, The double G D is obtained by taking the diagonals of the quad-graph G and adding a white vertex at the crossing of the diagonals. It is bipartite and has two kinds of black vertices corresponding to vertices of G and vertices of G * . The associated quad-graph (G D ) is obtained by dividing the quadrangular faces of G into four, see Figure 16. The graph G Q is the dual graph of the superimposition of the quad-graph G and of the double graph G D . It has three kinds of faces containing in their interior either a vertex of G, a vertex of G * , or a white vertex of G D ; the latter are quadrangles whose pair of parallel edges correspond to primal and dual edges of G. The quad-graph (G Q ) is that of G D with an additional quadrangular face for each edge of G , which should be thought of as "flat", see Figure 15 and the discussion below. Each train-track of G induces two train-tracks that are anti-parallel in G D and make antiparallel bigons in G Q , see Figures 16 and 15. Therefore, if G is an isoradial graph, i.e., if its train-tracks do not self-intersect and two train-tracks never intersect more than once, then the associated bipartite graphs G D and G Q are isoradial and minimal, respectively. Moreover, an isoradial embedding of G naturally induces an isoradial embedding of G D and a minimal immersion of G Q , as follows. As proved by Kenyon-Schlenker [KS05], an isoradial embedding is given by some half-angle map ? on the oriented train-tracks of G, such that the half-angles associated to any given oriented train-track and to the same train-track with the opposite orientation differ by ? 2 . One can then simply define the induced half-angle maps ? D and ? Q by associating to each oriented train-track of G D and G Q the half-angle of the unique oriented train-track of G it is parallel to. This is illustrated in Figures 16 and 15. If ? defines an isoradial embedding of G

, Connection to the dimer model on the graph G Q arising from the Ising model Consider a half-angle map ? Q ? X G Q as above. Following Section 2.3, let us compute the discrete Abel map on vertices of the quad-graph (G Q ) , taking as reference point a vertex v 0 of G

. Then, using the notation of Figure 15, and recalling that ? is defined as an element of R/?Z, we have the following equalities modulo ? ? v ? V, ?(v) = 0, ? f ? V * , ?(f) = ? 2 , ?(b 1 ) = ? + ? 2 , ?(b 2 ) = ?, ?

R. J. Baxter, Exactly solved models in statistical mechanics, 1982.

C. Boutillier, D. Cimasoni, and B. De-tilière, , 2019.

C. Boutillier, D. Cimasoni, and B. De-tilière, Dimers on minimal graphs and genus g Harnack curves, 2020.

C. Boutillier and . Béatrice-de-tilière, The critical Z-invariant Ising model via dimers: the periodic case. Probab. theory and related fields, vol.147, pp.379-413, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00494462

C. Boutillier and . Béatrice-de-tilière, The critical Z-invariant Ising model via dimers: locality property, Comm. Math. Phys, vol.301, issue.2, pp.473-516, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00563265

C. Boutillier, K. Béatrice-de-tilière, and . Raschel, The Z-invariant massive Laplacian on isoradial graphs. Invent. Math, vol.208, issue.1, pp.109-189, 2017.

C. Boutillier, K. Béatrice-de-tilière, and . Raschel, The Z-invariant Ising model via dimers. Probability Theory and Related Fields, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01423324

C. Boutillier, Pattern densities in non-frozen planar dimer models, Comm. Math. Phys, vol.271, issue.1, pp.55-91, 2007.

H. Cohn, R. Kenyon, and J. Propp, A variational principle for domino tilings, J. Amer. Math. Soc, vol.14, issue.2, pp.297-346, 2001.

D. Chelkak, B. Laslier, and M. Russkikh, Dimer model and holomorphic functions on t-embeddings of planar graphs. arXiv e-prints, 2020.
URL : https://hal.archives-ouvertes.fr/hal-02613085

R. L. Dobru?in, Description of a random field by means of conditional probabilities and conditions for its regularity, Teor. Verojatnost. i Primenen, vol.13, pp.201-229, 1968.

T. Béatrice-de, Quadri-tilings of the plane, vol.137, pp.487-518, 2007.

T. Béatrice-de, Scaling limit of isoradial dimer models and the case of triangular quadri-tilings, Ann. Inst. Henri Poincaré, vol.43, issue.6, pp.729-750, 2007.

