physics, vol.38, issue.4, pp.389-397, 1996.
DOI : 10.1007/bf01815521

URL : http://arxiv.org/pdf/solv-int/9512005

P. W. Kasteleyn, Graph theory and crystal physics, Graph theory and theoretical physics, pp.43-110, 1967.

R. Kenyon, Annales de l'Institut Henri Poincaré (B) Probability and Statistics, vol.33, pp.591-618, 1997.

R. Kenyon, The Laplacian and Dirac operators on critical planar graphs. Inventiones mathematicae, vol.150, pp.409-439, 2002.
DOI : 10.1007/s00222-002-0249-4

URL : http://arxiv.org/pdf/math-ph/0202018

R. Kenyon, A. Okounkov, and S. Sheffield, Dimers and amoebae, Ann. of Math, vol.163, issue.2, pp.1019-1056, 2006.

R. Kenyon and R. Pemantle, Double-dimers, the Ising model and the hexahedron recurrence, Journal of Combinatorial Theory, Series A, vol.137, pp.27-63, 2016.

R. Kenyon, J. Propp, and D. Wilson, Trees and matchings. The Electronic Journal of Combinatorics, vol.7, 2000.

R. Kenyon and J. Schlenker, Rhombic embeddings of planar quad-graphs, Transactions of the American Mathematical Society, vol.357, issue.9, pp.3443-3458, 2005.

R. Pemantle and M. C. Wilson, Analytic Combinatorics in Several Variables, 2013.

T. K. Petersen and D. Speyer, An arctic circle theorem for groves, Journal of Combinatorial Theory, Series A, vol.111, issue.1, pp.137-164, 2005.

J. Propp, The Many Faces of Alternating-Sign Matrices, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001), vol.AA, pp.43-58, 2001.
URL : https://hal.archives-ouvertes.fr/hal-01182972

D. E. Speyer, Perfect matchings and the octahedron recurrence, Journal of Algebraic Combinatorics, vol.25, issue.3, pp.309-348, 2007.

S. O. Warnaar and B. Nienhuis, Solvable lattice models labelled by Dynkin diagrams, Journal of Physics A: Mathematical and General, vol.26, issue.10, p.2301, 1993.

F. Wu and K. Lin, Staggered ice-rule vertex model-the pfaffian solution, Physical Review B, vol.12, issue.1, p.419, 1975.