By definition of $T$, see (4.4), we have $t_{w_1,v} = e^{-i\frac{\beta}{2}} \text{cn}(u_\beta) = e^{-i\frac{\beta_e}{2}} \text{cn}(u_{\beta_e})$, and we conclude that

$$ \left| \frac{r'_{w,w_2}}{t_{w_1,v}} \right| = \left[ \text{sn}(\theta) \text{cn}(\theta) \right]^{\frac{1}{2}} \left| \frac{\text{sn}(u_{\alpha_e})}{\text{cn}(u_{\beta_e})} \right| \left[ \text{nd}(u_{\alpha_e}) \text{dn}(u_{\beta_e}) \right]^{\frac{1}{2}}. $$

We now turn to the term $r'_{w,w_2}$ in the case where $x = f$. The rhombus vectors of $G^{\circ/2}$ assigned to the edge $(w, f)$ are $e^{i\bar{\alpha}_e} = e^{i(\bar{\beta}-\pi)}$, $e^{i\bar{\beta}_e} = e^{i\bar{\alpha}}$. Moreover, referring to Figure 19, we see that taking $x = f$ has the effect of exchanging $b_1$ and $b_2$ and leaving $w_1, w_2$ fixed. The quantity $r'_{w,w_2}$ being skew-symmetric in $b_1, b_2$, we have that $r'_{w,w_2}$ is equal to the opposite of (4.12). As a consequence,

$$ \begin{aligned} r'_{w,w_2} &= -e^{i\frac{\bar{\alpha}}{2}} \text{sn}(u_\alpha) [\text{sn}(\theta) \text{cn}(\theta) \text{nd}(u_\alpha) \text{dn}(u_\beta)]^{\frac{1}{2}} \\ &= -e^{i\frac{\bar{\beta}_e}{2}} \text{sn}(u_{\beta_e}) [\text{sn}(\theta) \text{cn}(\theta) \text{nd}(u_{\beta_e}) \text{dn}(u_{\alpha_e+2K})]^{\frac{1}{2}} \\ &= -(k')^{\frac{1}{2}} e^{i\frac{\bar{\beta}_e}{2}} \text{sn}(u_{\beta_e}) [\text{sn}(\theta) \text{cn}(\theta) \text{nd}(u_{\beta_e}) \text{nd}(u_{\alpha_e})]^{\frac{1}{2}}. \end{aligned} $$

using that $e^{i\bar{\alpha}_e} = e^{i(\bar{\beta}-\pi)}$, $e^{i\bar{\beta}_e} = e^{i\bar{\alpha}}$, and that $\text{dn}(u-K) = k' \text{nd}(u)$. By definition of $T$, we have $t_{w_1,f} = e^{-i\frac{\beta-\pi}{2}} \text{cd}(u_{\beta-2K}) = e^{-i\frac{\bar{\alpha}_e}{2}} \text{cd}(u_{\alpha_e})$, and we deduce that

$$ \left| \frac{r'_{w,w_2}}{t_{w_1,v}} \right| = (k')^{\frac{1}{2}} [\text{sn}(\theta) \text{cn}(\theta)]^{\frac{1}{2}} \left| \frac{\text{sn}(u_{\beta_e})}{\text{cn}(u_{\alpha_e})} \right| [\text{nd}(u_{\beta_e}) \text{dn}(u_{\alpha_e})]^{\frac{1}{2}}. $$

Summarizing, we have proved that, for every edge $wx \in M_1$ such that $w \in W^o$,

$$ \begin{aligned} \left| \frac{r'_{w,w_2}}{t_{w_1,x}} \right| &= [\text{sn}(\theta) \text{cn}(\theta)]^{\frac{1}{2}} \times \left\{ \begin{array}{ll} \left| \frac{\text{sn}(u_{\alpha_e})}{\text{cn}(u_{\beta_e})} \right| [\text{nd}(u_{\alpha_e}) \text{dn}(u_{\beta_e})]^{\frac{1}{2}}, & \text{if } x = v \\ (k')^{\frac{1}{2}} \left| \frac{\text{sn}(u_{\beta_e})}{\text{cn}(u_{\alpha_e})} \right| [\text{nd}(u_{\beta_e}) \text{dn}(u_{\alpha_e})]^{\frac{1}{2}}, & \text{if } x = f \end{array} \right. \\ &= [\text{sn}(\theta) \text{cn}(\theta)]^{\frac{1}{2}} \eta_{wx}; \end{aligned} $$

and thus,

$$ (II) = \prod_{w \in W^o} [\text{sn}(\theta) \text{cn}(\theta)]^{\frac{1}{2}} \prod_{wx \in M_1: w \in W^o} \eta_{wx}. \quad (4.13) $$

We now compute Part (I) involving boundary vertices of $W^o$. We will be using the notation of Figure 15 for vertices, rhombus vectors and angles of boundary rhombus pairs of $\mathcal{R}^o$, and add a superscript r when the pair is the root pair, i.e., the one where $v^c = r$. With our choice of perfect matching $M_1$, the boundary contribution (I) of (4.10) can be rewritten as, see also Figure 17 (right):

$$ (I) = \left( \prod_{w^r, w^\ell \in W^o \setminus \{w^{r,r}, w^{\ell,r}\}} \left| \frac{s_{b^r, w^r} s_{b^\ell, w^\ell}}{t_{w^r, v^r} t_{w^c, v^c}} \right| \right) \times \left| \frac{s_{b^{r,r}, w^{r,r}} s_{b^{\ell,r}, w^{\ell,r}}}{t_{w^{r,r}, v^{r,r}} t_{w^{c,r}, f^{c,r}}} \right| $$

Suppose first that $w^r, w^\ell \in W^o \setminus \{w^{r,r}, w^{\ell,r}\}$. Recalling the definition of $S$ and $T$ along the boundary, see (4.3) and (4.5), we have

$$ \begin{aligned} \left| \frac{s_{b^r, w^r} s_{b^\ell, w^\ell}}{t_{w^r, v^r} t_{w^c, v^c}} \right| &= \left| \frac{\text{sn}(\theta^\circ) \text{cn}(\theta^\circ) \text{cn}(u_{\beta^r}) \text{cn}(u_{\alpha^\ell}) [\text{nd}(u_{\alpha^r}) \text{nd}(u_{\beta^r}) \text{nd}(u_{\alpha^\ell}) \text{nd}(u_{\beta^\ell})]^{\frac{1}{2}}}{\text{cn}(u_{\beta^r}) \times k' \text{sn}(\theta^\circ) \text{nd}(u_{\alpha^r}) \text{dn}(u_{\beta^r})} \right| \\ &= \left| \frac{\text{cn}(\theta^\circ) \text{cn}(u_{\alpha^\ell})}{k' \text{cn}(u_{\beta^r})} [\text{dn}(u_{\alpha^r}) \text{dn}(u_{\beta^r}) \text{nd}(u_{\alpha^\ell}) \text{nd}(u_{\beta^\ell})]^{\frac{1}{2}} \right|. \end{aligned} $$

On the other hand, by definition of $\eta$, the product of weights of the edges $w^r v^r, w^\ell v^c$ of the perfect matching $M_1$ is equal to,

$$ \begin{aligned} \eta_{w^r v^r} \eta_{w^\ell v^c} &= \left| \frac{\text{sn}(u_{\alpha^r}) \text{sn}(u_{\alpha^\ell+2K})}{\text{cn}(u_{\beta^r}) \text{cn}(u_{\beta^\ell+2K})} [\text{nd}(u_{\alpha^r}) \text{dn}(u_{\beta^r}) \text{nd}(u_{\alpha^\ell+2K}) \text{dn}(u_{\beta^\ell+2K})]^{\frac{1}{2}} \right| \\ &= \left| \frac{\text{cn}(u_{\alpha^\ell})}{k' \text{cn}(u_{\beta^r})} [\text{dn}(u_{\alpha^r}) \text{dn}(u_{\beta^r}) \text{nd}(u_{\alpha^\ell}) \text{nd}(u_{\beta^\ell})]^{\frac{1}{2}} \right|, \end{aligned} $$
By definition of $T$, see (4.4), we have $t_{w_1,v} = e^{-i\frac{\bar{\beta}}{2}} \text{cn}(u_\beta) = e^{-i\frac{\bar{\beta}_e}{2}} \text{cn}(u_{\beta_e})$, and we conclude that

$$ \left| \frac{r'_{w,w_2}}{t_{w_1,v}} \right| = \left[ \text{sn}(\theta) \text{cn}(\theta) \right]^{\frac{1}{2}} \left| \frac{\text{sn}(u_{\alpha_e})}{\text{cn}(u_{\beta_e})} \right| \left[ \text{nd}(u_{\alpha_e}) \text{dn}(u_{\beta_e}) \right]^{\frac{1}{2}}. $$

We now turn to the term $r'_{w,w_2}$ in the case where $x = f$. The rhombus vectors of $G^{\circ/2}$ assigned to the edge $(w, f)$ are $e^{i\bar{\alpha}_e} = e^{i(\bar{\beta}-\pi)}$, $e^{i\bar{\beta}_e} = e^{i\bar{\alpha}}$. Moreover, referring to Figure 19, we see that taking $x = f$ has the effect of exchanging $b_1$ and $b_2$ and leaving $w_1, w_2$ fixed. The quantity $r'_{w,w_2}$ being skew-symmetric in $b_1, b_2$, we have that $r'_{w,w_2}$ is equal to the opposite of (4.12). As a consequence,

$$ \begin{aligned} r'_{w,w_2} &= -e^{i\frac{\bar{\alpha}}{2}} \text{sn}(u_\alpha) [\text{sn}(\theta) \text{cn}(\theta) \text{nd}(u_\alpha) \text{dn}(u_\beta)]^{\frac{1}{2}} \\ &= -e^{i\frac{\bar{\beta}_e}{2}} \text{sn}(u_{\beta_e}) [\text{sn}(\theta) \text{cn}(\theta) \text{nd}(u_{\beta_e}) \text{dn}(u_{\alpha_e+2K})]^{\frac{1}{2}} \\ &= -(k')^{\frac{1}{2}} e^{i\frac{\bar{\beta}_e}{2}} \text{sn}(u_{\beta_e}) [\text{sn}(\theta) \text{cn}(\theta) \text{nd}(u_{\beta_e}) \text{nd}(u_{\alpha_e})]^{\frac{1}{2}}. \end{aligned} $$

using that $e^{i\bar{\alpha}_e} = e^{i(\bar{\beta}-\pi)}$, $e^{i\bar{\beta}_e} = e^{i\bar{\alpha}}$, and that $\text{dn}(u-K) = k' \text{nd}(u)$. By definition of $T$, we have $t_{w_1,f} = e^{-i\frac{\bar{\beta}-\pi}{2}} \text{cd}(u_{\beta-2K}) = e^{-i\frac{\bar{\alpha}_e}{2}} \text{cd}(u_{\alpha_e})$, and we deduce that

$$ \left| \frac{r'_{w,w_2}}{t_{w_1,v}} \right| = (k')^{\frac{1}{2}} [\text{sn}(\theta) \text{cn}(\theta)]^{\frac{1}{2}} \left| \frac{\text{sn}(u_{\beta_e})}{\text{cn}(u_{\alpha_e})} \right| [\text{nd}(u_{\beta_e}) \text{dn}(u_{\alpha_e})]^{\frac{1}{2}}. $$

Summarizing, we have proved that, for every edge $wx \in M_1$ such that $w \in W^o$,

$$ \begin{aligned} \left| \frac{r'_{w,w_2}}{t_{w_1,x}} \right| &= [\text{sn}(\theta) \text{cn}(\theta)]^{\frac{1}{2}} \times \begin{cases} \left| \frac{\text{sn}(u_{\alpha_e})}{\text{cn}(u_{\beta_e})} \right| [\text{nd}(u_{\alpha_e}) \text{dn}(u_{\beta_e})]^{\frac{1}{2}}, & \text{if } x=v \\ (k')^{\frac{1}{2}} \left| \frac{\text{sn}(u_{\beta_e})}{\text{cn}(u_{\alpha_e})} \right| [\text{nd}(u_{\beta_e}) \text{dn}(u_{\alpha_e})]^{\frac{1}{2}}, & \text{if } x=f \end{cases} \\ &= [\text{sn}(\theta) \text{cn}(\theta)]^{\frac{1}{2}} \eta_{wx}; \end{aligned} $$

and thus,

$$ (II) = \prod_{w \in W^o} [\text{sn}(\theta) \text{cn}(\theta)]^{\frac{1}{2}} \prod_{wx \in M_1: w \in W^o} \eta_{wx}. \quad (4.13) $$

We now compute Part (I) involving boundary vertices of $W^o$. We will be using the notation of Figure 15 for vertices, rhombus vectors and angles of boundary rhombus pairs of $\mathcal{R}^o$, and add a superscript r when the pair is the root pair, i.e., the one where $v^c = r$. With our choice of perfect matching $M_1$, the boundary contribution (I) of (4.10) can be rewritten as, see also Figure 17 (right):

$$ (I) = \left( \prod_{w^r, w^\ell \in W^o \setminus \{w^{r,r}, w^{\ell,r}\}} \left| \frac{s_{b^r, w^r} s_{b^\ell, w^\ell}}{t_{w^r, v^r} t_{w^c, v^c}} \right| \right) \times \left| \frac{s_{b^{r,r}, w^{r,r}} s_{b^{\ell,r}, w^{\ell,r}}}{t_{w^{r,r}, v^{r,r}} t_{w^{c,r}, f^{c,r}}} \right| $$

Suppose first that $w^r, w^\ell \in W^o \setminus \{w^{r,r}, w^{\ell,r}\}$. Recalling the definition of $S$ and $T$ along the boundary, see (4.3) and (4.5), we have

$$ \begin{aligned} \left| \frac{s_{b^r, w^r} s_{b^\ell, w^\ell}}{t_{w^r, v^r} t_{w^c, v^c}} \right| &= \left| \frac{\text{sn}(\theta^\circ) \text{cn}(\theta^\circ) \text{cn}(u_{\beta^r}) \text{cn}(u_{\alpha^\ell}) [\text{nd}(u_{\alpha^r}) \text{nd}(u_{\beta^r}) \text{nd}(u_{\alpha^\ell}) \text{nd}(u_{\beta^\ell})]^{\frac{1}{2}}}{\text{cn}(u_{\beta^r}) \times k' \text{sn}(\theta^\circ) \text{nd}(u_{\alpha^r}) \text{dn}(u_{\beta^r})} \right| \\ &= \left| \frac{\text{cn}(\theta^\circ) \text{cn}(u_{\alpha^\ell})}{k' \text{cn}(u_{\beta^r})} [\text{dn}(u_{\alpha^r}) \text{dn}(u_{\beta^r}) \text{nd}(u_{\alpha^\ell}) \text{nd}(u_{\beta^\ell})]^{\frac{1}{2}} \right|. \end{aligned} $$

On the other hand, by definition of $\eta$, the product of weights of the edges $w^r v^r, w^\ell v^c$ of the perfect matching $M_1$ is equal to,

$$ \begin{aligned} \eta_{w^r v^r} \eta_{w^\ell v^c} &= \left| \frac{\text{sn}(u_{\alpha^r}) \text{sn}(u_{\alpha^\ell+2K})}{\text{cn}(u_{\beta^r}) \text{cn}(u_{\beta^\ell+2K})} [\text{nd}(u_{\alpha^r}) \text{dn}(u_{\beta^r}) \text{nd}(u_{\alpha^\ell+2K}) \text{dn}(u_{\beta^\ell+2K})]^{\frac{1}{2}} \right| \\ &= \left| \frac{\text{cn}(u_{\alpha^\ell})}{k' \text{cn}(u_{\beta^r})} [\text{dn}(u_{\alpha^r}) \text{dn}(u_{\beta^r}) \text{nd}(u_{\alpha^\ell}) \text{nd}(u_{\beta^\ell})]^{\frac{1}{2}} \right|, \end{aligned} $$
writing $u_{\alpha^\ell+2K} = u_{\alpha^\ell} - K$, using elliptic trigonometric identities and the fact that $\bar{\alpha}^r = \bar{\beta}^\ell[2\pi]$.

$$ \text{so that, } \left| \frac{s_{b^r, w^r} s_{b^\ell, w^\ell}}{t_{w^r, v^r} t_{w^c, v^c}} \right| = \text{cn}(\theta^\beta) \eta_{w^\ell v^c} \eta_{w^r v^r}. $$

Let us now consider the term $\left| \frac{s_{b^r, r, w^r, r} s_{b^\ell, r, w^\ell, r}}{t_{w^r, r, v^r, r} t_{w^c, r, v^c, r}} \right|$. For the purpose of this computation, it is useful to imagine that the vertex $v^c = r$ is present and that $t_{w^c, r, v^c, r}, \eta_{w^\ell, r v^c, r}$ are defined as for the other pairs of rhombi. We then have, omitting to write the superscript r,

$$ \left| \frac{s_{b^r, w^r} s_{b^\ell, w^\ell}}{t_{w^r, v^r} t_{w^c, v^c}} \frac{1}{\eta_{w^r v^r} \eta_{w^\ell v^c}} \right| = \left| \frac{s_{b^r, w^r} s_{b^\ell, w^\ell}}{t_{w^r, v^r} t_{w^c, v^c}} \times \frac{1}{\eta_{w^r v^r} \eta_{w^\ell v^c}} \times \frac{\eta_{w^\ell, v^c}}{\eta_{w^\ell f^c}} \times \frac{t_{w^c, v^c}}{t_{w^c, f^c}} \right| $$

The product of the first two terms is equal to $|\text{cn}(\theta^\beta)|$ by the above computation. Then, returning to the definition of the weight function $\eta$, we have

$$ \begin{aligned} \left| \frac{\eta_{w^\ell v^c}}{\eta_{w^\ell f}} \right| &= \left| \frac{\text{sn}(u_{\alpha^\ell+2K})}{\text{cn}(u_{\beta^\ell+2K})} \left[ \text{nd}(u_{\alpha^\ell+2K}) \text{dn}(u_{\beta^\ell+2K}) \right]^{\frac{1}{2}} \frac{\text{cn}(u_{\beta^\ell})}{\text{sn}(u_{\alpha^\ell+2K})} \left[ \text{nd}(u_{\beta^\ell}) \text{dn}(u_{\alpha^\ell+2K}) \right]^{\frac{1}{2}} (k')^{-\frac{1}{2}} \right| \\ &= \left| (k')^{-1} \text{cs}(u_{\beta^\ell}) \right|. \end{aligned} $$

Returning to the definition of the matrix $T$ along the boundary, we have

$$ \left| \frac{t_{w^c, v^c}}{t_{w^c, f^c}} \right| = \left| \frac{k' \text{sn}(\theta^\beta) \text{nd}(u_{\alpha^r}) \text{cd}(u_{\beta^r})}{\text{cd}(u_{\beta^\ell})} \right| = \left| \frac{k' \text{sn}(\theta^\beta) \text{cd}(u_{\beta^r})}{\text{cn}(u_{\alpha^r})} \right|, $$

using that $\bar{\beta}^\ell = \bar{\alpha}^r[2\pi]$. Putting the three computations together, and writing the superscript $r$ again, we deduce that $\left| \frac{s_{b^r, r, w^r, r} s_{b^\ell, r, w^\ell, r}}{t_{w^r, r, v^r, r} t_{w^c, r, v^c, r}} \frac{1}{\eta_{w^r, r, v^r, r} \eta_{w^\ell, r, v^c, r}} \right| = \left| \text{sn}(\theta^{\beta, r}) \text{cn}(\theta^{\beta, r}) \frac{\text{cd}(u_{\beta^r, r})}{\text{sn}(u_{\alpha^r, r})} \right|$, and thus

$$ \begin{aligned} (I) &= \left( \text{sn}(\theta^{\beta,r}) \text{cn}(\theta^{\beta,r}) \right) \left| \frac{\text{cd}(u_{\beta,r})}{\text{sn}(u_{\alpha,r})} \right| \left( \prod_{w^r, w^\ell \in W^{\partial} \setminus \{w^{r,r}, w^{\ell,r}\}} \text{cn}(\theta^\ell)^{\frac{1}{2}} \right) \left( \prod_{wx \in M_1: w \in W^\partial} \eta_{wx} \right) \\ &= \text{sn}(\theta^{\beta,r}) \left| \frac{\text{cd}(u_{\beta,r})}{\text{sn}(u_{\alpha,r})} \right| \left( \prod_{w \in W^\partial} \text{cn}(\theta_w)^{\frac{1}{2}} \right) \left( \prod_{wx \in M_1: w \in W^\partial} \eta_{wx} \right), \end{aligned} \quad (4.14) $$

using that $W^\partial = \{w_\ell, w_r \in \mathbb{R}^\partial\}$. Combining (4.13) and (4.14) allows to conclude the proof of Lemma 4.10. $\square$

The next lemma proves a simplified expression for the product of the weights $\eta_{wx}$ in Lemma 4.10.

**Lemma 4.11.** For every $u \in C''$, we have the following identity,

$$ \prod_{wx \in M_1: w \in W} \eta_{wx} = (k')^{\frac{|W^*|}{2}} \prod_{wx \in M_1: w \in W} \left| \text{sc}(u_{\alpha_e}) \text{sc}(u_{\beta_e}) \right|^{\frac{1}{2}}, $$

where the weight function $\eta$ is defined in (4.11).

*Proof.* We have the following identities:

$$ |\text{sn}(u_\alpha)| \text{nd}(u_\alpha)^{\frac{1}{2}} = |\text{sn}(u_\alpha) \text{sn}(u_{\alpha+2K})|^{\frac{1}{2}} |\text{sc}(u_\alpha)|^{\frac{1}{2}} $$

$$ |\text{cn}(u_\alpha)| \text{nd}(u_\alpha)^{\frac{1}{2}} = |\text{sn}(u_\alpha) \text{sn}(u_{\alpha+2K})|^{\frac{1}{2}} |\text{cs}(u_\alpha)|^{\frac{1}{2}}. $$

As a consequence, for every edge $wx$ of $G^{D,r}$, the weight function $\eta_{wx}$ can be rewritten as:

$$ \eta_{wx} = (k')^{\frac{1}{2}} \mathbb{1}_{\{x \in V^*\}} |\text{sc}(u_{\alpha_e}) \text{sc}(u_{\beta_e})| \eta'_{wx}, \quad \text{where } \eta'_{wx} = \begin{cases} \left| \frac{\text{sn}(u_{\alpha_e}) \text{sn}(u_{\alpha_e+2K})}{\text{sn}(u_{\beta_e}) \text{sn}(u_{\beta_e+2K})} \right|^{\frac{1}{2}} & \text{if } x = v \\ \left| \frac{\text{sn}(u_{\beta_e}) \text{sn}(u_{\beta_e+2K})}{\text{sn}(u_{\alpha_e}) \text{sn}(u_{\alpha_e+2K})} \right|^{\frac{1}{2}} & \text{if } x = f. \end{cases} $$
writing $u_{\alpha^\ell+2K} = u_{\alpha^\ell} - K$, using elliptic trigonometric identities and the fact that
$\bar{\alpha}^r = \bar{\beta}^\ell[2\pi]$.

$$
\text{so that, } \left| \frac{s_{b^r, w^r} s_{b^\ell, w^\ell}}{t_{w^r, v^r} t_{w^c, v^c}} \right| = \operatorname{cn}(\theta^\partial) \eta_{w^\ell v^c} \eta_{w^r v^r}.
$$

Let us now consider the term $\left| \frac{s_{b^r, r, w^r, r} s_{b^\ell, \ell, w^\ell, \ell}}{t_{w^r, r, v^r, r} t_{w^c, r, f^c, r}} \right|$. For the purpose of this computation, it is useful to imagine that the vertex $v^c = r$ is present and that $t_{w^c, r, v^c, r}, \eta_{w^\ell, r v^c, r}$ are defined as for the other pairs of rhombi. We then have, omitting to write the superscript r,

$$
\left| \frac{s_{b^r, w^r} s_{b^\ell, w^\ell}}{t_{w^r, v^r} t_{w^c, v^c}} \frac{1}{\eta_{w^r v^r} \eta_{w^\ell f^c}} \right| = \left| \frac{s_{b^r, w^r} s_{b^\ell, w^\ell}}{t_{w^r, v^r} t_{w^c, v^c}} \times \frac{1}{\eta_{w^r v^r} \eta_{w^\ell v^c}} \times \frac{\eta_{w^\ell, v^c}}{\eta_{w^\ell f^c}} \times \frac{t_{w^c, v^c}}{t_{w^c, f^c}} \right|.
$$

The product of the first two terms is equal to $|\operatorname{cn}(\theta^\partial)|$ by the above computation. Then,
returning to the definition of the weight function $\eta$, we have

$$
\begin{align*}
\left| \frac{\eta_{w^\ell v^c}}{\eta_{w^\ell f}} \right| &= \left| \frac{\operatorname{sn}(u_{\alpha^\ell+2K})}{\operatorname{cn}(u_{\beta^\ell+2K})} \left[ \operatorname{nd}(u_{\alpha^\ell+2K}) \operatorname{dn}(u_{\beta^\ell+2K}) \right]^{\frac{1}{2}} \frac{\operatorname{cn}(u_{\beta^\ell})}{\operatorname{sn}(u_{\alpha^\ell+2K})} \left[ \operatorname{nd}(u_{\beta^\ell}) \operatorname{dn}(u_{\alpha^\ell+2K}) \right]^{\frac{1}{2}} (k')^{-\frac{1}{2}} \right| \\
&= \left| (k')^{-1} \operatorname{cs}(u_{\beta^\ell}) \right|.
\end{align*}
$$

Returning to the definition of the matrix $T$ along the boundary, we have

$$
\left| \frac{t_{w^c, v^c}}{t_{w^c, f^c}} \right| = \left| \frac{k' \operatorname{sn}(\theta^\partial) \operatorname{nd}(u_{\alpha^r}) \operatorname{cd}(u_{\beta^r})}{\operatorname{cd}(u_{\beta^\ell})} \right| = \left| \frac{k' \operatorname{sn}(\theta^\partial) \operatorname{cd}(u_{\beta^r})}{\operatorname{cn}(u_{\alpha^r})} \right|,
$$

using that $\bar{\beta}^\ell = \bar{\alpha}^r[2\pi]$. Putting the three computations together, and writing the superscript r again, we deduce that
$$ \left| \frac{s_{b^r, r, w^r, r} s_{b^\ell, \ell, w^\ell, \ell}}{t_{w^r, r, v^r, r} t_{w^c, r, f^c, r}} \frac{1}{\eta_{w^r, r, v^r, r} \eta_{w^\ell, r, f^c}} \right| = \left| \operatorname{sn}(\theta^{\partial,r}) \operatorname{cn}(\theta^{\partial,r}) \frac{\operatorname{cd}(u_{\beta^r,r})}{\operatorname{sn}(u_{\alpha^r,r})} \right|, $$
and thus

$$
\begin{align}
(\mathrm{I}) &= (\mathrm{sn}(\theta^{\partial,r}) \mathrm{cn}(\theta^{\partial,r})) \left| \frac{\mathrm{cd}(u_{\beta^r,r})}{\mathrm{sn}(u_{\alpha^r,r})} \right| \left( \prod_{w^r, w^\ell \in W^{\partial} \setminus \{w^{r,r}, w^{\ell,r}\}} \mathrm{cn}(\theta^{\partial})^{\frac{1}{2}} \right) \left( \prod_{wx \in M_1: w \in W^{\partial}} \eta_{wx} \right) \nonumber \\
&= \mathrm{sn}(\theta^{\partial,r}) \left| \frac{\mathrm{cd}(u_{\beta^r,r})}{\mathrm{sn}(u_{\alpha^r,r})} \right| \left( \prod_{w \in W^{\partial}} \mathrm{cn}(\theta_w)^{\frac{1}{2}} \right) \left( \prod_{wx \in M_1: w \in W^{\partial}} \eta_{wx} \right), \tag{4.14}
\end{align}
$$

using that $W^\partial = \{w_\ell, w_r \in \mathbb{R}^\partial\}$. Combining (4.13) and (4.14) allows to conclude the proof of Lemma 4.10. $\square$

The next lemma proves a simplified expression for the product of the weights $\eta_{wx}$ in
Lemma 4.10.

**Lemma 4.11.** For every $u \in C'',$ we have the following identity,

$$
\prod_{wx \in M_1: w \in W} \eta_{wx} = (k')^{\frac{|W^*|}{2}} \prod_{wx \in M_1: w \in W} \left| \operatorname{sc}(u_{\alpha_e}) \operatorname{sc}(u_{\beta_e}) \right|^{\frac{1}{2}},
$$

where the weight function $\eta$ is defined in (4.11).

*Proof.* We have the following identities:

$$
| \operatorname{sn}(u_{\alpha}) | \operatorname{nd}(u_{\alpha})^{\frac{1}{2}} = | \operatorname{sn}(u_{\alpha}) | \operatorname{sn}(u_{\alpha+2K}) |^{\frac{1}{2}} | \operatorname{sc}(u_{\alpha}) |^{\frac{1}{2}}
$$

$$
| \operatorname{cn}(u_{\alpha}) | \operatorname{nd}(u_{\alpha})^{\frac{1}{2}} = | \operatorname{sn}(u_{\alpha}) | \operatorname{sn}(u_{\alpha+2K}) |^{\frac{1}{2}} | \operatorname{cs}(u_{\alpha}) |^{\frac{1}{2}}.
$$

As a consequence, for every edge $wx$ of $G^{D,r}$, the weight function $\eta_{wx}$ can be rewritten as:

$$
\eta_{wx} = (k')^{\frac{1}{2}} \mathbb{1}_{\{x \in V^*\}} |\operatorname{sc}(u_{\alpha_e}) \operatorname{sc}(u_{\beta_e})| \eta'_{wx}, \quad \text{where } \eta'_{wx} = \begin{cases} \left| \frac{\operatorname{sn}(u_{\alpha_e}) \operatorname{sn}(u_{\alpha_e+2K})}{\operatorname{sn}(u_{\beta_e}) \operatorname{sn}(u_{\beta_e+2K})} \right|^{\frac{1}{2}} & \text{if } x=v \\ \left| \frac{\operatorname{sn}(u_{\beta_e}) \operatorname{sn}(u_{\beta_e+2K})}{\operatorname{sn}(u_{\alpha_e}) \operatorname{sn}(u_{\alpha_e+2K})} \right|^{\frac{1}{2}} & \text{if } x=f. \end{cases}
$$
Now consider a vertex $w$ of $G^{D,r}$ and, using the notation of Figure 16, the corresponding rhombus $v_1, f_1, v_2, f_2$ of the diamond graph $G^\diamond$. Introduce the following notation for the rhombus vectors:

$$ (v_1, f_1) = e^{i\bar{\alpha}_1(w)}, \quad (v_2, f_2) = e^{i\bar{\alpha}_2(w)}, \quad (v_1, f_2) = e^{i\bar{\beta}_1(w)}, \quad (v_2, f_1) = e^{i\bar{\beta}_2(w)}, \quad (4.15) $$

that is, the notation $\alpha$, resp. $\beta$, is for vectors on the right, resp. left, of the primal edge of the rhombus. With this notation, the weight $\eta'_{wx}$ can be written as

$$ \eta'_{wx} = \left| \frac{\mathrm{sn}(u_{\alpha_1(w)}) \mathrm{sn}(u_{\alpha_2(w)})}{\mathrm{sn}(u_{\beta_1(w)}) \mathrm{sn}(u_{\beta_2(w)})} \right|^{\frac{1}{2}}, $$

and this, independently of whether $x = v$ or $f$. Let us prove that

$$ \prod_{wx \in M_1: w \in W} \eta'_{wx} = \prod_{w \in W} \left| \frac{\mathrm{sn}(u_{\alpha_1(w)}) \mathrm{sn}(u_{\alpha_2(w)})}{\mathrm{sn}(u_{\beta_1(w)}) \mathrm{sn}(u_{\beta_2(w)})} \right|^{\frac{1}{2}} = 1. $$

Because of (4.15), the product over white vertices $W$ can be seen as a product over rhombus vectors of $G^\diamond$. Then, every inner rhombus vector of $G^\diamond$ occurs twice exactly and contributes once to the numerator and once to the denominator, so that the contributions cancel. Boundary rhombus vectors of $G^\diamond$ occur once but, referring to Section 2.5.3 on train-tracks, we know that they come in parallel pairs and contribute once to the numerator and once to the denominator; the contributions thus also compensate ending the proof of this lemma. $\square$

Putting together Corollary 4.9, Lemmas 4.10 and 4.11 ends the proof of Theorem 4.3.

## 4.3 Dimer model on the graph $G^Q$ and inverse $Z^u$-Dirac operator

Using Theorem 4.2, in Corollaries 4.12 and 4.14, we prove linear relations satisfied by the inverse Kasteleyn operator $(K^Q)^{-1}$ and the inverse of the $Z^u$-Dirac operators $K(u)$ and $K^\theta(u)$. Section 4.3.2 is about applications of these results to the dimer model on $G^Q$. In particular, when the graph $G^Q$ is infinite, we prove an alternative way of obtaining a local formula for the inverse [12] which is seen as directly related to the Z-massive Green functions.