T. Béatrice-de, The Z-Dirac and massive Laplacian operators in the Z-invariant Ising model. arXiv e-prints, 2017.

J. D. Fay, Theta functions on Riemann surfaces, vol.352, 1973.

V. V. Fock, Inverse spectral problem for GK integrable system, 2015.

T. George, Spectra of biperiodic planar networks, 2019.

A. B. Goncharov and R. Kenyon, Dimers and cluster integrable systems, Ann. Sci.Éc. Norm. Supér, vol.46, issue.5, pp.747-813, 2013.

M. Israel, M. M. Gelfand, A. V. Kapranov, and . Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser Boston, 1994.

D. R. Gulotta, Properly ordered dimers, R-charges, and an efficient inverse algorithm, Journal of High Energy Physics, issue.10, p.14, 2008.

P. W. Kasteleyn, The statistics of dimers on a lattice: I. the number of dimer arrangements on a quadratic lattice, Physica, vol.27, pp.1209-1225, 1961.

P. W. Kasteleyn, Graph theory and crystal physics, Graph Theory and Theoretical Physics, pp.43-110, 1967.

R. Kenyon, Local statistics of lattice dimers, Ann. Inst. H. Poincaré Probab. Statist, vol.33, issue.5, pp.591-618, 1997.

R. Kenyon, Dominos and the Gaussian free field, Ann. Probab, vol.29, issue.3, pp.1128-1137, 2001.

R. Kenyon, The Laplacian and Dirac operators on critical planar graphs, Invent. Math, vol.150, issue.2, pp.409-439, 2002.

R. Kenyon, An introduction to the dimer model, School and Conference on Probability Theory, ICTP Lect. Notes, XVII, pp.267-304, 2004.

R. Kenyon, W. Y. Lam, S. Ramassamy, and M. Russkikh, Dimers and Circle patterns. arXiv e-prints, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01911855

R. Kenyon and A. Okounkov, Planar dimers and Harnack curves, Duke Math. J, vol.131, issue.3, pp.499-524, 2006.

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

R. Kenyon and J. Schlenker, Rhombic embeddings of planar quad-graphs, Trans. Amer. Math. Soc, vol.357, issue.9, pp.3443-3458, 2005.

G. Kuperberg, An exploration of the permanent-determinant method, Electron. J. Combin, vol.5, 1998.

D. F. Lawden, Elliptic functions and applications, Applied Mathematical Sciences, vol.80, 1989.

O. E. Lanford and D. Ruelle, Observables at infinity and states with short range correlations in statistical mechanics, Comm. Math. Phys, vol.13, pp.194-215, 1969.

G. Mikhalkin, Real algebraic curves, the moment map and amoebas, Annals of Mathematics-Second Series, vol.151, issue.1, pp.309-326, 2000.

M. Passare, The trigonometry of harnack curves, Journal of Siberian Federal University. Mathematics and Physics, vol.9, issue.3, pp.347-352, 2016.

J. K. Percus, One more technique for the dimer problem, J. Math. Phys, vol.10, p.1881, 1969.

A. Postnikov, Total positivity, Grassmannians, and networks. arXiv e-prints, 2006.

J. Propp, Generalized domino-shuffling, Theoret. Comput. Sci, vol.303, issue.2-3, pp.267-301, 2003.

M. Russkikh, Dominos in hedgehog domains. arXiv e-prints, 2018.

S. Sheffield, Random surfaces, Société Mathématique de France (SMF), vol.304, 2005.

N. V. Harold, M. E. Temperley, and . Fisher, Dimer problem in statistical mechanics-an exact result, Philosophical Magazine, vol.6, issue.68, pp.1061-1063, 1961.

D. P. Thurston, From dominoes to hexagons, Proceedings of the, 2014.

. Maui, Qinhuangdao conferences in honour of Vaughan F. R. Jones' 60th birthday, vol.46, pp.399-414, 2015.

K. Weierstrass, Zur Theorie der Jacobi'schen Functionen von mehreren Veränderlichen, Berl. Ber, vol.1882, pp.505-508, 1882.

Y. Fa, K. Wu, and . Lin, Staggered ice-rule vertex model -the Pfaffian solution, Phys. Rev. B, vol.12, pp.419-428, 1975.