Arctic Curve of the Free-Fermion Six-Vertex Model in an L-Shaped Domain

Abstract

We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall boundary conditions, in the case of free-fermion vertex weights. We describe how the recently developed ‘Tangent method’ can be used to determine the form of the arctic curve. The obtained result is in agreement with numerics.

Appendices

Appendix A: Comparison with Finite-Size Results

In this paper we have determined the arctic curve of a free-fermionic model. As a result of the simplifications occurring in this case, with respect to what it would be for a generic six-vertex model prediction, it is much easier to perform a comparison of the result with informations obtained by alternative methods.

In particular, through the correspondence with a model of dimer coverings on a bipartite planar graph, at finite size, a suitable 1-point function in the bulk can be calculated, either from the inverse Kasteleyn matrix, or, more efficiently, through a method, devised by Propp, as part of the Urban Renewal, or Generalised Domino Shuffling, algorithm for the exact sampling of configurations (see [44], Sect. 3).

Our geometry is particularly adapted to the use of Propp’s algorithm. With respect to the graphical notation in [44] (see in particular Sect. 1.2), we shall just initialise the weights as in a graph of the form shown in Fig. 10. Then, from the algorithm we obtain the edge-inclusion probabilities, that is, the probabilities $p_{ij}$, $q_{ij}$, $r_{ij}$ and $s_{ij}$ that the edges in the plaquette of coordinates (i, j), and position NW, NE, SW and SE, respectively, are occupied in an uniformly chosen perfect matching compatible with the domain shape (again notations are chosen as to match with those in [44]). Frozen regions correspond to coordinates (i, j) such that the quadruples $(p_{ij},q_{ij},r_{ij},s_{ij})$ are equal to (1, 0, 0, 0), (0, 1, 0, 0), etc., up to corrections exponentially small in the size of the domain. We represent graphically these four functions in a compact way, with two different strategies, aiming at representing the arctic curve, or, instead, the limit shape.

Fig. 10

The Aztec Diamond graph related to the L-shaped domain

In the first case, consider the combination

$$\begin{aligned} x_{ij}=\frac{1}{2}(1+p_{ij}-q_{ij}-r_{ij}+s_{ij}), \end{aligned}$$

(A.1)

associated to each plaquette, that is valued in [0, 1], and is near to 0 or to 1 in the frozen regions (it is the local fraction of dimers which are oriented diagonally, instead that anti-diagonally). We plot in gray the plaquettes (i, j) such that $x_{ij}$ is valued in $[\varepsilon ,1-\varepsilon ]$, where $\varepsilon =N^{-2/3}$. The scaling of this threshold marks the change of regime between typical and atypical local fluctuations of the arctic curve [38]. The choice of the multiplicative constant 1 is of no special significance, and any other finite constant would have produced similar results. The comparison with our analytic prediction, shown in Fig. 9, is remarkably good (everywhere within one lattice spacing).

In the second case, a more refined visualization of the edge-inclusion probabilities is obtained by associating to a plaquette the complex number

$$\begin{aligned} z_{ij}=\sqrt{p_{ij}} + \mathrm {i}\, \sqrt{q_{ij}}- \mathrm {i}\, \sqrt{r_{ij}} - \sqrt{s_{ij}}. \end{aligned}$$

(A.2)

This quantity is valued in the disk of radius 1, and is exponentially near to 1, $\mathrm {i}$, $-1$ or $-\mathrm {i}$, if the plaquette is in a frozen region. We make a coloured plot of the domain, with hue determined according to the argument of $z_{ij}$, and brightness determined according to the absolute value of $z_{ij}$ (so that the colour is near to white in the liquid region). The data are shown in Fig. 7.

