Returning to the definition of $K^Q$, this immediately gives,

$$P_{\text{dimer}}^{Q}(\text{bw}_1) = P_{\text{dimer}}^{D,\beta}(\text{wv}_2), \quad P_{\text{dimer}}^{Q}(\text{bw}_2) = P_{\text{dimer}}^{D,\beta}(\text{wf}_2).$$

We now use the computations of Example 3.21 evaluated at $u = \beta$. In the second line we use that $(u_{\alpha+2K})^*|_{\beta} = (2K - u_{\alpha})|_{\beta} = 2K - \theta$, $(u_{\beta})^*|_{\beta} = (K - u_{\beta})|_{\beta} = K$,

$$P_{\text{dimer}}^{Q}(\text{w}_1\text{b}) = P_{\text{dimer}}^{D,\beta}(\text{wv}_2) = H(2\theta) - H(0) = H(2\theta), \text{ since } H(0) = 0$$

$$P_{\text{dimer}}^{Q}(\text{w}_2\text{b}) = P_{\text{dimer}}^{D,\beta}(\text{wf}_2) = H(2(2K - \theta)) - H(2K) = \frac{1}{2} - H(2\theta),$$

where in the last equality we used [11, Lemma 45] to obtain:

$$H(4K - 2\theta) = H(-2\theta) + 1 = -H(2\theta) + 1$$

$$H(2K) = H(4K - 2K) = H(-2K) + 1 = -H(2K) + 1 \Rightarrow H(2K) = \frac{1}{2}.$$

Since probabilities around the vertex b sum to 1, we have $P_{\text{dimer}}^{Q}(\text{w}_3\text{b}) = \frac{1}{2}$.

## B.3 Dimers on $G^F$

We now compute single edge probabilities for dimer configurations of $G^F$ chosen with respect to the Boltzmann measure $P_{\text{dimer}}^F$.

<figure><img src="image_36.png" /><figcaption>Figure 32: Notation.</figcaption></figure>

Setting $a = a'$ in Formula (5.13) gives,

$$(K^F)_{a',a}^{-1} = -\frac{1}{2}(\tilde{K}^Q)_{w',b}^{-1}\varepsilon_{b,a} + \kappa_{a',a} = -\frac{1}{2\tanh(2J)}P_{\text{dimer}}^Q(\text{bw}')\varepsilon_{b,a'}\varepsilon_{b,a} - \frac{1}{4}\varepsilon_{a',a},$$

where we used that $P_{\text{dimer}}^{Q}(\text{bw}') = \tilde{K}_{b,w'}^{Q}(\tilde{K}_{w',b}^{Q})^{-1} = \varepsilon_{b,a'} \tanh(2J)(\tilde{K}_{w',b}^{Q})^{-1}$, since the orientation of the triangle is Kasteleyn.

We deduce that $P_{\text{dimer}}^{F}(aa') = K_{a,a'}^{F}(K^F)_{a',a}^{-1} = \frac{1}{4} - \frac{P_{\text{dimer}}^{Q}(\text{bw}')}{2\tanh(2J)}$.

Setting $a = a'$ and in Formula (5.11) gives,

$$\begin{aligned} (K^F)_{a',b}^{-1} &= \frac{1}{2} \left[ e^{2J} \cosh^{-1}(2J) (K^Q)_{w',b}^{-1} + \varepsilon_{b',b} \cosh^{-1}(2J) (K^Q)_{w',b'}^{-1} \right] \\ &= \frac{1}{2} \left[ (1 + \tanh(2J))(K^Q)_{w',b}^{-1} + \varepsilon_{b',b} \cosh^{-1}(2J) (K^Q)_{w',b'}^{-1} \right], \text{ using (5.9)} \\ &= \frac{1}{2} \left[ (1 + \tanh(2J)) \varepsilon_{b,a'} \frac{P_{\text{dimer}}^{Q}(\text{bw}')}{\tanh(2J)} + \varepsilon_{b,a'} P_{\text{dimer}}^{Q}(\text{b'w}') \right], \end{aligned}$$
Returning to the definition of $K^Q$, this immediately gives,

$$P_{\text{dimer}}^{Q}(\text{bw}_1) = P_{\text{dimer}}^{D,\beta}(\text{wv}_2), \quad P_{\text{dimer}}^{Q}(\text{bw}_2) = P_{\text{dimer}}^{D,\beta}(\text{wf}_2).$$

