Phase Diagram of the Ashkin–Teller Model

The Ashkin–Teller model is a pair of interacting Ising models and has two parameters: J is a coupling constant in the Ising models and U describes the strength of the interaction between them. In the ferromagnetic case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J,U>0$$\end{document}J,U>0 on the square lattice, we establish a complete phase diagram conjectured in physics in 1970s (by Kadanoff and Wegner, Wu and Lin, Baxter and others): when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J<U$$\end{document}J<U, the transitions for the Ising spins and their products occur at two distinct curves that are dual to each other; when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J\ge U$$\end{document}J≥U, both transitions occur at the self-dual curve. All transitions are shown to be sharp using the OSSS inequality. We use a finite-size criterion argument and continuity to extend the result of Glazman and Peled (Electron J Probab 28:1-53, 2023) from a self-dual point to its neighborhood. Our proofs go through the random-cluster representation of the Ashkin–Teller model introduced by Chayes–Machta and Pfister–Velenik and we rely on couplings to FK-percolation.


Introduction
The Ashkin-Teller (AT) model is named after two physicists who introduced it in 1943 [1] and can be viewed as a pair of interacting Ising models.For a finite subgraph Ω = (V, E) of Z 2 , the AT model is supported on pairs of spin configurations (τ, τ ) ∈ {±1} V × {±1} V and the distribution is defined by where J τ , J τ , U are real parameters and Z = Z(Ω, J τ , J τ , U ) is the unique constant (called partition function) that renders the above a probability measure.
In the current article, we consider the ferromagnetic symmetric (or isotropic) case J = J τ = J τ ≥ 0, and U ≥ 0 and denote the measure by AT Ω,J,U .Important particular cases: U = 0 gives two independent Ising models;for J = 0, τ reduces to a Bernoulli site percolation with parameter 1/2, and τ τ to an Ising model, independent of each other; the line U = J corresponds to the 4-Potts model.These models are very well-studied and their phase diagram is known; see [17,9] for excellent surveys.Henceforth in this article we assume that J, U > 0. A key observation in the analysis of the general AT model is its relation to the six-vertex model [15].This gives a non-staggered six-vertex model (i.e. with shift invariant local weights) only at the self-dual line of the AT model: it was found in [28] and is described by the equation (SD) Outside of this line, the corresponding six-vertex model is staggered and thus the seminal Baxter's solution [2] does not apply.Kadanoff and Wegner [24,35], Wu and Lin [36], and others conjectured that, when J < U , there are two distinct transition lines in the AT model: one for correlations of spins τ (or τ ) and the other for correlations of products τ τ .
In the current article, we prove this conjecture and establish a complete phase diagram of the AT model in the ferromagnetic regime.
It will be convenient to state the results in infinite volume and to consider also plus boundary conditions.Denote by ∂Ω the set of boundary vertices of Ω -these are all vertices in Ω that are adjacent to at least one vertex in Z 2 \ Ω.We define the measure with plus boundary conditions by conditioning all boundary vertices to have spin plus in τ and in τ : AT +,+ Ω,J,U := AT Ω,J,U (• | τ |∂Ω ≡ τ |∂Ω ≡ 1).Expectations with respect to the AT measures are denoted by brackets: • Ω,J,U := E Ω,J,U [•] and • +,+ Ω,J,U := E +,+ Ω,J,U [•].
