Arctic curves of the 20V model on a triangle

We apply the Tangent Method of Colomo and Sportiello to predict the arctic curves of the Twenty Vertex model with specific domain wall boundary conditions on a triangle, in the Disordered phase, leading to a phase diagram with six types of frozen phases and one liquid one. The result relies on a relation to the Six Vertex model with domain wall boundary conditions and suitable weights, as a consequence of integrability. We also perform the exact refined enumeration of configurations.


Introduction
Two-dimensional integrable lattice models such as the Six Vertex (6V) model have a long history first rooted in the physics of spin systems with local interaction, for which exact bulk thermodynamic properties were derived [Lie67], including continuum descriptions via Coulomb Gas [Nie84] or Conformal Field Theory [DFSZ87]. More recently, these models also entered the realm of combinatorics (by considering domains with 'domain-wall' boundary conditions (DWBC) [Kor82] and best illustrated by the correspondence between 6V-DWBC and Alternating Sign Matrices (ASM) [Kup96]), probability theory (by interpreting the configurations in terms of particle trajectories, see e.g. [BCG16]), and algebraic geometry (by interpreting partition functions as K-theoretic characters of certain varieties, see e.g. [GZJ22]). It was observed that in the presence of DWBC, the models behave quite differently and may display interesting scaling behavior, such as the arctic phenomenon. The latter is the emergence of sharp phase separations between ordered regions (crystal-like, near the boundaries of the domain) and disordered (liquid) ones.
Such a phenomenon had been already observed in 'free fermion' tiling or dimer models, where typically tiles/dimers choose a preferred crystalline orientation near boundaries while they tend to be disordered away from them. This was first observed in the uniform domino tilings of the Aztec diamond [JPS98], giving rise to an arctic circle, and a general theory was developed for dimers [KO07,KOS06]. The free fermion character of these models can be visualized in their formulation in terms of non-intersecting lattice paths, i.e. families of paths with fixed ends, and sharing no vertex (i.e. avoiding each-other), and can consequently be expressed in terms of free lattice fermions. A manifestation of the free fermion character of these models is that their arctic curves are always analytic.
The 6V model in its disordered phase 1 , while including a free fermion case, is generically a model of interacting fermions: it admits an 'osculating path' description, in which paths are non-intersecting, but are allowed to interact by 'kissing' i.e. sharing a vertex at which they bounce against each-other.
The 6V model on a square domain with DWBC exhibits an arctic phenomenon in its disordered phase, which was predicted via non-rigorous methods [CP10,CNP11], the latest of which being the Tangent Method introduced by Colomo and Sportiello [CS16]. The new feature arising from these studies is that the arctic curves are generically no longer analytic, but rather piecewise analytic. For instance, the arctic curve for large Alternating Sign Matrices (uniformly weighted 6V-DWBC) is made of four pieces of different ellipses as predicted in [CP10] and later proved in [Agg20].
The Tangent Method was validated in a number of cases, mostly in free fermion situations [CPS19, DFL18,DFG18,DFG19b,DFG19a,CKN21]. Beyond free fermions and the case of the 6V-DWBC model (see also [DF21a] for the case of U-turn reflective boundaries), the method was applied to another model of osculating paths: the Twenty Vertex (20V) model which is the triangular lattice version of the 6V model [Kel74,DFG20,DDFG20,DF21b,DF21a]. In [DFG20], four possible variations around DWBC were considered for the 20V model, denoted DWBC1,2,3,4. It turns out that for DWBC1,2 (on a square domain) and DWBC3 (on a quadrangular domain) the (refined) enumeration of configurations can be achieved exactly in terms of the (possibly U-turn) 6V-DWBC model. Moreover in these two cases the total number of configurations matches the number of domino tilings of Aztec-like domains (see [DFG20] and [DF21b]) and there is an intriguing correspondence between arctic curves of both tilings and Vertex models.
In the present paper, we define new boundary conditions on a triangular domain for the 20V model, and we show that these give rise to an arctic phenomenon. After performing an exact (refined) enumeration of the configurations, we derive exact Tangent Method predictions for the (outer) arctic curves for arbitrary integrable Boltzmann weights of the disordered phase, by relating the model to the 6V-DWBC model, in the spirit of [DDFG20,DF21a].
The paper is organized as follows. In section 2 we recall the definition of the 20V model, which is an ice-type model on the triangular lattice, and its integrable weights, and define a triangular domain and specific domain-wall boundary conditions for the model, which we coin 20V 3 . We also show simple transformations that allow to immediately obtain some other similar boundary conditions which we call 20V 1 and 20V 2 . In section 3, using integrability of the weights, we compute the fully inhomogeneous partition function of the 20V 3 model in terms of the inhomogeneous partition function of the Six Vertex (6V) model, leading in particular to the (refined) enumeration of the configurations of the 20V 3 model. section 4. is devoted to the computation of arctic curves. After recalling the principle of the Tangent Method, we give asymptotic estimates of the one-point function and path partition function determining the arctic curve: the main results obtained by applying the Tangent Method are theorems 4.2-4.7, respectively for three distinct portions of the outer arctic curve of the model, for arbitrary values of the integrable weights in the Disordered phase. We also briefly discuss the possible hidden structure for the inner part of the arctic curve, not predicted by our method, except in the free fermion case, where the arctic curve is expected to be analytic, i.e. all the branches are part of the same analytic curve. In section 5 we present examples of arctic curves, successively in the uniformly weighted case for which the NE portion of arctic curve is part of an algebraic curve of degree 10, in the free-fermion case (where the Boltzmann weights are expressed in terms of free-fermion 6V weights), and finally in the generic case. We gather a few concluding remarks in section 6.