Appendix B: Proof of Proposition 3.1

We present here the derivation of representation (3.4) for the generating function $h_{N,r,s}(w)$, which is defined by (2.4) and (2.5). It can be written as

$$\begin{aligned} h_{N,r,s}(w)= \frac{F_{N,r,s}(w)}{F_{N,r,s}(1)}, \end{aligned}$$

(B.1)

where

$$\begin{aligned} F_{N,r,s}(w)=\sum _{r_1=1}^{r} \left( G_{N,s}^{(r_1,r,\ldots ,r)}-G_{N,s}^{(r_1-1,r,\ldots ,r)}\right) w^{r_1-1}. \end{aligned}$$

(B.2)

Note that $F_{N,r,s}(1)=G_{N,s}^{(r,\ldots ,r)}$ is the EFP of the six-vertex model with domain wall boundary conditions ($F_{N,r,s}(1)\equiv F_N^{(r,s)}$, in the notation of [37]). Change of the integration variables $z_j\mapsto x_j=(\alpha z_j+1-\alpha )/z_j$, $j=1,\ldots ,s$, in (2.2) yields

$$\begin{aligned} G_{N,s}^{(r_1,\ldots ,r_s)}= & {} (-1)^{\frac{s(s-1)}{2}} \prod _{j=1}^{s} (1-\alpha )^{N-r_j} \nonumber \\&\times \oint _{C_\infty }^{} \cdots \oint _{C_\infty }^{} \prod _{j=1}^s \frac{x_j^{N-j}}{(x_j-\alpha )^{N-r_j}(x_j-1)^{s-j+1}} \prod _{1\le j<k\le s}(x_k-x_j)\, \frac{\mathrm {d}^s x}{(2\pi \mathrm {i})^s}, \end{aligned}$$

(B.3)

where $C_\infty$ denotes a circular contour of large radius around the origin (thus enclosing the points $x=\alpha$ and $x=1$). Hence,

$$\begin{aligned} F_{N,r,s}(w)= & {} (-1)^{\frac{s(s-1)}{2}}(1-\alpha )^{(N-r)s} w^{r-1} \nonumber \\&\times \oint _{C_\infty }^{} \cdots \oint _{C_\infty }^{} \frac{x_1^{N-1}}{(x_1-\alpha )^{N-r}(x_1-1)^{s-1}(x_1-u)} \nonumber \\&\times \prod _{j=2}^s \frac{x_j^{N-j}}{(x_j-\alpha )^{N-r}(x_j-1)^{s-j+1}} \prod _{1\le j<k\le s}(x_k-x_j)\, \frac{\mathrm {d}^s x}{(2\pi \mathrm {i})^s}, \end{aligned}$$

(B.4)

where $u=(\alpha w +1-\alpha )/w$. Using

$$\begin{aligned} \det \left[ (x_{s-k+1}-\alpha )^{s-j}\right] _{j,k=1,\ldots ,s} =\prod _{1\le j<k\le s}^{}(x_k-x_j) \end{aligned}$$

(B.5)

we can write $F_{N,r,s}(w)$ in the form of an $s\times s$ determinant

$$\begin{aligned} F_{N,r,s}(w)=(-1)^{\frac{s(s-1)}{2}}(1-\alpha )^{s(s+q)} w^{r-1} \det A(u), \end{aligned}$$

(B.6)

where the matrix A(u) contains dependence on u only in the last column

$$\begin{aligned} A_{jk}(u)= {\left\{ \begin{array}{ll} \displaystyle \oint _{C_\infty } \frac{x^{r+q+k-1}}{(x-\alpha )^{q+j}(x-1)^k}\, \frac{\mathrm {d}x}{2\pi \mathrm {i}} &{}\quad k\ne s \\ \displaystyle \oint _{C_\infty } \frac{x^{r+q+s-1}}{(x-\alpha )^{q+j}(x-1)^{s-1}(x-u)}\, \frac{\mathrm {d}x}{2\pi \mathrm {i}} &{}\quad k=s, \end{array}\right. } \end{aligned}$$

(B.7)

and where we have set $N=r+s+q$, $q\ge 0$.