The correlations satisfy the Griffiths-Kelly-Sherman (GKS) inequality [25], which states that for any A, B, C, D ⊂ V , one has where τ A := u∈A τ u and τ B := v∈B τ v .This in particular implies that any A, B ⊂ V , τ A • τ B Ω,βJ,βU and τ A • τ B +,+ Ω,βJ,βU are increasing in β > 0, A standard application of the GKS inequality implies that the weak limits over Ω n Z 2 exist and do not depend on {Ω n }: ) and when J < U , the transition occurs at two distinct curves γ τ and γ τ τ dual to each other (Theorems 1 and 2).There are three regimes: disorder in τ and in τ τ (gray), order in τ and in τ (white), disorder in τ and order in τ τ (dashed gray).Right: Domain Ω (in bold black) on L and its dual Ω * (in gray).Notice that Ω * is not a domain on L * .The even domain D Ω (dashed) on Z 2 .
The next result states that the transition lines are dual to each other (see Fig. 1) and that the critical points for the measures under the free and plus boundary conditions coincide.In order to make a precise statement, we define the critical curve γ τ := {(J, U ) ∈ R 2 : 0 < J < U, β τ c (J, U ) = 1}.
We define γ τ τ in a similar way.Given a pair of parameters (J, U ), we define the dual set of parameters (J * , U * ) as the unique solutions to the following equations Note that this duality relation is an involution.We refer the reader to Subsection 2.1 and Lemma 2.2 for more details on the duality in the ATRC model.
Theorem 2. Fix 0 < J < U .Then, the following holds: General approach [11] gives sharpness under plus boundary conditions and equality of the transition points β τ c = β τ τ c =: β c .By standard duality arguments, one deduces β c ≤ β sd .The bound β c ≥ β sd follows from Zhang-type arguments provided the transition points for the free and monochromatic measures coincide.The latter can be shown by applying the classical FK-percolation argument to the marginals of the ATRC.
Organisation of the article.Sections 2-6 treat the case J < U : in Section 2, we introduce the random-cluster representation of the AT model (ATRC) and derive Theorems 1 and 2 from Proposition 1.1;Sections 3-6 are dedicated to proving Proposition 1.1.In Section 3, we describe the six-vertex and FK-percolation models and give their background, including their relation to the AT model.In Section 4, we show that τ exhibits exponential decay of correlations in finite volume under the boundary conditions τ = τ .In Section 5, we show that τ exhibits no ordering under AT +,+ J,U .In Section 6, we derive Proposition 1.1.Section 7 deals with the case J ≥ U : we introduce the ATRC model and prove Theorem 3. Appendices provide details regarding sharpness for the AT (A), exponential relaxation for FK-percolation (B), stochastic ordering of the ATRC with respect to its local weights (C) and uniqueness of the infinite-volume ATRC measure (D).
From now on, we will consider the AT model on a rotated square lattice that we denote by L: its vertex set is {(x, y) ∈ Z 2 : x+y is even} and edges connect (x, y) to (x±1, y±1), see Figure 1.This is more convenient for the coupling with the six-vertex model (Section 3).
In this section, we fix J < U and drop them from the notation.In particular, we write AT Ω,β for the measure AT Ω,βJ,βU .
We start by defining the random-cluster representation of the AT model (ATRC) introduced by Chayes-Machta [7] and Pfister-Velenik [31].Using a ϕ β (S) argument, we prove that (β τ τ c J, β τ τ c U ) is strictly above the self-dual line.By duality, this implies that (β τ c J, β τ c U ) is strictly below the self-dual line which concludes the proof.