We now use the computations of Example 3.21 evaluated at $u = \beta$. In the second line we use that $(u_{\alpha+2K})^*|_{\beta} = (2K - u_{\alpha})|_{\beta} = 2K - \theta$, $(u_{\beta})^*|_{\beta} = (K - u_{\beta})|_{\beta} = K$,

$$P_{\text{dimer}}^{Q}(\text{w}_1\text{b}) = P_{\text{dimer}}^{D,\beta}(\text{wv}_2) = H(2\theta) - H(0) = H(2\theta), \text{ since } H(0) = 0$$

$$P_{\text{dimer}}^{Q}(\text{w}_2\text{b}) = P_{\text{dimer}}^{D,\beta}(\text{wf}_2) = H(2(2K - \theta)) - H(2K) = \frac{1}{2} - H(2\theta),$$

where in the last equality we used [11, Lemma 45] to obtain:

$$H(4K - 2\theta) = H(-2\theta) + 1 = -H(2\theta) + 1 \\ H(2K) = H(4K - 2K) = H(-2K) + 1 = -H(2K) + 1 \Rightarrow H(2K) = \frac{1}{2}.$$

Since probabilities around the vertex b sum to 1, we have $P_{\text{dimer}}^{Q}(\text{w}_3\text{b}) = \frac{1}{2}$.

## B.3 Dimers on $G^F$

We now compute single edge probabilities for dimer configurations of $G^F$ chosen with respect to the Boltzmann measure $P_{\text{dimer}}^F$.

<figure><img src="image_36.png" /><figcaption>Figure 32: Notation.</figcaption></figure>

Setting $a = a'$ in Formula (5.13) gives,

$$(K^F)_{a',a}^{-1} = -\frac{1}{2}(\tilde{K}^Q)_{w',b}^{-1}\varepsilon_{b,a} + \kappa_{a',a} = -\frac{1}{2\tanh(2J)}P_{\text{dimer}}^Q(\text{bw}')\varepsilon_{b,a'}\varepsilon_{b,a} - \frac{1}{4}\varepsilon_{a',a},$$

where we used that $P_{\text{dimer}}^{Q}(\text{bw}') = \tilde{K}_{b,w'}^{Q}(\tilde{K}_{w',b}^{Q})^{-1} = \varepsilon_{b,a'} \tanh(2J)(\tilde{K}_{w',b}^{Q})^{-1}$, since the orientation of the triangle is Kasteleyn.

We deduce that $P_{\text{dimer}}^{F}(aa') = K_{a,a'}^{F}(K^F)_{a',a}^{-1} = \frac{1}{4} - \frac{P_{\text{dimer}}^{Q}(\text{bw}')}{2\tanh(2J)}$.

Setting $a = a'$ and in Formula (5.11) gives,

$$(K^F)_{a',b}^{-1} = \frac{1}{2} \left[ e^{2J} \cosh^{-1}(2J) (K^Q)_{w',b}^{-1} + \varepsilon_{b',b} \cosh^{-1}(2J) (K^Q)_{w',b'}^{-1} \right] \\ = \frac{1}{2} \left[ (1 + \tanh(2J))(K^Q)_{w',b}^{-1} + \varepsilon_{b',b} \cosh^{-1}(2J) (K^Q)_{w',b'}^{-1} \right], \text{ using (5.9)} \\ = \frac{1}{2} \left[ (1 + \tanh(2J)) \varepsilon_{b,a'} \frac{P_{\text{dimer}}^{Q}(\text{bw}')}{\tanh(2J)} + \varepsilon_{b,a'} P_{\text{dimer}}^{Q}(\text{b'w}') \right],$$
where, as before, we returned to the definition of $K^Q$ and of single edge probabilities of $G^Q$

$$ 
\begin{aligned}
&= \frac{\epsilon_{b,a'}}{2} \left[ \mathbb{P}_{\text{dimer}}^{Q}(\text{bw}') + \mathbb{P}_{\text{dimer}}^{Q}(\text{b}'\text{w}') + \frac{\mathbb{P}_{\text{dimer}}^{Q}(\text{bw}')}{\tanh(2J)} \right] \\
&= \frac{\epsilon_{b,a'}}{2} \left[ 1 - \mathbb{P}_{\text{dimer}}^{Q}(\text{b}''\text{w}') + \frac{\mathbb{P}_{\text{dimer}}^{Q}(\text{bw}')}{\tanh(2J)} \right],
\end{aligned}
$$

using that the sum of probabilities is 1 around $w'$ for $\mathbb{P}_{\text{dimer}}^Q$. We deduce that $\mathbb{P}_{\text{dimer}}^F(ba') = K_{b,a'}^F(K_F^F)^{-1}$, $b = \frac{1}{2} - \frac{\mathbb{P}_{\text{dimer}}^Q(b''w')}{2} + \frac{\mathbb{P}_{\text{dimer}}^Q(bw')}{2\tanh(2J)}$.