20V model and integrable weights
The 20 Vertex (20V) model is the triangular lattice version of the square 'ice' model. Its configurations are choices of orientations of the edges of the triangular lattice, with the constraint that there are exactly three incoming and three outgoing edges adjacent to each vertex ('ice rule'). This gives rise to the 20 = 6 3 possible vertex environments depicted in figure 1 (top two rows): here and in the following, we represent the triangular lattice with vertices in Z 2 for convenience. A standard bijection allows to reformulate 20V configurations in terms of osculating Schröder paths, namely paths on Z 2 with horizontal, diagonal, vertical steps (1, 0), (1, −1), (0, −1), which are non-intersecting but are allowed to have contact ('kissing') points (see figure 1, two bottom rows). The vertices of the triangular lattice are at the intersection of three (horizontal, diagonal and vertical) lines. We may resolve these intersection by slightly moving up all diagonal lines. The resulting lattice is the Kagome lattice, with three times as many vertices, but simple intersections (either horizontal-vertical, horizontaldiagonal or diagonal-vertical), giving rise to three natural square sublattices which we label 1,2,3.
In [Kel74,Bax89], Boltzmann weights for the 20V model are constructed in terms of weights of the Kagome lattice, themselves decomposing into three sets of 6V weights for the vertices of type 1,2,3 respectively. Indeed as shown in [Kel74,Bax89,DDFG20], configurations of the 20V model may be obtained by considering oriented edges of the Kagome lattice satisfying the ice rule at all the vertices (of type 1,2,3). The correspondence is 1 to possibly 2. The weights of the 20V model are defined as the sum over the three inner arrow configurations that satisfy the three ice rules at the vertices of type 1,2,3, of the product of the three 6V Boltzmann weights of vertices 1,2,3. For illustration, we have represented in figure 2 the three types of 6V configurations together with their Boltzmann weights (a i , b i , c i ) for i = 1, 2, 3. Integrable weights for the 20-V model were obtained [Kel74,Bax89] by further requiring that the expression for the weights is independent of the resolution of the triple intersections. For instance we could have moved down slightly all diagonal lines, giving rise to different relations for the 20V weights in terms of 6V weights. Solving these algebraic equations leads to a 4-parameter (projective) family of integrable Boltzmann weights. The first condition is that the three 6V models must share the same quantum parameter q. Moreover, attaching spectral parameters z, t, w to horizontal, diagonal, vertical lines respectively, the integrable 6V weights can be written as where α i are constant normalization factors. These finally lead to the following expressions for the 20V integrable weights ω i ≡ ω i [z, t, w] displayed in figure 3: where ν 0 = α 1 α 2 α 3 .
By construction the above weights allow for freely moving around lines across intersections, a key property which we shall use extensively in the following to simplify the model.
With the following parametrization of the quantum and spectral parameters the Boltzmann weights of the three 6V models on the sublattices 1, 2, 3 with horizontal, diagonal, vertical parameters z, t, w respectively read: where β 1 = √ zw α 1 , β 2 = √ zt α 2 and β 3 = √ tw α 3 are assumed to be positive numbers. The corresponding 20V model weights read [DFG18,Kel74]: with ν = β 1 β 2 β 3 . We will finally restrict to the so-called Disordered Phase of the model, corresponding to real values of the parameters η, λ and µ and to a range of these parameters ensuring that all ω's are positive: Note as a consequence that λ + 3η > µ. Note also the existence of a 'combinatorial point' where the weights ω i are uniform and all equal to 1: (2.6)