ATRC: defintion and basic properties
The ATRC is reminiscent of the Edwards-Sokal [14] coupling between FK-percolation and the Potts model.Since the AT model is supported on a pair of spin configurations, the ATRC is supported on a pair of bond percolation configurations.
Percolation configurations.For a finite subgraph Ω ⊂ L, the sets of its vertices and edges are denoted by V Ω and E Ω , respectively.We view ω ∈ {0, 1} E Ω as a percolation configuration: we say that e is open in ω if ω(e) = 1, and otherwise e is closed.We identify ω with a spanning subgraph of Ω and edges that are open in ω.Define |ω| as the number of edges in ω.Boundary conditions for ω are given by a partition η of ∂Ω.We define k η (ω) as the number of connected components in ω when all vertices belonging to the same element of partition in η are identified.Two important special cases: 1 denotes wired b.c.given by a trivial partition consisting of one element ∂Ω; 0 denotes free b.c.given by a partition of ∂Ω into singletons.
In this context, we will refer to the measure as ATRC ητ ,η τ τ Ω,wτ ,w τ τ .In Section 4.2, we will encounter a version of this measure with non-homogeneous weights.
Coupling between ATRC and AT.For X, Y ⊂ L and a percolation configuration ω ∈ {0, 1} E Ω , we define X ω ← → Y as an event that X and Y are linked by a path of open edges in ω.If X = {x} and Y = {y}, we simply write x ω ← → y.We also use the notation x ω ← → ∞ for the event of x belonging to an infinite connected component of ω.
The key property of the ATRC is that connectivities in it describe correlations in the AT model [31]: for β > 0 and any finite subgraph Ω ⊂ L containing 0, We omit the proof as it is straightforward and similar to the classical Edwards-Sokal coupling; see [31,Proposition 3.1] for details.
Positive correlations and monotonicity.We first introduce the notion of stochastic domination and positive association.Given a partially ordered set P and a real-valued function f on P, f is said to be increasing if for any ω, ω ∈ P with ω ≤ ω , one has f (ω) ≤ f (ω ).A subset A ⊆ P is then called increasing if its indicator 1 {A} is increasing with respect to P. Given two probability measures µ and ν on P equipped with some σ-algebra A, we say that µ is stochastically dominated by ν (or ν stochastically dominates µ), and write µ ≤ st ν (or ν ≥ st µ), if for every increasing event A ∈ A, we have µ(A) ≤ ν(A).Moreover, µ is said to be positively associated or to satisfy the FKG inequality if, for all increasing non-negative functions f and g, we have We introduce a natural partial order on pairs of percolation configurations: we say that (ω τ , ω τ τ ) ≥ (ω τ , ωττ ) if and only if, for every edge e, we have ω τ (e) ≥ ωτ (e) and ω τ τ (e) ≥ ωττ (e).By [31,Proposition 4.1] (and its proof), the measures ATRC ητ ,η τ τ Ω,β are positively associated for any β > 0 and any boundary conditions η τ , η τ τ : for any increasing events A, B, one has (FKG) This can be used to compare different boundary conditions.For two partitions η and η of ∂Ω, we say that η ≥ η if any two vertices belonging to the same element of η also belong to the same element of η.Then, for any β > 0, and any boundary conditions such that η τ ≥ ητ and η τ τ ≥ ηττ , . (CBC) The Holley criterion [23] also allows to show stochastic ordering of the measures in the parameter β: if β 1 ≥ β 2 , then, for any boundary conditions η τ , η τ τ , See [22,Lemma 11.14] for a proof.In fact, this proof gives a little more.Indeed, it consists of checking inequalities for quantities that are continuous functions of (β i J, β i U ) and the inequalities are strict when β 1 > β 2 .This implies that the ATRC measure with parameters in a small neighbourhood of (β 1 J, β 1 U ) dominates that with parameters in a small neighbourhood of (β 2 J, β 2 U ).More precisely, for (x, y) ∈ R 2 , define B r (x, y) as the Euclidian ball of radius r centred at (x, y).If β 1 > β 2 , then there exists ε > 0 such that for any This extension will be useful for our proof of Theorem 2.