To proceed with (B.6), it is useful to consider first the case $w=1$, that corresponds to $u=1$. Using

$$\begin{aligned}&\oint _{C_\infty } \frac{x^{c}}{(x-\alpha )^{a}(x-\beta )^b}\,\frac{\mathrm {d}x}{2\pi \mathrm {i}} = \frac{1}{(a-1)!(b-1)!}\partial _\alpha ^{a-1} \partial _\beta ^{b-1} \oint _{C_\infty } \frac{x^{c}}{(x-\alpha )(x-\beta )}\, \frac{\mathrm {d}x}{2\pi \mathrm {i}} \nonumber \\&\quad = \sum _{m=a-1}^{c-b}\left( {\begin{array}{c}m\\ a-1\end{array}}\right) \left( {\begin{array}{c}c-m-1\\ b-1\end{array}}\right) \alpha ^{m-a+1} \beta ^{c-m-b}, \qquad a,b,c\in {\mathbb {N}}, \end{aligned}$$

(B.8)

for the entries of the matrix $A\equiv A(1)$, upon setting $\beta =1$, $a=q+j$, $b=k$, and $c=r+q+k-1$ and making the change $m\mapsto m+q$, we get

$$\begin{aligned} A_{jk}=\sum _{m=j-1}^{r-1}\left( {\begin{array}{c}m+q\\ q+j-1\end{array}}\right) \left( {\begin{array}{c}r+k-2-m\\ k-1\end{array}}\right) \alpha ^{m-j+1}. \end{aligned}$$

(B.9)

Consider now entries of a given column; since

$$\begin{aligned} \left( {\begin{array}{c}m+q\\ q+j-1\end{array}}\right) =\frac{q!}{(q+j-1)!}\left( {\begin{array}{c}m+q\\ q\end{array}}\right) (m)_{j-1}, \end{aligned}$$

(B.10)

where $(m)_{a}:=m (m-1)\cdots (m-a+1)$ denotes the falling factorial, we have

$$\begin{aligned} A_{jk}=\frac{q!}{(q+j-1)! \alpha ^{j-1}} \widetilde{A}_{jk} \end{aligned}$$

(B.11)

where

$$\begin{aligned} \widetilde{A}_{jk}=\sum _{m=0}^{r-1}\left( {\begin{array}{c}m+q\\ q\end{array}}\right) (m)_{j-1} \left( {\begin{array}{c}r+k-2-m\\ k-1\end{array}}\right) \alpha ^{m}, \end{aligned}$$

(B.12)

and hence

$$\begin{aligned} \det A= \frac{(q!)^s}{\prod _{j=0}^{s-1}(q+j)! } \alpha ^{-\frac{s(s-1)}{2}} \det \widetilde{A}. \end{aligned}$$

(B.13)

The determinant of $\widetilde{A}$ evaluates as follows

$$\begin{aligned} \det \widetilde{A}= & {} \sum _{m_1,\ldots ,m_s=0}^{r-1} \prod _{k=1}^{s}\left( {\begin{array}{c}m_k+q\\ q\end{array}}\right) \left( {\begin{array}{c}r+k-2-m_k\\ k-1\end{array}}\right) \prod _{l<k}^{}(m_k-m_l) \alpha ^{m_1+\ldots +m_s} \nonumber \\= & {} \frac{(-1)^{\frac{s(s-1)}{2}}}{\prod _{j=0}^{s} j!} \sum _{m_1,\ldots ,m_s=0}^{r-1} \prod _{k=1}^{s}\left( {\begin{array}{c}m_k+q\\ q\end{array}}\right) \prod _{l<k}^{}(m_k-m_l)^2 \alpha ^{m_1+\ldots +m_s}, \end{aligned}$$