### 4.3.1 Inverse Kasteleyn operator $(K^Q)^{-1}$ and inverse $Z^u$-Dirac operator

*Infinite case.* In the paper [12] we prove an explicit local expression for an inverse $(K^Q)^{-1}$ of the operator $K^Q$, which decreases to 0 exponentially fast in the distance when $k \neq 0$, and as the inverse distance when $k = 0$. When $k = 0$, the local expression is actually computed in [45]. The operator $(K^Q)^{-1}$ is the unique inverse decreasing to zero at infinity; this is established in [45] when $k = 0$ and in [12] when $k \neq 0$. In Corollary 4.12 below, we use the existence and uniqueness of this inverse operator but not the explicit expression; we also need the following notation.

**Notation for coefficients of Corollary 4.12** Let $w$ be a white vertex of $G^Q$ and $v, f$ be its adjacent vertices in the diamond graph $G^{\diamond/2}$, such that $v \in V$ and $f \in V^*$. Denote by $e^{i\bar{\beta}_1}$ the rhombus vector corresponding to the edge $(w, v)$. Let $w$ be a white vertex of $G^D$ and $b, b'$ be the black vertices of $G^Q$ of the corresponding quadrangle. To the vertex $b$, we assign the rhombus vectors $e^{i\bar{\alpha}_f}$, $e^{i\bar{\beta}_f}$ of $G^{\diamond/2}$ of the edge $(b, w)$, where the vertex $w$ is such that the edge $bw$ is parallel to an edge of $G$. Then, the rhombus vectors assigned to the vertex $b'$ are $e^{i\bar{\alpha}_{f+\pi}}$, $e^{i\bar{\beta}_{f+\pi}}$, see Figure 20; the subscripts "i" and "f" stand for "initial" and "final".

As a consequence of Theorem 4.2 (infinite case), we obtain the following.
Now consider a vertex $w$ of $G^{D,r}$ and, using the notation of Figure 16, the corresponding rhombus $v_1, f_1, v_2, f_2$ of the diamond graph $G^\diamond$. Introduce the following notation for the rhombus vectors:

$$ (v_1, f_1) = e^{i\bar{\alpha}_1(w)}, \quad (v_2, f_2) = e^{i\bar{\alpha}_2(w)}, \quad (v_1, f_2) = e^{i\bar{\beta}_1(w)}, \quad (v_2, f_1) = e^{i\bar{\beta}_2(w)}, \quad (4.15) $$

that is, the notation $\alpha$, resp. $\beta$, is for vectors on the right, resp. left, of the primal edge of the rhombus. With this notation, the weight $\eta'_{wx}$ can be written as

$$ \eta'_{wx} = \left| \frac{\mathrm{sn}(u_{\alpha_1(w)}) \mathrm{sn}(u_{\alpha_2(w)})}{\mathrm{sn}(u_{\beta_1(w)}) \mathrm{sn}(u_{\beta_2(w)})} \right|^{\frac{1}{2}}, $$

and this, independently of whether $x = v$ or $f$. Let us prove that

$$ \prod_{wx \in M_1: w \in W} \eta'_{wx} = \prod_{w \in W} \left| \frac{\mathrm{sn}(u_{\alpha_1(w)}) \mathrm{sn}(u_{\alpha_2(w)})}{\mathrm{sn}(u_{\beta_1(w)}) \mathrm{sn}(u_{\beta_2(w)})} \right|^{\frac{1}{2}} = 1. $$

Because of (4.15), the product over white vertices $W$ can be seen as a product over rhombus vectors of $G^\diamond$. Then, every inner rhombus vector of $G^\diamond$ occurs twice exactly and contributes once to the numerator and once to the denominator, so that the contributions cancel. Boundary rhombus vectors of $G^\diamond$ occur once but, referring to Section 2.5.3 on train-tracks, we know that they come in parallel pairs and contribute once to the numerator and once to the denominator; the contributions thus also compensate ending the proof of this lemma. $\square$

Putting together Corollary 4.9, Lemmas 4.10 and 4.11 ends the proof of Theorem 4.3.

## 4.3 Dimer model on the graph $G^Q$ and inverse $Z^u$-Dirac operator

Using Theorem 4.2, in Corollaries 4.12 and 4.14, we prove linear relations satisfied by the inverse Kasteleyn operator $(K^Q)^{-1}$ and the inverse of the $Z^u$-Dirac operators $K(u)$ and $K^\theta(u)$. Section 4.3.2 is about applications of these results to the dimer model on $G^Q$. In particular, when the graph $G^Q$ is infinite, we prove an alternative way of obtaining a local formula for the inverse [12] which is seen as directly related to the $Z$-massive Green functions.

### 4.3.1 Inverse Kasteleyn operator $(K^Q)^{-1}$ and inverse $Z^u$-Dirac operator

*Infinite case.* In the paper [12] we prove an explicit local expression for an inverse $(K^Q)^{-1}$ of the operator $K^Q$, which decreases to 0 exponentially fast in the distance when $k \neq 0$, and as the inverse distance when $k = 0$. When $k = 0$, the local expression is actually computed in [45]. The operator $(K^Q)^{-1}$ is the unique inverse decreasing to zero at infinity; this is established in [45] when $k = 0$ and in [12] when $k \neq 0$. In Corollary 4.12 below, we use the existence and uniqueness of this inverse operator but not the explicit expression; we also need the following notation.

**Notation for coefficients of Corollary 4.12** Let $w$ be a white vertex of $G^Q$ and $v, f$ be its adjacent vertices in the diamond graph $G^{\diamond/2}$, such that $v \in V$ and $f \in V^*$. Denote by $e^{i\bar{\alpha}_1}$ the rhombus vector corresponding to the edge $(w, v)$. Let $w$ be a white vertex of $G^D$ and $b, b'$ be the black vertices of $G^Q$ of the corresponding quadrangle. To the vertex $b$, we assign the rhombus vectors $e^{i\bar{\alpha}_f}$, $e^{i\bar{\beta}_f}$ of $G^{\diamond/2}$ of the edge $(b, w)$, where the vertex $w$ is such that the edge $bw$ is parallel to an edge of $G$. Then, the rhombus vectors assigned to the vertex $b'$ are $e^{i\bar{\alpha}_f + \pi}$, $e^{i\bar{\beta}_f + \pi}$, see Figure 20; the subscripts “i” and “f” stand for “initial” and “final”.

As a consequence of Theorem 4.2 (infinite case), we obtain the following.
<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /><figcaption>Figure 20: Notation for coefficients of Corollary 4.12 and 4.14.</figcaption></figure>

**Corollary 4.12.** For every $u \in C$, the inverse operators $(K^Q)^{-1}$, $K(u)^{-1}$, and the matrices $S(u)$, $T(u)$ satisfy the following identity.

* Matrix form.

$$(K^Q)^{-1} S(u) = T(u)K(u)^{-1}.$$

* Coefficients. For every $w, \bar{v}, \bar{f}$, and every $w, b, b'$ as in the notation above, we have:

$$(K^Q)_{w,b}^{-1} s(u)_{b,w} + (K^Q)_{w,b'}^{-1} s(u)_{b',w} = t(u)_{w,\bar{v}} K(u)_{\bar{v},w}^{-1} + t(u)_{w,\bar{f}} K(u)_{\bar{f},w}^{-1}.$$

Or equivalently,

$$(K^Q)_{w,b}^{-1} \mathrm{cn}(u_{\beta_f}) - i (K^Q)_{w,b'}^{-1} \mathrm{sn}(u_{\beta_f}) \mathrm{dn}(u_{\alpha_f}) = \frac{e^{i \frac{\beta_f - \bar{\beta}_f}{2}}}{\Lambda(u_{\alpha_f}, u_{\beta_f})} \left[ \mathrm{cn}(u_{\beta_i}) K(u)_{\bar{v},w}^{-1} - i \mathrm{sn}(u_{\beta_i}) K(u)_{\bar{f},w}^{-1} \right], \quad (4.16)$$

where $\Lambda(u_\alpha, u_\beta) = [\mathrm{sn}\,\theta \ \mathrm{cn}\,\theta \ \mathrm{nd}(u_\alpha) \ \mathrm{nd}(u_\beta)]^{\frac{1}{2}}$, and $\theta = u_\alpha - u_\beta$.

*Proof.* To obtain the matrix form, let us consider the matrix $(K^Q)^{-1} S(u) - T(u)K(u)^{-1}$ and left multiply it by $K^Q$. Since coefficients of the inverses tend to 0 at infinity and the other matrices involved only have finitely many non-zero terms per row and column, the matrix product is associative and we obtain

$$K^Q[(K^Q)^{-1}S(u) - T(u)K(u)^{-1}] = S(u) - K^Q T(u)K(u)^{-1}. \quad (4.17)$$

Now, right multiplying Equation (4.6) of Theorem (4.2) by $K(u)^{-1}$ and again using associativity of the matrix product we have that Equation (4.17) is equal to zero. As a consequence, each column of $(K^Q)^{-1}S(u) - T(u)K(u)^{-1}$ is a function in the kernel of $K^Q$, which moreover tends to zero at infinity since it is the case for coefficients of $(K^Q)^{-1}$ and $K(u)^{-1}$. Recall that by [45] in the case $k=0$ and by [12] in the case $k \neq 0$, $K^Q$ has a unique inverse tending to zero at infinity. This implies that 0 is the only function tending to zero at infinity in the kernel of $K^Q$, yielding that $(K^Q)^{-1}S(u) - T(u)K(u)^{-1} = 0$.

For coefficients, we return to the definition of the matrix $S(u)$, see (4.2), and obtain

$$s(u)_{b,w} = e^{-i \frac{\beta_f}{2}} [\mathrm{sn}(\theta_f) \mathrm{cn}(\theta_f) \mathrm{nd}(u_{\alpha_f}) \mathrm{nd}(u_{\beta_f})]^{\frac{1}{2}} \mathrm{cn}(u_{\beta_f}) = e^{-i \frac{\beta_f}{2}} \mathrm{cn}(u_{\beta_f}) \Lambda(u_{\alpha_f}, u_{\beta_f})$$

$$\begin{align*}
s(u)_{b',w} &= e^{-i \frac{\beta_f + \pi}{2}} [\mathrm{sn}(\theta_f) \mathrm{cn}(\theta_f) \mathrm{nd}(u_{\alpha_f+2K}) \mathrm{nd}(u_{\beta_f+2K})]^{\frac{1}{2}} \mathrm{cn}(u_{\beta_f+2K}) \\
&= -ie^{-i \frac{\beta_f}{2}} [\mathrm{sn}(\theta_f) \mathrm{cn}(\theta_f) \mathrm{nd}(u_{\alpha_f}) \mathrm{nd}(u_{\beta_f})]^{\frac{1}{2}} \mathrm{sn}(u_{\beta_f}) \mathrm{dn}(u_{\alpha_f}) \\
&= -ie^{-i \frac{\beta_f}{2}} \mathrm{sn}(u_{\beta_f}) \mathrm{dn}(u_{\alpha_f}) \Lambda(u_{\alpha_f}, u_{\beta_f}),
\end{align*}$$

using that $\mathrm{cn}(u-K) = k' \mathrm{sd}(u-K)$, $\mathrm{nd}(u-K) = (k')^{-1} \mathrm{dn}(u)$ in the penultimate line. Returning to the definition of coefficients of the matrix $T$, see (4.4), we have

$$t(u)_{w,\bar{v}} = e^{-i \frac{\bar{\beta}_i}{2}} \mathrm{cn}(u_{\beta_i})$$

$$t(u)_{w,\bar{f}} = -ie^{-i \frac{\bar{\beta}_i}{2}} \mathrm{cd}(u_{\beta_i+2K}) = -ie^{-i \frac{\bar{\beta}_i}{2}} \mathrm{sn}(u_{\beta_i}),$$
<figure><img src="image_22.png" /><figcaption>Figure 20: Notation for coefficients of Corollary 4.12 and 4.14.</figcaption></figure>

**Corollary 4.12.** For every $u \in C$, the inverse operators $(K^Q)^{-1}$, $K(u)^{-1}$, and the matrices $S(u)$, $T(u)$ satisfy the following identity.

* Matrix form.

$$(K^Q)^{-1} S(u) = T(u)K(u)^{-1}.$$

* Coefficients. For every $w, \underline{w}, \underline{f}$, and every $w, b, b'$ as in the notation above, we have:

$$(K^Q)^{-1}_{w,b} s(u)_{b,w} + (K^Q)^{-1}_{w,b'} s(u)_{b',w} = t(u)_{w,\underline{v}} K(u)_{\underline{v},w} + t(u)_{w,\underline{f}} K(u)_{\underline{f},w}.$$

Or equivalently,

$$(K^Q)^{-1}_{w,b} \mathrm{cn}(u_{\beta_f}) - i(K^Q)^{-1}_{w,b'} \mathrm{sn}(u_{\beta_f}) \mathrm{dn}(u_{\alpha_f}) = \frac{e^{i \frac{\beta_f - \beta_i}{2}}}{\Lambda(u_{\alpha_f}, u_{\beta_f})} \left[ \mathrm{cn}(u_{\beta_i}) K(u)_{\underline{v},w}^{-1} - i \mathrm{sn}(u_{\beta_i}) K(u)_{\underline{f},w}^{-1} \right], \quad (4.16)$$

$$\text{where } \Lambda(u_{\alpha}, u_{\beta}) = [\mathrm{sn}\,\theta \, \mathrm{cn}\,\theta \, \mathrm{nd}(u_{\alpha}) \, \mathrm{nd}(u_{\beta})]^{\frac{1}{2}}, \text{ and } \theta = u_{\alpha} - u_{\beta}.$$

*Proof.* To obtain the matrix form, let us consider the matrix $(K^Q)^{-1}S(u) - T(u)K(u)^{-1}$ and left multiply it by $K^Q$. Since coefficients of the inverses tend to 0 at infinity and the other matrices involved only have finitely many non-zero terms per row and column, the matrix product is associative and we obtain

$$K^Q[(K^Q)^{-1}S(u) - T(u)K(u)^{-1}] = S(u) - K^Q T(u)K(u)^{-1}. \quad (4.17)$$

Now, right multiplying Equation (4.6) of Theorem (4.2) by $K(u)^{-1}$ and again using associativity of the matrix product we have that Equation (4.17) is equal to zero. As a consequence, each column of $(K^Q)^{-1}S(u) - T(u)K(u)^{-1}$ is a function in the kernel of $K^Q$, which moreover tends to zero at infinity since it is the case for coefficients of $(K^Q)^{-1}$ and $K(u)^{-1}$. Recall that by [45] in the case $k=0$ and by [12] in the case $k \neq 0$, $K^Q$ has a unique inverse tending to zero at infinity. This implies that 0 is the only function tending to zero at infinity in the kernel of $K^Q$, yielding that $(K^Q)^{-1}S(u) - T(u)K(u)^{-1} = 0$.

For coefficients, we return to the definition of the matrix $S(u)$, see (4.2), and obtain

$$\begin{align*}
s(u)_{b,w} &= e^{-i \frac{\beta_f}{2}} [\mathrm{sn}(\theta_f) \mathrm{cn}(\theta_f) \mathrm{nd}(u_{\alpha_f}) \mathrm{nd}(u_{\beta_f})]^{\frac{1}{2}} \mathrm{cn}(u_{\beta_f}) = e^{-i \frac{\beta_f}{2}} \mathrm{cn}(u_{\beta_f}) \Lambda(u_{\alpha_f}, u_{\beta_f}) \\
s(u)_{b',w} &= e^{-i \frac{\beta_f + \pi}{2}} [\mathrm{sn}(\theta_f) \mathrm{cn}(\theta_f) \mathrm{nd}(u_{\alpha_f+2K}) \mathrm{nd}(u_{\beta_f+2K})]^{\frac{1}{2}} \mathrm{cn}(u_{\beta_f+2K}) \\
&= -ie^{-i \frac{\beta_f}{2}} [\mathrm{sn}(\theta_f) \mathrm{cn}(\theta_f) \mathrm{nd}(u_{\alpha_f}) \mathrm{nd}(u_{\beta_f})]^{\frac{1}{2}} \mathrm{sn}(u_{\beta_f}) \mathrm{dn}(u_{\alpha_f}) \\
&= -ie^{-i \frac{\beta_f}{2}} \mathrm{sn}(u_{\beta_f}) \mathrm{dn}(u_{\alpha_f}) \Lambda(u_{\alpha_f}, u_{\beta_f}),
\end{align*}$$

using that $\mathrm{cn}(u-K) = k' \mathrm{sd}(u-K)$, $\mathrm{nd}(u-K) = (k')^{-1} \mathrm{dn}(u)$ in the penultimate line. Returning to the definition of coefficients of the matrix $T$, see (4.4), we have

$$\begin{align*}
t(u)_{w,\underline{v}} &= e^{-i \frac{\beta_i}{2}} \mathrm{cn}(u_{\beta_i}) \\
t(u)_{w,\underline{f}} &= -ie^{-i \frac{\beta_i}{2}} \mathrm{cd}(u_{\beta_i+2K}) = -ie^{-i \frac{\beta_i}{2}} \mathrm{sn}(u_{\beta_i}),
\end{align*}$$
using that cd(u - K) = sn(u) in the last equality. This ends the proof of the formula for coefficients. □

**Remark 4.13.** Note that in the quadrangle bwb'w' on the right, the only role of the white vertex w is to fix the notation of the angles $\bar{\alpha}_f, \bar{\beta}_f$. Taking the white vertex w' instead would thus give an identity equivalent to (4.16). As a consequence, this trick cannot be used to extract coefficients of $(K^Q)^{-1}$, which is what we do in Section 4.3.2 using another technique.

*Finite case.* We restrict to $u \in C''$ so that the $Z^u$-Dirac operator $K^\partial(u)$ is invertible.

**Notation for coefficients of Corollary 4.14** As in the notation for coefficients of Corollary 4.12, we add a subscript i/f for rhombus vectors and half-angles of initial/final vertices. If $w \in \{w^c \in \mathbb{R}^\partial\}$, or $w \in \{w^\ell, w^r \in \mathbb{R}^\partial\}$, we thus have the notation of Figure 21.

<figure><img src="image_24.png" /><figcaption>Figure 21: Notation for boundary coefficients of Corollary 4.14.</figcaption></figure>

As a consequence of Theorem 4.2 (finite case), we obtain the following.

**Corollary 4.14.** For every $u \in C''$, the inverse operators $(K^Q)^{-1}, (K^Q)^{-1}, K^\partial(u)^{-1}$ and the matrices $S(u), T(u)$ satisfy the following identity.

* Matrix form.

$$(K^Q)^{-1} S(u) = T(u) K^\partial(u)^{-1} \Leftrightarrow (D^{Q,w})^{-1} (K^Q)^{-1} (D^{Q,b})^{-1} S(u) = T(u) K^\partial(u)^{-1}.$$

* Coefficients. We have two cases to consider.

1. For every $w, v, f$; for every $w, b, b'$ such that $w \notin \{w^\ell, w^r \in \mathbb{R}^\partial\}$, using the notation of Figure 20 and 21 (left), we have

$$(K^Q)^{-1}_{w,b} \operatorname{cn}(u_{\beta_f}) - i(K^Q)^{-1}_{w,b'} \operatorname{sn}(u_{\beta_f}) \operatorname{dn}(u_{\alpha_f}) = \frac{e^{i \frac{\beta_f}{2}} \operatorname{sn}(\theta_i^\partial)^{\mathbb{I}\{w \in \{w^c \in \mathbb{R}^\partial\}\}}}{\Lambda(u_{\alpha_f}, u_{\beta_f})} \left( \mathbb{I}_{\{w \neq w^{c,r}\}} t(u)_{w,v} K^\partial(u)_{v,w}^{-1} + t(u)_{w,f} K^\partial(u)_{f,w}^{-1} \right).$$

2. For every $w, v, f$, for every $w$ such that $w \in \{w^\ell, w^r\}$ for one of the rhombus pairs of $\mathbb{R}^\partial$, then using the notation of Figures 20 and 21, we have

$$(K^Q)^{-1}_{w,b^\ell} \operatorname{cn}(u_{\alpha_f^\ell}) = \frac{e^{i \frac{\alpha_f^\ell}{2}} \operatorname{sn}(\theta_i^\partial)^{-1} \operatorname{sn}(\theta_i^\partial)^{\mathbb{I}\{w \in \{w^c \in \mathbb{R}^\partial\}\}}}{\Lambda(u_{\alpha_f^\ell}, u_{\beta_f^\ell})} \left( \mathbb{I}_{\{w \neq w^{c,r}\}} t(u)_{w,v} K^\partial(u)_{v,w}^{-1} + t(u)_{w,f} K^\partial(u)_{f,w}^{-1} \right)$$

$$(K^Q)^{-1}_{w,b^r} \operatorname{cn}(u_{\beta_f^r}) = \frac{e^{i \frac{\beta_f^r}{2}} \operatorname{sn}(\theta_i^\partial)^{-1} \operatorname{sn}(\theta_i^\partial)^{\mathbb{I}\{w \in \{w^c \in \mathbb{R}^\partial\}\}}}{\Lambda(u_{\alpha_f^r}, u_{\beta_f^r})} \left( \mathbb{I}_{\{w \neq w^{c,r}\}} t(u)_{w,v} K^\partial(u)_{v,w^r}^{-1} + t(u)_{w,f} K^\partial(u)_{f,w^r}^{-1} \right)$$

where coefficients of $t(u)$ are given by Equation (4.4) or (4.5) when $w = w^c$ for some boundary rhombus pair.
using that cd(u - K) = sn(u) in the last equality. This ends the proof of the formula for coefficients. □

**Remark 4.13.** Note that in the quadrangle bwb'w' on the right, the only role of the white vertex w is to fix the notation of the angles $\bar{\alpha}_f, \bar{\beta}_f$. Taking the white vertex w' instead would thus give an identity equivalent to (4.16). As a consequence, this trick cannot be used to extract coefficients of $(K^Q)^{-1}$, which is what we do in Section 4.3.2 using another technique.

*Finite case.* We restrict to $u \in C''$ so that the $Z^u$-Dirac operator $K^\partial(u)$ is invertible.

**Notation for coefficients of Corollary 4.14** As in the notation for coefficients of Corollary 4.12, we add a subscript i/f for rhombus vectors and half-angles of initial/final vertices. If $w \in \{w^c \in \mathbb{R}^\partial\}$, or $w \in \{w^\ell, w^r \in \mathbb{R}^\partial\}$, we thus have the notation of Figure 21.

<figure><img src="image_23.png" /><figcaption>Figure 21: Notation for boundary coefficients of Corollary 4.14.</figcaption></figure>

As a consequence of Theorem 4.2 (finite case), we obtain the following.

**Corollary 4.14.** For every $u \in C''$, the inverse operators $(K^Q)^{-1}$, $(K^Q)^{-1}$, $K^\partial(u)^{-1}$ and the matrices $S(u)$, $T(u)$ satisfy the following identity.

* Matrix form.

$$(K^Q)^{-1} S(u) = T(u) K^\partial(u)^{-1} \Leftrightarrow (D^{Q,w})^{-1} (K^Q)^{-1} (D^{Q,b})^{-1} S(u) = T(u) K^\partial(u)^{-1}.$$

* Coefficients. We have two cases to consider.

1. For every $w, v, f$; for every $w, b, b'$ such that $w \notin \{w^\ell, w^r \in \mathbb{R}^\partial\}$, using the notation of Figure 20 and 21 (left), we have

$$(K^Q)_{w,b}^{-1} \mathrm{cn}(u_{\beta_f}) - i(K^Q)_{w,b'}^{-1} \mathrm{sn}(u_{\beta_f}) \mathrm{dn}(u_{\alpha_f}) = \frac{e^{i \frac{\beta_f}{2}} \mathrm{sn}(\theta_i^\partial)^{\mathrm{I}\{w \in \{w^c \in \mathbb{R}^\partial\}\}}}{\Lambda(u_{\alpha_f}, u_{\beta_f})} \left( \mathrm{I}_{\{w \neq w^{c,r}\}} t(u)_{w,v} K^\partial(u)_{v,w}^{-1} + t(u)_{w,f} K^\partial(u)_{f,w}^{-1} \right).$$

2. For every $w, v, f$, for every $w$ such that $w \in \{w^\ell, w^r\}$ for one of the rhombus pairs of $\mathbb{R}^\partial$, then using the notation of Figures 20 and 21, we have

$$(K^Q)_{w,b}^{-1} \mathrm{cn}(u_{\alpha_f}) = \frac{e^{i \frac{\alpha_f}{2}} \mathrm{sn}(\theta_i^\partial)^{-1} \mathrm{sn}(\theta_i^\partial)^{\mathrm{I}\{w \in \{w^c \in \mathbb{R}^\partial\}\}}}{\Lambda(u_{\alpha_f}, u_{\beta_f})} \left( \mathrm{I}_{\{w \neq w^{c,r}\}} t(u)_{w,v} K^\partial(u)_{v,w}^{-1} + t(u)_{w,f} K^\partial(u)_{f,w}^{-1} \right)$$

$$(K^Q)_{w,b}^{-1} \mathrm{cn}(u_{\beta_f}) = \frac{e^{i \frac{\beta_f}{2}} \mathrm{sn}(\theta_i^\partial)^{-1} \mathrm{sn}(\theta_i^\partial)^{\mathrm{I}\{w \in \{w^c \in \mathbb{R}^\partial\}\}}}{\Lambda(u_{\alpha_f}, u_{\beta_f})} \left( \mathrm{I}_{\{w \neq w^{c,r}\}} t(u)_{w,v} K^\partial(u)_{v,w}^{-1} + t(u)_{w,f} K^\partial(u)_{f,w}^{-1} \right)$$

where coefficients of $t(u)$ are given by Equation (4.4) or (4.5) when $w = w^c$ for some boundary rhombus pair.
**Example 4.15.** Let us compute the explicit values of Corollary 4.12 and of Point 1. of Corollary 4.14 in the case where $u = \mathbf{u}^f := \frac{\alpha_f + \beta_f}{2} + K$. This will be used again in Sections 4.3.2 and 5.3. In the finite case, we also need to evaluate it at $u = \mathbf{v}^f := \frac{\alpha_f + \beta_f}{2} - K$. Returning to Relations (3.20) of Example 3.18, we have,

$$ \mathrm{sn}(\mathbf{u}_{\beta_f}^f) \mathrm{dn}(\mathbf{u}_{\alpha_f}^f) = \mathrm{sn}(K - \mathbf{u}_{\alpha_f}^f) \mathrm{dn}(\mathbf{u}_{\alpha_f}^f) = \mathrm{cn}(\mathbf{u}_{\alpha_f}^f) $$

$$ \mathrm{sn}(\mathbf{v}_{\beta_f}^f) \mathrm{dn}(\mathbf{v}_{\alpha_f}^f) = \mathrm{sn}(-K - \mathbf{v}_{\alpha_f}^f) \mathrm{dn}(\mathbf{v}_{\alpha_f}^f) = -\mathrm{cn}(\mathbf{v}_{\alpha_f}^f) $$

$$ \Lambda(\mathbf{u}_{\alpha_f}^f, \mathbf{u}_{\beta_f}^f) = [(k')^{-1} \mathrm{sn}(\theta_f) \mathrm{cn}(\theta_f)]^{\frac{1}{2}} = \Lambda(\mathbf{v}_{\alpha_f}^f, \mathbf{v}_{\beta_f}^f) $$

$$ \mathbf{u}_{\beta_f}^f = \frac{K - \theta_f}{2}, \quad \mathbf{u}_{\alpha_f}^f = \frac{K + \theta_f}{2}, \quad \mathbf{v}_{\beta_f}^f = -\mathbf{u}_{\alpha_f}^f, \quad \mathbf{v}_{\alpha_f}^f = -\mathbf{u}_{\beta_f}^f. $$

Then,

* Infinite case. For every $w, v, f$; for every $w, b, b'$, with the notation of Figure 20, we have

$$ (K^Q)_{w,b}^{-1} \mathrm{cn}\left(\frac{K-\theta_f}{2}\right) - i(K^Q)_{w,b'}^{-1} \mathrm{sn}\left(\frac{K+\theta_f}{2}\right) = \\ = \frac{e^{i \frac{\beta_f - \beta_i}{2}} (k')^{\frac{1}{2}}}{[\mathrm{cn}(\theta_f) \mathrm{sn}(\theta_f)]^{\frac{1}{2}}} \left( \mathrm{cn}(\mathbf{u}_{\beta_i}^f) \mathrm{K}(\mathbf{u}^f)_{\underline{v},w}^{-1} - i \mathrm{sn}(\mathbf{u}_{\beta_i}^f) \mathrm{K}(\mathbf{u}^f)_{\underline{f},w}^{-1} \right). $$

* Finite case. Point 1.

For every $w, v, f$; for every $w, b, b'$ such that $w \notin \{w^l, w^r \in \mathbb{R}^\partial\}$, with the notation of Figures 20 and 21 (left), we have

$$ (K^Q)_{w,b}^{-1} \mathrm{cn}\left(\frac{K-\theta_f}{2}\right) - i(K^Q)_{w,b'}^{-1} \mathrm{sn}\left(\frac{K+\theta_f}{2}\right) = \\ = \frac{e^{i \frac{\beta_f - \beta_i}{2}} (k')^{\frac{1}{2}} \mathrm{sn}(\theta_f^l) \mathrm{cn}\left(\frac{w \in \{w^c \in \mathbb{R}^\partial\}}{[\mathrm{cn}(\theta_f) \mathrm{sn}(\theta_f)]^{\frac{1}{2}}}\right)}{[\mathrm{cn}(\theta_f) \mathrm{sn}(\theta_f)]^{\frac{1}{2}}} \left( \mathrm{I}_{\{w \neq w^c, r\}} t(\mathbf{u}^f)_{w,v} K^\partial(\mathbf{u}^f)_{\underline{v},w}^{-1} + t(\mathbf{u}^f)_{w,f} K^\partial(\mathbf{u}^f)_{\underline{f},w}^{-1} \right) $$

$$ (K^Q)_{w,b}^{-1} \mathrm{cn}\left(\frac{K+\theta_f}{2}\right) + i(K^Q)_{w,b'}^{-1} \mathrm{sn}\left(\frac{K-\theta_f}{2}\right) = \\ = \frac{e^{i \frac{\beta_f - \beta_i}{2}} (k')^{\frac{1}{2}} \mathrm{sn}(\theta_f^l) \mathrm{cn}\left(\frac{w \in \{w^c \in \mathbb{R}^\partial\}}{[\mathrm{cn}(\theta_f) \mathrm{sn}(\theta_f)]^{\frac{1}{2}}}\right)}{[\mathrm{cn}(\theta_f) \mathrm{sn}(\theta_f)]^{\frac{1}{2}}} \left( \mathrm{I}_{\{w \neq w^c, r\}} t(\mathbf{v}^f)_{w,v} K^\partial(\mathbf{v}^f)_{\underline{v},w}^{-1} + t(\mathbf{v}^f)_{w,f} K^\partial(\mathbf{v}^f)_{\underline{f},w}^{-1} \right). $$

### 4.3.2 The dimer model on an isoradial graph $G^Q$ and the $Z^u$-Dirac operator

**Infinite case.** In the paper [12], we prove an explicit local expression for an inverse $(K^Q)^{-1}$; as a byproduct we obtain a local formula for a Gibbs measure $P_{\mathrm{dimer}}^Q$ on $(M(G^Q), \mathcal{F})$, involving the operators $K^Q$ and $(K^Q)^{-1}$; we refer to the paper [12] for the explicit formula for $(K^Q)^{-1}$ and to Section 2.2.2 for the explicit formula of the Gibbs measure $P_{\mathrm{dimer}}^Q$. Recall also that the dimer model on $G^Q$ corresponds to a free-fermion, zero-field 6-vertex model [62, 76, 32].

Using Corollary 4.12 and Corollary 3.15, we provide an alternative direct way of finding the local formula for the inverse operator $(K^Q)^{-1}$ of [12], where the locality property is directly seen as inherited from that of the Z-massive and dual massive Green functions: we first express coefficients of $(K^Q)^{-1}$ using the inverse $Z^u$-Dirac operator for appropriate values of $u$, and then express the latter using the Z-massive and dual massive Green function of [11]. Note that it is not immediate to see equality between the formulas of [12] and Corollary 4.16; it probably requires to use elliptic trigonometric identities. The approach we propose here also extends to the finite case.