Domain Markov property.
As in the standard FK-percolation, one can interpret a configuration outside of a subdomain as boundary conditions.Indeed, let Ω ⊂ ∆ be two finite subgraphs of L and ξ ∈ {0, 1} E ∆ \E Ω a percolation configuration on ∆ \ Ω.Given boundary conditions η on ∆, define a partition η ∪ ξ of ∂Ω by first identifying vertices belonging to the same element of η and then identifying vertices belonging to the same cluster of ξ.Then, the following domain Markov property holds: Thus, by (CBC), for any increasing sequence of subgraphs Ω k L, the measures ATRC 1,1 form a stochastically decreasing sequence.Thus, the weak (or local) limit exists and is unique, by standard arguments.Denote it by ATRC 1,1 β .Define ATRC 0,0 β analogously.We write ATRC 1,1 J,U and ATRC 0,0 J,U for the corresponding measures with β = 1.

Dual
We want to emphasize that we are not considering two standard dual percolation configurations but we also swap the order of τ -edges and τ τ -edges.The measures ATRC 0,0 J,U =: ATRC 0,0 L,J,U and ATRC 1,1 J,U =: ATRC 1,1 L,J,U on L can be defined on L * in the same manner, and we denote them by ATRC 0,0 L * ,J,U and ATRC 1,1 L * ,J,U , respectively.Recall the mapping (J, U ) → (J * , U * ) defined by (3) and note its properties: it is continuous, an involution, identity on the self-dual line (SD), sends every point above (SD) to a point below (SD).The pushforward of the ATRC measure under the duality transformation is also an ATRC measure with the dual parameters: Lemma 2.2 (Prop 3.2 in [31]).Let 0 < J < U .Let (ω τ , ω τ τ ) be distributed according to ATRC 1,1 L,J,U .Then, the distribution of (ω τ , ωττ ) is given by ATRC 0,0 L * ,J * ,U * .
The proof goes by applying the classical FK-percolation argument to the marginals of ATRC on ω τ and ω τ τ , see Appendix D for more details.We are ready to prove part (i) of Theorem 2. Recall that B r (x, y) is the Euclidean ball of radius r centred at (x, y).Proof of Theorem 2(i).Fix J < U .By (CBC), we have β τ c ≤ β τ,f c .Assume for contradiction that the inequality is strict, and take β ∈ (β τ c , β τ,f c ).Then, by (MON+), there exists ε > 0 such that, for any (J , U ) ∈ B ε (βJ, βU ), ATRC 0,0 J ,U (0 This contradicts Lemma 2.3. Denote by H τ n (resp.H τ τ n ) the event that the box [0, 2n − 1] × [0, 2n − 1] is crossed horizontally by ω τ (resp.ω τ τ ).Note that the complement of H τ n is the event that the box [0, 2n − 1] × [0, 2n − 1] is crossed vertically by the dual ω * τ .The following lemma states a standard characterisation of non-transition points.It is a consequence of Lemma 2.3 and sharpness of the phase transition in the ATRC.The latter can be derived using a robust approach going through the OSSS inequality [11]; see Appendix A for more details.

ϕ β (S) argument: proof of Theorem 1
Following [33,27] (see also [12]), for a finite subgraph S ⊂ L containing 0, define The following lemma states a key property of ϕ β (S): if it is less than 1 for some S, then ω τ exhibits exponential decay of connection probabilities.This finite criterion allows to use continuity of ϕ β (S) and Proposition 1.1 to extend exponential decay of ω τ beyond (SD).Let Λ k be the box of size k in L, that is Lemma 2.6.Let β > 0. Assume that ϕ β (S) < 1, for some finite subgraph S ⊂ L containing 0. Then, there exists c := c(β, S) > 0 such that Remark 2.7.Note that the boundary conditions are free in [12] and wired in our case.
The reason is that an analogue of Lemma 2.6 is proven in [12] via a modified Simon-Lieb inequality [27,33] for the Ising model.Such inequalities are not available in our case.While Lemma 2.6 under wired conditions is elementary, proving exponential decay under wired boundary conditions in finite volume (Proposition 1.1) is the subject of Sections 3-6.
Proof of Lemma 2.6.Let S ⊂ L be a finite subgraph containing 0 such that ϕ β (S) < 1 and let k be such that where we also used translation invariance of ATRC 1,1 We are now ready to derive Theorem 1 from Proposition 1.1 and Theorem 2.
Proof of Theorem 1. Fix J < U .By Proposition 1.1, we can take n > 1 such that ) is increasing and continuous, there exists ε = ε(J, U ) > 0, such that ϕ β (Λ n ) < 1, for all β < β sd + ε.The latter implies exponential decay by Lemma 2.6 and hence β τ c > β sd .In other words, all points on γ τ are strictly above the self-dual curve.Hence their images under the duality mapping (3) are strictly below the self-dual curve.By Theorem 2, these points are exactly the points of γ τ τ and this finishes the proof.
Remark 2.8.Standard arguments similar to the proof of Lemma 2.6 show that exists and is right-continuous in β, which gives another way to argue that the exponential decay from Proposition 1.1 extends to an open neighbourhood of the self-dual line (SD).