(B.14)

where we have used the fact that $(m)_{j-1}$ is a monic polynomial of degree $j-1$ in m, and, similarly, that $\left( {\begin{array}{c}r+k-2-m\\ k-1\end{array}}\right)$ is a polynomial of degree $k-1$ in m, with the leading coefficient $(-1)^{k-1}/(k-1)!$. In total, our calculation amounts to

$$\begin{aligned} F_{N,r,s}(1)= & {} \frac{(q!)^s}{\prod _{j=0}^{s-1}(q+j)!\prod _{j=0}^{s}j!} \frac{(1-\alpha )^{s(s+q)}}{\alpha ^{\frac{s(s-1)}{2}}} \nonumber \\&\times \sum _{m_1,\ldots ,m_s=0}^{r-1} \prod _{j=1}^{s}\left( {\begin{array}{c}m_j+q\\ q\end{array}}\right) \prod _{l<k}(m_k-m_l)^2\alpha ^{m_1+\ldots +m_s}. \end{aligned}$$

(B.15)

Note that this representation may equivalently be written as

$$\begin{aligned} F_{N,r,s}(1) =\frac{(q!)^s}{\prod _{j=0}^{s-1}(q+j)!j!} \frac{(1-\alpha )^{s(s+q)}}{\alpha ^{\frac{s(s-1)}{2}}} \det \left[ \sum _{m=0}^{r-1}\left( {\begin{array}{c}m+q\\ q\end{array}}\right) m^{j+k-2}\alpha ^m\right] _{j,k=1,\ldots ,s}, \end{aligned}$$

(B.16)

in agreement with [38, 39].

Consider now the case of generic w. To apply the derivation above with a minimal modification, consider instead of the matrix A(u) some matrix B(u), which differs from A(u) only in the entries of the last column,

$$\begin{aligned} B_{js}(u) = \oint _{C_\infty }^{} \frac{x^{r+q}}{(x-\alpha )^{q+j}} \left( \frac{x^{s-1}}{(x-1)^{s-1}(x-u)}+\sum _{k=1}^{s-1}\gamma _k\frac{x^{k-1}}{(x-1)^k} \right) \, \frac{\mathrm {d}x}{2\pi \mathrm {i}}, \end{aligned}$$

(B.17)

where $\gamma _k$, $k=1,\ldots ,s-1$, are some constants in x. Note that $\det A(u) =\det B(u)$. For $\gamma _k=u^{s-1-k}/(u-1)^{s-k}$ the pole at $x=1$ disappears in the integral, since

$$\begin{aligned} \sum _{k=1}^{s-1}\gamma _k\frac{x^{k-1}}{(x-1)^k}= \frac{u^{s-1}}{(u-1)^{s-1}(x-u)}-\frac{x^{s-1}}{(x-1)^{s-1}(x-u)}. \end{aligned}$$

(B.18)

Therefore, with this choice of $\gamma _k$’s, and recalling (B.8), we have

$$\begin{aligned} B_{js}(u)=\frac{q!}{(q+j-1)!\alpha ^{j-1}} \frac{u^{r+s-2}}{(u-1)^{s-1}} \sum _{m=0}^{r-1}\left( {\begin{array}{c}m+q\\ q\end{array}}\right) (m)_{j-1} \left( \frac{\alpha }{u}\right) ^m. \end{aligned}$$

(B.19)

Similarly to (B.12), introduce matrix $\widetilde{B}(u)$, with entries

$$\begin{aligned} \widetilde{B}_{jk}(u)= {\left\{ \begin{array}{ll} \widetilde{A}_{jk}&{} k\ne s\\ \displaystyle \sum _{m=0}^{r-1} \left( {\begin{array}{c}q+m\\ q\end{array}}\right) (m)_{j-1} \left( \frac{\alpha }{u}\right) ^{m}&k=s. \end{array}\right. } \end{aligned}$$

(B.20)

We have

$$\begin{aligned} \det B(u)= \frac{(q!)^s}{\prod _{j=0}^{s-1}(q+j)! } \frac{u^{r+s-2}}{\alpha ^{\frac{s(s-1)}{2}}(u-1)^{s-1}} \det \widetilde{B}(u). \end{aligned}$$