As a consequence also, we have: $\mathbb{P}_{\text{dimer}}^F(ba') + \mathbb{P}_{\text{dimer}}^F(aa') = \frac{3}{4} - \frac{\mathbb{P}_{\text{dimer}}^Q(b''w')}{2}$. Using that the sum of probabilities around $a'$ is 1, we deduce that $\mathbb{P}_{\text{dimer}}^F(ab) = \frac{\mathbb{P}_{\text{dimer}}^Q(bw)}{2} + \frac{\mathbb{P}_{\text{dimer}}^Q(bw')}{2\tanh(2J)}$. Using that the sum of probabilities around the vertex $b$ is 1, we obtain

$$ \mathbb{P}_{\text{dimer}}^{F}(bb') = \frac{1}{2} + \frac{\mathbb{P}_{\text{dimer}}^{Q}(b''w')}{2} - \frac{\mathbb{P}_{\text{dimer}}^{Q}(bw)}{2} - \frac{\mathbb{P}_{\text{dimer}}^{Q}(bw')}{2} $$

By symmetry $\mathbb{P}_{\text{dimer}}^Q(bw) = \frac{1}{2} = \mathbb{P}_{\text{dimer}}^Q(b''w')$, implying the expressions of Example 5.6.

## References

[1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR-0167642

[2] H. Au-Yang and J. H. H. Perk, Critical correlations in a Z-invariant inhomogeneous Ising model, Physica A 144 (1987), 44-104. MR-0899036

[3] H. Au-Yang and J. H. H. Perk, Correlation functions and susceptibility in the Z-invariant Ising model, MathPhys Odyssey 2001: Integrable Models and Beyond (Birkhäuser Boston M. Kashiwara and T. Miwa, eds.), 2002, pp. 23-48. MR-1903971

[4] R. J. Baxter, Solvable eight-vertex model on an arbitrary planar lattice, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 289 (1978), no. 1359, 315-346. MR-0479213

[5] R. J. Baxter, Free-fermion, checkerboard and Z-invariant lattice models in statistical mechanics, Proc. Roy. Soc. London Ser. A 404 (1986), no. 1826, 1-33. MR-0836281

[6] R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1989, Reprint of the 1982 original. MR-0998375

[7] V. Beffara and H. Duminil-Copin, Smirnov's fermionic observable away from criticality, Ann. Probab. 40 (2012), no. 6, 2667-2689. MR-3050513

[8] C. Boutillier, D. Cimasoni, and B. de Tilière, Elliptic dimers on minimal graphs and genus 1 harnack curves, arXiv:2007.14699 (2020), 71 p.

[9] C. Boutillier and B. de Tilière, The critical Z-invariant Ising model via dimers: the periodic case, Probab. Theory Related Fields 147 (2010), 379-413. MR-2639710

[10] C. Boutillier and B. de Tilière, The critical Z-invariant Ising model via dimers: locality property, Comm. Math. Phys. 301 (2011), no. 2, 473-516. MR-2764995

[11] C. Boutillier, B. de Tilière, and K. Raschel, The Z-invariant massive Laplacian on isoradial graphs, Invent. Math. 208 (2017), no. 1, 109-189. MR-3621833

[12] C. Boutillier, B. de Tilière, and K. Raschel, The Z-invariant Ising model via dimers, Probab. Theory Related Fields. 174 (2019), no. 1-2, 235-305. MR-3947324

[13] C. Boutillier and B. de Tilière, Height representation of XOR-Ising loops via bipartite dimers, Electron. J. Probab. 19 (2014), no. 80, 1-33. MR-3256880