Models, couplings and required input
In this section, we introduce the six-vertex model together with its height and spin representations.We also state couplings of this model with the ATRC model and FK percolation that will be crucial to our arguments.A combination of these two couplings has been made explicit recently in the work of Peled and the third author [21] and we summarize the results of that work that we will rely on. -1 Figure 2: Top: The height representation of the six-vertex model in the four vertices of a unit square in Z 2 , normalized to equal 0 at the lower left vertex.Bottom: The spin representation is derived from the heights by setting the spin state at each vertex to +1 (resp.−1) if the height modulo 4 equals 0, 1 (resp.2, 3).

Graph notation
Dual subgraphs and configurations.For a finite subgraph Ω of L, define its dual graph Ω * in L * formed by edges dual to the edges of Ω.As for primal graphs, we denote the sets of its vertices and edges by V Ω * and E Ω * .The boundary ∂Ω * is defined in the same way as for subgraphs of L. Given a percolation configuration ω ∈ {0, ) is a domain if it is induced by vertices within a simple cycle (including the cycle itself).We denote the set of vertices on the surrounding cycle by ∂Ω and call it the domain-boundary of Ω.The set of edges on ∂Ω is called edge-boundary of Ω and is denoted by E ∂Ω .
Domains in Z 2 .Given a domain Ω in L, let D Ω be the subgraph of Z 2 induced by vertices in Ω ∪ Ω * .We call such a domain an even domain of Z 2 (see Figure 1).Define ∂D Ω = ∂Ω ∪ ∂Ω * .Given a domain Ω on L * , we define D Ω in the same manner and call it an odd domain of Z 2 .We emphasize that we only consider even and odd domains.

Six-vertex model and its representations
In this section, we define the six-vertex model (more precisely, the F-model ) and its different representations in terms of spins and height functions.For the whole subsection, fix a domain Ω in L (or L * ) and its corresponding even (odd) domain This constraint implies that, for each edge e, the value of h is constant either at the endpoints of e or at the endpoints of e * .Up to an even additive constant, this leaves six local possibilities (types), where types 5 and 6 correspond to h taking constant values along both e and e * , see Figure 2.
The six-vertex height function measure on D with parameters c, c b > 0 and boundary conditions t ∈ Z ∂D is supported on height functions h ∈ Z D that coincide with t on ∂D and is given by where Z t,c b hf,D,c is a normalizing constant and n as the probability measure given by (13) and supported on all height functions in Z D that have a fixed value 2n on ∂D ∩ L. Note that the value on ∂D ∩ L * is not fixed in this case, so the conditions can be viewed as semi-free.

From Ashkin-Teller to six-vertex
In this section, we describe the connection between the self-dual Ashkin-Teller model on a domain Ω in L and the spin representation of the six-vertex model on the corresponding even domain D Ω in Z 2 .We consider two types of boundary conditions that will play an important role in proving Proposition 1.1.

The compatibility relation on pairs of σ
The following is a consequence of [21, Proposition 8.1] and a remark after it, or may be proved along the same lines: Proposition 3.2.Let 0 < J < U be a point on the self-dual line (SD) and c = coth 2h.
2) If Ω * is a domain in L * , then we can couple σ ∼ Spin +,± D Ω * ,c and ω τ ∼ µ 1,1 Ω,J,U by Remark 3.3.Part 1) of Proposition 3.2 is a special case of [21, Proposition 8.1] while part 2) may be proved in the same way.The proof relies on the following identity: where ω = ω * τ for 1) and ω = ω τ for 2).This follows from Euler's formula using that either Ω or Ω * is a domain and our definition of the domain-boundary.