(B.21)

In this case, the analogue of (B.14) is

$$\begin{aligned} \det \widetilde{B}(u)= & {} \sum _{m_1,\ldots ,m_s=0}^{r-1} \prod _{k=1}^{s}\left( {\begin{array}{c}m_k+q\\ q\end{array}}\right) \nonumber \\&\times \prod _{k=1}^{s-1}\left( {\begin{array}{c}r+k-2-m_k\\ k-1\end{array}}\right) \prod _{l<k}^{}(m_k-m_l) \frac{\alpha ^{m_1+\ldots +m_s}}{u^{m_s}} \nonumber \\= & {} \frac{(-1)^{\frac{(s-1)(s-2)}{2}}}{\prod _{j=0}^{s-2} j!} \sum _{m_1,\ldots ,m_s=0}^{r-1} \prod _{k=1}^{s}\left( {\begin{array}{c}m_k+q\\ q\end{array}}\right) \prod _{l<k}^{}(m_k-m_l) \nonumber \\&\times \prod _{k=1}^{s-1} m_k^{k-1} \frac{\alpha ^{m_1+\ldots +m_s}}{u^{m_s}}. \end{aligned}$$

(B.22)

Symmetrizing the summand with respect to permutations of $m_1,\ldots ,m_s$ and substituting everything in (B.6), we get

$$\begin{aligned} F_{N,r,s}(w)= & {} \frac{(q!)^s}{s!\prod _{j=0}^{s-1}(q+j)!\prod _{j=0}^{s-2}j!} \frac{(1-\alpha )^{s(s+q)}}{\alpha ^{\frac{s(s-1)}{2}}} w^{r-1} \frac{u^{r+s-2}}{(u-1)^{s-1}} \nonumber \\&\times \sum _{m_1,\ldots ,m_s=0}^{r-1} \prod _{j=1}^{s}\left( {\begin{array}{c}m_j+q\\ q\end{array}}\right) \prod _{l<k}(m_k-m_l)\alpha ^{m_1+\ldots +m_s} \nonumber \\&\times \sum _{p=1}^{s} (-1)^{p-1} \prod _{\begin{array}{c} l<k\\ l,k\ne p \end{array}}(m_k-m_l) u^{-m_p}. \end{aligned}$$

(B.23)

Finally, rewriting the sum over p as a contour integral, we arrive at

$$\begin{aligned} F_{N,r,s}(w)=\frac{(q!)^s}{\prod _{j=0}^{s-1}(q+j)!\prod _{l=0}^{s}j!} \frac{(1-\alpha )^{s(N-r)}}{ \alpha ^{s(s-1)/2}}w^{r-1}I_{N,r,s}(u) \end{aligned}$$

(B.24)

where the quantity $I_{N,r,s}(u)$ is defined in (3.6). Recalling (B.1), the statement of the Proposition 3.1, representation (3.4), immediately follows.

We also mention that (B.23) can be written as

$$\begin{aligned} F_{N,r,s}(w) =\frac{(q!)^s}{\prod _{j=0}^{s-1}(q+j)!\prod _{j=0}^{s-2}j!} \frac{(1-\alpha )^{s(s+q)}}{\alpha ^{\frac{s(s-1)}{2}}} w^{r-1} \frac{u^{r+s-2}}{(1-u)^{s-1}} \det H, \end{aligned}$$

(B.25)

where the $s\times s$ matrix H is

$$\begin{aligned} H_{jk}= {\left\{ \begin{array}{ll} \displaystyle \sum _{m=0}^{r-1}\left( {\begin{array}{c}m+q\\ q\end{array}}\right) m^{j+k-2}\alpha ^m &{} k\ne s \\ \displaystyle \sum _{m=0}^{r-1}\left( {\begin{array}{c}m+q\\ q\end{array}}\right) m^{j-1} \left( \frac{\alpha }{u}\right) ^m&k=s. \end{array}\right. } \end{aligned}$$

(B.26)

Note that, as $w\rightarrow 1$ (that is, $u\rightarrow 1$), the expected result (B.16) is reproduced from (B.25) upon taking into account that $\det H$ has a zero of order $(s-1)$ at $u=1$.