[14] R. Burton and R. Pemantle, Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances, Ann. Probab. 21 (1993), no. 3, 1329-1371. MR-1235419
where, as before, we returned to the definition of $K^Q$ and of single edge probabilities of $G^Q$

$$ 
\begin{aligned}
&= \frac{\epsilon_{b,a'}}{2} \left[ \mathbb{P}_{\text{dimer}}^{Q}(\text{bw}') + \mathbb{P}_{\text{dimer}}^{Q}(\text{b}'\text{w}') + \frac{\mathbb{P}_{\text{dimer}}^{Q}(\text{bw}')}{\tanh(2J)} \right] \\
&= \frac{\epsilon_{b,a'}}{2} \left[ 1 - \mathbb{P}_{\text{dimer}}^{Q}(\text{b}''\text{w}') + \frac{\mathbb{P}_{\text{dimer}}^{Q}(\text{bw}')}{\tanh(2J)} \right],
\end{aligned}
$$

using that the sum of probabilities is 1 around $w'$ for $\mathbb{P}_{\text{dimer}}^Q$. We deduce that $\mathbb{P}_{\text{dimer}}^F(ba') = K_{b,a'}^F(K_F^F)^{-1}$, $b = \frac{1}{2} - \frac{\mathbb{P}_{\text{dimer}}^Q(b''w')}{2} + \frac{\mathbb{P}_{\text{dimer}}^Q(bw')}{2\tanh(2J)}$.

As a consequence also, we have: $\mathbb{P}_{\text{dimer}}^F(ba') + \mathbb{P}_{\text{dimer}}^F(aa') = \frac{3}{4} - \frac{\mathbb{P}_{\text{dimer}}^Q(b''w')}{2}$. Using that the sum of probabilities around $a'$ is 1, we deduce that $\mathbb{P}_{\text{dimer}}^F(ab) = \frac{\mathbb{P}_{\text{dimer}}^Q(bw)}{2} + \frac{\mathbb{P}_{\text{dimer}}^Q(bw')}{2\tanh(2J)}$. Using that the sum of probabilities around the vertex $b$ is 1, we obtain

$$ \mathbb{P}_{\text{dimer}}^{F}(bb') = \frac{1}{2} + \frac{\mathbb{P}_{\text{dimer}}^{Q}(b''w')}{2} - \frac{\mathbb{P}_{\text{dimer}}^{Q}(bw)}{2} - \frac{\mathbb{P}_{\text{dimer}}^{Q}(bw')}{2} $$

By symmetry $\mathbb{P}_{\text{dimer}}^Q(bw) = \frac{1}{2} = \mathbb{P}_{\text{dimer}}^Q(b''w')$, implying the expressions of Example 5.6.

## References

[1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR-0167642

[2] H. Au-Yang and J. H. H. Perk, Critical correlations in a Z-invariant inhomogeneous Ising model, Physica A 144 (1987), 44-104. MR-0899036

[3] H. Au-Yang and J. H. H. Perk, Correlation functions and susceptibility in the Z-invariant Ising model, MathPhys Odyssey 2001: Integrable Models and Beyond (Birkhäuser Boston M. Kashiwara and T. Miwa, eds.), 2002, pp. 23-48. MR-1903971

[4] R. J. Baxter, Solvable eight-vertex model on an arbitrary planar lattice, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 289 (1978), no. 1359, 315-346. MR-0479213

[5] R. J. Baxter, Free-fermion, checkerboard and Z-invariant lattice models in statistical mechanics, Proc. Roy. Soc. London Ser. A 404 (1986), no. 1826, 1-33. MR-0836281

[6] R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1989, Reprint of the 1982 original. MR-0998375

[7] V. Beffara and H. Duminil-Copin, Smirnov's fermionic observable away from criticality, Ann. Probab. 40 (2012), no. 6, 2667-2689. MR-3050513

[8] C. Boutillier, D. Cimasoni, and B. de Tilière, Elliptic dimers on minimal graphs and genus 1 harnack curves, arXiv:2007.14699 (2020), 71 p.