**Notation for Corollary 4.16** Let $w$ be a white vertex of $G^Q$ and $b$ be a black one. Consider $\underline{v}, \underline{f}, e^{i\beta_i}$, and $w, b, w, e^{i\bar{\alpha}_f}, e^{i\bar{\beta}_f}$ as in Figure 20. The quadrangle of the vertex
**Example 4.15.** Let us compute the explicit values of Corollary 4.12 and of Point 1. of Corollary 4.14 in the case where $u = \mathbf{u}^f := \frac{\alpha_f + \beta_f}{2} + K$. This will be used again in Sections 4.3.2 and 5.3. In the finite case, we also need to evaluate it at $u = \mathbf{v}^f := \frac{\alpha_f + \beta_f}{2} - K$. Returning to Relations (3.20) of Example 3.18, we have,

$$ \mathrm{sn}(\mathbf{u}_{\beta_f}^f) \mathrm{dn}(\mathbf{u}_{\alpha_f}^f) = \mathrm{sn}(K - \mathbf{u}_{\alpha_f}^f) \mathrm{dn}(\mathbf{u}_{\alpha_f}^f) = \mathrm{cn}(\mathbf{u}_{\alpha_f}^f) $$

$$ \mathrm{sn}(\mathbf{v}_{\beta_f}^f) \mathrm{dn}(\mathbf{v}_{\alpha_f}^f) = \mathrm{sn}(-K - \mathbf{v}_{\alpha_f}^f) \mathrm{dn}(\mathbf{v}_{\alpha_f}^f) = -\mathrm{cn}(\mathbf{v}_{\alpha_f}^f) $$

$$ \Lambda(\mathbf{u}_{\alpha_f}^f, \mathbf{u}_{\beta_f}^f) = [(k')^{-1} \mathrm{sn}(\theta_f) \mathrm{cn}(\theta_f)]^{\frac{1}{2}} = \Lambda(\mathbf{v}_{\alpha_f}^f, \mathbf{v}_{\beta_f}^f) $$

$$ \mathbf{u}_{\beta_f}^f = \frac{K - \theta_f}{2}, \quad \mathbf{u}_{\alpha_f}^f = \frac{K + \theta_f}{2}, \quad \mathbf{v}_{\beta_f}^f = -\mathbf{u}_{\alpha_f}^f, \quad \mathbf{v}_{\alpha_f}^f = -\mathbf{u}_{\beta_f}^f. $$

Then,

* Infinite case. For every w, v, f; for every w, b, b', with the notation of Figure 20, we have

$$ (K^Q)_{w,b}^{-1} \mathrm{cn}\left(\frac{K-\theta_f}{2}\right) - i(K^Q)_{w,b'}^{-1} \mathrm{sn}\left(\frac{K+\theta_f}{2}\right) = \\ = \frac{e^{i\frac{\beta_f-\beta_i}{2}}(k')^{\frac{1}{2}}}{[\mathrm{cn}(\theta_f) \mathrm{sn}(\theta_f)]^{\frac{1}{2}}} \left(\mathrm{cn}(\mathbf{u}_{\beta_i}^f)K(\mathbf{u}^f)_{\underline{v},w}^{-1} - i \mathrm{sn}(\mathbf{u}_{\beta_i}^f)K(\mathbf{u}^f)_{\underline{f},w}^{-1}\right). $$

* Finite case. Point 1.

For every w, v, f; for every w, b, b' such that $w \notin \{w^\ell, w^r \in \mathbb{R}^\partial\}$, with the notation of Figures 20 and 21 (left), we have

$$ (K^Q)_{w,b}^{-1} \mathrm{cn}\left(\frac{K-\theta_f}{2}\right) - i(K^Q)_{w,b'}^{-1} \mathrm{sn}\left(\frac{K+\theta_f}{2}\right) = \\ = \frac{e^{i\frac{\beta_f-\beta_i}{2}}(k')^{\frac{1}{2}} \mathrm{sn}(\theta_f^\partial) \mathbb{1}_{\{w \in \{w^c \in \mathbb{R}^\partial\}\}}}{[\mathrm{cn}(\theta_f) \mathrm{sn}(\theta_f)]^{\frac{1}{2}}} \left(\mathbb{I}_{\{w \neq w^c, r\}} t(\mathbf{u}^f)_{w,v} K^\partial(\mathbf{u}^f)_{\underline{v},w}^{-1} + t(\mathbf{u}^f)_{w,f} K^\partial(\mathbf{u}^f)_{\underline{f},w}^{-1}\right) $$

$$ (K^Q)_{w,b}^{-1} \mathrm{cn}\left(\frac{K+\theta_f}{2}\right) + i(K^Q)_{w,b'}^{-1} \mathrm{sn}\left(\frac{K-\theta_f}{2}\right) = \\ = \frac{e^{i\frac{\beta_f-\beta_i}{2}}(k')^{\frac{1}{2}} \mathrm{sn}(\theta_f^\partial) \mathbb{1}_{\{w \in \{w^c \in \mathbb{R}^\partial\}\}}}{[\mathrm{cn}(\theta_f) \mathrm{sn}(\theta_f)]^{\frac{1}{2}}} \left(\mathbb{I}_{\{w \neq w^c, r\}} t(\mathbf{v}^f)_{w,v} K^\partial(\mathbf{v}^f)_{\underline{v},w}^{-1} + t(\mathbf{v}^f)_{w,f} K^\partial(\mathbf{v}^f)_{\underline{f},w}^{-1}\right). $$

### 4.3.2 The dimer model on an isoradial graph $G^Q$ and the $Z^u$-Dirac operator

**Infinite case.** In the paper [12], we prove an explicit local expression for an inverse $(K^Q)^{-1}$; as a byproduct we obtain a local formula for a Gibbs measure $P_{\mathrm{dimer}}^Q$ on $(M(G^Q), \mathcal{F})$, involving the operators $K^Q$ and $(K^Q)^{-1}$; we refer to the paper [12] for the explicit formula for $(K^Q)^{-1}$ and to Section 2.2.2 for the explicit formula of the Gibbs measure $P_{\mathrm{dimer}}^Q$. Recall also that the dimer model on $G^Q$ corresponds to a free-fermion, zero-field 6-vertex model [62, 76, 32].

Using Corollary 4.12 and Corollary 3.15, we provide an alternative direct way of finding the local formula for the inverse operator $(K^Q)^{-1}$ of [12], where the locality property is directly seen as inherited from that of the Z-massive and dual massive Green functions: we first express coefficients of $(K^Q)^{-1}$ using the inverse $Z^u$-Dirac operator for appropriate values of u, and then express the latter using the Z-massive and dual massive Green function of [11]. Note that it is not immediate to see equality between the formulas of [12] and Corollary 4.16; it probably requires to use elliptic trigonometric identities. The approach we propose here also extends to the finite case.

**Notation for Corollary 4.16** Let w be a white vertex of $G^Q$ and b be a black one. Consider v, f, $e^{i\beta_i}$, and w, b, w, $e^{i\bar{\alpha}_f}$, $e^{i\bar{\beta}_f}$ as in Figure 20. The quadrangle of the vertex
b corresponds to a rhombus $v_1, f_1, v_2, f_2$ of the diamond graph $G^\diamond$, where vertices are labeled so that the edge $(v_1, v_2)$ of $G$ is parallel to the edge $(b, w)$ of $G^Q$, and $f_1$ is on the right of $(v_1, v_2)$, see Figure 22.

<figure><img src="image_25.png" /><figcaption>Figure 22: Notation for Corollary 4.16.</figcaption></figure>

**Corollary 4.16.** For every white vertex w and every black vertex b of $G^Q$, using the notation of Figure 22, we have

$$ (K^Q)_{w,b}^{-1} = \frac{e^{\frac{i}{2}(\beta_f - \beta_i)}}{[\operatorname{cn}(\theta_f) \operatorname{sn}(\theta_f) \operatorname{nd}(\theta_f)]^{\frac{1}{2}}} \left( \operatorname{cn}\left(\frac{\beta_f - \beta_i}{2}\right) \mathcal{K}(\beta_f) \frac{1}{v,w} - i \operatorname{sn}\left(\frac{\beta_f - \beta_i}{2}\right) \mathcal{K}(\beta_f) \frac{1}{f,w} \right) \\ (K^Q)_{w,b}^{-1} = e^{-i \frac{\beta_i + \beta_f}{2}} (k')^{-1} \left( \frac{\operatorname{cn}\left(\frac{\beta_f - \beta_i}{2}\right)}{\operatorname{cn}(\theta_f)} [\operatorname{dn}(\theta_f) G_{v,v_2}^m - k' G_{v,v_1}^m] - \frac{\operatorname{sn}\left(\frac{\beta_f - \beta_i}{2}\right)}{\operatorname{sn}(\theta_f)} [\operatorname{dn}(\theta_f) G_{f,f_2}^{m,*} - G_{f,f_1}^{m,*}] \right). $$

*Proof.* We set $u = \beta_f$ in Corollary 4.12. Then, $u_{\alpha_f} = \theta_f$, $u_{\beta_f} = 0$, and we have $\operatorname{cn}(u_{\beta_f}) = 1$, $\operatorname{sn}(u_{\beta_f}) = 0$, $\operatorname{dn}(u_{\beta_f}) = 1$. This gives $\Lambda(u_{\alpha_f}, u_{\beta_f})|_{\beta_f} = [\operatorname{sn} \theta_f \operatorname{cn} \theta_f \operatorname{nd}(\theta_f)]^{\frac{1}{2}}$, thus explaining the first equality of the Corollary.

We now set $u = \beta_f$ in Corollary 3.15. Using that $\operatorname{dn}(-u) = \operatorname{dn}(u)$, $\operatorname{dn}(K \pm u) = k' \operatorname{nd}(u)$, $\operatorname{dn}(K) = k'$, $\operatorname{sc}(\theta_f^*) = (k')^{-1} \operatorname{cs}(\theta_f)$, we obtain:

$$ K(\beta_f)_{v,w}^{-1} = e^{-i \frac{\alpha_f + \beta_f}{2}} (k')^{-1} \operatorname{sc}(\theta_f)^{\frac{1}{2}} \left( \operatorname{dn}(\theta_f)^{\frac{1}{2}} G_{v,v_2}^m - [(k')^2 \operatorname{nd}(\theta_f)]^{\frac{1}{2}} G_{v,v_1}^m \right) \\ K(\beta_f)_{f,w}^{-1} = -ie^{-i \frac{\alpha_f + \beta_f}{2}} (k')^{-1} \operatorname{sc}(\theta_f^*)^{\frac{1}{2}} \left( [\operatorname{dn}(K) \operatorname{dn}(\theta_f)]^{\frac{1}{2}} G_{f,f_2}^{m,*} - [\operatorname{dn}(0) \operatorname{dn}(\theta_f - K)]^{\frac{1}{2}} G_{f,f_1}^{m,*} \right) \\ = -ie^{-i \frac{\alpha_f + \beta_f}{2}} (k')^{-1} \operatorname{cs}(\theta_f)^{\frac{1}{2}} \left( \operatorname{dn}(\theta_f)^{\frac{1}{2}} G_{f,f_2}^{m,*} - \operatorname{nd}(\theta_f)^{\frac{1}{2}} G_{f,f_1}^{m,*} \right). $$

Plugging this into the first equality of the corollary yields the second and concludes the proof. □

**Example 4.17.** As an example of application we express the probability of single edges occurring in dimer configurations of $G^Q$ chosen with respect to the measure $P_{\text{dimer}}^Q$, as a function of single edge probabilities of the dimer model on $G^D$ with Gibbs measure $P_{\text{dimer}}^{D,\beta}$. We then use Example 3.21 evaluated at $u = \beta$ to obtain explicit expressions using the function $H$; details of computations are given in Appendix B.2. Using the notation of Figure 16, we have

$$ P_{\text{dimer}}^{Q}(bw_1) = P_{\text{dimer}}^{D,\beta}(wv_2) = H(2\theta) $$

$$ P_{\text{dimer}}^{Q}(bw_2) = P_{\text{dimer}}^{D,\beta}(wf_2) = \frac{1}{2} - H(2\theta) $$

$$ P_{\text{dimer}}^{Q}(bw_3) = \frac{1}{2}. $$
b corresponds to a rhombus $v_1, f_1, v_2, f_2$ of the diamond graph $G^\diamond$, where vertices are labeled so that the edge $(v_1, v_2)$ of $G$ is parallel to the edge $(b, w)$ of $G^Q$, and $f_1$ is on the right of $(v_1, v_2)$, see Figure 22.

<figure><img src="image_24.png" /><figcaption>Figure 22: Notation for Corollary 4.16.</figcaption></figure>

**Corollary 4.16.** For every white vertex w and every black vertex b of $G^Q$, using the notation of Figure 22, we have

$$ (K^Q)_{w,b}^{-1} = \frac{e^{\frac{i}{2}(\beta_f - \beta_i)}}{[\operatorname{cn}(\theta_f) \operatorname{sn}(\theta_f) \operatorname{nd}(\theta_f)]^{\frac{1}{2}}} \left( \operatorname{cn}\left(\frac{\beta_f - \beta_i}{2}\right) \mathcal{K}(\beta_f) \frac{1}{v,w} - i \operatorname{sn}\left(\frac{\beta_f - \beta_i}{2}\right) \mathcal{K}(\beta_f) \frac{1}{f,w} \right) \\ (K^Q)_{w,b}^{-1} = e^{-i \frac{\beta_i + \beta_f}{2}} (k')^{-1} \left( \frac{\operatorname{cn}\left(\frac{\beta_f - \beta_i}{2}\right)}{\operatorname{cn}(\theta_f)} [\operatorname{dn}(\theta_f) G_{v,v_2}^m - k' G_{v,v_1}^m] - \frac{\operatorname{sn}\left(\frac{\beta_f - \beta_i}{2}\right)}{\operatorname{sn}(\theta_f)} [\operatorname{dn}(\theta_f) G_{f,f_2}^{m,*} - G_{f,f_1}^{m,*}] \right). $$

*Proof.* We set $u = \beta_f$ in Corollary 4.12. Then, $u_{\alpha_f} = \theta_f$, $u_{\beta_f} = 0$, and we have $\operatorname{cn}(u_{\beta_f}) = 1$, $\operatorname{sn}(u_{\beta_f}) = 0$, $\operatorname{dn}(u_{\beta_f}) = 1$. This gives $\Lambda(u_{\alpha_f}, u_{\beta_f})|_{\beta_f} = [\operatorname{sn} \theta_f \operatorname{cn} \theta_f \operatorname{nd}(\theta_f)]^{\frac{1}{2}}$, thus explaining the first equality of the Corollary.

We now set $u = \beta_f$ in Corollary 3.15. Using that $\operatorname{dn}(-u) = \operatorname{dn}(u)$, $\operatorname{dn}(K \pm u) = k' \operatorname{nd}(u)$, $\operatorname{dn}(K) = k'$, $\operatorname{sc}(\theta_f^*) = (k')^{-1} \operatorname{cs}(\theta_f)$, we obtain:

$$ K(\beta_f)_{v,w}^{-1} = e^{-i \frac{\alpha_f + \beta_f}{2}} (k')^{-1} \operatorname{sc}(\theta_f)^{\frac{1}{2}} \left( \operatorname{dn}(\theta_f)^{\frac{1}{2}} G_{v,v_2}^m - [(k')^2 \operatorname{nd}(\theta_f)]^{\frac{1}{2}} G_{v,v_1}^m \right) \\ K(\beta_f)_{f,w}^{-1} = -i e^{-i \frac{\alpha_f + \beta_f}{2}} (k')^{-1} \operatorname{sc}(\theta_f^*)^{\frac{1}{2}} \left( [\operatorname{dn}(K) \operatorname{dn}(\theta_f)]^{\frac{1}{2}} G_{f,f_2}^{m,*} - [\operatorname{dn}(0) \operatorname{dn}(\theta_f - K)]^{\frac{1}{2}} G_{f,f_1}^{m,*} \right) \\ = -i e^{-i \frac{\alpha_f + \beta_f}{2}} (k')^{-1} \operatorname{cs}(\theta_f)^{\frac{1}{2}} \left( \operatorname{dn}(\theta_f)^{\frac{1}{2}} G_{f,f_2}^{m,*} - \operatorname{nd}(\theta_f)^{\frac{1}{2}} G_{f,f_1}^{m,*} \right). $$

Plugging this into the first equality of the corollary yields the second and concludes the proof. □

**Example 4.17.** As an example of application we express the probability of single edges occurring in dimer configurations of $G^Q$ chosen with respect to the measure $P_{\text{dimer}}^Q$, as a function of single edge probabilities of the dimer model on $G^D$ with Gibbs measure $P_{\text{dimer}}^{D,\beta}$. We then use Example 3.21 evaluated at $u = \beta$ to obtain explicit expressions using the function $H$; details of computations are given in Appendix B.2. Using the notation of Figure 16, we have

$$ P_{\text{dimer}}^{Q}(bw_1) = P_{\text{dimer}}^{D,\beta}(wv_2) = H(2\theta) $$

$$ P_{\text{dimer}}^{Q}(bw_2) = P_{\text{dimer}}^{D,\beta}(wf_2) = \frac{1}{2} - H(2\theta) $$

$$ P_{\text{dimer}}^{Q}(bw_3) = \frac{1}{2}. $$
We recover the results of the computations of Theorem 37 of [12] after using addition formulas.

Finite case. An explicit expression for the Boltzmann measure $P_{\text{dimer}}^{\text{Q}}$ as a function of the matrix $K^{\text{Q}}$ and its inverse $(K^{\text{Q}})^{-1}$ is given in Section 2.2. We now express the inverse Kasteleyn matrix $(K^{\text{Q}})^{-1}$ as a function of the $Z^u$-Dirac operator for some choices of $u$. Note that we cannot proceed as in the infinite case using $u = \beta_f$ since then the matrix $K^{\partial}(\beta_f)$ is not invertible.

**Corollary 4.18.**

1. For every white vertex w of $G^{\text{Q}}$ and every black vertex b $\notin \{b^\ell, b^r \in \mathbb{R}^\partial\}$, using the notation of Figures 20 and 21 (left), let $\mathbf{u}^f = \frac{\alpha_f + \beta_f}{2} + K$, $\mathbf{v}^f = \frac{\alpha_f + \beta_f}{2} - K$. Then,

$$ (K^{\text{Q}})_{w,b}^{-1} = e^{i \frac{\beta_f}{2}} (k')^{\frac{1}{2}} \text{sn}(\theta_i^{\partial}) \frac{1+(k')^{-1} \text{dn}(\theta_f)}{2[\text{cn}(\theta_f) \text{sn}(\theta_f)]^{\frac{1}{2}}} \left[ \text{cn}\left(\frac{K-\theta_f}{2}\right) \Gamma(\mathbf{u}^f) + \text{cn}\left(\frac{K+\theta_f}{2}\right) \Gamma(\mathbf{v}^f) \right], $$

where, $\Gamma(u) = \mathbb{I}_{\{w \neq w^{c,r}\}} t(u)_{w,v} K^{\partial}(u)_{v,w}^{-1} + t(u)_{w,f} K^{\partial}(u)_{f,w}^{-1}$, and coefficients of $t(u)$ are given by (4.4) and (4.5).

2. For every white vertex w and every black vertex b $\in \{b^\ell, b^r\}$ for one of the rhombus pairs of $\mathbb{R}^\partial$, using the notation of Figures 20 and 21, let $\mathbf{u}^{f,\ell} = \frac{\alpha_f^\ell + \beta_f^\ell}{2} + K$ and $\mathbf{u}^{f,r} = \frac{\alpha_f^r + \beta_f^r}{2} + K$. Then,

$$ (K^{\text{Q}})_{w,b}^{-1} = \frac{e^{i \frac{\alpha_f^\ell}{2}} (k')^{\frac{1}{2}} \text{sn}(\theta_i^{\partial})^{-1} \text{sn}(\theta_f^{\partial})^{\frac{1}{2}} \{w \in \{w^c \in \mathbb{R}^{\partial}\}\}}{\text{cn}\left(\frac{K+\theta_f}{2}\right) [\text{cn}(\theta_f^{\partial}) \text{sn}(\theta_f^{\partial})]^{\frac{1}{2}}} \times \\ \times \left( \mathbb{I}_{\{w \neq w^{c,r}\}} t(\mathbf{u}^{f,\ell})_{w,v} K^{\partial}(\mathbf{u}^{f,\ell})_{v,w}^{-1} + t(\mathbf{u}^{f,\ell})_{w,f} K^{\partial}(\mathbf{u}^{f,\ell})_{f,w}^{-1} \right) $$

$$ (K^{\text{Q}})_{w,b}^{-1} = \frac{e^{i \frac{\beta_f^\ell}{2}} (k')^{\frac{1}{2}} \text{sn}(\theta_i^{\partial})^{-1} \text{sn}(\theta_f^{\partial})^{\frac{1}{2}} \{w \in \{w^c \in \mathbb{R}^{\partial}\}\}}{\text{cn}\left(\frac{K-\theta_f}{2}\right) [\text{cn}(\theta_f^{\partial}) \text{sn}(\theta_f^{\partial})]^{\frac{1}{2}}} \times \\ \times \left( \mathbb{I}_{\{w \neq w^{c,r}\}} t(\mathbf{u}^{f,r})_{w,v} K^{\partial}(\mathbf{u}^{f,r})_{v,w}^{-1} + t(\mathbf{u}^{f,r})_{w,f} K^{\partial}(\mathbf{u}^{f,r})_{f,w}^{-1} \right). $$

**Remark 4.19.** Example 3.18 expresses coefficients of $K(\mathbf{u}^f)^{-1}$, $K(\mathbf{v}^f)^{-1}$ using the Z-massive and dual massive Green function. Plugging this into Corollary 4.18 allows to write coefficients of $(K^{\text{Q}})^{-1}$ using the Green functions $G^{m,\partial}(\mathbf{u}^f)$, $G^{m,\partial}(\mathbf{v}^f)$, $G^{m,*}$.

*Proof.* Consider $w, v, f$ and $w, b, b'$ such that $w \notin \{w^\ell, w^r \in \mathbb{R}^\partial\}$ as in the statement of Corollary 4.14. Then, from Example 4.15, we have the following linear system of equations:

$$ (K^{\text{Q}})_{w,b}^{-1} \text{cn}\left(\frac{K-\theta_f}{2}\right) - i(K^{\text{Q}})_{w,b'}^{-1} \text{sn}\left(\frac{K+\theta_f}{2}\right) = C \times \Gamma(\mathbf{u}^f) $$

$$ (K^{\text{Q}})_{w,b}^{-1} \text{cn}\left(\frac{K+\theta_f}{2}\right) + i(K^{\text{Q}})_{w,b'}^{-1} \text{sn}\left(\frac{K-\theta_f}{2}\right) = C \times \Gamma(\mathbf{v}^f), $$

$$ \text{where } C = \frac{e^{i \frac{\beta_f^\ell}{2}} (k')^{\frac{1}{2}} \text{sn}(\theta_i^{\partial})^{\frac{1}{2}} \{w \in \{w^c \in \mathbb{R}^{\partial}\}\}}{[\text{cn}(\theta_f) \text{sn}(\theta_f)]^{\frac{1}{2}}}, \Gamma(u) = \mathbb{I}_{\{w \neq w^{c,r}\}} t(u)_{w,v} K^{\partial}(u)_{v,w}^{-1} + t(u)_{w,f} K^{\partial}(u)_{f,w}^{-1}. $$

Solving for $(K^{\text{Q}})_{w,b}^{-1}$ gives,

$$ (K^{\text{Q}})_{w,b}^{-1} = \frac{C}{\text{cn}^2\left(\frac{K-\theta_f}{2}\right) + \text{cn}^2\left(\frac{K+\theta_f}{2}\right)} [\text{cn}\left(\frac{K-\theta_f}{2}\right) \Gamma(\mathbf{u}^f) + \text{cn}\left(\frac{K+\theta_f}{2}\right) \Gamma(\mathbf{v}^f)]. $$

Now, using [54, 2.4.8] and the fact that $\text{dn}(2K - u) = \text{dn}(u)$, $\text{cn}(2K - u) = -\text{cn}(u)$, we have for every $u \in \mathbb{T}(k)$,

$$ \text{cn}^2(u) + \text{cn}^2(K - u) = \frac{\text{cn}(2u) + \text{dn}(2u)}{1 + \text{dn}(2u)} + \frac{\text{cn}(2K - 2u) + \text{dn}(2K - 2u)}{1 + \text{dn}(2K - 2u)} = \frac{2 \text{dn}(2u)}{1 + \text{dn}(2u)} = \frac{2}{1 + \text{nd}(2u)}. $$

$$ \text{This implies that, } \text{cn}^2\left(\frac{K-\theta_f}{2}\right) + \text{cn}^2\left(\frac{K+\theta_f}{2}\right) = \frac{2}{1+\text{nd}\left(2\frac{K-\theta_f}{2}\right)} = \frac{2}{1+(k')^{-1}\text{dn}(\theta_f)}. \quad (4.18) $$

Putting everything together ends the proof of Point 1. Point 2. directly follows from Point 2. of Corollary 4.14. $\square$
We recover the results of the computations of Theorem 37 of [12] after using addition formulas.

Finite case. An explicit expression for the Boltzmann measure P<sub>dimer</sub><sup>Q</sup> as a function of the matrix K<sup>Q</sup> and its inverse (K<sup>Q</sup>)<sup>-1</sup> is given in Section 2.2. We now express the inverse Kasteleyn matrix (K<sup>Q</sup>)<sup>-1</sup> as a function of the Z<sup>u</sup>-Dirac operator for some choices of u. Note that we cannot proceed as in the infinite case using u = β<sub>f</sub> since then the matrix K<sup>Q</sup>(β<sub>f</sub>) is not invertible.

**Corollary 4.18.**

1. For every white vertex w of G<sup>Q</sup> and every black vertex b ∉ {b<sup>l</sup>, b<sup>r</sup> ∈ ℝ<sup>θ</sup>}, using the notation of Figures 20 and 21 (left), let **u**<sup>f</sup> = <sup>α<sub>f</sub>+β<sub>f</sub></sup>&frasl;<sub>2</sub> + K, **v**<sup>f</sup> = <sup>α<sub>f</sub>+β<sub>f</sub></sup>&frasl;<sub>2</sub> − K. Then,

$$ (K^Q)^{-1}_{w,b} = e^{i \frac{\beta_f}{2}} (k')^{\frac{1}{2}} \text{sn}(\theta_i^\beta) \frac{\text{I}_{\{w \in \{w^c \in \mathbb{R}^\theta\}\}} (1+(k')^{-1} \text{dn}(\theta_f))}{2[\text{cn}(\theta_f) \text{sn}(\theta_f)]^{\frac{1}{2}}} \left[ \text{cn}\left(\frac{K-\theta_f}{2}\right) \Gamma(\mathbf{u}^f) + \text{cn}\left(\frac{K+\theta_f}{2}\right) \Gamma(\mathbf{v}^f) \right], $$

where, Γ(u) = I<sub>{w≠w<sup>c,r</sup>}</sub>t(u)<sub>w,v</sub>K<sup>θ</sup>(u)<sub>v,w</sub><sup>-1</sup>+t(u)<sub>w,f</sub>K<sup>θ</sup>(u)<sub>f,w</sub><sup>-1</sup>, and coefficients of t(u) are given by (4.4) and (4.5).

2. For every white vertex w and every black vertex b ∈ {b<sup>l</sup>, b<sup>r</sup>} for one of the rhombus pairs of ℝ<sup>θ</sup>, using the notation of Figures 20 and 21, let **u**<sup>f,l</sup> = <sup>α<sub>f</sub><sup>l</sup>+β<sub>f</sub><sup>l</sup></sup>&frasl;<sub>2</sub> + K and **u**<sup>f,r</sup> = <sup>α<sub>f</sub><sup>r</sup>+β<sub>f</sub><sup>r</sup></sup>&frasl;<sub>2</sub> + K. Then,

$$ (K^Q)^{-1}_{w,b^l} = \frac{e^{i \frac{\alpha_f^l}{2}} (k')^{\frac{1}{2}} \text{sn}(\theta_i^\beta)^{-1} \text{sn}(\theta_i^\beta)^{\frac{1}{|\{w \in \{w^c \in \mathbb{R}^\theta\}\}|}}}{\text{cn}\left(\frac{K+\theta_f^l}{2}\right) [\text{cn}(\theta_i^\beta) \text{sn}(\theta_i^\beta)]^{\frac{1}{2}}} \times \\ \times \left( \text{I}_{\{w \neq w^{c,r}\}} t(\mathbf{u}^{f,l})_{w,v} K^{\theta}(\mathbf{u}^{f,l})_{v,w^l}^{-1} + t(\mathbf{u}^{f,l})_{w,f} K^{\theta}(\mathbf{u}^{f,l})_{f,w^r}^{-1} \right) $$

$$ (K^Q)^{-1}_{w,b^r} = \frac{e^{i \frac{\beta_f^r}{2}} (k')^{\frac{1}{2}} \text{sn}(\theta_i^\beta)^{-1} \text{sn}(\theta_i^\beta)^{\frac{1}{|\{w \in \{w^c \in \mathbb{R}^\theta\}\}|}}}{\text{cn}\left(\frac{K-\theta_f^r}{2}\right) [\text{cn}(\theta_i^\beta) \text{sn}(\theta_i^\beta)]^{\frac{1}{2}}} \times \\ \times \left( \text{I}_{\{w \neq w^{c,r}\}} t(\mathbf{u}^{f,r})_{w,v} K^{\theta}(\mathbf{u}^{f,r})_{v,w^r}^{-1} + t(\mathbf{u}^{f,r})_{w,f} K^{\theta}(\mathbf{u}^{f,r})_{f,w^r}^{-1} \right). $$

**Remark 4.19.** Example 3.18 expresses coefficients of K(**u**<sup>f</sup>)<sup>-1</sup>, K(**v**<sup>f</sup>)<sup>-1</sup> using the Z-massive and dual massive Green function. Plugging this into Corollary 4.18 allows to write coefficients of (K<sup>Q</sup>)<sup>-1</sup> using the Green functions G<sup>m,θ</sup>(**u**<sup>f</sup>), G<sup>m,θ</sup>(**v**<sup>f</sup>), G<sup>m,*</sup>.

*Proof.* Consider w, v, f and w, b, b' such that w ∉ {w<sup>l</sup>, w<sup>r</sup> ∈ ℝ<sup>θ</sup>} as in the statement of Corollary 4.14. Then, from Example 4.15, we have the following linear system of equations:

$$ (K^Q)^{-1}_{w,b} \text{cn}\left(\frac{K-\theta_f}{2}\right) - i(K^Q)^{-1}_{w,b'} \text{sn}\left(\frac{K+\theta_f}{2}\right) = C \times \Gamma(\mathbf{u}^f) $$

$$ (K^Q)^{-1}_{w,b} \text{cn}\left(\frac{K+\theta_f}{2}\right) + i(K^Q)^{-1}_{w,b'} \text{sn}\left(\frac{K-\theta_f}{2}\right) = C \times \Gamma(\mathbf{v}^f), $$

$$ \text{where } C = \frac{e^{i \frac{\beta_f}{2}} (k')^{\frac{1}{2}} \text{sn}(\theta_i^\beta)^{\frac{1}{|\{w \in \{w^c \in \mathbb{R}^\theta\}\}|}}}{[\text{cn}(\theta_f) \text{sn}(\theta_f)]^{\frac{1}{2}}}, \Gamma(u) = \text{I}_{\{w \neq w^{c,r}\}} t(u)_{w,v} K^\theta(u)^{-1} + t(u)_{w,f} K^\theta(u)^{-1} $$

Solving for (K<sup>Q</sup>)<sup>-1</sup><sub>w,b</sub> gives,

$$ (K^Q)^{-1}_{w,b} = \frac{C}{\text{cn}^2\left(\frac{K-\theta_f}{2}\right) + \text{cn}^2\left(\frac{K+\theta_f}{2}\right)} [\text{cn}\left(\frac{K-\theta_f}{2}\right) \Gamma(\mathbf{u}^f) + \text{cn}\left(\frac{K+\theta_f}{2}\right) \Gamma(\mathbf{v}^f)]. $$

Now, using [54, 2.4.8] and the fact that dn(2K − u) = dn(u), cn(2K − u) = −cn(u), we have for every u ∈ T(k),

$$ \text{cn}^2(u) + \text{cn}^2(K - u) = \frac{\text{cn}(2u) + \text{dn}(2u)}{1 + \text{dn}(2u)} + \frac{\text{cn}(2K - 2u) + \text{dn}(2K - 2u)}{1 + \text{dn}(2K - 2u)} = \frac{2 \text{dn}(2u)}{1 + \text{dn}(2u)} = \frac{2}{1 + \text{nd}(2u)}. $$

$$ \text{This implies that, } \text{cn}^2\left(\frac{K-\theta_f}{2}\right) + \text{cn}^2\left(\frac{K+\theta_f}{2}\right) = \frac{2}{1+\text{nd}\left(2\frac{K-\theta_f}{2}\right)} = \frac{2}{1+(k')^{-1}\text{dn}(\theta_f)}. \quad (4.18) $$

Putting everything together ends the proof of Point 1. Point 2. directly follows from Point 2. of Corollary 4.14. □
## 5 Dimer model on the Fisher graph $G^F$ and the Kasteleyn matrix $K^Q$

In the whole of this section, $G$ is a planar, embedded graph defined as in Section 2.1; it is infinite or finite in which case it has an additional vertex on every boundary edge. Unless specified, definitions and results hold for the finite and infinite cases; until Section 5.3, we do not suppose that $G$ is isoradial.

Consider the dimer model on the Fisher graph $G^F$ with weight function $\mu^j$ arising from the LTE of the Ising model on $G$ with coupling constants $J$ and in the finite case, + boundary conditions, see Section 2.3.2. Consider also the dimer model on the bipartite graph $G^Q$ with weight function $\nu^j$ arising for the XOR-Ising model, see Section 2.3.3.