Appendix C: Arctic Curve for Regime II, Symmetric Domain

Here we report explicit expression for the polynomial ${\mathcal {A}}(z_1,z_2)$ describing the arctic curve, for the case $Q=0$ of the model in Regime II (symmetric L-shaped domain). The curve is given by the equation ${\mathcal {A}}(z_1,z_2)=0$ and it is of degree 6.

We first introduce properly scaled diagonal coordinates $Z_1$ and $Z_2$, defining them by

$$\begin{aligned} z_1=\sqrt{\alpha }Z_1,\qquad z_2=\sqrt{1-\alpha }Z_2. \end{aligned}$$

(C.1)

Recall that the original diagonal coordinates are defined by (4.18). Note, that in terms of the new coordinates the Arctic ellipse (2.9) just reads

$$\begin{aligned} Z_1^2+Z_2^2=1. \end{aligned}$$

(C.2)

Next, we introduce the following parameterization for the scaling parameter $R\in [1,R_\mathrm {c}]$:

$$\begin{aligned} R=\frac{1+\sqrt{\alpha }\beta }{1-\sqrt{\alpha }\beta },\qquad \beta \in [0,1]. \end{aligned}$$

(C.3)

The meaning of this re-parametrization is to simplify further expressions for the coefficients of the arctic curve, making them polynomials in $\alpha$ and $\beta$.

At last, we introduce coefficients $C_{n_1n_2}$ which describe the polynomial $A(z_1,z_2)$ appearing in (4.22), in terms of the coordinates (C.1)

$$\begin{aligned} {\mathcal {A}}(z_1,z_2)=(1-\alpha )^2\alpha ^6\sum _{0\le n_1+n_2\le 6} C_{n_1 n_2} Z_1^{n_1} Z_2^{n_2}. \end{aligned}$$

(C.4)

Note that because of the symmetry of the L-shaped domain under reflection with respect to the North-West / South-East diagonal, the arctic curve possesses the symmetry $A(z_1,-z_2)=A(z_1,z_2)$, that is, it depends only on even powers of $z_2$, i.e., $C_{n_1n_2}=0$ if $n_2$ is odd ($n_2=1,3,5$). This excludes 12 coefficients out of 28 in total, which describe a generic degree 6 curve.

The nonzero 16 coefficients have the following expressions:

$$\begin{aligned} C_{60}&=64 (1-\alpha )^2\left( 1-2\alpha \beta +\alpha \beta ^2\right) ^2, \nonumber \\ C_{50}&=64 (1-\alpha )^2 \left[ 1-(5+2\alpha )\beta +18\alpha \beta ^2 -2\alpha (4+7\alpha )\beta ^3+13\alpha ^2\beta ^4-3\alpha ^2\beta ^5\right] , \nonumber \\ C_{42}&=128(1-\alpha )^2\left[ 1+(2-6\alpha )\beta -2\left( 1-\alpha -3\alpha ^2\right) \beta ^2+2(1-3\alpha )\alpha \beta ^3 +\alpha ^2\beta ^4\right] , \nonumber \\ C_{40}&=16(1-\alpha )\big [1+\alpha -\left( 22-18\alpha +8\alpha ^2\right) \beta +\left( 41+13\alpha -32\alpha ^2+8\alpha ^3\right) \beta ^2 \nonumber \\&\quad -4\alpha \left( 36-25\alpha -\alpha ^2\right) \beta ^3 +\alpha \left( 52+63\alpha -85\alpha ^2\right) \beta ^4 -6\alpha ^2(13-11\alpha )\beta ^5 \nonumber \\&\quad +(15-13\alpha )\alpha ^2\beta ^6\big ], \nonumber \\ C_{32}&=128(1-\alpha )^2\big [(1-2(2+\alpha )\beta -(4-18\alpha )\beta ^2 +\left( 4-6\alpha -14\alpha ^2\right) \beta ^3 \nonumber \\&\quad -(4-13 \alpha )\alpha \beta ^4-2\alpha ^2 \beta ^5\big ], \nonumber \\ C_{30}&=32 (1-\alpha )(1-\beta )\big [\alpha -\left( 2+3\alpha +2\alpha ^2\right) \beta +\left( 22-21\alpha +19\alpha ^2\right) \beta ^2 \nonumber \\&\quad -\alpha \left( 59-48\alpha +19\alpha ^2\right) \beta ^3 +\alpha \left( 22+18\alpha -15\alpha ^2\right) \beta ^4 \nonumber \\&\quad -\alpha ^2(28-17\alpha )\beta ^5+\alpha ^2(5-3\alpha )\beta ^6\big ], \nonumber \\ C_{24}&=64 (1-\alpha )^2 \big [1+(8-12\alpha )\beta -2\left( 4-\alpha -6\alpha ^2\right) \beta ^2 +4\alpha (2-3\alpha )\beta ^3+\alpha ^2\beta ^4\big ], \nonumber \\ C_{22}&=-32(1-\alpha )\big [1+\left( 12-26\alpha +8\alpha ^2\right) \beta -\left( 15-3\alpha -35\alpha ^2+8\alpha ^3\right) \beta ^2 \nonumber \\&\quad -\left( 22-96\alpha +90\alpha ^2+4\alpha ^3\right) \beta ^3 +\left( 12-25\alpha -35\alpha ^2+63\alpha ^3\right) \beta ^4 \nonumber \\&\quad -2\alpha \left( 6-26\alpha +23\alpha ^2\right) \beta ^5 -\alpha ^2(6-7\alpha )\beta ^6\big ], \nonumber \\ C_{20}&=4(2-\alpha )\alpha -16\alpha \left( 9-8\alpha +\alpha ^2\right) \beta +4\left( 24+78\alpha -36\alpha ^2-42\alpha ^3+4\alpha ^4\right) \beta ^2 \nonumber \\&\quad -32\left( 26-41\alpha +64\alpha ^2-45\alpha ^3+3\alpha ^4\right) \beta ^3 \nonumber \\&\quad +4\left( 104+286\alpha -438\alpha ^2+322\alpha ^3-204\alpha ^4\right) \beta ^4 \nonumber \\&\quad -4\alpha \left( 105-96\alpha +33\alpha ^2-28\alpha ^3\right) \beta ^5 +2\alpha \left( 41+102\alpha -163\alpha ^2+34\alpha ^3\right) \beta ^6 \nonumber \\&\quad -8\alpha ^2\left( 16-20\alpha +5\alpha ^2\right) \beta ^7 +\alpha ^2\left( 15-18\alpha +4\alpha ^2\right) \beta ^8, \nonumber \\ C_{14}&=64(1-\alpha )^2\big [1-(3+2\alpha )\beta -(8-18\alpha )\beta ^2 +2\left( 4-2\alpha -7\alpha ^2\right) \beta ^3 \nonumber \\&\quad -\alpha (8-13\alpha )\beta ^4-\alpha ^2\beta ^5\big ], \nonumber \\ C_{12}&=-32(1-\alpha )\big [2-\alpha -\left( 6+3\alpha -2\alpha ^2\right) \beta -\left( 16-58\alpha +21\alpha ^2\right) \beta ^2 \nonumber \\&\quad +\left( 14-20\alpha -48\alpha ^2+19\alpha ^3\right) \beta ^3 +\left( 14-53\alpha +78\alpha ^2-4\alpha ^3\right) \beta ^4 \nonumber \\&\quad -\left( 4-3\alpha -16\alpha ^2+36\alpha ^3\right) \beta ^5 +\alpha \left( 4-17\alpha +20\alpha ^2\right) \beta ^6 +\alpha ^2(2-3\alpha )\beta ^7\big ], \nonumber \\ C_{10}&=4\big [4(1-\alpha )^2\beta ^2+\alpha (1-\beta )^4\big ] \big [\alpha -\left( 4-\alpha +2\alpha ^2\right) \beta +\left( 28-30 \alpha +12 \alpha ^2\right) \beta ^2 \nonumber \\&\quad -\left( 8+22 \alpha -20\alpha ^2\right) \beta ^3 +\alpha (21-16\alpha )\beta ^4-\alpha (3-2\alpha )\beta ^5\big ], \nonumber \\ C_{06}&=256(1-\alpha )^3(1-\beta )\beta (1-\alpha \beta ), \nonumber \\ C_{04}&=16 (1-\alpha )^2 \big [1-(26-8 \alpha ) \beta +\left( 41+30\alpha -8\alpha ^2\right) \beta ^2 -4 \left( 1+21\alpha +\alpha ^2\right) \beta ^3 \nonumber \\&\quad -\left( 8-30 \alpha -41 \alpha ^2\right) \beta ^4 +2 (4-13 \alpha ) \alpha \beta ^5+\alpha ^2 \beta ^6\big ], \nonumber \\ C_{02}&=-8(1-\alpha )(1-\beta )\big [2-\alpha -\left( 18-7 \alpha +2 \alpha ^2\right) \beta +\left( 32+16 \alpha +\alpha ^2+2 \alpha ^3\right) \beta ^2 \nonumber \\&\quad +\left( 24-138\alpha +29\alpha ^2-10\alpha ^3\right) \beta ^3 +\left( 10-29\alpha +138\alpha ^2-24\alpha ^3\right) \beta ^4 \nonumber \\&\quad -\left( 2+\alpha +16 \alpha ^2+32\alpha ^3\right) \beta ^5 +\alpha \left( 2-7\alpha +18\alpha ^2\right) \beta ^6 +\alpha ^2(1-2\alpha )\beta ^7\big ], \nonumber \\ C_{00}&=\left( 1-6\beta +\beta ^2\right) \left[ 4(1-\alpha )^2\beta ^2+\alpha (1-\beta )^4\right] ^2. \end{aligned}$$