The 20V-DWBC3 model on the triangle Tm
We consider the partition function Z  spectral parameters z = z 1 , z 2 , . . . , z m (from top to bottom), t = t 1 , t 2 , . . . , t m (from top to bottom) and w = w 1 , w 2 , . . . , w m (from right to left): each vertex v is weighted by the weight ω(v) of (2.5) corresponding to its local configuration, with parameters λ, µ, η corresponding to the three (horizontal, diagonal, vertical) lines meeting at v, while the partition function is the sum over the 20V configurations of the product of their local vertex weights.

Transformations of the 20V model
In this section, we use transformations of the 20V model previously exploited in references [DDFG20,DF21a] to map the 20V 3 configurations on T m onto configurations of the 20V model on the same domain but with different boundary conditions. For convenience, we place the origin at the bottom right vertex of T m , so that the extremal vertices of T m have coordinates We introduce the following sequence of transformations of the path configurations of the 20V 3 model, as illustrated in figure 5 (top line). Assume that k is the position of the topmost vertex visited by a path in the rightmost vertical line. By inspection we find that the seven classes of 20V local vertex environments depicted in figure 3 are mapped bijectively under S • R • VF as follows: (2.7) For further use, we denote by π the permutation π = (01)(24) corresponding to this mapping of configurations.
Examining the effect on the boundary conditions, it is easy to see that 20V 3 configurations are mapped to 20V configurations on the same triangle T m but with new boundary conditions depicted in figures 6(a) and (b) for odd and even size m: we call this model 20V 2 . Each step being invertible, S • R • VF is clearly a bijection.
Denoting by Z 20V2 m [z, t, w] the partition function for the inhomogeneous 20V 2 model, with the same labeling of spectral parameters as for the 20V 3 model, and following the horizontal, diagonal and vertical lines throughout the transformations, we deduce the following relation.
Theorem 2.1. The partition functions of the fully inhomogeneous 20V 3 and 20V 2 models are related via: Indeed, the mapping of 20V weights under the transformation S • R • VF amounts to the simple transformation ω i [z, t, w] → ω π(i) [t, z, w], as diagonal and horizontal lines are interchanged in the process.
From a purely enumerative point of view, it is clear from figure 5 that S • R • VF maps (refined) configurations of the 20V 3 model to those of the 20V 2 model (up to k → m + 1 − k).
Corollary 2.2. The refined numbers of configurations in the 20V 3 and 20V 2 models are related via: We may repeat the above as indicated in figure 5 (bottom line) with instead of VF a horizontal flip (HF), namely erasing all horizontal path edges, and promoting all empty horizontal edges to new path edges. The reflection R is now replaced with a reflectionR w.r.t. a vertical line and the shear S with a horizontal shearS: (x, y) → (m + 1 + x − y, y). The net result is the same as applying an extra diagonal reflection R * w.r.t. a direction perpendicular to that of the diagonal lines with parameters t i , after the previous transformation S • R • VF, resulting in the relationS •R • HF = R * • S • R • VF. We denote by 20V 1 the corresponding model. Its configurations and boundary conditions are simply obtained from those of the 20V 2 model by applying the diagonal reflection R * , under which the seven weight classes are mapped as follows: (2.8) Denoting byπ the corresponding permutation of labelsπ = (16)(25), the mapping of weights is ω i [z, t, w] → ωπ (i) [w, t, z] as horizontal and vertical lines are interchanged. This give the following relation between fully inhomogeneous partition functions: .
Examples of 20V 1 configurations are easily obtained from those of figure 6 by applying the diagonal reflection R * . Finally, denoting by Z 20V1 m,k the refined number of 20V 1 configurations such that the topmost path leaves the topmost horizontal line at position k counted from the left, then we have the following.
Corollary 2.3. The refined numbers of configurations in the 20V 2 and 20V 1 models are related via:

Enumerative results
This section is devoted to exact results on the enumeration of the configurations of various 20V models on the triangle T m . This includes the refined enumeration of configurations according to certain statistics.

Inhomogeneous case: relation between 20V and 6V partition functions
[z, t, w] may be expressed simply in terms of the partition function Z 6V n [z, w] of the 6V model on a square of size n = m+1 2 with DWBC. As already noted in [DFG19c,DDFG20,DF21b], the integrability of the 20V weights allows to freely move around the spectral lines across intersections and to transform the model.
In the 20V 3 case, moving the diagonal lines up results in the transformations depicted in figure 7 for the even and odd size cases respectively. Once the diagonal lines are moved up, the ice rule allows to propagate the orientations of the boundary edges to all the edges of the diagonal lines (all oriented upward) and those of the horizontal and vertical lines in their domain of intersection. The vertex at the intersection of the jth diagonal line and ith horizontal one is in the configuration a 2 and receives the weight a 2 (z i , t j ), while each vertex at the intersection of the jth diagonal line and kth vertical one is in the configuration a 3 and receives the weight a 3 (t j , w k ). Finally the vertices at the intersections between the ith horizontal and jth vertical lines not inside the marked squares are all in the configuration b 1 and receive weights b 1 (z i , w j ). The remaining vertices (inside the marked squares) form the n × n square grid of a 6V model with the weights of the sublattice 1, and with boundary edges oriented horizontally inward and vertically outward, that is exactly the Domain Wall boundary conditions. We denote by Z 6V1 n [z, w] the corresponding partition function. Collecting all weights, the partition functions of the 20V model on the triangle of even and odd sizes read: (3.1)