Suppose that edges of $G^F$ are oriented according to a Kasteleyn orientation, and denote by $K^F$ the corresponding Kasteleyn matrix, as defined in Section 2.2. Following Dubédat [32], we partition vertices of $G^F$ as $V^F = A \cup B$, where $A$ consists of vertices incident to four internal edges of $G^F$, and $B$ consists of those incident to either two internal and one external edges or to two internal edges (this possibility only occurs in the finite case). Vertices of type $A$, resp. $B$, will be denoted by $a$, resp. $b$, with or without sub/super-scripts. Up to a reordering of the rows and columns, the matrix $K^F$ can be written in block form as

$$K^F = \begin{pmatrix} K_{B,B}^F & K_{B,A}^F \\ K_{A,B}^F & K_{A,A}^F \end{pmatrix}.$$

Recall that in the finite case, boundary quadrangles of $G^Q$ are degenerate and consist of edges in bijection with boundary edges of $G$. Boundary $B$-vertices of $G^F$ are defined to be $B$-vertices incident to two internal edges only; they are in natural bijection with black, resp. white, vertices of boundary quadrangles of $G^Q$, and also with boundary white vertices of the double graph $G^D$.

In [32] Dubédat shows how, in the case where $G$ is $Z^2$, a Kasteleyn orientation on $G^F$ induces a Kasteleyn orientation on $G^Q$; this generalizes to the case where $G$ is planar: using the notation of Figure 23 below, define

$$\varepsilon_{b,w} = \varepsilon_{b,a}, \quad \varepsilon_{b,w'} = \varepsilon_{b,a'}, \quad \varepsilon_{b,w''} = \varepsilon_{b,b'}\varepsilon_{b',a''}, \qquad (5.1)$$

then it is straightforward to check that the orientation so defined on $G^Q$ is Kasteleyn. Note that when $G$ is finite, the case $\varepsilon_{b,w''}$ is not present when $b$ belongs to a boundary quadrangle.

<figure><img src="image_26.png" /><figcaption>Figure 23: Notation used to relate the Kasteleyn orientations on $G^F$ and $G^Q$.</figcaption></figure>

Denote by $\tilde{K}^Q$ the bipartite Kasteleyn matrix corresponding to the weight function $\nu^j$ and to the Kasteleyn orientation constructed in (5.1) with rows indexed by black vertices.
# 5 Dimer model on the Fisher graph $G^F$ and the Kasteleyn matrix $K^Q$

In the whole of this section, $G$ is a planar, embedded graph defined as in Section 2.1; it is infinite or finite in which case it has an additional vertex on every boundary edge. Unless specified, definitions and results hold for the finite and infinite cases; until Section 5.3, we do not suppose that $G$ is isoradial.

Consider the dimer model on the Fisher graph $G^F$ with weight function $\mu^j$ arising from the LTE of the Ising model on $G$ with coupling constants $J$ and in the finite case, + boundary conditions, see Section 2.3.2. Consider also the dimer model on the bipartite graph $G^Q$ with weight function $\nu^j$ arising for the XOR-Ising model, see Section 2.3.3.

Suppose that edges of $G^F$ are oriented according to a Kasteleyn orientation, and denote by $K^F$ the corresponding Kasteleyn matrix, as defined in Section 2.2. Following Dubédat [32], we partition vertices of $G^F$ as $V^F = A \cup B$, where $A$ consists of vertices incident to four internal edges of $G^F$, and $B$ consists of those incident to either two internal and one external edges or to two internal edges (this possibility only occurs in the finite case). Vertices of type $A$, resp. $B$, will be denoted by $a$, resp. $b$, with or without sub/super-scripts. Up to a reordering of the rows and columns, the matrix $K^F$ can be written in block form as

$$K^F = \begin{pmatrix} K_{B,B}^F & K_{B,A}^F \\ K_{A,B}^F & K_{A,A}^F \end{pmatrix}.$$

Recall that in the finite case, boundary quadrangles of $G^Q$ are degenerate and consist of edges in bijection with boundary edges of $G$. Boundary $B$-vertices of $G^F$ are defined to be $B$-vertices incident to two internal edges only; they are in natural bijection with black, resp. white, vertices of boundary quadrangles of $G^Q$, and also with boundary white vertices of the double graph $G^D$.

In [32] Dubédat shows how, in the case where $G$ is $Z^2$, a Kasteleyn orientation on $G^F$ induces a Kasteleyn orientation on $G^Q$; this generalizes to the case where $G$ is planar: using the notation of Figure 23 below, define

$$\varepsilon_{b,w} = \varepsilon_{b,a}, \quad \varepsilon_{b,w'} = \varepsilon_{b,a'}, \quad \varepsilon_{b,w''} = \varepsilon_{b,b'}\varepsilon_{b',a''}, \qquad (5.1)$$

then it is straightforward to check that the orientation so defined on $G^Q$ is Kasteleyn. Note that when $G$ is finite, the case $\varepsilon_{b,w''}$ is not present when $b$ belongs to a boundary quadrangle.

<figure><img src="image_25.png" /><figcaption>Figure 23: Notation used to relate the Kasteleyn orientations on $G^F$ and $G^Q$.</figcaption></figure>

Denote by $\tilde{K}^Q$ the bipartite Kasteleyn matrix corresponding to the weight function $\nu^j$ and to the Kasteleyn orientation constructed in (5.1) with rows indexed by black vertices.
Wrapping up and using the notation of Figure 23, we have: for the Kasteleyn matrix $K^F$,

$$ 
\begin{aligned}
K_{u,v}^F &= \varepsilon_{u,v}, && \text{if } u \sim v, \text{ and } u \text{ or } v \text{ is of type A (non exlusive "or")} \\
K_{b,b'}^{F} &= \varepsilon_{b,b'} e^{-2J_e}, && \text{if the external edge } bb' \text{ corresponds to a dual edge } e^* \text{ of } G^*
\end{aligned} 
\qquad (5.2) 
$$

and for the bipartite Kasteleyn matrix $\tilde{K}^Q$,

<table>
  <thead>
    <tr>
      <th> Infinite case &amp; finite case when b<br> does not belong to a boundary quadrangle </th>
      <th> Finite case when b belongs to a<br> boundary quadrangle </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        <div> K<sup>Q</sup><sub>b,w</sub> = ε<sub>b,w</sub> = ε<sub>b,a</sub> </div>
        <div> ˜K<sup>Q</sup><sub>b,w'</sub> = ε<sub>b,w'</sub> tanh(2J<sub>e</sub>) = ε<sub>b,a'</sub> tanh(2J<sub>e</sub>) </div>
        <div> ˜K<sup>Q</sup><sub>b,w''</sub> = ε<sub>b,w''</sub> cosh<sup>-1</sup>(2J<sub>e</sub>) = ε<sub>b,b'</sub>ε<sub>b',a''</sub> cosh<sup>-1</sup>(2J<sub>e</sub>) </div>
      </td>
      <td>
        <div> ˜K<sup>Q</sup><sub>b,w</sub> = ε<sub>b,w</sub> = ε<sub>b,a</sub> </div>
        <div> ˜K<sup>Q</sup><sub>b,w'</sub> = ε<sub>b,w'</sub> = ε<sub>b,a'</sub> </div>
        <div> n.a. </div>
      </td>
    </tr>
  </tbody>
</table>
\qquad (5.3)

The first contribution of this section is Theorem 5.4 of Section 5.2 expressing the inverse Kasteleyn operator $(K^F)^{-1}$ using the inverse bipartite Kasteleyn operator $(\tilde{K}^Q)^{-1}$; proving that the contour Ising Boltzmann/Gibbs measures can be computed from the bipartite dimer model on $G^Q$; note that this result is not restricted to the Z-invariant case. The proof of Theorem 5.4 builds on matrix relations of [32]; this is the subject of Section 5.1. Note that this result can also be derived from the paper [16], see Remark 1.9.

In Section 5.3 we restrict to the Z-invariant case and obtain Corollary 5.8, one of the main results of this paper, expressing the inverse Kasteleyn operator $(K^F)^{-1}$ using the inverse $Z^u$-Dirac operator and also using the Z-massive and dual Green functions. This shows that the contour Ising Boltzmann/Gibbs measures can be computed using information from random walks only (with specific boundary conditions in the finite case). As written in the introduction of this paper, this directly translates into results on the spin-Ising observable of [18] and on the fermionic spinor observable of [39] which is also the FK-Ising observable of [68, 18] (up to normalization) [32, 63]. Note that in the infinite case, this also gives an alternative direct way of finding the local formula for $(K^F)^{-1}$ of [12], explicitly relating it to the Green functions.

## 5.1 Relating Kasteleyn matrices of the Fisher graph $G^F$ and the bipartite graph $G^Q$

Dubédat [32] establishes a matrix relation between the matrix $K^F$ and a block triangular matrix containing $\tilde{K}^Q$ as one of the diagonal blocks. Using this matrix relation, he proves that the squared dimer partition function of $G^F$ is equal, up to a constant, to the dimer partition function of $G^Q$ in the finite case, and that the characteristic polynomials of the two models are equal in the case of infinite $Z^2$-periodic graphs. By adding defects to Ising coupling constants, this allows him to prove bosonization identities.

We attribute the forthcoming Proposition 5.2, consisting of two matrix relations, to Dubédat. The first is the actual identity of [32]; it is appropriate for comparing determinants of the matrices $K^F$ and $\tilde{K}^Q$ (related matrix relations can also be found in [16]). The second proves an identity between $K^F$ and a block diagonal matrix containing $\tilde{K}^Q$ in both diagonal blocks; it is not present in the paper [32] but does not require much more work; it is useful for comparing matrix inverses. For convenience of the reader we provide a proof because: we write weights in a different way, directly write the proof for all planar graphs (and not $Z^2$), handle the boundary conditions very carefully, and the second identity needs an additional argument.

In order to state Proposition 5.2, we need to introduce the following matrices, all of which are “square”.
The Z-Dirac and massive Laplacian operators in the Z-invariant Ising model

Wrapping up and using the notation of Figure 23, we have: for the Kasteleyn matrix $K^F$,

$$K_{u,v}^F = \varepsilon_{u,v}, \quad \text{if } u \sim v, \text{ and } u \text{ or } v \text{ is of type } A \text{ (non exlusive "or")}$$

$$K_{b,b'}^{F} = \varepsilon_{b,b'} e^{-2J_e}, \quad \text{if the external edge } bb' \text{ corresponds to a dual edge } e^* \text{ of } G^*$$

(5.2)

and for the bipartite Kasteleyn matrix $\tilde{K}^Q$,

<table><thead><tr><th> Infinite case & finite case when b<br> does not belong to a boundary quadrangle </th><th> Finite case when b belongs to a<br> boundary quadrangle </th></tr></thead><tbody><tr><td> K<sup>Q</sup><sub>b,w</sub> = ε<sub>b,w</sub> = ε<sub>b,a</sub><br> K̃<sup>Q</sup><sub>b,w'</sub> = ε<sub>b,w'</sub> tanh(2J<sub>e</sub>) = ε<sub>b,a'</sub> tanh(2J<sub>e</sub>)<br> K̃<sup>Q</sup><sub>b,w''</sub> = ε<sub>b,w''</sub> cosh<sup>-1</sup>(2J<sub>e</sub>) = ε<sub>b,b'</sub>ε<sub>b',a''</sub> cosh<sup>-1</sup>(2J<sub>e</sub>) </td><td> K̃<sup>Q</sup><sub>b,w</sub> = ε<sub>b,w</sub> = ε<sub>b,a</sub><br> K̃<sup>Q</sup><sub>b,w'</sub> = ε<sub>b,w'</sub> = ε<sub>b,a'</sub><br> n.a. </td></tr></tbody></table>

(5.3)

The first contribution of this section is Theorem 5.4 of Section 5.2 expressing the inverse Kasteleyn operator $(K^F)^{-1}$ using the inverse bipartite Kasteleyn operator $(\tilde{K}^Q)^{-1}$; proving that the contour Ising Boltzmann/Gibbs measures can be computed from the bipartite dimer model on $G^Q$; note that this result is not restricted to the Z-invariant case. The proof of Theorem 5.4 builds on matrix relations of [32]; this is the subject of Section 5.1. Note that this result can also be derived from the paper [16], see Remark 1.9.

In Section 5.3 we restrict to the Z-invariant case and obtain Corollary 5.8, one of the main results of this paper, expressing the inverse Kasteleyn operator $(K^F)^{-1}$ using the inverse $Z^u$-Dirac operator and also using the Z-massive and dual Green functions. This shows that the contour Ising Boltzmann/Gibbs measures can be computed using information from random walks only (with specific boundary conditions in the finite case). As written in the introduction of this paper, this directly translates into results on the spin-Ising observable of [18] and on the fermionic spinor observable of [39] which is also the FK-Ising observable of [68, 18] (up to normalization) [32, 63]. Note that in the infinite case, this also gives an alternative direct way of finding the local formula for $(K^F)^{-1}$ of [12], explicitly relating it to the Green functions.

## 5.1 Relating Kasteleyn matrices of the Fisher graph $G^F$ and the bipartite graph $G^Q$

Dubédat [32] establishes a matrix relation between the matrix $K^F$ and a block triangular matrix containing $\tilde{K}^Q$ as one of the diagonal blocks. Using this matrix relation, he proves that the squared dimer partition function of $G^F$ is equal, up to a constant, to the dimer partition function of $G^Q$ in the finite case, and that the characteristic polynomials of the two models are equal in the case of infinite $Z^2$-periodic graphs. By adding defects to Ising coupling constants, this allows him to prove bosonization identities.

We attribute the forthcoming Proposition 5.2, consisting of two matrix relations, to Dubédat. The first is the actual identity of [32]; it is appropriate for comparing determinants of the matrices $K^F$ and $\tilde{K}^Q$ (related matrix relations can also be found in [16]). The second proves an identity between $K^F$ and a block diagonal matrix containing $\tilde{K}^Q$ in both diagonal blocks; it is not present in the paper [32] but does not require much more work; it is useful for comparing matrix inverses. For convenience of the reader we provide a proof because: we write weights in a different way, directly write the proof for all planar graphs (and not $Z^2$), handle the boundary conditions very carefully, and the second identity needs an additional argument.

In order to state Proposition 5.2, we need to introduce the following matrices, all of which are “square”.
The matrix $I_{W,A}$ has rows indexed by white vertices of $G^Q$ and columns by $A$-vertices of $G^F$. It is the identity matrix associated to the following bijection between $W$ and $A$. Using the notation of Figure 23, a vertex $w$ of $G^Q$ is incident to a unique external edge $wb$ of $G^Q$; the edge $wb$ is naturally “parallel” to a path from $\tilde{b}$ to $b$ in the external cycle of the closest decoration of $G^F$; then the vertex $a$ in bijection with $w$ is the unique vertex of type $A$ in the path from $\tilde{b}$ to $b$.

The matrix $X$ has rows indexed by $B$-vertices of $G^F$ and columns by black vertices of $G^Q$. It is block diagonal, with blocks of size $2 \times 2$ corresponding to edges of $G^*$. For each such edge, the rows are indexed by $b$ and $b'$, the two corresponding adjacent $B$-vertices, and the columns are indexed by the two black vertices $b, b'$ of the quadrangle of $G^Q$ traversed by the edge $bb'$, with $b$ closest to $b$, see Figure 23. The non-zero coefficients of the row corresponding to the vertex $b$ are,

$$ (x_{b,b} \quad x_{b,b'}) = (1 \quad K_{b',b}^F). $$

In the finite case, the matrix $X$ also has size 1 blocks corresponding to boundary $B$-vertices of $G^F$. For such a vertex $b$, let $b$ be the closest black vertex of $G^Q$. Then, the only non-zero coefficient of the row corresponding to $b$ is:

$$ x_{b,b} = 1. $$

The matrix $M$ has rows indexed by $B$-vertices and columns by $A$-vertices of $G^F$. It is block diagonal, with blocks corresponding to decorations, each block having per size the number of $B$-vertices times the number of $A$-vertices of the decoration. The matrix $M$ is the matrix $K_{B,A}^F$ with some signs reversed. That is, for a $B$-vertex $b$, denote by $a, a'$ its two neighbors of type $A$ so that in cclw order around the triangle we have $a, a', b$, see Figure 23. Then the non-zero coefficients of the row corresponding to the vertex $b$ are,

$$ (m_{b,a} \quad m_{b,a'}) = (-\varepsilon_{b,a} \quad \varepsilon_{b,a'}) $$

The matrix $M'$ has rows indexed by $A$-vertices and columns by $B$-vertices of $G^F$. It is defined as,

$$ M' = -(K_{B,A}^F)^{-1} K_{B,B}^F. $$

**Remark 5.1.** The matrix $K_{B,A}^F = -K_{A,B}^F$ is invertible. Indeed, it is block diagonal, with blocks corresponding to decorations; for each decoration, the block is a directed adjacency matrix of the bipartite graph consisting of the outer cycle of the decoration. The orientation on this cycle is Kasteleyn because it is on the whole graph and this cycle contains no vertex in its interior [42]. As a consequence, the determinant of this block is equal to $\pm$ the number of dimer configurations of this cycle, that is $\pm 2$, and the block is thus invertible.

In the sequel, it would have been tempting to sometimes use the inverse of the matrix $K_{A,A'}^F$, but this matrix is not always invertible. Indeed, it is block diagonal with blocks corresponding to decorations, and when the decoration is associated to a dual vertex of odd degree, then the corresponding block of $K_{A,A'}^F$ is not invertible.

**Proposition 5.2.** [32] The Kasteleyn matrix $K^F$ of the Fisher graph $G^F$ and the bipartite Kasteleyn matrix $\tilde{K}^Q$ of the graph $G^Q$ are related by the following identities:

$$ K^F \begin{pmatrix} I & M \\ 0 & I \end{pmatrix} = \begin{pmatrix} K_{B,B}^F & (X \tilde{K}^Q I_{W,A}) \\ K_{A,B}^F & 0 \end{pmatrix}, \quad K^F \begin{pmatrix} I & M \\ M' & I \end{pmatrix} = \begin{pmatrix} 0 & (X \tilde{K}^Q I_{W,A}) \\ -(X \tilde{K}^Q I_{W,A})^t & 0 \end{pmatrix}. $$

*Proof.* The first identity is an easy consequence of the second, so let us prove the second; unless specified, the arguments hold in the infinite and finite cases. We need to show
The matrix $I_{W,A}$ has rows indexed by white vertices of $G^Q$ and columns by $A$-vertices of $G^F$. It is the identity matrix associated to the following bijection between $W$ and $A$. Using the notation of Figure 23, a vertex $w$ of $G^Q$ is incident to a unique external edge $wb$ of $G^Q$; the edge $wb$ is naturally “parallel” to a path from $\tilde{b}$ to $b$ in the external cycle of the closest decoration of $G^F$; then the vertex $a$ in bijection with $w$ is the unique vertex of type $A$ in the path from $\tilde{b}$ to $b$.

The matrix $X$ has rows indexed by $B$-vertices of $G^F$ and columns by black vertices of $G^Q$. It is block diagonal, with blocks of size $2 \times 2$ corresponding to edges of $G^*$. For each such edge, the rows are indexed by $b$ and $b'$, the two corresponding adjacent $B$-vertices, and the columns are indexed by the two black vertices $b, b'$ of the quadrangle of $G^Q$ traversed by the edge $bb'$, with $b$ closest to $b$, see Figure 23. The non-zero coefficients of the row corresponding to the vertex $b$ are,

$$ (x_{b,b} \quad x_{b,b'}) = (1 \quad K_{b',b}^F). $$

In the finite case, the matrix $X$ also has size 1 blocks corresponding to boundary $B$-vertices of $G^F$. For such a vertex $b$, let $b$ be the closest black vertex of $G^Q$. Then, the only non-zero coefficient of the row corresponding to $b$ is:

$$ x_{b,b} = 1. $$

The matrix $M$ has rows indexed by $B$-vertices and columns by $A$-vertices of $G^F$. It is block diagonal, with blocks corresponding to decorations, each block having per size the number of $B$-vertices times the number of $A$-vertices of the decoration. The matrix $M$ is the matrix $K_{B,A}^F$ with some signs reversed. That is, for a $B$-vertex $b$, denote by $a, a'$ its two neighbors of type $A$ so that in cclw order around the triangle we have $a, a', b$, see Figure 23. Then the non-zero coefficients of the row corresponding to the vertex $b$ are,

$$ (m_{b,a} \quad m_{b,a'}) = (-\epsilon_{b,a} \quad \epsilon_{b,a'}) $$

The matrix $M'$ has rows indexed by $A$-vertices and columns by $B$-vertices of $G^F$. It is defined as,

$$ M' = -(K_{B,A}^F)^{-1} K_{B,B}^F. $$

**Remark 5.1.** The matrix $K_{B,A}^F = -K_{A,B}^F$ is invertible. Indeed, it is block diagonal, with blocks corresponding to decorations; for each decoration, the block is a directed adjacency matrix of the bipartite graph consisting of the outer cycle of the decoration. The orientation on this cycle is Kasteleyn because it is on the whole graph and this cycle contains no vertex in its interior [42]. As a consequence, the determinant of this block is equal to $\pm$ the number of dimer configurations of this cycle, that is $\pm 2$, and the block is thus invertible.

In the sequel, it would have been tempting to sometimes use the inverse of the matrix $K_{A,A'}^F$, but this matrix is not always invertible. Indeed, it is block diagonal with blocks corresponding to decorations, and when the decoration is associated to a dual vertex of odd degree, then the corresponding block of $K_{A,A'}^F$ is not invertible.

**Proposition 5.2.** [32] The Kasteleyn matrix $K^F$ of the Fisher graph $G^F$ and the bipartite Kasteleyn matrix $\tilde{K}^Q$ of the graph $G^Q$ are related by the following identities:

$$ K^F \begin{pmatrix} I & M \\ 0 & I \end{pmatrix} = \begin{pmatrix} K_{B,B}^F & (X \tilde{K}^Q I_{W,A}) \\ K_{A,B}^F & 0 \end{pmatrix}, \quad K^F \begin{pmatrix} I & M \\ M' & I \end{pmatrix} = \begin{pmatrix} 0 & (X \tilde{K}^Q I_{W,A}) \\ -(X \tilde{K}^Q I_{W,A})^t & 0 \end{pmatrix}. $$

*Proof.* The first identity is an easy consequence of the second, so let us prove the second; unless specified, the arguments hold in the infinite and finite cases. We need to show
the four identities below.

$$ K_{B,B}^{F} + K_{B,A}^{F} M' = 0 \qquad (5.4) $$

$$ K_{A,B}^{F} M + K_{A,A}^{F} = 0 \qquad (5.5) $$

$$ K_{B,B}^{F} M + K_{B,A}^{F} = X \tilde{K}^{Q} I_{W,A} \qquad (5.6) $$

$$ K_{A,B}^{F} + K_{A,A}^{F} M' = -(X \tilde{K}^{Q} I_{W,A})^{t}. \qquad (5.7) $$

Note that even in the infinite case, they all make sense since matrices involved have finitely many non-zero coefficients per rows and columns.

Identity (5.4) is immediate by definition of $M'$. We now prove (5.5) and (5.6) and then show that (5.7) follows.

*Proof of (5.5).* Let us show that $K_{A,B}^{F} M = -K_{A,A}^{F}$. Consider an A-vertex $a$ of $G^F$, and let $a_1, a_2, b_1, b_2$ be its four neighbors in $G^F$ with the notation of Figure 24. Then

<figure><img src="image_27.png" /><figcaption>Figure 24: Notation for the proof of (5.5).</figcaption></figure>

the coefficient $(K_{A,B}^F M)_{a,a'}$ is a priori non zero when $a' \in \{a, a_1, a_2\}$. Returning to the definition of $K^F$ and $M$, we have

$$ (K_{A,B}^{F} M)_{a,a} = K_{a,b_1}^{F} m_{b_1,a} + K_{a,b_2}^{F} m_{b_2,a} = \varepsilon_{a,b_1} \varepsilon_{b_1,a} - \varepsilon_{a,b_2} \varepsilon_{b_2,a} = 0 = -K_{a,a}^{F}. $$

When $a' \in \{a_1, a_2\}$, using moreover that the orientation around the triangles $a, b_1, a_1$ and $a, a_2, b$ is Kasteleyn, we have

$$ (K_{A,B}^{F} M)_{a,a_1} = K_{a,b_1}^{F} m_{b_1,a_1} = -\varepsilon_{a,b_1} \varepsilon_{b_1,a_1} = -\varepsilon_{a,a_1} = -K_{a,a_1}^{F} $$

$$ (K_{A,B}^{F} M)_{a,a_2} = K_{a,b_2}^{F} m_{b_2,a_2} = \varepsilon_{a,b_2} \varepsilon_{b_2,a_2} = -\varepsilon_{a,a_2} = -K_{a,a_2}^{F}, $$

thus ending the proof of (5.5).

*Proof of (5.6).* *Infinite case.* Figure 25 (left) below sets the notation and labeling: $b, b'$ are adjacent B-vertices of $G^F$, and $a_1, a_2, a_3, a_4$ are their neighbors of type A in $G^F$; $b, b'$ are the two black vertices of the quadrangle of $G^Q$ traversed by the edge $bb'$, and $w_1, \dots, w_4$ are their neighboring white vertices in $G^Q$.

<figure><img src="image_28.png" /><figcaption>Figure 25: Notation for the proof of (5.6).</figcaption></figure>
the four identities below.

$$ K_{B,B}^{F} + K_{B,A}^{F} M' = 0 \qquad (5.4) $$

$$ K_{A,B}^{F} M + K_{A,A}^{F} = 0 \qquad (5.5) $$

$$ K_{B,B}^{F} M + K_{B,A}^{F} = X \tilde{K}^{Q} I_{W,A} \qquad (5.6) $$

$$ K_{A,B}^{F} + K_{A,A}^{F} M' = -(X \tilde{K}^{Q} I_{W,A})^{t}. \qquad (5.7) $$

Note that even in the infinite case, they all make sense since matrices involved have finitely many non-zero coefficients per rows and columns.

Identity (5.4) is immediate by definition of $M'$. We now prove (5.5) and (5.6) and then show that (5.7) follows.

*Proof of (5.5).* Let us show that $K_{A,B}^{F} M = -K_{A,A}^{F}$. Consider an A-vertex a of $G^F$, and let $a_1, a_2, b_1, b_2$ be its four neighbors in $G^F$ with the notation of Figure 24. Then

<figure><img src="image_26.png" /><figcaption>Figure 24: Notation for the proof of (5.5).</figcaption></figure>

the coefficient $(K_{A,B}^F M)_{a,a'}$ is a priori non zero when $a' \in \{a, a_1, a_2\}$. Returning to the definition of $K^F$ and $M$, we have

$$ (K_{A,B}^{F} M)_{a,a} = K_{a,b_1}^{F} m_{b_1,a} + K_{a,b_2}^{F} m_{b_2,a} = \varepsilon_{a,b_1} \varepsilon_{b_1,a} - \varepsilon_{a,b_2} \varepsilon_{b_2,a} = 0 = -K_{a,a}^{F}. $$

When $a' \in \{a_1, a_2\}$, using moreover that the orientation around the triangles $a, b_1, a_1$ and $a, a_2, b$ is Kasteleyn, we have

$$ (K_{A,B}^{F} M)_{a,a_1} = K_{a,b_1}^{F} m_{b_1,a_1} = -\varepsilon_{a,b_1} \varepsilon_{b_1,a_1} = -\varepsilon_{a,a_1} = -K_{a,a_1}^{F} $$

$$ (K_{A,B}^{F} M)_{a,a_2} = K_{a,b_2}^{F} m_{b_2,a_2} = \varepsilon_{a,b_2} \varepsilon_{b_2,a_2} = -\varepsilon_{a,a_2} = -K_{a,a_2}^{F}, $$

thus ending the proof of (5.5).

*Proof of (5.6).* *Infinite case.* Figure 25 (left) below sets the notation and labeling: $b, b'$ are adjacent B-vertices of $G^F$, and $a_1, a_2, a_3, a_4$ are their neighbors of type A in $G^F$; $b, b'$ are the two black vertices of the quadrangle of $G^Q$ traversed by the edge $bb'$, and $w_1, \dots, w_4$ are their neighboring white vertices in $G^Q$.

<figure><img src="image_27.png" /><figcaption>Figure 25: Notation for the proof of (5.6).</figcaption></figure>
Consider a B-vertex b of G<sup>F</sup>. Then the coefficient (K<sup>F</sup><sub>B,B</sub>M + K<sup>F</sup><sub>B,A</sub>)<sub>b,a</sub> of the LHS of (5.6) is non-zero when a ∈ {a<sub>1</sub>, ..., a<sub>4</sub>}, and

$$ (K_{B,B}^{F} M + K_{B,A}^{F})_{b,a} = \begin{cases} K_{b,a}^{F} = \varepsilon_{b,a} & \text{if } a \in \{a_1, a_2\} \\ K_{b,b'}^{F} m_{b',a} & \text{if } a \in \{a_3, a_4\}, \end{cases} $$

where recall $m_{b',a_3} = -\varepsilon_{b',a_3}$ and $m_{b',a_4} = \varepsilon_{b',a_4}$.

The coefficient $(X\tilde{K}^Q I_{W,A})_{b,a}$ of the RHS of (5.6) is non-zero for the same choices of a. Returning to the definition of X and $\tilde{K}^Q$ we have,

$$ \begin{align} (X\tilde{K}^Q I_{W,A})_{b,a_1} &= x_{b,b} \tilde{K}_{b,w_1}^Q = 1 \cdot \varepsilon_{b,a_1}, \notag \\ (X\tilde{K}^Q I_{W,A})_{b,a_3} &= x_{b,b'} \tilde{K}_{b',w_3}^Q = K_{b',b}^F \varepsilon_{b',a_3} = -K_{b,b'}^F \varepsilon_{b',a_3}, \notag \\ (X\tilde{K}^Q I_{W,A})_{b,a_2} &= x_{b,b} \tilde{K}_{b,w_2}^Q + x_{b,b'} \tilde{K}_{b',w_2}^Q \notag \\ &= 1 \cdot \varepsilon_{b,a_2} \tanh(2J) + K_{b',b}^F \varepsilon_{b',b} \varepsilon_{b,a_2} \cosh^{-1}(2J) \notag \\ &= \varepsilon_{b,a_2} (\tanh(2J) + |K_{b',b}^F| \cosh^{-1}(2J)) \notag \\ &= \varepsilon_{b,a_2}, \quad \text{since } |K_{b',b}^F| = e^{-2J}, \tag{5.8} \end{align} $$

$$ \begin{align} (X\tilde{K}^Q I_{W,A})_{b,a_4} &= x_{b,b} \tilde{K}_{b,w_4}^Q + x_{b,b'} \tilde{K}_{b',w_4}^Q \notag \\ &= 1 \cdot \varepsilon_{b,b'} \varepsilon_{b',a_4} \cosh^{-1}(2J) + K_{b',b}^F \varepsilon_{b',a_4} \tanh(2J) \notag \\ &= K_{b,b'}^F \varepsilon_{b',a_4} \left( \frac{\varepsilon_{b,b'}}{K_{b,b'}^F} \cosh^{-1}(2J) - \tanh(2J) \right) \notag \\ &= K_{b,b'}^F \varepsilon_{b',a_4} (e^{2J} \cosh^{-1}(2J) - \tanh(2J)) = K_{b,b'}^F \varepsilon_{b',a_4}. \tag{5.9} \end{align} $$

Finite case. The proof is as in the infinite case as long as b is not a boundary B-vertex of G<sup>F</sup>, so let b be a boundary B-vertex and refer to Figure 25 (right) for notation. The coefficient (K<sup>F</sup><sub>B,B</sub>M + K<sup>F</sup><sub>B,A</sub>)<sub>b,a</sub> of the LHS is non-zero when a ∈ {a<sub>1</sub>, a<sub>2</sub>}.

$$ (K_{B,B}^{F} M + K_{B,A}^{F})_{b,a} = K_{b,a}^{F} = \varepsilon_{b,a}, \quad \text{if } a \in \{a_1, a_2\} $$

The coefficient $(X\tilde{K}^Q I_{W,A})_{b,a}$ of the RHS is non-zero for the same choices of a. Returning to the definition of X and $\tilde{K}^Q$ (boundary case) we have:

$$ \begin{align*} (X\tilde{K}^Q I_{W,A})_{b,a_1} &= x_{b,b} \tilde{K}_{b,w_1}^Q = 1 \cdot \varepsilon_{b,a_1}, \\ (X\tilde{K}^Q I_{W,A})_{b,a_2} &= x_{b,b} \tilde{K}_{b,w_2}^Q = 1 \cdot \varepsilon_{b,a_2}. \end{align*} $$

This ends the proof of (5.6).

*Proof of (5.7).* From Remark 5.1 the matrix $K_{A,B}$ is invertible, thus from (5.5) we have $M = -(K_{A,B}^F)^{-1}K_{A,A}^F$. Plugging this into the LHS of (5.6) gives that it is equal to:

$$ \text{LHS (5.6)} = -K_{B,B}^{F} (K_{A,B}^{F})^{-1} K_{A,A}^{F} + K_{B,A}^{F}. $$

Returning to the definition of $M'$ (or to (5.4)), we have that the LHS of (5.7) is

$$ \text{LHS(5.7)} = K_{A,B}^{F} - K_{A,A}^{F} (K_{B,A}^{F})^{-1} K_{B,B}^{F}. $$

Using that the matrix K<sup>F</sup> is skew-symmetric, we deduce that LHS (5.7) = -[LHS (5.6)]<sup>t</sup>. The same clearly holds for the RHS of the two equations; they are thus equivalent and we have proved (5.6). □
Consider a B-vertex b of G<sup>F</sup>. Then the coefficient (K<sup>F</sup><sub>B,B</sub>M + K<sup>F</sup><sub>B,A</sub>)<sub>b,a</sub> of the LHS of (5.6) is non-zero when a ∈ {a<sub>1</sub>, ..., a<sub>4</sub>}, and

$$ (K_{B,B}^{F} M + K_{B,A}^{F})_{b,a} = \begin{cases} K_{b,a}^{F} = \varepsilon_{b,a} & \text{if } a \in \{a_1, a_2\} \\ K_{b,b'}^{F} m_{b',a} & \text{if } a \in \{a_3, a_4\}, \end{cases} $$

where recall $m_{b',a_3} = -\varepsilon_{b',a_3}$ and $m_{b',a_4} = \varepsilon_{b',a_4}$.