Input from the six-vertex model
In this section, we mention basic properties of six-vertex measures and state some results from [21].The following proposition is a combination of Theorem 2, Proposition 6.1 and Lemma 6.2 in [21].We remark that we only consider even and odd domains in Z 2 .
For u ∈ L, S ⊂ L, define u h =0 ← − → S to be the event that u is connected (in L) to S by a path of heights different from 0. We similarly define u * h =1 ← − → S * for u * ∈ L * , S * ⊂ L * .
Proposition 3.5 ([21]).Fix c > 2, and let λ be the unique positive solution of c = e λ/2 + e −λ/2 .Then, for any sequence of domains D k Z 2 , the measures HF 0,1 D k ,c and HF 0,1;e λ/2 D k ,c converge weakly to the same limit that we denote by HF 0,1 c .Moreover, HF 0,1 c -a.s.exist unique infinite clusters in L of height 0 and in L * of height 1.Finally, clusters of other heights are exponentially small: for some Let us emphasize that, while existence of subsequential limits is a straightforward consequence of discontinuity of the phase transition in FK-percolation, the ordering of both even and odd heights is non-trivial.This also implies that the weak limit of HF 0,1;e λ/2 D k ,c remains the same, whether it is taken along even or odd domains.Analogously, for any n ∈ Z, one obtains limit measures HF 2n,2n+1 ) satisfying the corresponding properties.Since the modulo 4 mapping (Section 3.2) is local, Propositon 3.5 directly implies the following corollary.It has been established in [21,Theorem 4] that the marginals of Spin +,+ D k ,c on σ • (resp.σ • ) satisfy the FKG inequality with respect to the pointwise order on {±1} V (Ω) (resp.{±1} V (Ω * ) ).Though [21] deals only with boundary conditions specified on the whole boundary, the extension to free or semi-free conditions is straightforward.Indeed, the statement for σ • holds as long as spins σ • on the boundary are not forced to disagree.converge to some infinite volume state µ 0,1 J,U that admits exponential decay of connection probabilities.Proposition 3.9 ( [21]).Let 0 < J < U be on the self-dual line (SD) and Ω k be a sequence of domains increasing to L. The measures µ 0,1 Ω k ,J,U converge weakly to some measure µ 0,1 L) which is independent of the sequence Ω k and admits exponential decay of ω τ -connection probabilities: there exist M, α > 0 such that, for any u, v ∈ L, We sketch the argument given in [21].
Sketch of proof of Proposition 3.9.The couplings in Proposition 3.2 and Corollary 4.3 imply convergence of µ 0,1 Ω,J,U , as Ω L, to some µ 0,1 J,U that satisfies FKG and is invariant to translations.Thus, it is enough show that that it is exponentially unlikely that ω τ contains a circuit surrounding Λ n .Indeed, on this event, the marginal of Spin +,+ c at vertices in (Λ n ) * is invariant to the spin flip.By Proposition 4.2, radii of clusters of minuses have exponential tails and the claim follows.
We emphasise a difference between Propositions 3.9 and 1.1: the latter proves exponential decay under the largest boundary conditions and in finite volume.As we saw in Section 2.3, this is necessary for the proof of Theorem 1.

FK-percolation
Fortuin-Kasteleyn (FK) percolation [16] is an archetypical dependent percolation model.It is well-understood thanks to recent remarkable works; see [9,22] for background.We will transfer some known results from FK-percolation to the six-vertex model via the BKW coupling (Section 3.6) and further to the self-dual ATRC via the coupling in Proposition 3.2.