(C.5)

Note that the coefficients are polynomials in $\alpha$ of the degree at most 4, and in $\beta$ they are all, but $C_{00}$, of the degree at most 8; the latter is of the degree 10.

In the limit $\beta \rightarrow 1$, that is $R\rightarrow R_\mathrm {c}$, the arctic curve factorizes onto two straight lines $Z_1=1$, the usual Arctic ellipse (C.2), as expected, and the point $(Z_1,Z_2)=(1,0)$ belonging to the Arctic ellipse:

$$\begin{aligned} A(z_1,z_2)\Big |_{\beta =1}= 64(1-\alpha )^6\alpha ^6 \left( Z_1-1\right) ^2 \left( Z_1^2+Z_2^2-1\right) \left[ (Z_1-1)^2+Z_2^2\right] . \end{aligned}$$

(C.6)

In the limit $\beta \rightarrow 0$, that is $R\rightarrow 1$, the arctic curve factorizes onto two straight lines $Z_1=-1$, and two Arctic ellipses of radii 1 / 2:

$$\begin{aligned} A(z_1,z_2)\Big |_{\beta =0}= & {} 16(1-\alpha )^4\alpha ^6 (Z_1+1)^2 \left[ \left( Z_2-\frac{1}{2\sqrt{1-\alpha }}\right) ^2+Z_1^2-\frac{1}{4}\right] \nonumber \\&\times \left[ \left( Z_2+\frac{1}{2\sqrt{1-\alpha }}\right) ^2+Z_1^2-\frac{1}{4}\right] , \end{aligned}$$

(C.7)

as expected.