The coefficient $(X\tilde{K}^Q I_{W,A})_{b,a}$ of the RHS of (5.6) is non-zero for the same choices of a. Returning to the definition of X and $\tilde{K}^Q$ we have,

$$ \begin{align} (X\tilde{K}^Q I_{W,A})_{b,a_1} &= x_{b,b} \tilde{K}_{b,w_1}^Q = 1 \cdot \varepsilon_{b,a_1}, \notag \\ (X\tilde{K}^Q I_{W,A})_{b,a_3} &= x_{b,b'} \tilde{K}_{b',w_3}^Q = K_{b',b}^F \varepsilon_{b',a_3} = -K_{b,b'}^F \varepsilon_{b',a_3}, \notag \\ (X\tilde{K}^Q I_{W,A})_{b,a_2} &= x_{b,b} \tilde{K}_{b,w_2}^Q + x_{b,b'} \tilde{K}_{b',w_2}^Q \notag \\ &= 1 \cdot \varepsilon_{b,a_2} \tanh(2J) + K_{b',b}^F \varepsilon_{b',b} \varepsilon_{b,a_2} \cosh^{-1}(2J) \notag \\ &= \varepsilon_{b,a_2} (\tanh(2J) + |K_{b',b}^F| \cosh^{-1}(2J)) \notag \\ &= \varepsilon_{b,a_2}, \quad \text{since } |K_{b',b}^F| = e^{-2J}, \tag{5.8} \end{align} $$

$$ \begin{align} (X\tilde{K}^Q I_{W,A})_{b,a_4} &= x_{b,b} \tilde{K}_{b,w_4}^Q + x_{b,b'} \tilde{K}_{b',w_4}^Q \notag \\ &= 1 \cdot \varepsilon_{b,b'} \varepsilon_{b',a_4} \cosh^{-1}(2J) + K_{b',b}^F \varepsilon_{b',a_4} \tanh(2J) \notag \\ &= K_{b,b'}^F \varepsilon_{b',a_4} \left( \frac{\varepsilon_{b,b'}}{K_{b,b'}^F} \cosh^{-1}(2J) - \tanh(2J) \right) \notag \\ &= K_{b,b'}^F \varepsilon_{b',a_4} (e^{2J} \cosh^{-1}(2J) - \tanh(2J)) = K_{b,b'}^F \varepsilon_{b',a_4}. \tag{5.9} \end{align} $$

Finite case. The proof is as in the infinite case as long as b is not a boundary B-vertex of G<sup>F</sup>, so let b be a boundary B-vertex and refer to Figure 25 (right) for notation. The coefficient (K<sup>F</sup><sub>B,B</sub>M + K<sup>F</sup><sub>B,A</sub>)<sub>b,a</sub> of the LHS is non-zero when a ∈ {a<sub>1</sub>, a<sub>2</sub>}.

$$ (K_{B,B}^{F} M + K_{B,A}^{F})_{b,a} = K_{b,a}^{F} = \varepsilon_{b,a}, \quad \text{if } a \in \{a_1, a_2\} $$

The coefficient $(X\tilde{K}^Q I_{W,A})_{b,a}$ of the RHS is non-zero for the same choices of a. Returning to the definition of X and $\tilde{K}^Q$ (boundary case) we have:

$$ \begin{align} (X\tilde{K}^Q I_{W,A})_{b,a_1} &= x_{b,b} \tilde{K}_{b,w_1}^Q = 1 \cdot \varepsilon_{b,a_1}, \notag \\ (X\tilde{K}^Q I_{W,A})_{b,a_2} &= x_{b,b} \tilde{K}_{b,w_2}^Q = 1 \cdot \varepsilon_{b,a_2}. \notag \end{align} $$

This ends the proof of (5.6).

*Proof of (5.7).* From Remark 5.1 the matrix $K_{A,B}$ is invertible, thus from (5.5) we have $M = -(K_{A,B}^F)^{-1}K_{A,A}^F$. Plugging this into the LHS of (5.6) gives that it is equal to:

$$ \text{LHS (5.6)} = -K_{B,B}^{F} (K_{A,B}^{F})^{-1} K_{A,A}^{F} + K_{B,A}^{F}. $$

Returning to the definition of $M'$ (or to (5.4)), we have that the LHS of (5.7) is

$$ \text{LHS(5.7)} = K_{A,B}^{F} - K_{A,A}^{F} (K_{B,A}^{F})^{-1} K_{B,B}^{F}. $$

Using that the matrix K<sup>F</sup> is skew-symmetric, we deduce that LHS (5.7) = -[LHS (5.6)]<sup>t</sup>. The same clearly holds for the RHS of the two equations; they are thus equivalent and we have proved (5.6). □
**Corollary 5.3 ([32]).** Suppose that the graph G is finite. Then,

$$[Z_{\text{dimer}}(G^F, \mu^j)]^2 = 2^{|V^*|} \left( \prod_{e^* \in E^*} (1 + e^{-4J_e}) \right) Z_{\text{dimer}}(G^Q, \nu^j),$$

$$\text{where } Z_{\text{dimer}}(G^F, \mu^j) = |\det K^F|, \quad Z_{\text{dimer}}(G^Q, \nu^j) = |\det \tilde{K}^Q|.$$

*Proof.* By the first identity of Proposition 5.2, we have $|\det K^F| = |\det K^F_{A,B} \det X \det K^Q|$. By Remark 5.1, we know that $|\det K^F_{A,B}| = 2^{|V^*|}$. The determinant of X is computed by calculating that of its blocks. Let $bb'$ be an edge of $G^F$ corresponding to an edge $e^*$ of $G^*$ and let b, b' be the black vertices of the quadrangle of $G^Q$ traversed by the edge $bb'$. Then by definition, the corresponding block of X is:

$$\begin{pmatrix} 1 & K^F_{b',b} \\ K^F_{b,b'} & 1 \end{pmatrix}_{(b,b')}^{(b,b')},$$

Its determinant is equal to $1+|K^F_{b,b'}|^2 = 1+e^{-4J_e}$, thus ending the proof of the corollary. □

## 5.2 Relating inverse Kasteleyn matrices of $G^F$ and $G^Q$

In this section we consider the inverse operators $(K^F)^{-1}$ and $(\tilde{K}^Q)^{-1}$. When the graph G is infinite, we assume that these inverses exist, decrease to zero at infinity and are unique, referring to these assumptions as $(*)$. This is proved to hold when the graph G is $Z^2$-periodic [47, 9], and when it is isoradial (not necessarily periodic) and the corresponding dimers weights on $G^F$ and $G^Q$ are Z-invariant [45, 12], in which case inverse operators furthermore have local expressions.

From Proposition 5.2 and proving additional relations (not present in the paper [32]) we show, in Theorem 5.4 below, identities relating the inverse operator $(K^F)^{-1}$ to the inverse operator $(\tilde{K}^Q)^{-1}$. As mentioned in Remark 1.9 of the introduction, these relations can also be derived from the paper [16]. Using Section 2.2.2, Theorem 5.4 allows to express the dimer Boltzmann measure (finite case) and the Gibbs measure (infinite case) of the Fisher graph $G^F$, denoted $P^F_{\text{dimer}}$, using coefficients of the matrix $K^F$ and of the inverse bipartite Kasteleyn operator $(\tilde{K}^Q)^{-1}$, see also Example 5.6. To state Theorem 5.4, we need to define two additional matrices $D_{B,A}$ and $\kappa$.

The matrix $D_{B,A}$ has rows indexed by black vertices of $G^Q$ and columns by A-vertices of $G^F$. It is a diagonal matrix associated to the following bijection between B and A. Using the notation of Figure 23, a vertex b of $G^Q$ belongs to a unique external edge bw of $G^Q$; then the vertex a in bijection with b is the vertex in bijection with w in the construction of the matrix $I_{w,A}$. The corresponding diagonal coefficient $d_{b,a}$ is,

$$d_{b,a} = \varepsilon_{b,a}.$$

The matrix $\kappa$ has rows and columns indexed by A-vertices of $G^F$. It is block diagonal with blocks corresponding to decorations, each block having per size the number of A-vertices of the decoration. Given two vertices $a, a'$ of a decoration of $G^F$, we have,

$$\kappa_{a,a'} = -\frac{1}{4} \begin{cases} -1 & \text{if } a = a' \\ (-1)^{n(a,a')} & \text{if } a \neq a' \end{cases}$$

where $n(a, a')$ is the number of edges oriented cw in the cclw path going from a to a' in the A-cycle of the decoration.
**Corollary 5.3 ([32]).** Suppose that the graph G is finite. Then,

$$[Z_{\text{dimer}}(G^F, \mu^j)]^2 = 2^{|V^*|} \left( \prod_{e^* \in E^*} (1 + e^{-4J_e}) \right) Z_{\text{dimer}}(G^Q, \nu^j),$$

$$\text{where } Z_{\text{dimer}}(G^F, \mu^j) = |\det K^F|, \quad Z_{\text{dimer}}(G^Q, \nu^j) = |\det \tilde{K}^Q|.$$

*Proof.* By the first identity of Proposition 5.2, we have $|\det K^F| = |\det K^F_{A,B} \det X \det K^Q|$. By Remark 5.1, we know that $|\det K^F_{A,B}| = 2^{|V^*|}$. The determinant of X is computed by calculating that of its blocks. Let $bb'$ be an edge of $G^F$ corresponding to an edge $e^*$ of $G^*$ and let b, b' be the black vertices of the quadrangle of $G^Q$ traversed by the edge $bb'$. Then by definition, the corresponding block of X is:

$$\begin{pmatrix} 1 & K_{b',b}^F \\ K_{b,b'}^F & 1 \end{pmatrix}_{(b,b')}^{(b,b')}. $$

Its determinant is equal to $1+|K_{b,b'}^F|^2 = 1+e^{-4J_e}$, thus ending the proof of the corollary. $\square$

## 5.2 Relating inverse Kasteleyn matrices of $G^F$ and $G^Q$

In this section we consider the inverse operators $(K^F)^{-1}$ and $(\tilde{K}^Q)^{-1}$. When the graph G is infinite, we assume that these inverses exist, decrease to zero at infinity and are unique, referring to these assumptions as $(*)$. This is proved to hold when the graph G is $Z^2$-periodic [47, 9], and when it is isoradial (not necessarily periodic) and the corresponding dimers weights on $G^F$ and $G^Q$ are Z-invariant [45, 12], in which case inverse operators furthermore have local expressions.

From Proposition 5.2 and proving additional relations (not present in the paper [32]) we show, in Theorem 5.4 below, identities relating the inverse operator $(K^F)^{-1}$ to the inverse operator $(\tilde{K}^Q)^{-1}$. As mentioned in Remark 1.9 of the introduction, these relations can also be derived from the paper [16]. Using Section 2.2.2, Theorem 5.4 allows to express the dimer Boltzmann measure (finite case) and the Gibbs measure (infinite case) of the Fisher graph $G^F$, denoted $P_{\text{dimer}}^F$, using coefficients of the matrix $K^F$ and of the inverse bipartite Kasteleyn operator $(\tilde{K}^Q)^{-1}$, see also Example 5.6. To state Theorem 5.4, we need to define two additional matrices $D_{B,A}$ and $\kappa$.

The matrix $D_{B,A}$ has rows indexed by black vertices of $G^Q$ and columns by A-vertices of $G^F$. It is a diagonal matrix associated to the following bijection between B and A. Using the notation of Figure 23, a vertex b of $G^Q$ belongs to a unique external edge bw of $G^Q$; then the vertex a in bijection with b is the vertex in bijection with w in the construction of the matrix $I_{w,A}$. The corresponding diagonal coefficient $d_{b,a}$ is,

$$d_{b,a} = \varepsilon_{b,a}.$$

The matrix $\kappa$ has rows and columns indexed by A-vertices of $G^F$. It is block diagonal with blocks corresponding to decorations, each block having per size the number of A-vertices of the decoration. Given two vertices $a, a'$ of a decoration of $G^F$, we have,

$$\kappa_{a,a'} = -\frac{1}{4} \begin{cases} -1 & \text{if } a = a' \\ (-1)^{n(a,a')} & \text{if } a \neq a' \end{cases}$$

where $n(a, a')$ is the number of edges oriented cw in the cclw path going from a to a' in the A-cycle of the decoration.
**Notation for coefficients** $(K^F)_{u,v}^{-1}$ **of Theorem 5.4** If $u = a$ is an A-vertex, then $w$ is the white vertex of $G^Q$ corresponding to $a$ in the bijection defining $I_{w,A}$. If $u = b$ is a B-vertex, then $a_1$ and $a_2$ are the two A-vertices belonging to the same triangle, with $b, a_1, a_2$ in cclw order around the triangle. Note that this definition also holds if $b$ is a boundary vertex.

If $v = b$ is a B-vertex, we let $b$ be the closest black vertex of $G^Q$. When moreover $b$ is not a boundary vertex, we let $b'$ be the B-vertex such that $bb'$ defines an edge $e^*$ of $G^*$; we let $b'$ be the black vertex of $G^Q$ closest to $b'$ (b and $b'$ are the black vertices of the quadrangle of $G^Q$ traversed by the edge $bb'$); the coupling constant of the edge $e$, dual of $e^*$, is denoted $J_e^f$, where "f" stands for "final". If $v = a$ is an A-vertex, we let $b$ and $b$ be as defined in the matrix $D_{B,A}$, see Figure 26.

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /><figcaption>Figure 26: Left: notation for initial vertices. Right: notation for final vertices when $b$ is not a boundary vertex and when it is.</figcaption></figure>

**Theorem 5.4.** Under assumption $(*)$, the inverse Kasteleyn operator $(K^F)^{-1}$ can be expressed using the inverse bipartite Kasteleyn operator $(\tilde{K}^Q)^{-1}$ as follows.

* Matrix form.

$$ (K^F)^{-1} = \begin{pmatrix} (K^F)_{B,B}^{-1} & (K^F)_{B,A}^{-1} \\ (K^F)_{A,B}^{-1} & (K^F)_{A,A}^{-1} \end{pmatrix} = \begin{pmatrix} M(X\tilde{K}^Q I_{W,A})^{-1} & -[(X\tilde{K}^Q I_{W,A})^t]^{-1} \\ (X\tilde{K}^Q I_{W,A})^{-1} & -\frac{1}{2}I_{A,W}(\tilde{K}^Q)^{-1}D_{B,A} + \kappa_{A,A} \end{pmatrix}. \quad (5.10) $$

* Coefficients. We have four cases to consider and use the notation of Figure 26.

1. For every $a \in A$ and every $b \in B$ such that, when the graph $G^F$ is moreover finite, $b$ is not a boundary vertex:

$$ (K^F)_{a,b}^{-1} = \frac{1}{1 + e^{-4J_e^f}} [(\tilde{K}^Q)_{w,b}^{-1} + (\tilde{K}^Q)_{w,b'}^{-1} \varepsilon_{b',b} e^{-2J_e^f}]. \quad (5.11) $$

2. When the graph $G^F$ is finite, for every $a \in A$ and every boundary vertex $b$ of $B$, we have

$$ (K^F)_{a,b}^{-1} = (\tilde{K}^Q)_{w,b}^{-1}. \quad (5.12) $$

3. For every $a, a \in A$,

$$ (K^F)_{a,a}^{-1} = -\frac{1}{2}(\tilde{K}^Q)_{w,b}^{-1}\varepsilon_{b,a} + \kappa_{a,a}. \quad (5.13) $$

4. For every $b, b \in B$,

$$ (K^F)_{b,b}^{-1} = -\varepsilon_{b,a_1}(K^F)_{a_1,b}^{-1} + \varepsilon_{b,a_2}(K^F)_{a_2,b}^{-1}, \quad (5.14) $$

where $(K^F)_{a_1,b}^{-1}, (K^F)_{a_2,b}^{-1}$ are given by (5.11).

**Remark 5.5.**

* When proving the local formula for $(K^F)_{a,a}^{-1}$ [10, 12] in the Z-invariant case, we obtained a formula of the form (5.13) – with the constant $\kappa_{a,a}$ – without explicitly
**Notation for coefficients** $(K^F)_{u,v}^{-1}$ **of Theorem 5.4** If $u = a$ is an A-vertex, then $w$ is the white vertex of $G^Q$ corresponding to $a$ in the bijection defining $I_{w,A}$. If $u = b$ is a B-vertex, then $a_1$ and $a_2$ are the two A-vertices belonging to the same triangle, with $b, a_1, a_2$ in cclw order around the triangle. Note that this definition also holds if $b$ is a boundary vertex.

If $v = b$ is a B-vertex, we let $b$ be the closest black vertex of $G^Q$. When moreover $b$ is not a boundary vertex, we let $b'$ be the B-vertex such that $bb'$ defines an edge $e^*$ of $G^*$; we let $b'$ be the black vertex of $G^Q$ closest to $b'$ (b and $b'$ are the black vertices of the quadrangle of $G^Q$ traversed by the edge $bb'$); the coupling constant of the edge $e$, dual of $e^*$, is denoted $J_e^f$, where "f" stands for "final". If $v = a$ is an A-vertex, we let $b$ and $b$ be as defined in the matrix $D_{B,A}$, see Figure 26.

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /><figcaption>Figure 26: Left: notation for initial vertices. Right: notation for final vertices when $b$ is not a boundary vertex and when it is.</figcaption></figure>

**Theorem 5.4.** Under assumption $(*)$, the inverse Kasteleyn operator $(K^F)^{-1}$ can be expressed using the inverse bipartite Kasteleyn operator $(\tilde{K}^Q)^{-1}$ as follows.

* Matrix form.

$$ (K^F)^{-1} = \begin{pmatrix} (K^F)_{B,B}^{-1} & (K^F)_{B,A}^{-1} \\ (K^F)_{A,B}^{-1} & (K^F)_{A,A}^{-1} \end{pmatrix} = \begin{pmatrix} M(X\tilde{K}^Q I_{w,A})^{-1} & -[(X\tilde{K}^Q I_{w,A})^t]^{-1} \\ (X\tilde{K}^Q I_{w,A})^{-1} & -\frac{1}{2}I_{A,W}(\tilde{K}^Q)^{-1}D_{B,A} + \kappa_{A,A} \end{pmatrix}. \quad (5.10) $$

* Coefficients. We have four cases to consider and use the notation of Figure 26.

1. For every $a \in A$ and every $b \in B$ such that, when the graph $G^F$ is moreover finite, $b$ is not a boundary vertex:

$$ (K^F)_{a,b}^{-1} = \frac{1}{1 + e^{-4J_e^f}} [(\tilde{K}^Q)_{w,b}^{-1} + (\tilde{K}^Q)_{w,b'}^{-1} \varepsilon_{b',b} e^{-2J_e^f}]. \quad (5.11) $$

2. When the graph $G^F$ is finite, for every $a \in A$ and every boundary vertex $b$ of $B$, we have

$$ (K^F)_{a,b}^{-1} = (\tilde{K}^Q)_{w,b}^{-1}. \quad (5.12) $$

3. For every $a, a \in A$,

$$ (K^F)_{a,a}^{-1} = -\frac{1}{2}(\tilde{K}^Q)_{w,b}^{-1}\varepsilon_{b,a} + \kappa_{a,a}. \quad (5.13) $$

4. For every $b, b \in B$,

$$ (K^F)_{b,b}^{-1} = -\varepsilon_{b,a_1}(K^F)_{a_1,b}^{-1} + \varepsilon_{b,a_2}(K^F)_{a_2,b}^{-1}, \quad (5.14) $$

where $(K^F)_{a_1,b}^{-1}, (K^F)_{a_2,b}^{-1}$ are given by (5.11).

**Remark 5.5.**

* When proving the local formula for $(K^F)_{a,a}^{-1}$ [10, 12] in the Z-invariant case, we obtained a formula of the form (5.13) – with the constant $\kappa_{a,a}$ – without explicitly
relating it to a coefficient of $(K^Q)^{-1}$. It is quite remarkable that this formula holds in the full planar case (without assuming Z-invariance), in the finite and infinite cases.

* In the finite case, we do not need positivity of the coupling constants $J$. In particular if the coupling constants are all negative, Theorem 5.4 expresses probabilities of the dimer model on the non-bipartite graph $G^F$ with positive weights $e^{-2J_e} > 1$ on external edges, as a function of the inverse Kasteleyn operator of a "dimer model" on the bipartite graph $G^Q$ with negative weights ($\tanh(J_e) < 0$) on quadrangle edges parallel to edges of $G$. Having a negative weight for an edge amounts to reversing its orientation. From a physics point of view, this amounts to adding defects or creating vortices at each face where this new orientation is not Kasteleyn. The physics paper [61] considers bipartite models with negative weights and somehow describes the non-Harnacity of the associated spectral curves.

* Coefficients of the inverse Kasteleyn operator $(K^F)^{-1}$ not only allow to express the dimer Boltzmann/Gibbs measure $P_{\text{dimer}}^F$, but are also related to important observables of the Ising model. By [16], the coefficient $(K^F)_{b,b}^{-1}$ is essentially the spin-observable of [18] when fixing one vertex to be on the boundary of the domain; by [32] the coefficient $(K^F)_{a,a}^{-1}$ is the fermionic spinor correlator of [39] and by [63], it is related to the FK-Ising observable [67, 68, 18] when fixing one vertex on the boundary of the domain, taking appropriate boundary conditions, up to normalization.

* Consider a fixed A-vertex $a$ and $a_1, a_2, a_3$ as in Figure 27 below, such that the decorations of $a_1, a_2, a_3$ are distinct from that of $a$. Then using the identity $[\tilde{K}^Q(\tilde{K}^Q)^{-1}]_{b,b} = 0$, from (5.13) we immediately obtain the Dotsenko three-terms relation [31, 59, 16], see also Definition 2.1. of [15].

$$ \varepsilon_{b,a_1} (K^F)_{a_1,a}^{-1} + \varepsilon_{b,a_2} \tanh(2J_e) (K^F)_{a_2,a}^{-1} + \varepsilon_{b,b'} \varepsilon_{b',a_3} \cosh^{-1}(2J_e) (K^F)_{a_3,a}^{-1} = 0. $$

<figure><img src="image_31.png" /><figcaption>Figure 27: Notation for the Dotsenko three-terms relation.</figcaption></figure>

*Proof.* The expressions for coefficients are obtained from Formula (5.10) and by returning to the definition of the matrices $I_{W,A}$, $D_{b,A}$, $M$ and $X$. The inverse of $X$ is computed by blocks. For an edge $bb'$ of $G^F$ corresponding to an edge $e^*$ of $G^*$, let $b, b'$ be the black vertices of the quadrangle of $G^Q$ traversed by the edge $bb'$. Then the inverse of the corresponding block is,

$$ \left( \begin{pmatrix} 1 & K_{b',b}^F \\ K_{b,b'}^F & 1 \end{pmatrix} \right)_{(b,b')}^{-1} = \frac{1}{1 + |K_{b,b'}^F|^2} \left( \begin{pmatrix} 1 & K_{b,b'}^F \\ K_{b',b}^F & 1 \end{pmatrix} \right)_{(b,b')}^{(b,b)}. \quad (5.15) $$

In the finite case, the matrix $X$ also has a size 1, identity block for each boundary B-vertex $b$ of $G^F$ and its closest black vertices $b$ of $G^Q$. This ends the proof of formulas (5.11), (5.13), (5.14) for coefficients and we now turn to the proof of (5.10).
relating it to a coefficient of $(K^Q)^{-1}$. It is quite remarkable that this formula holds in the full planar case (without assuming Z-invariance), in the finite and infinite cases.

* In the finite case, we do not need positivity of the coupling constants $J$. In particular if the coupling constants are all negative, Theorem 5.4 expresses probabilities of the dimer model on the non-bipartite graph $G^F$ with positive weights $e^{-2J_e} > 1$ on external edges, as a function of the inverse Kasteleyn operator of a "dimer model" on the bipartite graph $G^Q$ with negative weights ($\tanh(J_e) < 0$) on quadrangle edges parallel to edges of $G$. Having a negative weight for an edge amounts to reversing its orientation. From a physics point of view, this amounts to adding defects or creating vortices at each face where this new orientation is not Kasteleyn. The physics paper [61] considers bipartite models with negative weights and somehow describes the non-Harnacity of the associated spectral curves.

* Coefficients of the inverse Kasteleyn operator $(K^F)^{-1}$ not only allow to express the dimer Boltzmann/Gibbs measure $P_{\text{dimer}}^F$, but are also related to important observables of the Ising model. By [16], the coefficient $(K^F)_{b,b}^{-1}$ is essentially the spin-observable of [18] when fixing one vertex to be on the boundary of the domain; by [32] the coefficient $(K^F)_{a,a}^{-1}$ is the fermionic spinor correlator of [39] and by [63], it is related to the FK-Ising observable [67, 68, 18] when fixing one vertex on the boundary of the domain, taking appropriate boundary conditions, up to normalization.

* Consider a fixed A-vertex $a$ and $a_1, a_2, a_3$ as in Figure 27 below, such that the decorations of $a_1, a_2, a_3$ are distinct from that of $a$. Then using the identity $[\tilde{K}^Q(\tilde{K}^Q)^{-1}]_{b,b} = 0$, from (5.13) we immediately obtain the Dotsenko three-terms relation [31, 59, 16], see also Definition 2.1. of [15].

$$ \varepsilon_{b,a_1} (K^F)_{a_1,a}^{-1} + \varepsilon_{b,a_2} \tanh(2J_e) (K^F)_{a_2,a}^{-1} + \varepsilon_{b,b'} \varepsilon_{b',a_3} \cosh^{-1}(2J_e) (K^F)_{a_3,a}^{-1} = 0. $$

<figure><img src="image_30.png" /><figcaption>Figure 27: Notation for the Dotsenko three-terms relation.</figcaption></figure>

*Proof.* The expressions for coefficients are obtained from Formula (5.10) and by returning to the definition of the matrices $I_{W,A}$, $D_{b,A}$, $M$ and $X$. The inverse of $X$ is computed by blocks. For an edge $bb'$ of $G^F$ corresponding to an edge $e^*$ of $G^*$, let $b, b'$ be the black vertices of the quadrangle of $G^Q$ traversed by the edge $bb'$. Then the inverse of the corresponding block is,

$$ \left( \begin{pmatrix} 1 & K_{b',b}^F \\ K_{b,b'} & 1 \end{pmatrix} \right)_{(b,b')}^{-1} = \frac{1}{1 + |K_{b,b'}^F|^2} \left( \begin{pmatrix} 1 & K_{b,b'}^F \\ K_{b',b} & 1 \end{pmatrix} \right)_{(b,b')}^{(b,b)}. \quad (5.15) $$

In the finite case, the matrix $X$ also has a size 1, identity block for each boundary B-vertex $b$ of $G^F$ and its closest black vertices $b$ of $G^Q$. This ends the proof of formulas (5.11), (5.13), (5.14) for coefficients and we now turn to the proof of (5.10).
The expressions for $(K^F)_{B,A}^{-1}$, $(K^F)_{B,A}^{-1}$, $(K^F)_{A,B}^{-1}$ are a direct consequence of Proposition 5.2. This is not the case of $(K^F)_{A,A}^{-1}$ which requires proving additional identities. From Proposition 5.2 we know that,

$$(K^F)_{A,A}^{-1} = -M'[(X\tilde{K}^Q I_{W,A})^t]^{-1} = (X\tilde{K}^Q I_{W,A})^{-1}(M')^t = -I_{A,W}(\tilde{K}^Q)^{-1}X^{-1}K_{B,B}^F(K_{A,B}^F)^{-1},$$

where in the second and third equalities we used skew-symmetry of $K^F$, and in the third the definition of $M'$. We thus need to prove that

$$ \begin{align} -I_{A,W}(\tilde{K}^Q)^{-1}X^{-1}K_{B,B}^F(K_{A,B}^F)^{-1} &= -\frac{1}{2}I_{A,W}(\tilde{K}^Q)^{-1}D_{B,A} + \kappa_{A,A} \nonumber \\ \Leftrightarrow -I_{A,W}(\tilde{K}^Q)^{-1}X^{-1}K_{B,B}^F(K_{A,B}^F)^{-1} &= -I_{A,W}(\tilde{K}^Q)^{-1}\left[\frac{1}{2}D_{B,A} - \tilde{K}^Q I_{W,A}\kappa_{A,A}\right] \nonumber \\ \Leftrightarrow X^{-1}K_{B,B}^F(K_{A,B}^F)^{-1} &= \frac{1}{2}D_{B,A} - \tilde{K}^Q I_{W,A}\kappa_{A,A} \nonumber \\ \Leftrightarrow 2X^{-1}K_{B,B}^F &= D_{B,A}K_{A,B}^F - 2\tilde{K}^Q I_{W,A}\kappa_{A,A}K_{A,B}^F \tag{5.16} \end{align} $$

so let us prove (5.16). We will be using the notation of Figure 28 below.

<figure><img src="image_32.png" /><figcaption>Figure 28: Notation for the proof of Equation (5.16).</figcaption></figure>

We need to introduce an additional matrix, the matrix $D_{A,B}$, which has rows indexed by A-vertices and columns by B-vertices of $G^F$. It is diagonal: to an A-vertex a corresponds the unique B-vertex b such that b comes before a in the cw ordering of the triangle containing a and b. The diagonal coefficient $d_{a,b}$ is:

$$d_{a,b} = \frac{1}{2}\varepsilon_{b,a}.$$

For example to $a_2$ of Figure 28 corresponds the vertex b, and the coefficient $d_{a_2,b} = \frac{1}{2}\varepsilon_{b,a_2}$. Let us first show that

$$\kappa_{A,A}K_{A,B}^F = -D_{A,B}. \tag{5.17}$$

Consider a B-vertex b of $G^F$. Then, the coefficient $(\kappa_{A,A}K_{A,B}^F)_{a,b}$ of the LHS of (5.17) is a priori non-zero for all A-vertices a belonging to the same decoration as b. We have,

$$(\kappa_{A,A}K_{A,B}^F)_{a,b} = \kappa_{a,a_1}K_{a_1,b}^F + \kappa_{a,a_2}K_{a_2,b}^F = \kappa_{a,a_1}\varepsilon_{a_1,b} + \kappa_{a,a_2}\varepsilon_{a_2,b}.$$

Returning to the definition of the matrix $\Lambda$, as long as $a \neq a_2$, we have that $\kappa_{a,a_1} = \kappa_{a,a_2}\varepsilon_{a_2,a_1}$ implying,

$$(\kappa_{A,A}K_{A,B}^F)_{a,b} = \kappa_{a,a_2}[\varepsilon_{a_2,a_1}\varepsilon_{a_1,b} + \varepsilon_{a_2,b}] = 0 = -d_{a,b},$$

since the orientation around the triangle $a_1, a_2, b$ is Kasteleyn and using the definition of $D_{A,B}$. When $a = a_2$, then $\kappa_{a_2,a_1} = -\frac{1}{4}\varepsilon_{a_2,a_1}$, $\kappa_{a_2,a_2} = \frac{1}{4}$; thus

$$(\kappa_{A,A}K_{A,B}^F)_{a_2,b} = -\frac{1}{4}\varepsilon_{a_2,a_1}\varepsilon_{a_1,b} + \frac{1}{4}\varepsilon_{a_2,b} = -\frac{1}{2}\varepsilon_{b,a_2} = -d_{b,a_2},$$
The expressions for $(K^F)_{B,A}^{-1}$, $(K^F)_{B,A}^{-1}$, $(K^F)_{A,B}^{-1}$ are a direct consequence of Proposition 5.2. This is not the case of $(K^F)_{A,A}^{-1}$ which requires proving additional identities. From Proposition 5.2 we know that,

$$(K^F)_{A,A}^{-1} = -M'[(X\tilde{K}^Q I_{W,A})^t]^{-1} = (X\tilde{K}^Q I_{W,A})^{-1}(M')^t = -I_{A,W}(\tilde{K}^Q)^{-1}X^{-1}K_{B,B}^F(K_{A,B}^F)^{-1},$$

where in the second and third equalities we used skew-symmetry of $K^F$, and in the third the definition of $M'$. We thus need to prove that

$$ \begin{align} -I_{A,W}(\tilde{K}^Q)^{-1}X^{-1}K_{B,B}^F(K_{A,B}^F)^{-1} &= -\frac{1}{2}I_{A,W}(\tilde{K}^Q)^{-1}D_{B,A} + \kappa_{A,A} \nonumber \\ \Leftrightarrow -I_{A,W}(\tilde{K}^Q)^{-1}X^{-1}K_{B,B}^F(K_{A,B}^F)^{-1} &= -I_{A,W}(\tilde{K}^Q)^{-1}\left[\frac{1}{2}D_{B,A} - \tilde{K}^Q I_{W,A}\kappa_{A,A}\right] \nonumber \\ \Leftrightarrow X^{-1}K_{B,B}^F(K_{A,B}^F)^{-1} &= \frac{1}{2}D_{B,A} - \tilde{K}^Q I_{W,A}\kappa_{A,A} \nonumber \\ \Leftrightarrow 2X^{-1}K_{B,B}^F &= D_{B,A}K_{A,B}^F - 2\tilde{K}^Q I_{W,A}\kappa_{A,A}K_{A,B}^F \tag{5.16} \end{align} $$

so let us prove (5.16). We will be using the notation of Figure 28 below.