Combinatorial point
Setting all spectral parameters in (3.1) to the uniform values of the combinatorial point (2.6), √ 2, 1, 1), we deduce the following.
In the even case of the 20V 3 model, we have from the inhomogeneous relation (3.1): The relation for odd m follows similarly: A direct consequence of this theorem is the following: In [DFG19c] it was shown that the integer Z 6V n counts the number of Quarter-Turn symmetric domino tilings of the Holey Aztec Square of size 2 n. A compact formula for this number is [DFG19c]: where f(x, y)| x i y j stands for the coefficient of x i y j in the series expansion of f around (0, 0). This leads to the following sequence for m = 1, 2, . . . , 10, . . .:  , denote the corresponding partition function. All local vertex weights ω(v) are 1 except in the last column (corresponding to the spectral parameter w 1 ), where they read: Recall the refined number Z 20V3 m,k of 20V 3 configurations on T m with prescribed position k ∈ {1, 2, . . . , m} of the vertex where the top path hits the last column for the first time. The have trivial vertex weights except in the last column, whose total weight is Indeed, in the last column, the vertex at position k may either be in the configuration ω 2 or ω 4 but the two weights are the same, while the m − k top vertices are in the configuration ω 0 (empty) and the k − 1 bottom ones are in configuration ω 1 (two vertical steps of path). We conclude that Analogously let Z 6V1 n [ξ] denote the partition function of the 6V-DWBC model on the square grid of size n with all spectral parameters z, w fixed to the combinatorial values (2.6), except for the last column (where we set w 1 = e −i(λ+η+2ξ) as before). The weights are (a 1 , b 1 , c 1 2, 1) for all vertices except in the last column, where they read respectively: A decomposition of the 6V configurations similar to the above leads to the relation: Here we decompose the 6V 1 configurations according to the position k where the topmost path hits the last column for the first time, and denote by Z 6V1 n,k their partition function with uniform weights (a, b, c) = (1, √ 2, 1): indeed the prefactor (β 1 / √ 2) n 2 allows to replace the initial weights (a 1 , b 1 , c 1 ) There are analogous relations for the 20V 1,2 models defined in section 2.3, respectively related to 6V-DWBC partition functions on the sublattices 2 and 3 respectively, easily obtained by following the effect of the transformations S • R • VF and R * respectively. More precisely, we must consider the partition function Z 6V3 n [ξ] in which all spectral parameters t i , w i are set to their combinatorial point value except for w 1 → e 2iξ w 1 , and the partition function Z 6V2 n [ξ] in which all spectral parameters z i , t i are set to their combinatorial point value except for z 1 → e 2iξ z 1 . These lead to the following.
Moreover, we have for all m ⩾ 1: In the 20V 3 case, we use equations (3.3), (3.4) for i = 1, and the relation σ 1 = τ +1 2 to rewrite: The refined partition functions for the cases of 20V 2 and 20V 1 follow from corollaries 2.2 and 2.3 respectively. The Theorem follows.

Tangent method
The tangent method, devised by F. Colomo and A. Sportiello [CS16] is a non-rigorous method to determine the arctic curve for various statistical models of paths exhibiting crystalline phases (with ordered paths) and liquid phases (with disordered paths) separated by a sharp transition curve usually called arctic curve. Starting from a path formulation of the model (such as the osculating Schröder paths in the present case), one identifies one of the phase separations with the outermost path which is the natural boundary between the empty crystalline phase and a disordered path phase. To study the asymptotic shape of this separation, the method consists in moving the endpoint of this outermost path away from the original domain. The path is expected to still determine the arctic separation, until it detaches itself from the other paths and essentially follows a geodesic (here a straight line) until its new endpoint. For large sizes, the latter becomes tangent to the arctic curve. This line is determined by its endpoint and the point at which it exits the original domain. We represent in figure 8 the setting for the tangent method applied to the 20V 3 model on the triangle T m . The total partition function of the new model (with moved endpoint) decomposes into a sum over the exit position k of the outer path of a product of two partition functions (corresponding respectively to the pink and blue shaded areas). The first piece is a new partition function 2Z20V3 m,k [z, t, w] identical to Z 20V3 m [z, t, w] except for the last endpoint, moved from its original position (black dot) to position k (white dot). (In this section, the weights of the 20V model are uniform but arbitrary: they correspond to choosing all z i = z, t i = t, w i = w; to make the dependence on z, t, w explicit, we write them as arguments from now on.). The latter partition function is usually normalized by dividing it with the original partition function Z 20V3 m [z, t, w], and is called the one-point function The second piece is the partition function Y k,r [z, t, w] of a single path between the white dot and the green dot, namely of single Schröder paths in Z 2 + from point (0, k) to (r, 0), weighted by the product on their vertices of the local 20V weights (2.2) functions of z, t, w, whereas all empty vertices in the blue domain receive the weight ω 0 [z, t, w].
The total partition function reads: The tangent method is based on the remark that for large size m, the most probable exit position corresponds to the dominant contribution to the sum, and is therefore determined as the solution of an extremization problem. The latter involves first estimating the large m 2 The notationZ 20V3 m,k is to distinguish this quantity from the refined partition function Z 20V3 m,k , which has k additional vertical steps joining the white dot to the black one. (k/m, r/m finite) behavior of the two quantities H 20V3 m,k [z, t, w] and Y k,r [z, t, w], which will be done in the two following sections.
Finally, having determined a family of tangent lines parameterized by the displaced position r, we will deduce the arctic curve as the envelope of this family.