[9] C. Boutillier and B. de Tilière, The critical Z-invariant Ising model via dimers: the periodic case, Probab. Theory Related Fields 147 (2010), 379-413. MR-2639710

[10] C. Boutillier and B. de Tilière, The critical Z-invariant Ising model via dimers: locality property, Comm. Math. Phys. 301 (2011), no. 2, 473-516. MR-2764995

[11] C. Boutillier, B. de Tilière, and K. Raschel, The Z-invariant massive Laplacian on isoradial graphs, Invent. Math. 208 (2017), no. 1, 109-189. MR-3621833

[12] C. Boutillier, B. de Tilière, and K. Raschel, The Z-invariant Ising model via dimers, Probab. Theory Related Fields. 174 (2019), no. 1-2, 235-305. MR-3947324

[13] C. Boutillier and B. de Tilière, Height representation of XOR-Ising loops via bipartite dimers, Electron. J. Probab. 19 (2014), no. 80, 1-33. MR-3256880

[14] R. Burton and R. Pemantle, Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances, Ann. Probab. 21 (1993), no. 3, 1329-1371. MR-1235419
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[19] D. Chelkak and Y. Wan, *On the convergence of massive loop-erased random walks to massive SLE(2) curves*, arXiv preprint arXiv:1903.08045 (2019).

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[21] D. Cimasoni, Kac-Ward operators, kasteleyn operators, and s-holomorphicity on arbitrary surface graphs, Ann. Inst. Henri Poincaré, D **2** (2015), 113–168. MR-3354329

[22] D. Cimasoni and H. Duminil-Copin, *The critical temperature for the Ising model on planar doubly periodic graphs*, Electron. J. Probab. **18** (2013), no. 44, 1–18. MR-3040554

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[26] N. G. de Bruijn, *Algebraic theory of Penrose's non-periodic tilings of the plane. I*, Indagationes Mathematicae (Proceedings) **84** (1981), no. 1, 39–52. MR-0609465

[27] N. G. de Bruijn, *Algebraic theory of Penrose's non-periodic tilings of the plane. II*, Indagationes Mathematicae (Proceedings) **84** (1981), no. 1, 53–66. MR-0609465

[28] B. de Tilière, *Quadri-tilings of the plane*, Probab. Theory Related Fields **137** (2007), no. 3-4, 487–518. MR-2278466

[29] B. de Tilière, *From cycle rooted spanning forests to the critical Ising model: an explicit construction*, Comm. Math. Phys. **319** (2013), no. 1, 69–110. MR-3034026

[30] B. de Tilière, *Critical Ising model and spanning trees partition functions*, Ann. Inst. Henri Poincaré Probab. Stat. **52** (2016), no. 3, 1382–1405. MR-3531713

[31] V. S. Dotsenko and V. S. Dotsenko, *Critical behaviour of the phase transition in the 2d ising model with impurities*, Adv. in Phys. **32** (1983), no. 2, 129–172. MR-0710708

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[48] R. Kenyon and J.-M. Schlenker, *Rhombic embeddings of planar quad-graphs*, Trans. Amer. Math. Soc. **357** (2005), no. 9, 3443–3458 (electronic). MR-2146632

[49] R. W. Kenyon, J. G. Propp, and D. B. Wilson, *Trees and matchings*, Electron. J. Combin **7** (2000), no. 1, R25. MR-1756162

[50] G. Kirchhoff, *Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird*, Annalen der Physik **148** (1847), 497–508.

[51] H. A. Kramers and G. H. Wannier, *Statistics of the two-dimensional ferromagnet. Part I*, Phys. Rev. **60** (1941), no. 3, 252–262. MR-0004803

[52] H. A. Kramers and G. H. Wannier, *Statistics of the two-dimensional ferromagnet. Part II*, Phys. Rev. **60** (1941), no. 3, 263–276. MR-0004804

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[56] M. Lis, *The fermionic observables in the ising model and the inverse kac-ward operator*, Annales Henri Poincaré **15** (2014), no. 10, 1945–1965. MR-3257455

[57] M. Lis, *Phase transition free regions in the Ising model via the Kac-Ward operator*, Comm. Math. Phys. (2014), 1–16 (English). MR-3248059

[58] N. Makarov and S. Smirnov, *Off-critical lattice models and massive sles*, XVIth International Congress on Mathematical Physics, 2010, pp. 362–371. MR-2730811

[59] C. Mercat, *Discrete Riemann surfaces and the Ising model*, Comm. Math. Phys. **218** (2001), no. 1, 177–216. MR-1824204

[60] R. J. Messikh, *The surface tension near criticality of the 2d-Ising model*, arXiv:math/0610636 (2006).