<figure><img src="image_31.png" /><figcaption>Figure 28: Notation for the proof of Equation (5.16).</figcaption></figure>

We need to introduce an additional matrix, the matrix $D_{A,B}$, which has rows indexed by A-vertices and columns by B-vertices of $G^F$. It is diagonal: to an A-vertex a corresponds the unique B-vertex b such that b comes before a in the cw ordering of the triangle containing a and b. The diagonal coefficient $d_{a,b}$ is:

$$d_{a,b} = \frac{1}{2}\varepsilon_{b,a}.$$

For example to $a_2$ of Figure 28 corresponds the vertex b, and the coefficient $d_{a_2,b} = \frac{1}{2}\varepsilon_{b,a_2}$. Let us first show that

$$\kappa_{A,A}K_{A,B}^F = -D_{A,B}. \tag{5.17}$$

Consider a B-vertex b of $G^F$. Then, the coefficient $(\kappa_{A,A}K_{A,B}^F)_{a,b}$ of the LHS of (5.17) is a priori non-zero for all A-vertices a belonging to the same decoration as b. We have,

$$(\kappa_{A,A}K_{A,B}^F)_{a,b} = \kappa_{a,a_1}K_{a_1,b}^F + \kappa_{a,a_2}K_{a_2,b}^F = \kappa_{a,a_1}\varepsilon_{a_1,b} + \kappa_{a,a_2}\varepsilon_{a_2,b}.$$

Returning to the definition of the matrix $\Lambda$, as long as $a \neq a_2$, we have that $\kappa_{a,a_1} = \kappa_{a,a_2}\varepsilon_{a_2,a_1}$ implying,

$$(\kappa_{A,A}K_{A,B}^F)_{a,b} = \kappa_{a,a_2}[\varepsilon_{a_2,a_1}\varepsilon_{a_1,b} + \varepsilon_{a_2,b}] = 0 = -d_{a,b},$$

since the orientation around the triangle $a_1, a_2, b$ is Kasteleyn and using the definition of $D_{A,B}$. When $a = a_2$, then $\kappa_{a_2,a_1} = -\frac{1}{4}\varepsilon_{a_2,a_1}$, $\kappa_{a_2,a_2} = \frac{1}{4}$; thus

$$(\kappa_{A,A}K_{A,B}^F)_{a_2,b} = -\frac{1}{4}\varepsilon_{a_2,a_1}\varepsilon_{a_1,b} + \frac{1}{4}\varepsilon_{a_2,b} = -\frac{1}{2}\varepsilon_{b,a_2} = -d_{b,a_2},$$
using again the Kasteleyn orientation around the triangle and the definition of $D_{A,B}$, thus ending the proof of (5.17). Plugging (5.17) into (5.16) leaves us with showing the equivalent

$$2X^{-1}K_{B,B}^{F} = D_{B,A}K_{A,B}^{F} + 2\tilde{K}^{Q}I_{W,A}D_{A,B}. \qquad (5.18)$$

**Infinite case.** Let b be a vertex of $G^Q$, then the coefficient $(2X^{-1}K_{B,B}^F)_{b,\tilde{b}}$ of the LHS of (5.18) is non-zero when $\tilde{b} \in \{b, b'\}$. Recalling the computation of $X^{-1}$ given in (5.15), we have

$$\begin{aligned} 2(X^{-1}K_{B,B}^{F})_{b,b} &= \frac{2}{1 + |K_{b,b'}^{F}|^2} \times (K_{b,b'}^{F})_{K_{b',b}'} = e^{2J} \cosh^{-1}(2J) \epsilon_{b,b'} \epsilon_{b',b} e^{-4J} \\ &= -e^{-2J} \cosh^{-1}(2J) = -1 + \tanh(2J), \text{ using (5.8).} \end{aligned}$$
$$\begin{aligned} 2(X^{-1}K_{B,B}^{F})_{b,b'} &= \frac{2}{1 + |K_{b,b'}^{F}|^2} \times 1 \times K_{b,b'}^{F} = e^{2J} \cosh^{-1}(2J) \epsilon_{b,b'} e^{-2J} \\ &= \epsilon_{b,b'} \cosh^{-1}(2J). \end{aligned}$$

The RHS $(D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,\tilde{b}}$ a priori has non-zero coefficients when $\tilde{b} \in \{b, b', b_1\}$, and

$$\begin{aligned} (D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,b} &= \epsilon_{b,a_1}\epsilon_{a_1,b} + 2\tilde{K}_{b,w_2}^Q I_{w,a_2}\epsilon_{b,a_2} \\ &= -1 + \epsilon_{b,a_2}\tanh(2J_e)\epsilon_{b,a_2} = -1 + \tanh(2J_e). \end{aligned}$$
$$\begin{aligned} (D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,b'} &= 0 + 2\tilde{K}_{b,w_4}^Q I_{w_4,a_4}\epsilon_{b',a_4} \\ &= 0 + \epsilon_{b,b'}\epsilon_{b',a_4}\cosh^{-1}(2J_e)\epsilon_{b',a_4} = \epsilon_{b,b'}\cosh^{-1}(2J_e). \end{aligned}$$
$$\begin{aligned} (D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,b_1} &= \epsilon_{b,a_1}\epsilon_{a_1,b_1} + \tilde{K}_{b,w_1}^Q I_{w_1,a_1}\epsilon_{b_1,a_1} \\ &= -\epsilon_{b,a_1}\epsilon_{a_1,b_1} + \epsilon_{b,a_1}\epsilon_{b_1,a_1} = 0, \end{aligned}$$

and hence Equation (5.18) is proved in the infinite case.

**Finite, boundary case.** Consider a black vertex b of $G^Q$. As long as b is not a boundary vertex of $G^Q$, the argument is as in the infinite case, so we suppose that b is a boundary vertex. The coefficient $(2X^{-1}K_{B,B}^F)_{b,\tilde{b}}$ of the LHS of (5.18) is zero for all $\tilde{b} \in B$. The coefficient $(D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,\tilde{b}}$ of the RHS of (5.18) is a priori non-zero when $\tilde{b} \in \{b, b_1\}$, and we have

$$\begin{aligned} (D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,b} &= \epsilon_{b,a_1}\epsilon_{a_1,b} + 2\tilde{K}_{b,w_2}^Q I_{w,a_2}\epsilon_{b,a_2} \\ &= -1 + \epsilon_{b,a_2}\epsilon_{b,a_2} = 0, \end{aligned}$$
$$\begin{aligned} (D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,b_1} &= 0, \end{aligned}$$

where in the computation for $\tilde{b} = b$ we have used that the boundary coefficient $\tilde{K}_{b,w_2}^Q = \epsilon_{b,a_2}$. The computation for $\tilde{b} = b_1$ is as in the infinite case. This ends the proof of the finite, boundary case of (5.18) and the proof of Theorem 5.4. $\square$

**Example 5.6.** As an example we give the probability of single edges occurring in dimer configurations of $G^F$ chosen with respect to the Boltzmann measure $P_{\text{dimer}}^F$ in the finite case, or the Gibbs measure $P_{\text{dimer}}^F$ in the infinite case, as a function of edge probabilities of the dimer model on $G^Q$. Details of computations are given in Appendix B.3. Using the
using again the Kasteleyn orientation around the triangle and the definition of $D_{A,B}$, thus ending the proof of (5.17). Plugging (5.17) into (5.16) leaves us with showing the equivalent

$$2X^{-1}K_{B,B}^{F} = D_{B,A}K_{A,B}^{F} + 2\tilde{K}^{Q}I_{W,A}D_{A,B}. \qquad (5.18)$$

**Infinite case.** Let b be a vertex of $G^Q$, then the coefficient $(2X^{-1}K_{B,B}^F)_{b,\tilde{b}}$ of the LHS of (5.18) is non-zero when $\tilde{b} \in \{b, b'\}$. Recalling the computation of $X^{-1}$ given in (5.15), we have

$$\begin{aligned} 2(X^{-1}K_{B,B}^{F})_{b,b} &= \frac{2}{1 + |K_{b,b'}^{F}|^2} \times (K_{b,b'}^{F})K_{b',b}^{F} = e^{2J} \cosh^{-1}(2J)\epsilon_{b,b'}\epsilon_{b',b}e^{-4J} \\ &= -e^{-2J} \cosh^{-1}(2J) = -1 + \tanh(2J), \text{ using (5.8).} \end{aligned}$$

$$\begin{aligned} 2(X^{-1}K_{B,B}^{F})_{b,b'} &= \frac{2}{1 + |K_{b,b'}^{F}|^2} \times 1 \times K_{b,b'}^{F} = e^{2J} \cosh^{-1}(2J)\epsilon_{b,b'} e^{-2J} \\ &= \epsilon_{b,b'} \cosh^{-1}(2J). \end{aligned}$$

The RHS $(D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,\tilde{b}}$ a priori has non-zero coefficients when $\tilde{b} \in \{b, b', b_1\}$, and

$$(D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,b} = \epsilon_{b,a_1}\epsilon_{a_1,b} + 2\tilde{K}_{b,w_2}^Q I_{w,a_2}\epsilon_{b,a_2} \\ = -1 + \epsilon_{b,a_2}\tanh(2J_e)\epsilon_{b,a_2} = -1 + \tanh(2J_e).$$

$$(D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,b'} = 0 + 2\tilde{K}_{b,w_4}^Q I_{w_4,a_4}\epsilon_{b',a_4} \\ = 0 + \epsilon_{b,b'}\epsilon_{b',a_4}\cosh^{-1}(2J_e)\epsilon_{b',a_4} = \epsilon_{b,b'}\cosh^{-1}(2J_e).$$

$$(D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,b_1} = \epsilon_{b,a_1}\epsilon_{a_1,b_1} + \tilde{K}_{b,w_1}^Q I_{w_1,a_1}\epsilon_{b_1,a_1} \\ = -\epsilon_{b,a_1}\epsilon_{a_1,b_1} + \epsilon_{b,a_1}\epsilon_{b_1,a_1} = 0,$$

and hence Equation (5.18) is proved in the infinite case.

**Finite, boundary case.** Consider a black vertex b of $G^Q$. As long as b is not a boundary vertex of $G^Q$, the argument is as in the infinite case, so we suppose that b is a boundary vertex. The coefficient $(2X^{-1}K_{B,B}^F)_{b,\tilde{b}}$ of the LHS of (5.18) is zero for all $\tilde{b} \in B$. The coefficient $(D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,\tilde{b}}$ of the RHS of (5.18) is a priori non-zero when $\tilde{b} \in \{b, b_1\}$, and we have

$$(D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,b} = \epsilon_{b,a_1}\epsilon_{a_1,b} + 2\tilde{K}_{b,w_2}^Q I_{w,a_2}\epsilon_{b,a_2} \\ = -1 + \epsilon_{b,a_2}\epsilon_{b,a_2} = 0,$$

$$(D_{B,A}K_{A,B}^F + 2\tilde{K}^Q I_{W,A}D_{A,B})_{b,b_1} = 0,$$

where in the computation for $\tilde{b} = b$ we have used that the boundary coefficient $\tilde{K}_{b,w_2}^Q = \epsilon_{b,a_2}$. The computation for $\tilde{b} = b_1$ is as in the infinite case. This ends the proof of the finite, boundary case of (5.18) and the proof of Theorem 5.4. $\square$

**Example 5.6.** As an example we give the probability of single edges occurring in dimer configurations of $G^F$ chosen with respect to the Boltzmann measure $P_{\text{dimer}}^F$ in the finite case, or the Gibbs measure $P_{\text{dimer}}^F$ in the infinite case, as a function of edge probabilities of the dimer model on $G^Q$. Details of computations are given in Appendix B.3. Using the
notation of Figure 23, we have

$$P_{\text{dimer}}^{\text{F}}(aa') = \frac{1}{4} - \frac{P_{\text{dimer}}^{\text{Q}}(bw')}{2 \tanh(2J)}$$

$$P_{\text{dimer}}^{\text{F}}(ab) = P_{\text{dimer}}^{\text{F}}(a'b) = \frac{1}{4} + \frac{P_{\text{dimer}}^{\text{Q}}(bw')}{2 \tanh(2J)}$$

$$P_{\text{dimer}}^{\text{F}}(bb') = \frac{1}{2} - \frac{P_{\text{dimer}}^{\text{Q}}(bw')}{\tanh(2J)}.$$

## 5.3 In the Z-invariant case

We restrict to the case where the graph G is isoradial, finite or infinite, with Ising coupling constants J given by (2.6), dimer weights on $G^F$ by (2.7) and dimer weights on $G^Q$ by (2.8). The main result of this section is Corollary 5.8 relating the inverse Kasteleyn operator $(K^F)^{-1}$ to the inverse $Z^u$-Dirac operator using Theorem 5.4, Example 4.15, Corollaries 4.18 and 4.16, and to the Z-massive Green functions. This proves one of the main results of this paper, namely that the contour Ising Boltzmann/Gibbs measures can be computed from the inverse $Z^u$-Dirac operator, and also from the Z-massive Green functions. As a byproduct, in the infinite this also gives a direct alternative way of proving the local formula of [12] for $(K^F)^{-1}$, where the locality is seen as directly inherited from the Green function. Note that as for the local expression of $(K^Q)^{-1}$, it is not immediate to see equality with the expression of [12].

We first relate the real and complex bipartite Kasteleyn matrices $\tilde{K}^Q$ and $K^Q$ of the graph $G^Q$; we use Appendix A.2. Define the following function $q$ on pairs of vertices of $G^Q$, inductively on edges. For every edge bw of $G^Q$, let

$$q_{b,w} = \frac{K_{b,w}^{Q}}{\tilde{K}_{b,w}^{Q}} = \frac{e^{i \frac{\alpha + \beta}{2}}}{\varepsilon_{b,w}}, \quad q_{w,b} = q_{b,w}^{-1}.$$

For every pair of vertices x, y of $G^Q$, let $q_{x,y} = \prod_{(x',y') \in \gamma_{x,y}} q_{x',y'}$, where $\gamma_{x,y}$ is an edge-path of $G^Q$ from x to y. Then, since the matrices $\tilde{K}^Q$ and $K^Q$ satisfy the alternating product condition around every face/inner face of $G^Q$ if the graph is infinite/finite [53, 45], the function $q$ is well defined.

Consider a fixed vertex $x_0$ of $G^Q$, and define the diagonal matrices $D^{x_0,B}$, $D^{x_0,W}$ on black, resp. white vertices, of $G^Q$ by:

$$\forall b \in B, \quad D_{b,b}^{x_0,B} = q_{x_0,b}, \quad \forall w \in W, \quad D_{w,w}^{x_0,W} = q_{x_0,w}^{-1}.$$

By [53, 45], see also Appendix A.2, we have

$$\tilde{K}^{Q} = D^{x_0,B} K^{Q} D^{x_0,W},$$

thus implying the following lemma:

**Lemma 5.7** ([53, 45]). Consider the function $q$ as defined above. Then, coefficients of the inverse of the matrices $\tilde{K}^Q$ and $K^Q$ are related by the following, for every white vertex w and every black vertex b of $G^Q$,

$$(\tilde{K}^Q)_{w,b}^{-1} = q_{b,w} (K^Q)_{w,b}^{-1}.$$

Note that the above matrix relation holds in the finite and infinite cases because coefficients of the diagonal matrices are finite and uniformly bounded away from 0.

Cases 2. and 3. of Theorem 5.4 directly relate coefficients of $(K^F)^{-1}$ to $(\tilde{K}^Q)_{w,b}^{-1}$. Using Lemma 5.7, Corollary 4.16 (infinite case), Corollary 4.18 and Remark 4.19 (finite case),
notation of Figure 23, we have

$$P_{\text{dimer}}^{\text{F}}(aa') = \frac{1}{4} - \frac{P_{\text{dimer}}^{\text{Q}}(bw')}{2 \tanh(2J)}$$

$$P_{\text{dimer}}^{\text{F}}(ab) = P_{\text{dimer}}^{\text{F}}(a'b) = \frac{1}{4} + \frac{P_{\text{dimer}}^{\text{Q}}(bw')}{2 \tanh(2J)}$$

$$P_{\text{dimer}}^{\text{F}}(bb') = \frac{1}{2} - \frac{P_{\text{dimer}}^{\text{Q}}(bw')}{\tanh(2J)}.$$

## 5.3 In the Z-invariant case

We restrict to the case where the graph G is isoradial, finite or infinite, with Ising coupling constants J given by (2.6), dimer weights on $G^F$ by (2.7) and dimer weights on $G^Q$ by (2.8). The main result of this section is Corollary 5.8 relating the inverse Kasteleyn operator $(K^F)^{-1}$ to the inverse $Z^u$-Dirac operator using Theorem 5.4, Example 4.15, Corollaries 4.18 and 4.16, and to the Z-massive Green functions. This proves one of the main results of this paper, namely that the contour Ising Boltzmann/Gibbs measures can be computed from the inverse $Z^u$-Dirac operator, and also from the Z-massive Green functions. As a byproduct, in the infinite this also gives a direct alternative way of proving the local formula of [12] for $(K^F)^{-1}$, where the locality is seen as directly inherited from the Green function. Note that as for the local expression of $(K^Q)^{-1}$, it is not immediate to see equality with the expression of [12].

We first relate the real and complex bipartite Kasteleyn matrices $\tilde{K}^Q$ and $K^Q$ of the graph $G^Q$; we use Appendix A.2. Define the following function $q$ on pairs of vertices of $G^Q$, inductively on edges. For every edge bw of $G^Q$, let

$$q_{b,w} = \frac{K_{b,w}^{Q}}{\tilde{K}_{b,w}^{Q}} = \frac{e^{i \frac{\alpha + \beta}{2}}}{\varepsilon_{b,w}}, \quad q_{w,b} = q_{b,w}^{-1}.$$

For every pair of vertices x, y of $G^Q$, let $q_{x,y} = \prod_{(x',y') \in \gamma_{x,y}} q_{x',y'}$, where $\gamma_{x,y}$ is an edge-path of $G^Q$ from x to y. Then, since the matrices $\tilde{K}^Q$ and $K^Q$ satisfy the alternating product condition around every face/inner face of $G^Q$ if the graph is infinite/finite [53, 45], the function $q$ is well defined.

Consider a fixed vertex $x_0$ of $G^Q$, and define the diagonal matrices $D^{x_0,B}$, $D^{x_0,W}$ on black, resp. white vertices, of $G^Q$ by:

$$\forall b \in B, \quad D_{b,b}^{x_0,B} = q_{x_0,b}, \quad \forall w \in W, \quad D_{w,w}^{x_0,W} = q_{x_0,w}^{-1}.$$

By [53, 45], see also Appendix A.2, we have

$$\tilde{K}^Q = D^{x_0,B} K^Q D^{x_0,W},$$

thus implying the following lemma:

**Lemma 5.7 ([53, 45]).** Consider the function $q$ as defined above. Then, coefficients of the inverse of the matrices $\tilde{K}^Q$ and $K^Q$ are related by the following, for every white vertex w and every black vertex b of $G^Q$,

$$(\tilde{K}^Q)_{w,b}^{-1} = q_{b,w} (K^Q)_{w,b}^{-1}.$$

Note that the above matrix relation holds in the finite and infinite cases because coefficients of the diagonal matrices are finite and uniformly bounded away from 0.

Cases 2. and 3. of Theorem 5.4 directly relate coefficients of $(K^F)^{-1}$ to $(\tilde{K}^Q)_{w,b}^{-1}$. Using Lemma 5.7, Corollary 4.16 (infinite case), Corollary 4.18 and Remark 4.19 (finite case),
these coefficients of $(K^F)^{-1}$ are easily expressed using the inverse $Z^u$-Dirac operator and the Z-massive Green functions.

Case 4. of Theorem 5.4 uses Case 1. so we are left with considering Case 1. The notation used are summarized in Figure 29 below.

<figure><img src="image_33.png" /><figcaption>Figure 29: Left: notation for initial vertices. Right: notation for final vertices.</figcaption></figure>

**Corollary 5.8 (Case 1).**

* For every $a \in A$ and every $b \in B$ such that, when the graph $G^F$ is moreover finite, $b$ is not a boundary vertex:

$$ (K^F)_{a,b}^{-1} = q_{b,w} \frac{\mathrm{cn}\left(\frac{K-\theta_f}{2}\right) \frac{1+(k')^{-1} \mathrm{dn}(\theta_f)}{2}}{\left[(K^Q)_{w,b}^{-1} \mathrm{cn}\left(\frac{K-\theta_f}{2}\right) - i(K^Q)_{w,b'}^{-1} \mathrm{cn}\left(\frac{K-\theta_f}{2}\right)\right]} $$

* Moreover, as a function of the inverse $Z^u$-Dirac operator, where $u^f = \frac{\alpha_f + \beta_f}{2} + K$, we have:

Infinite case.

$$ \begin{aligned} (K^F)_{a,b}^{-1} &= q_{b,w} e^{i \frac{\beta_f - \beta_i}{2} (k') \frac{1}{2} \frac{\mathrm{cn}\left(\frac{K-\theta_f}{2}\right) (1+(k')^{-1} \mathrm{dn}(\theta_f))}{2[\mathrm{cn}(\theta_f) \mathrm{sn}(\theta_f)]^{\frac{1}{2}}}} \left( \mathrm{cn}(\mathbf{u}_{\beta_i}^f) K(\mathbf{u}^f)_{\bar{v},w}^{-1} - i \mathrm{sn}(\mathbf{u}_{\beta_i}^f) K(\mathbf{u}^f)_{\bar{f},w}^{-1} \right) \\ &= q_{b,w} e^{-i \frac{\alpha_f + \beta_i}{2} \frac{\mathrm{cn}\left(\frac{K-\theta_f}{2}\right) (1+(k')^{-1} \mathrm{dn}(\theta_f))}{2}} \times \\ & \quad \left( \frac{\mathrm{cn}(\mathbf{u}_{\beta_i}^f)}{\mathrm{cn}(\theta_f)} (G_{\bar{v},v_2}^m - G_{\bar{v},v_1}^m) - \frac{\mathrm{sn}(\mathbf{u}_{\beta_i}^f)}{\mathrm{sn}(\theta_f)} \left( \mathrm{nd}\left(\frac{K-\theta_f}{2}\right) G_{\bar{f},f_2}^{m,*} - \mathrm{nd}\left(\frac{K+\theta_f}{2}\right) G_{\bar{f},f_1}^{m,*} \right) \right). \end{aligned} $$

Finite case. If $a$ corresponds to a vertex $w^c$ for some rhombus pair of $\mathcal{R}^\partial$, we then use the notation of Figure 29 (2nd quadrant on the left):

$$ (K^F)_{a,b}^{-1} = q_{b,w} e^{i \frac{\beta_f}{2} (k') \frac{1}{2} \mathrm{sn}(\theta_i^\partial) \mathbb{I}_{\{w \in \mathcal{W} \in \mathcal{R}^\partial\}} \frac{\mathrm{cn}\left(\frac{K-\theta_f}{2}\right) (1 + (k')^{-1} \mathrm{dn}(\theta_f))}{2[\mathrm{cn}(\theta_f) \mathrm{sn}(\theta_f)]^{\frac{1}{2}}}} \times \\ \times \left( \mathbb{I}_{\{w \neq w^{c,r}\}} t(\mathbf{u}^f)_{w,\bar{v}} K^\partial(\mathbf{u}^f)_{\bar{v},w}^{-1} + t(\mathbf{u}^f)_{w,\bar{f}} K^\partial(\mathbf{u}^f)_{\bar{f},w}^{-1} \right), $$

where the coefficients $K^\partial(\mathbf{u}^f)_{\bar{v},w}^{-1}$, $K^\partial(\mathbf{u}^f)_{\bar{f},w}^{-1}$ are expressed using the Green functions $G^{m,\partial}(u)$ and $G^{m,*}(u)$ in Example 3.18.

*Proof.* Let us prove the first point. We first compare $(\tilde{K}^Q)_{w,b}^{-1}$ and $(\tilde{K}^Q)_{w,b'}^{-1}$ of Theorem 5.4. Using the notation of Figure 29 (right), we have

$$ (\tilde{K}^Q)_{w,b'}^{-1} \varepsilon_{b',b} = -i q_{b,w} (K^Q)_{w,b'}^{-1}, $$

that is because, omitting the subscript “f”,

$$ q_{b',w} \varepsilon_{b',b} = q_{b',w} q_{w,b} q_{b,w} \varepsilon_{b',b} = \frac{e^{i \frac{\beta - \pi + \alpha}{2}} \varepsilon_{b,w}}{\varepsilon_{b',w}} \frac{\varepsilon_{b',b} q_{b,w}}{e^{i \frac{\beta + \alpha}{2}} \varepsilon_{b',b} \varepsilon_{b,a'}} = -i \frac{\varepsilon_{b,a'}}{\varepsilon_{b',b'} \varepsilon_{b,a'}} \varepsilon_{b',b} q_{b,w} = -iq_{b,w}. $$
these coefficients of $(K^F)^{-1}$ are easily expressed using the inverse $Z^u$-Dirac operator and the Z-massive Green functions.

Case 4. of Theorem 5.4 uses Case 1. so we are left with considering Case 1. The notation used are summarized in Figure 29 below.

<figure><img src="image_32.png" /><figcaption>Figure 29: Left: notation for initial vertices. Right: notation for final vertices.</figcaption></figure>

**Corollary 5.8 (Case 1).**

* For every $a \in A$ and every $b \in B$ such that, when the graph $G^F$ is moreover finite, $b$ is not a boundary vertex:

$$ (K^F)_{a,b}^{-1} = q_{b,w} \frac{\mathrm{cn}\left(\frac{K-\theta_f}{2}\right) \frac{1+(k')^{-1} \mathrm{dn}(\theta_f)}{2}}{\left[(K^Q)_{w,b}^{-1} \mathrm{cn}\left(\frac{K-\theta_f}{2}\right) - i(K^Q)_{w,b'}^{-1} \mathrm{cn}\left(\frac{K-\theta_f}{2}\right)\right]} $$

* Moreover, as a function of the inverse $Z^u$-Dirac operator, where $u^f = \frac{\alpha_f + \beta_f}{2} + K$, we have:

Infinite case.

$$ (K^F)_{a,b}^{-1} = q_{b,w} e^{i \frac{\beta_f - \beta_i}{2} (k') \frac{1}{2} \frac{\mathrm{cn}\left(\frac{K-\theta_f}{2}\right) (1+(k')^{-1} \mathrm{dn}(\theta_f))}{2[\mathrm{cn}(\theta_f) \mathrm{sn}(\theta_f)]^{\frac{1}{2}}}} \left( \mathrm{cn}(\mathbf{u}_{\beta_i}^f) K(\mathbf{u}^f)_{\bar{v},w}^{-1} - i \mathrm{sn}(\mathbf{u}_{\beta_i}^f) K(\mathbf{u}^f)_{\bar{f},w}^{-1} \right) \\ = q_{b,w} e^{-i \frac{\alpha_f + \beta_i}{2} \frac{\mathrm{cn}\left(\frac{K-\theta_f}{2}\right) (1+(k')^{-1} \mathrm{dn}(\theta_f))}{2}} \times \\ \times \left( \frac{\mathrm{cn}(\mathbf{u}_{\beta_i}^f)}{\mathrm{cn}(\theta_f)} (G_{\bar{v},v_2}^m - G_{\bar{v},v_1}^m) - \frac{\mathrm{sn}(\mathbf{u}_{\beta_i}^f)}{\mathrm{sn}(\theta_f)} \left( \mathrm{nd}\left(\frac{K-\theta_f}{2}\right) G_{\bar{f},f_2}^{m,*} - \mathrm{nd}\left(\frac{K+\theta_f}{2}\right) G_{\bar{f},f_1}^{m,*} \right) \right). $$

Finite case. If $a$ corresponds to a vertex $w^c$ for some rhombus pair of $\mathcal{R}^\partial$, we then use the notation of Figure 29 (2nd quadrant on the left):

$$ (K^F)_{a,b}^{-1} = q_{b,w} e^{i \frac{\beta_f}{2} (k') \frac{1}{2} \mathrm{sn}(\theta_i^\partial) \mathbb{I}_{\{w \in \mathcal{W}^c \in \mathcal{R}^\partial\}} \frac{\mathrm{cn}\left(\frac{K-\theta_f}{2}\right) (1 + (k')^{-1} \mathrm{dn}(\theta_f))}{2[\mathrm{cn}(\theta_f) \mathrm{sn}(\theta_f)]^{\frac{1}{2}}} \times \\ \times \left( \mathbb{I}_{\{w \neq w^c, r\}} t(\mathbf{u}^f)_{w,\bar{v}} K^\partial(\mathbf{u}^f)_{\bar{v},w}^{-1} + t(\mathbf{u}^f)_{w,\bar{f}} K^\partial(\mathbf{u}^f)_{\bar{f},w}^{-1} \right), $$

where the coefficients $K^\partial(\mathbf{u}^f)_{\bar{v},w}^{-1}$, $K^\partial(\mathbf{u}^f)_{\bar{f},w}^{-1}$ are expressed using the Green functions $G^{m,\partial}(u)$ and $G^{m,*}(u)$ in Example 3.18.

*Proof.* Let us prove the first point. We first compare $(\tilde{K}^Q)_{w,b}^{-1}$ and $(\tilde{K}^Q)_{w,b'}^{-1}$ of Theorem 5.4. Using the notation of Figure 29 (right), we have

$$ (\tilde{K}^Q)_{w,b'}^{-1} \varepsilon_{b',b} = -i q_{b,w} (K^Q)_{w,b'}^{-1}, $$

that is because, omitting the subscript “f”,

$$ q_{b',w} \varepsilon_{b',b} = q_{b',w} q_{w,b} q_{b,w} \varepsilon_{b',b} = \frac{e^{i \frac{\beta - \pi + \alpha}{2}} \varepsilon_{b,w}}{\varepsilon_{b',w}} \frac{\varepsilon_{b',b} q_{b,w}}{e^{i \frac{\beta + \alpha}{2}} \varepsilon_{b',b} \varepsilon_{b,a'}} = -i \frac{\varepsilon_{b,a'}}{\varepsilon_{b',b'} \varepsilon_{b,a'}} \varepsilon_{b',b} q_{b,w} = -iq_{b,w}. $$
We thus have,

$$ (K^f)_{a,b}^{-1} = q_{b,w} \frac{1}{1 + e^{-4J_e^f}} \left[ (K^Q)_{w,b}^{-1} - i (K^Q)_{w,b'}^{-1} e^{-2J_e^f} \right]. \quad (5.19) $$

We are left with computing the terms involving the coupling constants $J_e^f$ in the Z-invariant case. By definition, $J_e^f = \frac{1}{2} \ln \frac{1+sn \theta_f}{cn \theta_f}$. Set $u = \frac{K-\theta_f}{2}$, $v = \frac{K+\theta_f}{2}$, then $u+v = K$, $u-v = -\theta_f$, so that

$$ e^{-2J_e^f} = \frac{cn \theta_f}{1+sn \theta_f} = \frac{cn(u-v)-cn(u+v)}{sn(u+v)-sn(u-v)} = sc(u) dn(v) = \frac{sn(K-v) dn(v)}{cn(u)} = \frac{cn\left(\frac{K+\theta_f}{2}\right)}{cn\left(\frac{K-\theta_f}{2}\right)}, \quad (5.20) $$

where in the third equality we used [54, chap.2, ex.14 (iii)] and in the fourth that $sn(v+K) = cd(v)$. From this and Identity (4.18), we obtain,

$$ \frac{1}{1 + e^{-4J_e^f}} = \frac{cn^2\left(\frac{K-\theta_f}{2}\right)}{cn^2\left(\frac{K-\theta_f}{2}\right)+cn^2\left(\frac{K+\theta_f}{2}\right)} = cn^2\left(\frac{K-\theta_f}{2}\right) \frac{1+(k')^{-1}}{2} dn(\theta_f). \quad (5.21) $$

Putting together Equation (5.19), (5.20) and (5.21) ends the proof of the first point; let us now prove the second.

In the infinite case, by Example 4.15, we have

$$ (K^Q)_{w,b}^{-1} \ln \left( \frac{K+\theta_f}{2} \right) - i (K^Q)_{w,b'}^{-1} \ln \left( \frac{K-\theta_f}{2} \right) = \\ = \frac{e^{i \frac{\beta_f - \beta_i}{2}} (k')^{\frac{1}{2}}}{[cn(\theta_f) \ln(\theta_f)]^{\frac{1}{2}}} \left( \ln(\mathbf{u}_{\beta_i}^f) K(\mathbf{u}^f)_{\underline{v},w}^{-1} - i \ln(\mathbf{u}_{\beta_i}^f) K(\mathbf{u}^f)_{\underline{f},w}^{-1} \right), $$

thus proving the first line. The second line is obtained by using Example 3.18 to express $K(\mathbf{u}^f)_{\underline{v},w}^{-1}$ and $K(\mathbf{u}^f)_{\underline{f},w}^{-1}$ using the Z-massive Green functions $G^m$ and $G^{m,*}$.