20V one-point function and asymptotics
As explained above, the first ingredient of the Tangent Method is the one-point function of the 20V model defined as the ratio , of the refined 20V partition function obtained by conditioning the topmost path to end at its first visit to the rightmost vertical, at position k ∈ {1, 2, . . . , m}, by the total partition function Z 20V m [z, t, w]. The prefactor replaces the weight ω k−1 1 of the original refined partition function Z 20V m,k [z, t, w] by ω k−1 0 as we omit the last k − 1 (vertical) steps of the path. As before, we may compute the quantities Z 20V m,k [z, t, w] for arbitrary homogeneous weights by modifying the rightmost vertical parameter to w 1 = we −2iξ . By the relation (3.1) applied to parameters z 1 = z 2 = · · · = z m = z, t 1 = t 2 = · · · = t m = t, w 1 = we −2iξ and w 2 = w 3 = · · · = w m = w, the corresponding partition function Z 20V m [z, t, w, ξ] of the 20V model is related to that Z 6V1 n [z, w, ξ] of the 6V model with horizontal spectral parameters z and vertical spectral parameters w except in the last column where w → w 1 , via Similarly for the 6V 1 model, we get By a slight abuse of notation, we still denote by Z 20V3 m (τ ) and Z 6V1 n (σ) the refined partition functions with arbitrary homogeneous weights These are related via the following Theorem 4.1. We have the relations: Combining the even case of (4.1) with (4.2) and (4.4), we find: Theorem 4.1 may be rephrased as: where the contour integral around the origin picks up the coefficient of τ k−1 ( by interpreting γ = γ

[ξ] and σ = σ[ξ] of equations (4.3) and (4.5) as implicit series expansions of τ = τ [ξ]).
We are now ready to estimate H 20V m,k in the scaling limit where k, m are large, while κ = 2k/m remains finite. We resort to the result 3 of [CP10] for the large n = m 2 asymptotics of the 6V where the function f is the following implicit function of σ = σ[ξ] parametrized by ξ: and where α = π π−2η . This exponential growth of H 6V1 n (σ) induces an exponential growth for the combination t, w] in (4.6). As the parameter γ is independent of m, this combination is dominated by either term or both at large m.
have the same leading behavior as m, k, n → ∞ with m ∼ 2n and k ∼ κn: where we have introduced the Vertex model action The leading contribution to the integral occurs at the saddle-point ∂ ξ S V (κ, ξ, τ [ξ]) = 0, which is solved as a parametric expression for κ: (4.9)

Schröder path partition function and asymptotics
Consider the partition function Y k,r [z, t, w] for a single Schröder path with 20V weights in the positive quadrant of Z 2 , starting form the point (0, k) and ending at point (r, 0). In the uniform combinatorial case (where all ω i = 1), we have a trinomial coefficient sum: In the scaling limit where n → ∞ with κ = k/n and ρ = r/n fixed (κ ∈ [0, 2] and ρ ∈ [0, ∞)), and replacing the sum by an integral over θ = ℓ/n we have the estimate: For arbitrary weights, a simple transfer matrix calculation [DDFG20] allows to compute Y k,r [z, t, w] as a function of the quantities (4.10) and to derive the large n scaling estimate for (k, r)/n = (κ, ρ) fixed: Y nκ,nρ ∼ˆ1 0 dp 3 dp 4 dp 5 dp 6 e n SP(κ,ρ,p3,p4,p 5 ,p 6 ) , where the path action S P reads As discussed in [DDFG20], the estimate remains valid if any subset of the ω i , i = 3, 4, 5, 6 vanishes, in which case we must set the corresponding p i to zero and integrate over the leftover ones. In particular we recover the uniform case where α 4 = α 5 = α 6 = 0 while α 1 = α 2 = α 3 = 1, and by identifying the leftover integration variable p 3 = θ.