Definition.
Let Ω ⊂ L be a finite subgraph and ξ a partition of ∂Ω.FK-percolation on Ω with parameters p ∈ [0, 1] and q > 0 is supported on percolation configurations η ∈ {0, 1} E Ω and is given by where Z = Z(Ω, p, q, ξ) is a normalizing constant and k ξ (η) was defined in Section 2.1.
The free and wired FK-percolation measures FK f Ω,p,q and FK w Ω,p,q are defined by free and wired boundary conditions, respectively (as in Section 2.1).
We now review several fundamental results about FK-percolation.
Proposition 3.10.Let p ∈ [0, 1], q > 1 and Ω k L be a sequence of subgraphs.Then, the weak limits of FK f Ω k ,p,q and FK w Ω k ,p,q exist and do not depend on the chosen sequence: and FK w p,q := lim k→∞ FK w Ω k ,p,q .
Moreover, these measures are extremal, invariant to translations and satisfy the following ordering, for any finite subgraph Ω ⊂ L, FK f Ω,p,q ≤ st FK f p,q ≤ st FK w p,q ≤ st FK w Ω,p,q .
As we will see below, the self-dual AT model with J < U corresponds to FKpercolation with q > 4 at p = p sd , where This model is self-dual: if ω has law FK f p sd ,q , then ω * (e * ) := 1 − ω(e) has law FK w p sd ,q .
The second item of this theorem implies exponential relaxation at p sd : Lemma 3.11.Let q > 4.Then, there exists α > 0 such that, for n ≥ 1 and any finite subgraph Ω ⊂ L that contains Λ 2n , The proof is standard and goes through the monotone coupling; see Appendix B.

Baxter-Kelland-Wu (BKW) coupling
FK-percolation and the six-vertex model were related to each other for the first time by Temperley and Lieb [34] on the level of partition functions.BKW [3] turned this relation into a probabilistic coupling when c > 2. We follow [21] and describe this coupling using a modified boundary coupling constant c b .Take q > 4, p = p sd .Let λ > 0 be the unique positive solution to e λ + e −λ = √ q, and set c := e λ/2 + e −λ/2 .
Let Ω be a domain in L and recall the notations introduced in Section 3.1.The measure FK w Ω,p sd ,q refers to the FK measure with wired boundary conditions on ∂Ω.Note that the statements of Proposition 3.10 and Lemma 3.11 remain valid if we replace w by w.
Consider η ∼ FK w Ω,p sd ,q and draw loops separating primal and dual clusters within Ω as in Figure 4. Given this loop configuration, we define a height function h ∈ Z D Ω by: H2 Assign constant heights to primal and dual clusters by going from ∂D Ω inside of D Ω and tossing a coin when crossing a loop: the height increases by 1 with probability e λ / √ q and decreases by 1 with probability e −λ / √ q, independently of one another.
The following result is classical; see e.g.[21,Chapter 3] for a proof in this setup.
Proposition 3.12 (BKW coupling).The resulting height function is distributed according to HF 0,1;e λ/2 D Ω ,c .Odd domains.Note that, by symmetry, the whole procedure also works on odd domains with the difference that one needs to replace H2 by H2' each time one crosses a loop, the height decreases by 1 with probability e λ / √ q and increases by 1 with probability e −λ / √ q, independently of one another.
For a domain Ω in L * , this gives a coupling of FK w Ω ,p sd ,q and HF 0,1;e λ/2 D Ω ,c .

Exponential decay for ATRC in finite volume
The goal of this section is to derive exponential decay of connection probabilities for µ 0,1 Ω,J,U , which is the marginal of a finite-volume ATRC measure on ω τ , see Section 3.3.
Recall that Λ n = {u ∈ L : u 1 ≤ 2n} is the box of size n in L.
Proposition 4.1.Let 0 < J < U be on the self-dual line (SD).There exists α > 0 such that, for any n ≥ 1 and any domain Ω in L containing Λ 4n , The proof consists of several steps.We first transfer the exponential relaxation property from FK-percolation (Lemma 3.11) to the six-vertex height function (Proposition 4.2) and then to the marginal ν Ω of the ATRC model with modified edge weights on the boundary.Using Proposition 3.5, we also show that the limit of ν Ω is given by µ 0,1 J,U (Lemma 4.4), and that ν Ω dominates µ 0,1 Ω,J,U (Lemma 4.7).The statement then follows from exponential decay in µ 0,1 J,U (Proposition 3.9).
Proposition 4.2.The convergence of HF 0,1;e λ/2 D,c towards HF 0,1 c admits exponential relaxation: there exists α > 0 such that, for any n ≥ 1 and any even domain D ⊃ ∆ 8n , Proof.We omit q, p sd from the notation for brevity.We first construct the limiting measure HF 0,1 c .Consider η ∼ FK w on L. Using known results about η ∼ FK w (Section 3.5) we can sample a height function h as follows.Set h = 0 on the unique infinite cluster of η and sample h in its holes according to H2 in the BKW coupling (Section 3.6).
Define C n to be the outermost circuit in η surrounding Λ n and contained in Λ 2n (if it does not exist, we set C n := ∅).Exponential decay in η * and stochastic ordering of FK measures imply existence of α > 0 such that, for any n ≥ 1 and any domain Ω ⊃ Λ 2n , By exponential relaxation of the wired FK measures (Lemma 3.11), there exists α > 0 such that, for any n ≥ 1 and any domain Ω ⊃ Λ 4n , Now, given where Ω(C) is the domain in L induced by the vertices within C (including C).Note that Ω(C) contains Λ n , whence D Ω(C) contains D Λn = ∆ 2n .
Recall that the six-vertex spin measures introduced in Sections 3.2 and 3.4 are the push-forwards of the height function measures under the local modulo 4 mapping.