In the finite case, by Example 4.15, we have

$$ (K^Q)_{w,b}^{-1} \ln \left( \frac{K-\theta_f}{2} \right) - i (K^Q)_{w,b'}^{-1} \ln \left( \frac{K+\theta_f}{2} \right) = \\ = \frac{e^{i \frac{\beta_f}{2}} (k')^{\frac{1}{2}} \ln(\theta_f) \Gamma_{\{w \in \{w^c \in \mathbb{R}^c\} \cup \{w^c, w^c, w^c\}}}}{[cn(\theta_f) \ln(\theta_f)]^{\frac{1}{2}}} \left( \Gamma_{\{w \neq w^{c,r}\}} t(\mathbf{u}^f)_{w,v} K^\partial(\mathbf{u}^f)_{\underline{v},w}^{-1} + t(\mathbf{u}^f)_{w,f} K^\partial(\mathbf{u}^f)_{\underline{f},w}^{-1} \right), $$

thus concluding the proof. $\square$

# 6 Examples

In this section we specify some of our results to two cases of interest: the critical Z-invariant Ising model, and the full Z-invariant Ising model when the underlying isoradial graph G is $\mathbb{Z}^2$ with the regular embedding.

## 6.1 Z-invariant critical case

The Z-invariant Ising model is critical when $k=0$ ($k'=1$) [55, 22, 57]. In this case, the elliptic functions $sn$, $cn$ are the trigonometric functions sin, cos and $dn \equiv 1$.

Returning to Section 3.1, the finite $Z^u$-Dirac operator $K^\partial(u)$ with boundary conditions arising from the Ising model has coefficients defined by, for every edge $wx$ of $G^D$,

$$ K^{\partial}(u)_{w,x} = e^{i \frac{\bar{\alpha}_e + \bar{\beta}_e}{2}} \begin{cases} \tan(\theta)^{\frac{1}{2}} & \text{if } x \in V \text{ and } (w,x) \notin (w^\ell, v^c) \in \mathbb{R}^{\partial,r} \\ \cot(\theta)^{\frac{1}{2}} & \text{if } x \in V^* \\ \tan(\theta)^{\frac{\cos(u_{\beta r})}{\cos(u_{\alpha \ell})}} & \text{if } (w,x) \in (w^\ell, v^c) \in \mathbb{R}^{\partial,r} \end{cases} $$

Away from the boundary, we recover the Dirac operator of [45].
We thus have,

$$ (K^f)_{a,b}^{-1} = q_{b,w} \frac{1}{1 + e^{-4J_e^f}} [(K^Q)_{w,b}^{-1} - i(K^Q)_{w,b'}^{-1} e^{-2J_e^f}]. \quad (5.19) $$

We are left with computing the terms involving the coupling constants $J_e^f$ in the Z-invariant case. By definition, $J_e^f = \frac{1}{2} \ln \frac{1+sn \theta_f}{cn \theta_f}$. Set $u = \frac{K-\theta_f}{2}$, $v = \frac{K+\theta_f}{2}$, then $u+v = K$, $u-v = -\theta_f$, so that

$$ e^{-2J_e^f} = \frac{cn \theta_f}{1+sn \theta_f} = \frac{cn(u-v)-cn(u+v)}{sn(u+v)-sn(u-v)} = sc(u) dn(v) = \frac{sn(K-v) dn(v)}{cn(u)} = \frac{cn\left(\frac{K+\theta_f}{2}\right)}{cn\left(\frac{K-\theta_f}{2}\right)}, \quad (5.20) $$

where in the third equality we used [54, chap.2, ex.14 (iii)] and in the fourth that $sn(v+K) = cd(v)$. From this and Identity (4.18), we obtain,

$$ \frac{1}{1 + e^{-4J_e^f}} = \frac{cn^2\left(\frac{K-\theta_f}{2}\right)}{cn^2\left(\frac{K-\theta_f}{2}\right)+cn^2\left(\frac{K+\theta_f}{2}\right)} = cn^2\left(\frac{K-\theta_f}{2}\right) \frac{1+(k')^{-1}}{2} dn(\theta_f). \quad (5.21) $$

Putting together Equation (5.19), (5.20) and (5.21) ends the proof of the first point; let us now prove the second.

In the infinite case, by Example 4.15, we have

$$ (K^Q)_{w,b}^{-1} \ln \left( \frac{K+\theta_f}{2} \right) - i (K^Q)_{w,b'}^{-1} \ln \left( \frac{K+\theta_f}{2} \right) = \\ = \frac{e^{i \frac{\beta_f - \beta_i}{2}} (k')^{\frac{1}{2}}}{[cn(\theta_f) \ln(\theta_f)]^{\frac{1}{2}}} \left( \ln(\mathbf{u}_{\beta_i}^f) K(\mathbf{u}^f)_{\underline{v},w}^{-1} - i \ln(\mathbf{u}_{\beta_i}^f) K(\mathbf{u}^f)_{\underline{f},w}^{-1} \right), $$

thus proving the first line. The second line is obtained by using Example 3.18 to express $K(\mathbf{u}^f)_{\underline{v},w}^{-1}$ and $K(\mathbf{u}^f)_{\underline{f},w}^{-1}$ using the Z-massive Green functions $G^m$ and $G^{m,*}$.

In the finite case, by Example 4.15, we have

$$ (K^Q)_{w,b}^{-1} \ln \left( \frac{K-\theta_f}{2} \right) - i (K^Q)_{w,b'}^{-1} \ln \left( \frac{K+\theta_f}{2} \right) = \\ = \frac{e^{i \frac{\beta_f}{2}} (k')^{\frac{1}{2}} \ln(\theta_f) \mathbf{1}_{\{w \in \{w^c \in \mathbb{R}^c\} \cap \{\theta_f \in \mathbb{R}^c\}}}}{[cn(\theta_f) \ln(\theta_f)]^{\frac{1}{2}}} \left( \mathbf{1}_{\{w \neq w^{c,r}\}} t(\mathbf{u}^f)_{w,v} K^\partial(\mathbf{u}^f)_{\underline{v},w}^{-1} + t(\mathbf{u}^f)_{w,f} K^\partial(\mathbf{u}^f)_{\underline{f},w}^{-1} \right), $$

thus concluding the proof. □

# 6 Examples

In this section we specify some of our results to two cases of interest: the critical Z-invariant Ising model, and the full Z-invariant Ising model when the underlying isoradial graph G is $\mathbb{Z}^2$ with the regular embedding.

## 6.1 Z-invariant critical case

The Z-invariant Ising model is critical when $k=0$ ($k'=1$) [55, 22, 57]. In this case, the elliptic functions $sn$, $cn$ are the trigonometric functions sin, cos and $dn \equiv 1$.

Returning to Section 3.1, the finite $Z^u$-Dirac operator $K^\partial(u)$ with boundary conditions arising from the Ising model has coefficients defined by, for every edge $wx$ of $G^D$,

$$ K^{\partial}(u)_{w,x} = e^{i \frac{\bar{\alpha}_e + \bar{\beta}_e}{2}} \begin{cases} \tan(\theta)^{\frac{1}{2}} & \text{if } x \in \mathcal{V} \text{ and } (w,x) \notin (w^\ell, v^c) \in \mathcal{R}^{\partial,r} \\ \cot(\theta)^{\frac{1}{2}} & \text{if } x \in \mathcal{V}^* \\ \tan(\theta)^{\frac{\cos(u_{\beta r})}{\cos(u_{\alpha \ell})}} & \text{if } (w,x) \in (w^\ell, v^c) \in \mathcal{R}^{\partial,r}. \end{cases} $$

Away from the boundary, we recover the Dirac operator of [45].
**Dimer model on the bipartite graph G<sup>Q</sup>, finite case** Corollary 4.18 expresses coefficients of (K<sup>Q</sup>)<sup>-1</sup> as a function of the inverse Z<sup>u</sup>-Dirac operator. When b does not belong to a boundary quadrangle (Case 1.), we have

$$ (K^Q)_{\underline{w},b}^{-1} = e^{i \frac{\bar{\beta}_f}{2} (k') \frac{1}{2} \sin(\theta_i^\partial) \mathbb{I}_{\{w \in \{w^c \in \mathcal{R}^\partial\}\}} \frac{1}{[\cos(\theta_f) \sin(\theta_f)]^{\frac{1}{2}}} [\cos(\frac{K-\theta_f}{2}) \Gamma(\mathbf{u}^f) + \cos(\frac{K+\theta_f}{2}) \Gamma(\mathbf{v}^f)] } $$

where, $\Gamma(u) = \mathbb{I}_{\{w \neq w^c, r\}} t(u)_{\underline{w}, \underline{v}} K^\partial(u)^{-1}_{\underline{v}, \underline{w}} + t(u)_{\underline{w}, \underline{f}} K^\partial(u)^{-1}_{\underline{f}, \underline{w}}$, and coefficients of $t(u)$ are given by (4.4) and (4.5). The mass of the Z-massive Laplacian is equal to 0 away from the boundary [11]; then Example 3.18 expresses coefficients of $K^\partial(\mathbf{u}^f)^{-1}$ using the Z-Green function $G^{0, \partial}(\mathbf{u}^f)$ and dual Z-Green function $G^{0, *}$; we have

$$ \begin{aligned} K^\partial(\mathbf{u}^f)^{-1}_{\underline{v}, \underline{w}} &= e^{-i \frac{\bar{\beta}_f + \bar{\beta}_f}{2}} \tan(\theta_f)^{\frac{1}{2}} (G^{0, \partial}(\mathbf{u}^f)_{\underline{v}, \underline{v}} - G^{0, \partial}(\mathbf{u}^f)_{\underline{v}, \underline{v}}) \\ K^\partial(\mathbf{u}^f)^{-1}_{\underline{f}, \underline{w}} &= -ie^{-i \frac{\bar{\beta}_f + \bar{\beta}_f}{2}} \cot(\theta_f)^{\frac{1}{2}} (\mathbb{I}_{\{w \notin \{w^c, w^r \in \mathcal{R}^\partial\}}}) G^{0, *}_{\underline{f}, \underline{f}} - G^{0, *}_{\underline{f}_1, \underline{f}_1} + \\ & \qquad + i \sum_{(v^c, f^c) \in \mathcal{R}^\partial, r} \frac{\cos(\mathbf{u}^f_{\underline{\beta}^r}) - \cos(\mathbf{u}^f_{\underline{\alpha}^r})}{\cos(\mathbf{u}^f_{\underline{\alpha}^r})} K^\partial(\mathbf{u}^f)^{-1}_{\underline{v}^c, \underline{w}} \cdot G^{0, *}_{\underline{f}^c, \underline{f}}. \end{aligned} $$

A similar expression holds for $K^\partial(\mathbf{v}^f)^{-1}$.

**Dimer model on the Fisher graph G<sup>F</sup>, finite case** Corollary 5.8 (finite case) simply becomes:

$$ (K^F)_{\underline{a}, \underline{b}}^{-1} = e^{i \frac{\bar{\beta}_f}{2} \sin(\theta_i^\partial) \mathbb{I}_{\{w \in \{w^c \in \mathcal{R}^\partial\}\}} \frac{\cos(\frac{K+\theta_f}{2})}{[\cos(\theta_f) \sin(\theta_f)]^{\frac{1}{2}}} \times \left( \mathbb{I}_{\{w \neq w^c, r\}} t(\mathbf{u}^f)_{\underline{w}, \underline{v}} K^\partial(\mathbf{u}^f)^{-1}_{\underline{v}, \underline{w}} + t(\mathbf{u}^f)_{\underline{w}, \underline{f}} K^\partial(\mathbf{u}^f)^{-1}_{\underline{f}, \underline{w}} \right). $$

**Partition function of the Ising model with + boundary conditions** Corollary 4.5, expressing the squared Z-invariant Ising partition function, holds for every $u \in C''$. When $k=0$, a nice expression is obtained by setting $u = iu'$ and taking the limit $u' \to -\infty$. We have

$$ \lim_{u' \to -\infty} \frac{\cos((iu')_\alpha)}{\cos((iv)_\beta)} = e^{i \frac{\beta - \alpha}{2}}, \quad \lim_{u' \to -\infty} \tan((iu')_\alpha) = 1. $$

As a consequence,

$$ \Delta^{m, \partial}(-i\infty)_{v,v'} = \begin{cases} -\tan(\theta) & \text{if } (v, v') \notin \{(v^c, v^\ell) \in \mathcal{R}^{\partial,r}\} \\ -e^{i \frac{\alpha^\ell - \beta^r}{2}} \tan(\theta^\partial) & \text{if } (v, v') \in \{(v^c, v^\ell) \in \mathcal{R}^{\partial,r}\} \\ \sum_{j=1}^d \tan(\theta_j) & \text{if } v' = v = v^\partial \in \{v^r, v^\ell \in \mathcal{R}^\partial\} \\ \tan(\theta^\partial)(e^{i \frac{\alpha^\ell - \beta^r}{2}} - 1) & \text{if } v' = v = v^c \in \{v^c \in \mathcal{R}^{\partial,r}\}, \end{cases} $$

where we recover the critical Laplacian of [45] away from the boundary. Corollary 4.5 becomes:

$$ [Z_{\text{Ising}}^{+}(G, J)]^2 = 2^{|V|-1} \left( \prod_{w \in W^{\partial}} \frac{1 + \sin(\theta_w)}{2 \sin(\theta_w)} \right) \sin(\theta^{\partial,r}) |\det \Delta^{m, \partial}(-i\infty)|. $$

We essentially recover the main result of [30] proving that the squared critical Z-invariant Ising model partition function is equal, up to an explicit constant, to the partition function of spanning trees with specific boundary conditions. The difference is that we here consider + boundary conditions instead of free ones, and more importantly, we use the boundary trick of Chelkak and Smirnov [18] allowing us to remove all contributions from dual spanning trees, which we could not do in [30].
**Dimer model on the bipartite graph G<sup>Q</sup>, finite case** Corollary 4.18 expresses coefficients of (K<sup>Q</sup>)<sup>-1</sup> as a function of the inverse Z<sup>u</sup>-Dirac operator. When b does not belong to a boundary quadrangle (Case 1.), we have

$$ (K^Q)_{w,b}^{-1} = e^{i \frac{\bar{\beta}_f}{2} (k') \frac{1}{2} \sin(\theta_i^\partial) \mathbb{I}_{\{w \in \{w^c \in \mathcal{R}^\partial\}\}} \frac{1}{[\cos(\theta_f) \sin(\theta_f)]^{\frac{1}{2}}} [\cos(\frac{K-\theta_f}{2}) \Gamma(\mathbf{u}^f) + \cos(\frac{K+\theta_f}{2}) \Gamma(\mathbf{v}^f)] $$

where, $\Gamma(u) = \mathbb{I}_{\{w \neq w^c, r\}} t(u)_{w,v} K^\partial(u)^{-1}_{v,w} + t(u)_{w,f} K^\partial(u)^{-1}_{f,w}$, and coefficients of $t(u)$ are given by (4.4) and (4.5). The mass of the Z-massive Laplacian is equal to 0 away from the boundary [11]; then Example 3.18 expresses coefficients of $K^\partial(\mathbf{u}^f)^{-1}$ using the Z-Green function $G^{0,\partial}(\mathbf{u}^f)$ and dual Z-Green function $G^{0,*}$; we have

$$ \begin{aligned} K^\partial(\mathbf{u}^f)^{-1}_{v,w} &= e^{-i \frac{\bar{\beta}_f + \bar{\beta}_f}{2}} \tan(\theta_f)^{\frac{1}{2}} (G^{0,\partial}(\mathbf{u}^f)_{v_2,v} - G^{0,\partial}(\mathbf{u}^f)_{v_1,v}) \\ K^\partial(\mathbf{u}^f)^{-1}_{f,w} &= -i e^{-i \frac{\bar{\beta}_f + \bar{\beta}_f}{2}} \cot(\theta_f)^{\frac{1}{2}} (\mathbb{I}_{\{w \notin \{w^c, w^r \in \mathcal{R}^\partial\}} G^{0,*}_{f_2,f} - G^{0,*}_{f_1,f}) + \\ & \quad + i \sum_{(v^c, f^c) \in \mathcal{R}^\partial, r} \frac{\cos(\mathbf{u}^f_{\beta^r}) - \cos(\mathbf{u}^f_{\alpha^\ell})}{\cos(\mathbf{u}^f_{\alpha^\ell})} K^\partial(\mathbf{u}^f)^{-1}_{v^c, w} \cdot G^{0,*}_{f^c, f}. \end{aligned} $$

A similar expression holds for $K^\partial(\mathbf{v}^f)^{-1}$.

**Dimer model on the Fisher graph G<sup>F</sup>, finite case** Corollary 5.8 (finite case) simply becomes:

$$ (K^F)_{\underline{a},b}^{-1} = e^{i \frac{\bar{\beta}_f}{2} \sin(\theta_i^\partial) \mathbb{I}_{\{w \in \{w^c \in \mathcal{R}^\partial\}\}} \frac{\cos(\frac{K+\theta_f}{2})}{[\cos(\theta_f) \sin(\theta_f)]^{\frac{1}{2}}} \times \\ \times (\mathbb{I}_{\{w \neq w^c, r\}} t(\mathbf{u}^f)_{w,v} K^\partial(\mathbf{u}^f)^{-1}_{v,w} + t(\mathbf{u}^f)_{w,f} K^\partial(\mathbf{u}^f)^{-1}_{f,w}). $$

**Partition function of the Ising model with + boundary conditions** Corollary 4.5, expressing the squared Z-invariant Ising partition function, holds for every $u \in C''$. When $k=0$, a nice expression is obtained by setting $u = iu'$ and taking the limit $u' \to -\infty$. We have

$$ \lim_{u' \to -\infty} \frac{\cos((iu')_\alpha)}{\cos((iv)_\beta)} = e^{i \frac{\beta - \alpha}{2}}, \quad \lim_{u' \to -\infty} \tan((iu')_\alpha) = 1. $$

As a consequence,

$$ \Delta^{m,\partial}(-i\infty)_{v,v'} = \begin{cases} -\tan(\theta) & \text{if } (v, v') \notin \{(v^c, v^\ell) \in \mathcal{R}^{\partial,r}\} \\ -e^{i \frac{\alpha^\ell - \beta^r}{2}} \tan(\theta^\partial) & \text{if } (v, v') \in \{(v^c, v^\ell) \in \mathcal{R}^{\partial,r}\} \\ \sum_{j=1}^d \tan(\theta_j) & \text{if } v' = v = v^\partial \in \{v^r, v^\ell \in \mathcal{R}^\partial\} \\ \tan(\theta^\partial)(e^{i \frac{\alpha^\ell - \beta^r}{2}} - 1) & \text{if } v' = v = v^c \in \{v^c \in \mathcal{R}^{\partial,r}\}, \end{cases} $$

where we recover the critical Laplacian of [45] away from the boundary. Corollary 4.5 becomes:

$$ [Z_{\text{Ising}}^{+}(G, J)]^2 = 2^{|V|-1} \left( \prod_{w \in W^{\partial}} \frac{1 + \sin(\theta_w)}{2 \sin(\theta_w)} \right) \operatorname{sn}(\theta^{\partial,r}) |\det \Delta^{m,\partial}(-i\infty)|. $$

We essentially recover the main result of [30] proving that the squared critical Z-invariant Ising model partition function is equal, up to an explicit constant, to the partition function of spanning trees with specific boundary conditions. The difference is that we here consider + boundary conditions instead of free ones, and more importantly, we use the boundary trick of Chelkak and Smirnov [18] allowing us to remove all contributions from dual spanning trees, which we could not do in [30].
## 6.2 Full Z-invariant case when the isoradial graph G is $\mathbb{Z}^2$

The goal of this example is to relate our results to the papers [60, 7]. When the isoradial graph $G = \mathbb{Z}^2$ in its regular embedding, all rhombus half-angles $\bar{\theta}_e$ are equal to $\frac{\pi}{4}$. Using that $\operatorname{sn}(\frac{K}{2}) = \frac{1}{(1+k')^{\frac{1}{2}}}$, $\operatorname{cn}(\frac{K}{2}) = \frac{(k')^{\frac{1}{2}}}{(1+k')^{\frac{1}{2}}}$, $\operatorname{dn}(\frac{K}{2}) = (k')^{\frac{1}{2}}$ [1, 16.5.2], we obtain the following parameterization of the Ising coupling constants:

$$ \forall e \in E, \quad J_e = J_e(k') = \frac{1}{2} \ln \left( \frac{1 + (1 + k')^{\frac{1}{2}}}{(k')^{\frac{1}{2}}} \right), $$

which is a $\mathbb{C}^\infty$-bijection from $(0, \infty)$ to $(\infty, 0)$.

The masses of the Z-massive Laplacian $\Delta^m$ are equal to [11], $\forall v \in V$, $m_v = \frac{2(k')^{\frac{1}{2}}}{1+k'}$, and the survival probabilities $s$ of the random walk associated to the Z-invariant conductances $\rho$ and masses $m$ are

$$ \forall v \in V, \quad s_v = \frac{4 \operatorname{sc}\left(\frac{K}{2}\right)}{4 \operatorname{sc}\left(\frac{K}{2}\right) + m_v} = \frac{2(k')^{\frac{1}{2}}}{1 + k'}. $$

As in the papers [60, 7] the survival probabilities satisfy

$$ \forall v \in V, \quad s_v = \frac{2}{\sinh(2J_e) + \sinh^{-1}(2J_e)}. $$

In the infinite case, by Theorem 5.4 and Corollary 4.16, the coefficient $(K^F)_{a,a}^{-1}$ of the inverse Kasteleyn operator $(K^F)^{-1}$ is equal to, as long as $a$ and $a$ do not belong to the same decoration,

$$ |(K^F)_{a,a}^{-1}| = (k')^{-1}(1+(k'))^{\frac{1}{2}} \left( \operatorname{cn}\left(\frac{\beta_f - \beta_i}{2}\right) [G_{v,v_2}^m - (k')^{\frac{1}{2}} G_{v,v_1}^m] - \operatorname{sn}\left(\frac{\beta_f - \beta_i}{2}\right) [(k')^{\frac{1}{2}} G_{f,f_2}^{m,*} - G_{f,f_1}^{m,*}] \right). $$

A similar expression holds in the finite case, see Corollary 4.18 and Example 3.18. According to [32, 63], this coefficient is the fermionic spinor observable of [39] and, up to normalization, the FK-Ising observable of [67, 68, 18]. Now, the proof of [7] for showing the occurrence of large deviation estimates of the massive random walk in the correlation length of the spin correlations [60] consists in proving that, in the super-critical regime, spin correlations can be approximated by the FK-Ising observable, and then using massive harmonicity of the latter to relate it to the massive random walk. Our explicit expression for $|(K^F)_{a,a}^{-1}|$ in the finite case gives a direct explanation of the occurrence of these large deviation estimates, and up to handling boundary terms, should give a rather direct proof valid in the whole super-critical Z-invariant case.

# A Gauge equivalence revisited

We consider gauge equivalence of weighted adjacency matrices of digraphs and rephrase gauge equivalence of bipartite weighted adjacency matrices as defined in [53, 47] in this context.

## A.1 Definitions

In the whole of this section, we consider square matrices of size $n \times n$; let $M$ be such a matrix. We associate two graphs to $M$, a non-directed one $G(M) = (V(M), E(M))$ and a directed one $D(M) = (V(M), A(M))$, both having the same vertex set of cardinality $n$ in bijection with rows/columns of $M$. An edge $x_j x_\ell$ is in $E(M)$ iff $M_{x_j, x_\ell} \neq 0$ or $M_{x_\ell, x_j} \neq 0$. A directed edge (or simply an edge) $(x_j, x_\ell)$ is in $A(M)$ iff the coefficient $M_{x_j, x_\ell} \neq 0$. The
## 6.2 Full Z-invariant case when the isoradial graph G is $\mathbb{Z}^2$

The goal of this example is to relate our results to the papers [60, 7]. When the isoradial graph $G = \mathbb{Z}^2$ in its regular embedding, all rhombus half-angles $\bar{\theta}_e$ are equal to $\frac{\pi}{4}$. Using that $\operatorname{sn}(\frac{K}{2}) = \frac{1}{(1+k')^{\frac{1}{2}}}$, $\operatorname{cn}(\frac{K}{2}) = \frac{(k')^{\frac{1}{2}}}{(1+k')^{\frac{1}{2}}}$, $\operatorname{dn}(\frac{K}{2}) = (k')^{\frac{1}{2}}$ [1, 16.5.2], we obtain the following parameterization of the Ising coupling constants:

$$ \forall e \in E, \quad J_e = J_e(k') = \frac{1}{2} \ln \left( \frac{1 + (1 + k')^{\frac{1}{2}}}{(k')^{\frac{1}{2}}} \right), $$

which is a $\mathbb{C}^\infty$-bijection from $(0, \infty)$ to $(\infty, 0)$.

The masses of the Z-massive Laplacian $\Delta^m$ are equal to [11], $\forall v \in V$, $m_v = \frac{2(k')^{\frac{1}{2}}}{1+k'}$, and the survival probabilities $s$ of the random walk associated to the Z-invariant conductances $\rho$ and masses $m$ are

$$ \forall v \in V, \quad s_v = \frac{4 \operatorname{sc}\left(\frac{K}{2}\right)}{4 \operatorname{sc}\left(\frac{K}{2}\right) + m_v} = \frac{2(k')^{\frac{1}{2}}}{1 + k'}. $$

As in the papers [60, 7] the survival probabilities satisfy

$$ \forall v \in V, \quad s_v = \frac{2}{\sinh(2J_e) + \sinh^{-1}(2J_e)}. $$

In the infinite case, by Theorem 5.4 and Corollary 4.16, the coefficient $(K^F)_{a,a}^{-1}$ of the inverse Kasteleyn operator $(K^F)^{-1}$ is equal to, as long as $a$ and $a$ do not belong to the same decoration,

$$ |(K^F)_{a,a}^{-1}| = (k')^{-1}(1+(k'))^{\frac{1}{2}} \left( \operatorname{cn}\left(\frac{\beta_f - \beta_i}{2}\right) [G_{v,v_2}^m - (k')^{\frac{1}{2}} G_{v,v_1}^m] - \operatorname{sn}\left(\frac{\beta_f - \beta_i}{2}\right) [(k')^{\frac{1}{2}} G_{f,f_2}^{m,*} - G_{f,f_1}^{m,*}] \right). $$

A similar expression holds in the finite case, see Corollary 4.18 and Example 3.18. According to [32, 63], this coefficient is the fermionic spinor observable of [39] and, up to normalization, the FK-Ising observable of [67, 68, 18]. Now, the proof of [7] for showing the occurrence of large deviation estimates of the massive random walk in the correlation length of the spin correlations [60] consists in proving that, in the super-critical regime, spin correlations can be approximated by the FK-Ising observable, and then using massive harmonicity of the latter to relate it to the massive random walk. Our explicit expression for $|(K^F)_{a,a}^{-1}|$ in the finite case gives a direct explanation of the occurrence of these large deviation estimates, and up to handling boundary terms, should give a rather direct proof valid in the whole super-critical Z-invariant case.

# A Gauge equivalence revisited

We consider gauge equivalence of weighted adjacency matrices of digraphs and rephrase gauge equivalence of bipartite weighted adjacency matrices as defined in [53, 47] in this context.

## A.1 Definitions

In the whole of this section, we consider square matrices of size $n \times n$; let $M$ be such a matrix. We associate two graphs to $M$, a non-directed one $G(M) = (V(M), E(M))$ and a directed one $D(M) = (V(M), A(M))$, both having the same vertex set of cardinality $n$ in bijection with rows/columns of $M$. An edge $x_j x_\ell$ is in $E(M)$ iff $M_{x_j, x_\ell} \neq 0$ or $M_{x_\ell, x_j} \neq 0$. A directed edge (or simply an edge) $(x_j, x_\ell)$ is in $A(M)$ iff the coefficient $M_{x_j, x_\ell} \neq 0$. The
matrix $M$ is a weighted adjacency matrix of the digraph $D(M)$. Whenever no confusion occurs, we remove the argument $M$ of the graphs.

Let us recall a few definitions. A di-path of a digraph $D = (V, A)$ is a sequence $(x_0, \dots, x_n)$ of vertices such that for every $j \in \{0, \dots, n-1\}$, $(x_j, x_{j+1})$ is an edge of $A$. A simple di-path is a path with pairwise disjoint vertices. A di-cycle is a di-path such that the first and last vertices are the same. A simple di-cycle is a cycle whose only common vertices are the first and the last. Note that loops and length-two di-cycles are simple. A digraph is strongly connected if any two pairs of vertices are joined by a di-path. The above definitions are easily adapted in the case of non-directed graphs.

**Definition A.1.** Consider two matrices $M$ and $N$ having the same associated digraph $D$. The matrices $M$ and $N$ are said to be gauge equivalent if,

$$ \forall \text{ simple di-cycle } c \text{ of } D, \quad \prod_{e=(x,y) \in c} M_{x,y} = \prod_{e=(x,y) \in c} N_{x,y}. $$

**Remark A.2.**

* Having the same associated digraph is equivalent to asking that $M_{x,y} \neq 0 \Leftrightarrow N_{x,y} \neq 0$.

* Since loops are simple di-cycles, if $M$ and $N$ are gauge equivalent, they have equal diagonal coefficients.

* Definition A.1 holds if and only if the product condition holds for every di-cycle of $D$.

**Lemma A.3.** Let $M, N$ be two gauge equivalent matrices, then

$$ \det M = \det N. $$

*Proof.* This is proved by writing the determinant as a sum over permutations, and doing the cyclic decomposition of permutations. $\square$

Let us suppose that $M$ and $N$ are gauge equivalent and that the associated digraph $D$ is strongly connected. Define the function $q \in C^{V \times V}$ on pairs of vertices of $V$ taking values in $C$ as follows. For vertices $x, y$ such that $(x, y)$ is an edge of $D$, set

$$ q_{x,y} = \frac{N_{x,y}}{M_{x,y}}. $$

For vertices $x, y$ of $D$, since the digraph is strongly connected, there exists a di-path $\gamma$ from $x$ to $y$; set

$$ q_{x,y} = \prod_{e=(x',y') \in \gamma} q_{x',y'}. $$

Note that if $y = x$, then $\gamma$ is a di-cycle and we have $q_{x,x} = 1$.

**Remark A.4.** The function $q$ is well defined, i.e., independent of the choice of path from $x$ to $y$. If $y = x$, then $q_{x,x} = 1$ independently of the choice of di-cycle from $x$ to $x$. If $y \neq x$, consider two di-paths $\gamma_1, \gamma_2$ from $x$ to $y$. Since the digraph $D$ is strongly connected, there exists a simple di-path $\tilde{\gamma}$ from $y$ to $x$. Then, $\gamma_1$ (resp. $\gamma_2$) followed by $\tilde{\gamma}$ is a di-cycle and by definition of gauge equivalence we have,

$$ \left( \prod_{e=(x',y') \in \gamma_1} q_{x',y'} \right) \left( \prod_{e=(x',y') \in \tilde{\gamma}} q_{x',y'} \right) = 1 = \left( \prod_{e=(x',y') \in \gamma_2} q_{x',y'} \right) \left( \prod_{e=(x',y') \in \tilde{\gamma}} q_{x',y'} \right), $$

implying that

$$ \prod_{e=(x',y') \in \gamma_1} q_{x',y'} = \prod_{e=(x',y') \in \gamma_2} q_{x',y'}. $$

and the function $q$ is thus well defined.
matrix $M$ is a weighted adjacency matrix of the digraph $D(M)$. Whenever no confusion occurs, we remove the argument $M$ of the graphs.

Let us recall a few definitions. A di-path of a digraph $D = (V, A)$ is a sequence $(x_0, \dots, x_n)$ of vertices such that for every $j \in \{0, \dots, n-1\}$, $(x_j, x_{j+1})$ is an edge of $A$. A simple di-path is a path with pairwise disjoint vertices. A di-cycle is a di-path such that the first and last vertices are the same. A simple di-cycle is a cycle whose only common vertices are the first and the last. Note that loops and length-two di-cycles are simple. A digraph is strongly connected if any two pairs of vertices are joined by a di-path. The above definitions are easily adapted in the case of non-directed graphs.

**Definition A.1.** Consider two matrices $M$ and $N$ having the same associated digraph $D$. The matrices $M$ and $N$ are said to be gauge equivalent if,

$$ \forall \text{ simple di-cycle } c \text{ of } D, \quad \prod_{e=(x,y) \in c} M_{x,y} = \prod_{e=(x,y) \in c} N_{x,y}. $$

**Remark A.2.**

* Having the same associated digraph is equivalent to asking that $M_{x,y} \neq 0 \Leftrightarrow N_{x,y} \neq 0$.

* Since loops are simple di-cycles, if $M$ and $N$ are gauge equivalent, they have equal diagonal coefficients.

* Definition A.1 holds if and only if the product condition holds for every di-cycle of $D$.