Arctic curve II: other branches
In this section, we show how to use the transformations of section 2.3 to derive other branches of the arctic curve for the 20V 3 model. On the other hand, the path partition function Y (2) k,r is obtained by substituting the new path weights w π(i) [t, z, w] into the expression for the original path partition function Y m+1−k,r . The following shows how to implement the transformation (2.7) on the 20V weights. (4.14) namely we have ω i [z, t, w, ξ] (λ,µ,ξ)→(λ,μ,ξ) = ω π(i) [t, z, w, ξ] with the permutation π = (01)(24).
From the expression (4.3), and the fact that the involution interchanges the weights ω 0 [ξ] ↔ ω 1 [ξ], we see thatτ = τ −1 as well asα 1 = α −1 1 . This allows to rewrite the last equation as ∂ κ2ŜP − Log(τα 1 ) = 0 henceforth the new saddle point equations are simply the transform of the original ones (4.12). The new solution is therefore the transform of the previous one, and in particular s 2 = κ 2 /ρ 2 =ŝ[ξ] with s[ξ] given by (4.13). Defining as before the inverse slope A 2 = s −1 2 , we end up with the family of tangent lines y , and the NE branch of the arctic curve for the 20V 2 model is given in parametric form by (4.15) Note finally that the range of ξ is mapped onto ξ ∈ [− λ−η+µ 2 , 0]. We must finally apply the inverse transformation R • S −1 : (x, y) → (x, 2 − x − y) (in rescaled variables) to recover the SE branch for the 20V 3 model, resulting in the following.
Theorem 4.4. The SE branch of the arctic curve for the 20V 3 model is given in parametric form by: The latter can be directly derived from that of the 20V 2 model, by applying the diagonal reflection R * .  Proof. By inspection, using the weights (2.5).
As a consequence, we may deduce the NE branch of the 20V 1 model from that of the 20V 2 model by the diagonal reflection R * .
Theorem 4.7. The NW branch of the arctic curve for the 20V 3 model is given in parametric form by: By the very definition of the tangent method we applied, it is clear that the NE branch of the arctic curves in the previous sections separates the liquid (disordered) phase below the curve from the empty frozen phase F 0 above the curve. The frozen region below the NW branch is by continuity from the boundary condition of type F 1 , while that below the SE branch is of type F 2 .
We now argue that there exist phase separation segments originating at the center point (−1, 1) of the diagonal boundary. Figure 10. Emergence of frozen regions near the diagonal S boundary. Left: we apply a total flip to the 20V 3 model (of even size here), and show the corresponding boundary conditions on the path configurations. The rightmost path linking the N boundary to the diagonal and the topmost path from the diagonal to the E boundary (in red), as well as the bottom-most path connecting the N boundary to the E boundary (in blue) delimit a frozen empty region (type F 0 ), and two frozen zones with vertical (type F 2 ) and horizontal (type F 1 ) paths. We also indicate the frozen NE corner which is fully packed with paths (type F 5 ), NW corner (type F 4 ) and SE corner (type F 3 ). Right: after flipping back all edges, predicted complete arctic curve with six frozen regions and the central liquid one (L). We expect three more portions of arctic curve in addition to the NW, NE, SE ones.
Starting from the 20V 3 model configurations, let us apply the total flip i.e. replace each path edge by an empty edge and vice-versa. The boundary conditions are then changed into those of figure 10 (left). There are 2 m paths entering along the N boundary, n of which exit along the SW diagonal (above the central point) while the remaining ones exit on the (top 3/4 of) vertical E boundary. There are also n paths entering along the SW diagonal boundary (below the central point), exiting on the (bottom 1/4 of) vertical E boundary.
In particular, the n leftmost paths starting on the N boundary must exit at the n topmost positions along the diagonal SW boundary. The rightmost of those ends up at the center on the diagonal. Similarly the n paths that enter the SE diagonal beyond the center exit at the n bottom-most positions along the vertical E boundary. In particular, the topmost one starts at the center. This clearly creates an empty region (of type F 0 , see figure 10 (left)), namely in the NE corner (positive quadrant) from the center of the diagonal SW boundary. Moreover the bottommost path connecting the N to the E boundaries closes up this empty region from above. The vertical and horizontal boundaries of this empty region separate it from regions with parallel paths (vertical on the left, type F 2 and horizontal on the right, type F 1 ), as dictated by the new SW boundary conditions. We have also indicated in figure 10 (left) the flipped corner phases (respectively F 4 , F 5 , F 3 delimited respectively by the NW, NE, SE branches of the previous sections).
Applying the total flip back, and noting that this interchanges regions F 0 ↔ F 5 , F 1 ↔ F 4 and F 2 ↔ F 3 , this produces a corner central region of type F 5 , NE of the center of the SW diagonal boundary, with a region of type F 3 on its left, and of type F 4 on its right. This shows that there are missing portions of the arctic curve (that encompasses the liquid phase), which should look like figure 10 (right).    Arctic curves of the 20V 3 model for η = π/6, µ = π/8. We represent values λ = 5π 6 − .001, 3π 4 7π 12 , 5π 12 , π 3 , 7π 24 + .01.