[61] C. Nash and D. O'Connor, *Dimer geometry, amoeba and a vortex dimer model*, J. Phys. A: Math. Theor. **50** (2017). MR-3688099

[62] B. Nienhuis, *Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas*, J. Statist. Phys. **34** (1984), no. 5-6, 731–761. MR-0751711

[63] B. Nienhuis and H. J. F. Knops, *Spinor exponents for the two-dimensional Potts model*, Phys. Rev. B. **32** (1985), 1872–1875.

[64] L. Onsager, *Crystal statistics. I. A two-dimensional model with an order-disorder transition*, Phys. Rev. **65** (1944), no. 3-4, 117–149. MR-0010315

[65] J. H. H. Perk and H. Au-Yang, *Yang Baxter equations*, Encyclopedia of Mathematical Physics (Jean-Pierre Françoise, Gregory L. Naber, and Tsou Sheung Tsun, eds.), Academic Press, Oxford, 2006, pp. 465–473.

[66] S. Sheffield, *Random surfaces*, Astérisque (2005), no. 304, vi+175. MR-2251117
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[43] A. E. Kennelly, *The equivalence of triangles and three-pointed stars in conducting networks*, Electrical World and Engineer **34** (1899), 413–414.

[44] R. Kenyon, *Local statistics of lattice dimers*, Ann. Inst. H. Poincaré Probab. Statist. **33** (1997), no. 5, 591–618. MR-99b:82039

[45] R. Kenyon, *The Laplacian and Dirac operators on critical planar graphs*, Invent. Math. **150** (2002), no. 2, 409–439. MR-2004c:31015

[46] R. Kenyon, *Determinantal spanning forests on planar graphs*, Ann. Probab. **47** (2019), no. 2, 952–988. MR-3916939

[47] R. Kenyon, A. Okounkov, and S. Sheffield, *Dimers and amoebae*, Ann. of Math. (2) **163** (2006), no. 3, 1019–1056. MR-2215138

[48] R. Kenyon and J.-M. Schlenker, *Rhombic embeddings of planar quad-graphs*, Trans. Amer. Math. Soc. **357** (2005), no. 9, 3443–3458 (electronic). MR-2146632

[49] R. W. Kenyon, J. G. Propp, and D. B. Wilson, *Trees and matchings*, Electron. J. Combin **7** (2000), no. 1, R25. MR-1756162

[50] G. Kirchhoff, *Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird*, Annalen der Physik **148** (1847), 497–508.

[51] H. A. Kramers and G. H. Wannier, *Statistics of the two-dimensional ferromagnet. Part I*, Phys. Rev. **60** (1941), no. 3, 252–262. MR-0004803

[52] H. A. Kramers and G. H. Wannier, *Statistics of the two-dimensional ferromagnet. Part II*, Phys. Rev. **60** (1941), no. 3, 263–276. MR-0004804

[53] G. Kuperberg, *An exploration of the permanent-determinant method*, Electron. J. Combin. **5** (1998), no. 1, R46. MR-1663576

[54] D. F. Lawden, *Elliptic functions and applications*, Applied Mathematical Sciences, vol. 80, Springer-Verlag, New York, 1989. MR-1007595

[55] Z. Li, *Critical temperature of periodic Ising models*, Comm. Math. Phys. **315** (2012), 337–381 (English). MR-2971729

[56] M. Lis, *The fermionic observables in the ising model and the inverse kac-ward operator*, Annales Henri Poincaré **15** (2014), no. 10, 1945–1965. MR-3257455

[57] M. Lis, *Phase transition free regions in the Ising model via the Kac-Ward operator*, Comm. Math. Phys. (2014), 1–16 (English). MR-3248059

[58] N. Makarov and S. Smirnov, *Off-critical lattice models and massive sles*, XVIth International Congress on Mathematical Physics, 2010, pp. 362–371. MR-2730811

[59] C. Mercat, *Discrete Riemann surfaces and the Ising model*, Comm. Math. Phys. **218** (2001), no. 1, 177–216. MR-1824204

[60] R. J. Messikh, *The surface tension near criticality of the 2d-Ising model*, arXiv:math/0610636 (2006).

[61] C. Nash and D. O'Connor, *Dimer geometry, amoeba and a vortex dimer model*, J. Phys. A: Math. Theor. **50** (2017). MR-3688099

[62] B. Nienhuis, *Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas*, J. Statist. Phys. **34** (1984), no. 5-6, 731–761. MR-0751711

[63] B. Nienhuis and H. J. F. Knops, *Spinor exponents for the two-dimensional Potts model*, Phys. Rev. B. **32** (1985), 1872–1875.