**Lemma A.3.** Let $M, N$ be two gauge equivalent matrices, then

$$ \det M = \det N. $$

*Proof.* This is proved by writing the determinant as a sum over permutations, and doing the cyclic decomposition of permutations. $\square$

Let us suppose that $M$ and $N$ are gauge equivalent and that the associated digraph $D$ is strongly connected. Define the function $q \in C^{V \times V}$ on pairs of vertices of $V$ taking values in $C$ as follows. For vertices $x, y$ such that $(x, y)$ is an edge of $D$, set

$$ q_{x,y} = \frac{N_{x,y}}{M_{x,y}}. $$

For vertices $x, y$ of $D$, since the digraph is strongly connected, there exists a di-path $\gamma$ from $x$ to $y$; set

$$ q_{x,y} = \prod_{e=(x',y') \in \gamma} q_{x',y'}. $$

Note that if $y = x$, then $\gamma$ is a di-cycle and we have $q_{x,x} = 1$.

**Remark A.4.** The function $q$ is well defined, i.e., independent of the choice of path from $x$ to $y$. If $y = x$, then $q_{x,x} = 1$ independently of the choice of di-cycle from $x$ to $x$. If $y \neq x$, consider two di-paths $\gamma_1, \gamma_2$ from $x$ to $y$. Since the digraph $D$ is strongly connected, there exists a simple di-path $\tilde{\gamma}$ from $y$ to $x$. Then, $\gamma_1$ (resp. $\gamma_2$) followed by $\tilde{\gamma}$ is a di-cycle and by definition of gauge equivalence we have,

$$ \left( \prod_{e=(x',y') \in \gamma_1} q_{x',y'} \right) \left( \prod_{e=(x',y') \in \tilde{\gamma}} q_{x',y'} \right) = 1 = \left( \prod_{e=(x',y') \in \gamma_2} q_{x',y'} \right) \left( \prod_{e=(x',y') \in \tilde{\gamma}} q_{x',y'} \right), $$

implying that

$$ \prod_{e=(x',y') \in \gamma_1} q_{x',y'} = \prod_{e=(x',y') \in \gamma_2} q_{x',y'}. $$

and the function $q$ is thus well defined.
The following lemma proves that if the associated digraph is strongly connected, gauge equivalence amounts to having the two matrices related through a diagonal matrix.

**Lemma A.5.** Let $M, N$ be two matrices having the same associated digraph $D$ and suppose that $D$ is strongly connected. Then $M$ and $N$ are gauge equivalent if and only if there exists an invertible diagonal matrix $D$ such that,

$$M = DND^{-1}.$$

*Proof.* Suppose that $M$ and $N$ are gauge equivalent. Fix a vertex $x_0$ of $D$, and define $D^{x_0}$ to be the diagonal matrix whose diagonal coefficient $D_{x,x}^{x_0}$ corresponding to the vertex $x$ is $q_{x_0,x}$. Since the digraph $D$ is strongly connected, the matrix $D^{x_0}$ is invertible. Let us prove that $M = D^{x_0}N(D^{x_0})^{-1}$. Non zero coefficients of $M$ and $N$ correspond to edges of $D$; let $(x, y)$ be such an edge. Consider a di-path $\gamma$ from $x_0$ to $x$, then $\gamma' = \gamma \cup \{(x, y)\}$ is a di-path from $x_0$ to $y$. Using $\gamma$ to compute $q_{x_0,x}$ and $\gamma'$ to compute $q_{x_0,y}$, we deduce that,

$$(D^{x_0}N(D^{x_0})^{-1})_{x,y} = \frac{q_{x_0,x}}{q_{x_0,y}}N_{x,y} = \frac{1}{q_{x,y}}N_{x,y} = \frac{M_{x,y}}{N_{x,y}}N_{x,y} = M_{x,y}.$$

Suppose that $M = DND^{-1}$, with $D$ an invertible diagonal matrix, and let us prove that $M$ and $N$ are gauge equivalent. Consider a simple di-cycle $c$ of $D$, then

$$\prod_{e=(x,y) \in c} M_{x,y} = \prod_{e=(x,y) \in c} D_{x,x} N_{x,y} D_{y,y}^{-1} = \prod_{e=(x,y) \in c} N_{x,y} \text{ (telescopic product)},$$

thus concluding the proof. □

**Remark A.6.** Consider two matrices $M, N$ having the same associated graphs $G$ and $D$, such that $G$ is connected and such that for every undirected edge $e$ of $G$, the two possible oriented edges are present in $D$; then $D$ is strongly connected. If moreover $G$ is planar and embedded, then every simple di-cycle $c$ of length $\ge 3$ is the union of face di-cycles, where edges not in $c$ are traversed in both directions. As a consequence, in this case proving gauge equivalence for $M$ and $N$ is equivalent to proving that,

$$\forall x \in V, \quad M_{x,x} = N_{x,x},$$

$$\forall \text{ edge } e \text{ of } G, \quad M_{x,y}M_{y,x} = N_{x,y}N_{y,x},$$

$$\forall \text{ face di-cycle } c, \quad \prod_{e=(x,y) \in c} M_{x,y} = \prod_{e=(x,y) \in c} N_{x,y}.$$

## A.2 The bipartite case

Consider a non-directed, finite, bipartite graph $G = (V = W \cup B, E)$, such that $|W| = |B| = n$, having at least one perfect matching. Note that $G$ being bipartite, it cannot have loops; we furthermore suppose that it has no multiple edges, i.e., that it is simple.

Fix a perfect matching $M_0 = \{b_1w_1, \dots, b_nw_n\}$ of $G$. From $G$ and $M_0$, construct a digraph $D^0$ in the following way: vertices $b_j$ and $w_j$ are merged into a single vertex $x_j$, and the corresponding edge becomes a loop. The vertex set of $D^0$ is $V^0 = \{x_1, \dots, x_n\}$. Edges not in the perfect matching $M_0$ remain in $D^0$ and are directed from their black vertex to the white one, defining the directed edges of $D^0$.

A bipartite, weighted adjacency matrix $K$ associated to the graph $G$ has rows indexed by vertices of $B$, columns by those of $W$, and non-zero coefficients correspond to edges of $G$. Up to a reordering of the rows and columns, we can suppose that rows of $K$ are indexed by $b_1, \dots, b_n$ and columns by $w_1, \dots, w_n$.
The following lemma proves that if the associated digraph is strongly connected, gauge equivalence amounts to having the two matrices related through a diagonal matrix.

**Lemma A.5.** Let $M, N$ be two matrices having the same associated digraph $D$ and suppose that $D$ is strongly connected. Then $M$ and $N$ are gauge equivalent if and only if there exists an invertible diagonal matrix $D$ such that,

$$M = DND^{-1}.$$

*Proof.* Suppose that $M$ and $N$ are gauge equivalent. Fix a vertex $x_0$ of $D$, and define $D^{x_0}$ to be the diagonal matrix whose diagonal coefficient $D_{x,x}^{x_0}$ corresponding to the vertex $x$ is $q_{x_0,x}$. Since the digraph $D$ is strongly connected, the matrix $D^{x_0}$ is invertible. Let us prove that $M = D^{x_0}N(D^{x_0})^{-1}$. Non zero coefficients of $M$ and $N$ correspond to edges of $D$; let $(x, y)$ be such an edge. Consider a di-path $\gamma$ from $x_0$ to $x$, then $\gamma' = \gamma \cup \{(x, y)\}$ is a di-path from $x_0$ to $y$. Using $\gamma$ to compute $q_{x_0,x}$ and $\gamma'$ to compute $q_{x_0,y}$, we deduce that,

$$(D^{x_0}N(D^{x_0})^{-1})_{x,y} = \frac{q_{x_0,x}}{q_{x_0,y}}N_{x,y} = \frac{1}{q_{x,y}}N_{x,y} = \frac{M_{x,y}}{N_{x,y}}N_{x,y} = M_{x,y}.$$

Suppose that $M = DND^{-1}$, with $D$ an invertible diagonal matrix, and let us prove that $M$ and $N$ are gauge equivalent. Consider a simple di-cycle $c$ of $D$, then

$$\prod_{e=(x,y) \in c} M_{x,y} = \prod_{e=(x,y) \in c} D_{x,x} N_{x,y} D_{y,y}^{-1} = \prod_{e=(x,y) \in c} N_{x,y} \text{ (telescopic product)},$$

thus concluding the proof.

<figure><img src="image_33.png" /></figure>

**Remark A.6.** Consider two matrices $M, N$ having the same associated graphs $G$ and $D$, such that $G$ is connected and such that for every undirected edge $e$ of $G$, the two possible oriented edges are present in $D$; then $D$ is strongly connected. If moreover $G$ is planar and embedded, then every simple di-cycle $c$ of length $\ge 3$ is the union of face di-cycles, where edges not in $c$ are traversed in both directions. As a consequence, in this case proving gauge equivalence for $M$ and $N$ is equivalent to proving that,

$$\forall x \in V, \quad M_{x,x} = N_{x,x},$$

$$\forall \text{ edge } e \text{ of } G, \quad M_{x,y}M_{y,x} = N_{x,y}N_{y,x},$$

$$\forall \text{ face di-cycle } c, \quad \prod_{e=(x,y) \in c} M_{x,y} = \prod_{e=(x,y) \in c} N_{x,y}.$$

## A.2 The bipartite case

Consider a non-directed, finite, bipartite graph $G = (V = W \cup B, E)$, such that $|W| = |B| = n$, having at least one perfect matching. Note that $G$ being bipartite, it cannot have loops; we furthermore suppose that it has no multiple edges, i.e., that it is simple.

Fix a perfect matching $M_0 = \{b_1w_1, \dots, b_nw_n\}$ of $G$. From $G$ and $M_0$, construct a digraph $D^0$ in the following way: vertices $b_j$ and $w_j$ are merged into a single vertex $x_j$, and the corresponding edge becomes a loop. The vertex set of $D^0$ is $V^0 = \{x_1, \dots, x_n\}$. Edges not in the perfect matching $M_0$ remain in $D^0$ and are directed from their black vertex to the white one, defining the directed edges of $D^0$.

A bipartite, weighted adjacency matrix $K$ associated to the graph $G$ has rows indexed by vertices of $B$, columns by those of $W$, and non-zero coefficients correspond to edges of $G$. Up to a reordering of the rows and columns, we can suppose that rows of $K$ are indexed by $b_1, \dots, b_n$ and columns by $w_1, \dots, w_n$.
Instead of seeing K as a bipartite adjacency matrix, we can interpret it as an adjacency matrix of the digraph D⁰. In this interpretation, rows and columns are indexed by vertices of V⁰ and diagonal coefficients correspond to edges of the perfect matching, they now represent loops.

Consider the diagonal matrix D⁰K whose j-th diagonal coefficients is the coefficient K<sub>b<sub>j</sub>w<sub>j</sub></sub> corresponding to the j-th edge of M<sub>0</sub>. Define the matrix K⁰ to be,

$$K^0 = (D_K^0)^{-1}K.$$

Note that the matrix K⁰ has ones on the diagonal.

**Definition A.7.** Let G be a finite, bipartite graph, and let K, L be two associated bipartite, weighted adjacency matrices. Fix a perfect matching M<sub>0</sub> of G, and let D⁰ be the directed graph constructed from G and M<sub>0</sub>. Then, K and L are said to be gauge equivalent if the matrices K⁰ and L⁰, seen as weighted adjacency matrices of the digraph D⁰, are gauge equivalent.

Rephrasing Lemma A.3 in the context of bipartite graphs, we obtain

**Corollary A.8.** Let K, L be two gauge equivalent bipartite, weighted adjacency matrices, then

$$\det K = \left( \prod_{e=\text{bw} \in M_0} \frac{K_{b,w}}{L_{b,w}} \right) \det L.$$

We now rephrase Definition A.7 in the more usual form [53, 47]. Consider the bipartite graph G together with the reference perfect matching M<sub>0</sub>. An alternating cycle of G and M<sub>0</sub> is a simple cycle of G whose edges alternate between edges in M<sub>0</sub> and edges in E \ M<sub>0</sub>.

**Lemma A.9.** The matrices K and L are gauge equivalent iff for every alternating cycle c of G and M<sub>0</sub> of length > 2, we have

$$\frac{\prod_{e=\text{bw} \in c \setminus M_0} K_{b,w}}{\prod_{e=\text{bw} \in c \cap M_0} K_{b,w}} = \frac{\prod_{e=\text{bw} \in c \setminus M_0} L_{b,w}}{\prod_{e=\text{bw} \in c \cap M_0} L_{b,w}}.$$

*Proof.* By definition, K and L are gauge equivalent if K⁰ and L⁰ seen as adjacency matrices of the digraph G⁰ are gauge equivalent, see Definition A.1. Length one di-cycles of D⁰ are loops corresponding to diagonal coefficients of K⁰, L⁰. The latter are all equal to 1 by definition, and thus equal. Consider a simple di-cycle c of D⁰ of length m ≥ 2. Up to a relabeling of the vertices, it can be denoted as c = {x₁, ..., x<sub>l</sub> = x<sub>1</sub>} . Then, ∀j, x<sub>j</sub> corresponds to an edge b<sub>j</sub>w<sub>j</sub> of the perfect matching M<sub>0</sub>. By construction of D⁰, the cycle c is in correspondence with an alternating cycle {w₁, b₁, w₂, ..., w<sub>m</sub>, b<sub>m</sub>, w₁} of length 2m of G. By definition of K⁰ we have,

$$\prod_{j=1}^{m} K_{x_j, x_{j+1}}^{0} = \frac{\prod_{j=1}^{m} K_{b_j, w_{j+1}}}{\prod_{j=1}^{m} K_{b_j, w_j}},$$

with cyclic notation for indices. A similar equality holds for the matrices L⁰ and L, thus ending the proof. □

**Remark A.10.** Definition A.7 is independent of the choice of M<sub>0</sub>. To prove this, use the fact that if M<sub>1</sub> is another reference perfect matching, then the superimposition of M<sub>0</sub> and M<sub>1</sub> consists of alternating cycles of length > 2 and doubled edges.

From now on we suppose that the bipartite graph G is finite, planar and embedded, and we let K, L be two bipartite, weighted adjacency matrices of G.
Instead of seeing K as a bipartite adjacency matrix, we can interpret it as an adjacency matrix of the digraph $D^0$. In this interpretation, rows and columns are indexed by vertices of $V^0$ and diagonal coefficients correspond to edges of the perfect matching, they now represent loops.

Consider the diagonal matrix $D_K^0$ whose j-th diagonal coefficients is the coefficient $K_{b_j,w_j}$ corresponding to the j-th edge of $M_0$. Define the matrix $K^0$ to be,

$$K^0 = (D_K^0)^{-1}K.$$

Note that the matrix $K^0$ has ones on the diagonal.

**Definition A.7.** Let G be a finite, bipartite graph, and let K, L be two associated bipartite, weighted adjacency matrices. Fix a perfect matching $M_0$ of G, and let $D^0$ be the directed graph constructed from G and $M_0$. Then, K and L are said to be gauge equivalent if the matrices $K^0$ and $L^0$, seen as weighted adjacency matrices of the digraph $D^0$, are gauge equivalent.

Rephrasing Lemma A.3 in the context of bipartite graphs, we obtain

**Corollary A.8.** Let K, L be two gauge equivalent bipartite, weighted adjacency matrices, then

$$\det K = \left( \prod_{e=\text{bw} \in M_0} \frac{K_{b,w}}{L_{b,w}} \right) \det L.$$

We now rephrase Definition A.7 in the more usual form [53, 47]. Consider the bipartite graph G together with the reference perfect matching $M_0$. An alternating cycle of G and $M_0$ is a simple cycle of G whose edges alternate between edges in $M_0$ and edges in $E \setminus M_0$.

**Lemma A.9.** The matrices K and L are gauge equivalent iff for every alternating cycle c of G and $M_0$ of length > 2, we have

$$\frac{\prod_{e=\text{bw} \in c \setminus M_0} K_{b,w}}{\prod_{e=\text{bw} \in c \cap M_0} K_{b,w}} = \frac{\prod_{e=\text{bw} \in c \setminus M_0} L_{b,w}}{\prod_{e=\text{bw} \in c \cap M_0} L_{b,w}}.$$

*Proof.* By definition, K and L are gauge equivalent if $K^0$ and $L^0$ seen as adjacency matrices of the digraph $G^0$ are gauge equivalent, see Definition A.1. Length one di-cycles of $D^0$ are loops corresponding to diagonal coefficients of $K^0$, $L^0$. The latter are all equal to 1 by definition, and thus equal. Consider a simple di-cycle c of $D^0$ of length $m \ge 2$. Up to a relabeling of the vertices, it can be denoted as $c = \{x_1, \dots, x_l = x_1\}$. Then, $\forall j, x_j$ corresponds to an edge $b_j w_j$ of the perfect matching $M_0$. By construction of $D^0$, the cycle c is in correspondence with an alternating cycle $\{w_1, b_1, w_2, \dots, w_m, b_m, w_1\}$ of length $2m$ of G. By definition of $K^0$ we have,

$$\prod_{j=1}^{m} K_{x_j, x_{j+1}}^{0} = \frac{\prod_{j=1}^{m} K_{b_j, w_{j+1}}}{\prod_{j=1}^{m} K_{b_j, w_j}},$$

with cyclic notation for indices. A similar equality holds for the matrices $L^0$ and L, thus ending the proof. □

**Remark A.10.** Definition A.7 is independent of the choice of $M_0$. To prove this, use the fact that if $M_1$ is another reference perfect matching, then the superimposition of $M_0$ and $M_1$ consists of alternating cycles of length > 2 and doubled edges.

From now on we suppose that the bipartite graph G is finite, planar and embedded, and we let K, L be two bipartite, weighted adjacency matrices of G.
**Lemma A.11** ([53]). If the alternating products of the matrices K and L are equal around every inner face-cycle of G, then K and L are gauge equivalent.

*Proof.* This is proved by induction on the number of faces contained in an alternating cycle of length > 2. □

Suppose that K and L satisfy the alternating product condition around every inner face-cycle of G. Similarly to the directed case, define the function $q \in \mathbb{C}^{V \times V}$ as follows. For every edge bw of G,

$$q_{b,w} = \frac{L_{b,w}}{K_{b,w}}, \quad q_{w,b} = q_{b,w}^{-1}.$$

For every pair of vertices x, y of G, let $\gamma$ be a path from x to y and set $q_{x,y} = \prod_{(x',y') \in \gamma} q_{x',y'}$. The function q is well defined [53] and

**Lemma A.12** ([53]). The matrices K and L satisfy the alternating product condition around every inner face-cycle if and only if there exist diagonal matrices $D^B$, $D^W$ such that

$$K = D^B L D^W.$$

*Proof.* The proof can be found in [53]. For the purpose of Section 5.3, we make the diagonal matrices explicit assuming the alternating product condition is satisfied. Fix a vertex $x_0$ of G and set,

$$\forall b \in B, \quad D_{b,b}^{x_0,B} = q_{x_0,b}, \quad \forall w \in W, \quad D_{b,b}^{x_0,W} = q_{x_0,w}^{-1},$$

where q is given above. Then, $K = D^{x_0,B} L D^{x_0,W}$. □

# B Computations of probabilities of single edges.

## B.1 Dimers model on an infinite isoradial double graph $G^D$

We compute the probability of single edges occurring in dimer configurations of $G^D$ chosen with respect to the measure $P_{\text{dimer}}^{D,u}$. Notation are recalled in Figure 30 below; since no confusion occurs, we omit the subscripts f from $\alpha, \beta, \theta$.

<figure><img src="image_34.png" /><figcaption>Figure 30: Notation for computing $P_{\text{dimer}}^{D,u}(wx)$ when $x = v_2$ or $f_2$.</figcaption></figure>

For every edge $wx$ of $G^D$, we have $P_{\text{dimer}}^{D,u}(wx) = K(u)_{w,x}K(u)_{x,w}^{-1}$, where $x$ is a vertex of G or $G^*$. Setting $v = v_2$ in $K(u)_{v,w}^{-1}$ and $f = f_2$ in $K(u)_{f,w}^{-1}$ of Corollary 3.15, and using that $\text{dn}(u - K) = k' \text{nd}(u)$, gives

$$K(u)_{v_2,w}^{-1} = e^{-i \frac{\hat{\alpha} + \hat{\beta}}{2}} (k')^{-1} [\text{sc}(\theta) \text{nd}(u_{\alpha}) \text{nd}(u_{\beta})]^{\frac{1}{2}} [\text{dn}(u_{\alpha}) \text{dn}(u_{\beta}) G_{v_2,v_2}^{m} - k' G_{v_2,v_1}^{m}]$$

$$K(u)_{f_2,w}^{-1} = -ie^{-i \frac{\hat{\alpha} + \hat{\beta}}{2}} (k')^{-1} [\text{sc}(\theta^*) \text{nd}((u_{\beta})^*) \text{nd}((u_{\alpha+2K})^*)]^{\frac{1}{2}} \times [\text{dn}((u_{\beta})^*) \text{dn}((u_{\alpha+2K})^*) G_{f_2,f_2}^{m,*} - k' G_{f_2,f_1}^{m,*}].$$
**Lemma A.11** ([53]). If the alternating products of the matrices K and L are equal around every inner face-cycle of G, then K and L are gauge equivalent.

*Proof.* This is proved by induction on the number of faces contained in an alternating cycle of length > 2. □

Suppose that K and L satisfy the alternating product condition around every inner face-cycle of G. Similarly to the directed case, define the function $q \in C^{V \times V}$ as follows. For every edge bw of G,

$$q_{b,w} = \frac{L_{b,w}}{K_{b,w}}, \quad q_{w,b} = q_{b,w}^{-1}.$$

For every pair of vertices $x, y$ of G, let $\gamma$ be a path from $x$ to $y$ and set $q_{x,y} = \prod_{(x',y') \in \gamma} q_{x',y'}$. The function $q$ is well defined [53] and

**Lemma A.12** ([53]). The matrices K and L satisfy the alternating product condition around every inner face-cycle if and only if there exist diagonal matrices $D^B, D^W$ such that

$$K = D^B L D^W.$$

*Proof.* The proof can be found in [53]. For the purpose of Section 5.3, we make the diagonal matrices explicit assuming the alternating product condition is satisfied. Fix a vertex $x_0$ of G and set,

$$\forall b \in B, \quad D_{b,b}^{x_0,B} = q_{x_0,b}, \quad \forall w \in W, \quad D_{b,b}^{x_0,W} = q_{x_0,w}^{-1},$$

where $q$ is given above. Then, $K = D^{x_0,B} L D^{x_0,W}$. □

# B Computations of probabilities of single edges.

## B.1 Dimers model on an infinite isoradial double graph $G^D$

We compute the probability of single edges occurring in dimer configurations of $G^D$ chosen with respect to the measure $P_{\text{dimer}}^{D,u}$. Notation are recalled in Figure 30 below; since no confusion occurs, we omit the subscripts f from $\alpha, \beta, \theta$.

<figure><img src="image_34.png" /><figcaption>Figure 30: Notation for computing $P_{\text{dimer}}^{D,u}(wx)$ when $x = v_2$ or $f_2$.</figcaption></figure>

For every edge $wx$ of $G^D$, we have $P_{\text{dimer}}^{D,u}(wx) = K(u)_{w,x}K(u)_{x,w}^{-1}$, where $x$ is a vertex of G or $G^*$. Setting $v = v_2$ in $K(u)_{v,w}^{-1}$ and $f = f_2$ in $K(u)_{f,w}^{-1}$ of Corollary 3.15, and using that $\text{dn}(u - K) = k' \text{nd}(u)$, gives

$$K(u)_{v_2,w}^{-1} = e^{-i \frac{\hat{\alpha} + \hat{\beta}}{2}} (k')^{-1} [\text{sc}(\theta) \text{nd}(u_{\alpha}) \text{nd}(u_{\beta})]^{\frac{1}{2}} [\text{dn}(u_{\alpha}) \text{dn}(u_{\beta}) G_{v_2,v_2}^{m} - k' G_{v_2,v_1}^{m}]$$

$$K(u)_{f_2,w}^{-1} = -ie^{-i \frac{\hat{\alpha} + \hat{\beta}}{2}} (k')^{-1} [\text{sc}(\theta^*) \text{nd}((u_{\beta})^*) \text{nd}((u_{\alpha+2K})^*)]^{\frac{1}{2}} \times [\text{dn}((u_{\beta})^*) \text{dn}((u_{\alpha+2K})^*) G_{f_2,f_2}^{m,*} - k' G_{f_2,f_1}^{m,*}].$$
Multiplying by $K(u)_{w,v_2}$ and $K(u)_{w,f_2}$ respectively yields the first equalities of Example 3.21:

$$K(u)_{w,v_2} K(u)_{v_2,w}^{-1} = (k')^{-1} \mathrm{sc}(\theta) [\mathrm{dn}(u_\alpha) \mathrm{dn}(u_\beta) G_{v_2,v_2}^m - k' G_{v_2,v_1}^m]$$

$$K(u)_{w,f_2} K(u)_{f_2,w}^{-1} = (k')^{-1} \mathrm{sc}(\theta^*) [\mathrm{dn}((u_\beta)^*) \mathrm{dn}((u_{\alpha+2K})^*) G_{f_2,f_2}^{m,*} - k' G_{f_2,f_1}^{m,*}]$$

Now by [11, Lemma 46], for every vertex $v$ of $G$, $G_{v,v}^m = \frac{k'K'}{\pi}$ and, for every $u \in \mathcal{C}$,

$$G_{v_2,v_1}^m = G_{v_1,v_2}^m = \frac{H(2u_\beta) - H(2u_\alpha)}{\mathrm{sc}(\theta)} + \frac{K'}{\pi} \mathrm{dn}(u_\alpha) \mathrm{dn}(u_\beta)$$

$$G_{f_2,f_1}^{m,*} = G_{f_1,f_2}^{m,*} = \frac{H(2(u_\beta)^*) - H(2(u_{\alpha+2K})^*)}{\mathrm{sc}(\theta^*)} + \frac{K'}{\pi} \mathrm{dn}((u_\beta)^*) \mathrm{dn}((u_{\alpha+2K})^*)$$

Combining this with the expressions of $K(u)_{x,w}^{-1}$ concludes the proof of the second equalities of Example 3.21.

In the critical case, by [11, Lemma 45] we have $\lim_{k \to 0} H(u) = \frac{u}{2\pi}$, and we recover that $\mathbb{P}_{\mathrm{dimer}}^{D,u}(e) = \frac{\theta}{\pi}$ independently of $u$ [45].

## B.2 Dimers on $G^Q$

We compute the probability of single edges occurring in dimer configurations of $G^Q$ chosen with respect to the measure $\mathbb{P}_{\mathrm{dimer}}^Q$. Using the mapping between the dimer model on $G^Q$ and the free-fermion, zero-field 6-vertex model [62, 76, 32], this also gives results for the corresponding 6-vertex Boltzmann measure. Notation used are those of Figure 31 below.

<figure><img src="image_35.png" /><figcaption>Figure 31: Notation around a vertex b of $G^Q$.</figcaption></figure>

For every edge $wb$ of $G^Q$ we have, $\mathbb{P}_{\mathrm{dimer}}^Q(bw) = K_{b,w}^Q(K^Q)_{w,b}^{-1}$. We compute $(K^Q)_{w_1,b}^{-1}$ using Corollary 4.16, $i \in \{1, 2\}$. In both cases we have $\bar{\beta}_f = \bar{\beta}$; when $w = w_1$ then $\bar{\beta}_i = \bar{\beta}$ and when $w = w_2$ then $\bar{\beta}_i = \bar{\beta} - \pi$. Thus,

$$(K^Q)_{w_1,b}^{-1} = \frac{1}{[\mathrm{cn}(\theta) \mathrm{sn}(\theta) \mathrm{nd}(\theta)]^{\frac{1}{2}}} K(\beta)_{v_2,w}^{-1} = \frac{e^{-i \frac{\bar{\alpha}+\bar{\beta}}{2}}}{\mathrm{sn}(\theta)} K(\beta)_{w,v_2} K(\beta)_{v_2,w}^{-1} = \frac{e^{-i \frac{\bar{\alpha}+\bar{\beta}}{2}}}{\mathrm{sn}(\theta)} \mathbb{P}_{\mathrm{dimer}}^{D,\beta}(wv_2)$$

$$(K^Q)_{w_2,b}^{-1} = \frac{i}{[\mathrm{cn}(\theta) \mathrm{sn}(\theta) \mathrm{nd}(\theta)]^{\frac{1}{2}}} (-i) K(\beta)_{f_2,w}^{-1} = \frac{-ie^{-i \frac{\bar{\alpha}+\bar{\beta}}{2}}}{\mathrm{cn}(\theta)} K(\beta)_{w,f_2} K(\beta)_{f_2,w}^{-1} \\ = \frac{-ie^{-i \frac{\bar{\alpha}+\bar{\beta}}{2}}}{\mathrm{cn}(\theta)} \mathbb{P}_{\mathrm{dimer}}^{D,\beta}(wf_2),$$

where in the second equalities of each line we used the definition of $K(\beta)$, see Section 3.1 and in the third equalities we used the formulas for edge-probabilities on $G^D$.
Multiplying by $K(u)_{w,v_2}$ and $K(u)_{w,f_2}$ respectively yields the first equalities of Example 3.21:

$$K(u)_{w,v_2} K(u)_{v_2,w}^{-1} = (k')^{-1} \mathrm{sc}(\theta) [\mathrm{dn}(u_\alpha) \mathrm{dn}(u_\beta) G_{v_2,v_2}^m - k' G_{v_2,v_1}^m]$$

$$K(u)_{w,f_2} K(u)_{f_2,w}^{-1} = (k')^{-1} \mathrm{sc}(\theta^*) [\mathrm{dn}((u_\beta)^*) \mathrm{dn}((u_{\alpha+2K})^*) G_{f_2,f_2}^{m,*} - k' G_{f_2,f_1}^{m,*}]$$

Now by [11, Lemma 46], for every vertex $v$ of $G$, $G_{v,v}^m = \frac{k'K'}{\pi}$ and, for every $u \in \mathcal{C}$,

$$G_{v_2,v_1}^m = G_{v_1,v_2}^m = \frac{H(2u_\beta) - H(2u_\alpha)}{\mathrm{sc}(\theta)} + \frac{K'}{\pi} \mathrm{dn}(u_\alpha) \mathrm{dn}(u_\beta)$$

$$G_{f_2,f_1}^{m,*} = G_{f_1,f_2}^{m,*} = \frac{H(2(u_\beta)^*) - H(2(u_{\alpha+2K})^*)}{\mathrm{sc}(\theta^*)} + \frac{K'}{\pi} \mathrm{dn}((u_\beta)^*) \mathrm{dn}((u_{\alpha+2K})^*)$$

Combining this with the expressions of $K(u)_{x,w}^{-1}$ concludes the proof of the second equalities of Example 3.21.

In the critical case, by [11, Lemma 45] we have $\lim_{k \to 0} H(u) = \frac{u}{2\pi}$, and we recover that $P_{\mathrm{dimer}}^{D,u}(e) = \frac{\theta}{\pi}$ independently of $u$ [45].

## B.2 Dimers on $G^Q$

We compute the probability of single edges occurring in dimer configurations of $G^Q$ chosen with respect to the measure $P_{\mathrm{dimer}}^Q$. Using the mapping between the dimer model on $G^Q$ and the free-fermion, zero-field 6-vertex model [62, 76, 32], this also gives results for the corresponding 6-vertex Boltzmann measure. Notation used are those of Figure 31 below.

<figure><img src="image_35.png" /><figcaption>Figure 31: Notation around a vertex b of $G^Q$.</figcaption></figure>

For every edge $wb$ of $G^Q$ we have, $P_{\mathrm{dimer}}^Q(bw) = K_{b,w}^Q(K^Q)_{w,b}^{-1}$. We compute $(K^Q)_{w_1,b}^{-1}$ using Corollary 4.16, $i \in \{1,2\}$. In both cases we have $\bar{\beta}_f = \bar{\beta}$; when $w = w_1$ then $\bar{\beta}_i = \bar{\beta}$ and when $w = w_2$ then $\bar{\beta}_i = \bar{\beta} - \pi$. Thus,

$$(K^Q)_{w_1,b}^{-1} = \frac{1}{[\mathrm{cn}(\theta) \mathrm{sn}(\theta) \mathrm{nd}(\theta)]^{\frac{1}{2}}} K(\beta)_{v_2,w}^{-1} = \frac{e^{-i \frac{\bar{\alpha}+\bar{\beta}}{2}}}{\mathrm{sn}(\theta)} K(\beta)_{w,v_2} K(\beta)_{v_2,w}^{-1} = \frac{e^{-i \frac{\bar{\alpha}+\bar{\beta}}{2}}}{\mathrm{sn}(\theta)} P_{\mathrm{dimer}}^{D,\beta}(wv_2)$$

$$(K^Q)_{w_2,b}^{-1} = \frac{i}{[\mathrm{cn}(\theta) \mathrm{sn}(\theta) \mathrm{nd}(\theta)]^{\frac{1}{2}}} (-i) K(\beta)_{f_2,w}^{-1} = \frac{-ie^{-i \frac{\bar{\alpha}+\bar{\beta}}{2}}}{\mathrm{cn}(\theta)} K(\beta)_{w,f_2} K(\beta)_{f_2,w}^{-1} \\ = \frac{-ie^{-i \frac{\bar{\alpha}+\bar{\beta}}{2}}}{\mathrm{cn}(\theta)} P_{\mathrm{dimer}}^{D,\beta}(wf_2),$$

where in the second equalities of each line we used the definition of $K(\beta)$, see Section 3.1 and in the third equalities we used the formulas for edge-probabilities on $G^D$.