Discussion/conclusion
In this paper we have derived exact predictions for the arctic curves of the 20V 3 model on a triangle with suitable DWBC, using the tangent method. This was done in the same spirit as [DDFG20] by using the integrability of the lattice model which allows to relate various asymptotic enumerations of the 20V 3 model to those of the 6V model. Our results include the complete 'outer' arctic curves for the full range of parameters describing integrable weights. We also offered a peek into the inner structure of the arctic curves in the (analytic) case of free fermions. Note that we only considered the disordered regime of the model corresponding to real values of the parameters η, λ, µ (in particular the 6V parameter ∆ = (q 2 + q −2 )/2 ∈ [−1, 1]). A similar study of arctic curves can be done in the anti-ferroelectric regime ∆ < −1, however the occurrence of elliptic functions for the 6V solution (see [CPZJ10]) makes calculations more cumbersome. We found that the NE portion of the arctic curve for the uniform case is algebraic of degree 10, and that the SE and NW portions are simple shears applied to other portions of this algebraic curve. In reference [DDFG20], the NE part of the arctic curve of the same 20V model for another (square) domain with domain-wall type boundary conditions (called DWBC1,2) was also found to be a degree 10 algebraic curve, with a similar 'shear' phenomenon occurring for other parts. Interestingly, the algebraic curve appears to be the arctic curve for different objects: the quarter-turn symmetric domino tilings of a holey Aztec square domain. In view of this we have represented in figure 16 the algebraic curve of section 5.1, together with a domain made of two overlapping squares in which it is inscribed. We conjecture that there should exist a domino tiling problem on a domain whose shape approximates the two overlapping squares (possibly with some holes of small size in its interior), and for which the arctic curve is our degree 10 algebraic curve. One could even hope for an identity between the numbers of configurations of the 20V 3 model and of such a tiling problem, possibly with a half-turn rotational symmetry.
As to the numbers of configurations themselves, we showed that for even/odd size triangles: Z 20V3 2n = 2 n(n+1)/2 Z 6V n and Z 20V3 2n−1 = 2 n(n−1)/2 Z 6V n , where Z 6V n counts the number of quarter turn symmetric domino tilings of the holey Aztec square of size n. This mysterious relation calls for some combinatorial explanation. In particular the presence of overall powers of 2 may point towards tilings of symmetric domains (see [Ciu97]), unless it is simply a factor by the number of domino tilings of some Aztec diamond (equal to 2 n(n+1)/2 for a diamond of size n).

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).