[64] L. Onsager, *Crystal statistics. I. A two-dimensional model with an order-disorder transition*, Phys. Rev. **65** (1944), no. 3-4, 117–149. MR-0010315

[65] J. H. H. Perk and H. Au-Yang, *Yang Baxter equations*, Encyclopedia of Mathematical Physics (Jean-Pierre Françoise, Gregory L. Naber, and Tsou Sheung Tsun, eds.), Academic Press, Oxford, 2006, pp. 465–473.

[66] S. Sheffield, *Random surfaces*, Astérisque (2005), no. 304, vi+175. MR-2251117
[67] S. Smirnov, *Towards conformal invariance of 2D lattice models*, Proceedings of the ICM, Madrid, vol. 2, 2006, pp. 1421–1452. MR-2275653

[68] S. Smirnov, *Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model*, Ann. Math. **172** (2010), no. 2, 1435–1467. MR-2680496

[69] S. Smirnov, *Discrete complex analysis and probability*, Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II–IV: Invited Lectures, World Scientific, 2010, pp. 595–621. MR-1957512

[70] W. Sun, *Toroidal dimer model and temperley’s bijection*, arXiv:1603.00690 (2016).

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[72] H. N. V. Temperley and M. E. Fisher, *Dimer problem in statistical mechanics-an exact result*, Philosophical Magazine **6** (1961), no. 68, 1061–1063. MR-0136398

[73] W. T. Tutte, *The dissection of equilateral triangles into equilateral triangles*, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 44, Cambridge Univ Press, 1948, pp. 463–482. MR-0027521

[74] D. B. Wilson, *XOR-Ising loops and the Gaussian free field*, arXiv:1102.3782 (2011).

[75] F. W. Wu, *Ising model with four-spin interactions*, Phys. Rev. B **4** (1971), 2312–2314.

[76] F. Y. Wu and K. Y. Lin, *Staggered ice-rule vertex model – The Pfaffian solution*, Phys. Rev. B **12** (1975), 419–428.

**Acknowledgments.** We thank Cédric Boutillier, Dima Chelkak, David Cimasoni, Adrien Kassel, Marcin Lis, Paul Melotti, Sanjay Ramassamy and Kilian Raschel for interesting conversations in the course of writing this paper. We also thank both anonymous referees for their valuable comments which have helped increase the quality of this paper.
[67] S. Smirnov, *Towards conformal invariance of 2D lattice models*, Proceedings of the ICM, Madrid, vol. 2, 2006, pp. 1421–1452. MR-2275653

[68] S. Smirnov, *Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model*, Ann. Math. **172** (2010), no. 2, 1435–1467. MR-2680496

[69] S. Smirnov, *Discrete complex analysis and probability*, Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II–IV: Invited Lectures, World Scientific, 2010, pp. 595–621. MR-1957512

[70] W. Sun, *Toroidal dimer model and temperley’s bijection*, arXiv:1603.00690 (2016).

[71] H. N. V. Temperley, *In Combinatorics: Proceedings of the British combinatorial conference 1973*, (1974), 202–204.

[72] H. N. V. Temperley and M. E. Fisher, *Dimer problem in statistical mechanics-an exact result*, Philosophical Magazine **6** (1961), no. 68, 1061–1063. MR-0136398

[73] W. T. Tutte, *The dissection of equilateral triangles into equilateral triangles*, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 44, Cambridge Univ Press, 1948, pp. 463–482. MR-0027521

[74] D. B. Wilson, *XOR-Ising loops and the Gaussian free field*, arXiv:1102.3782 (2011).

[75] F. W. Wu, *Ising model with four-spin interactions*, Phys. Rev. B **4** (1971), 2312–2314.

[76] F. Y. Wu and K. Y. Lin, *Staggered ice-rule vertex model – The Pfaffian solution*, Phys. Rev. B **12** (1975), 419–428.

**Acknowledgments.** We thank Cédric Boutillier, Dima Chelkak, David Cimasoni, Adrien Kassel, Marcin Lis, Paul Melotti, Sanjay Ramassamy and Kilian Raschel for interesting conversations in the course of writing this paper. We also thank both anonymous referees for their valuable comments which have helped increase the quality of this paper.