A modified ATRC marginal
Fix J < U on the self-dual line (SD), take c = coth 2J and the unique λ > 0 such that c = e λ/2 + e −λ/2 .Let Ω be a domain on L and D Ω be the corresponding even domain on Z 2 .Recall the definition of the edge-boundary D Ω ,c . Define ν Ω as the distribution of ω ∈ {0, 1} E Ω sampled independently for each edge e as follows: if the endpoints of e * have opposite values in σ • , then ω e = 1; if the endpoints of e have opposite values in σ • , then ω e = 0; if σ • agrees on e * and σ • agrees on e, then We call ν Ω a modified ATRC marginal as it converges to µ 0,1 J,U as Ω L.Moreover, this convergence admits exponential relaxation, which is the content of the next lemma.Lemma 4.4.For any sequence of domains Ω k increasing to L, the measures ν Ω k converge to µ 0,1 J,U .Moreover, this convergence admits exponential relaxation: there exists α > 0 such that, for any n ≥ 1 and any domain Ω ⊃ Λ 4n , Recall the representation (6) of the ATRC measures.The previous lemma becomes more clear once we identify ν Ω as the marginal of ATRC 0,1 wτ ,w τ τ on ω τ where the weights are as in (7) except that w τ is modified on the edge-boundary E ∂Ω .
Proof of Proposition 4.1.Fix n ≥ 1 and a domain Ω ⊃ Λ 4n in L. We have where we used (CBC) for the first inequality and Lemma 4.7 for the second one.Now, by Lemma 4.4 and Proposition 3.9, there exist α, M > 0 such that It remains to show Lemmata 4.4 and 4.5.
Proof of Lemma 4.5.Fix a domain Ω in L, and take c b := e λ/2 .Recall that E σ • denotes the set of disagreement edges in σ • (Section 3.3).
Step 1: The measure ν Ω can be written in the following form: For brevity, we write E for E Ω and ∂E for E ∂Ω .In a slight abuse of notation, we also set The law of (σ, ω) defined by ( 19) satisfies: Note that Summing over σ then gives Finally, by Euler's formula (or induction), Step 2: The marginal of ATRC 0,1 Ω,wτ ,w τ τ on ω τ with weights w τ , w τ τ given by ( 21) coincides with the right-hand side of (22).
5 No infinite cluster in the wired self-dual ATRC The proof of Proposition 5.1 again relies on the coupling with the six-vertex model, Proposition 3.2.First of all, by the non-coextistence theorem [32,11], it is sufficient to show that ATRC 1,1 J,U admits an infinite ω * τ -cluster.If the latter is not the case, the infinite-volume limit of the marginals of Spin +,± D,c on {±1} L * can be shown to be tailtrivial.Exploring clusters of 1 and −1 (in T-connectivity) and using the non-coexistence theorem, we obtain that the limit of HF 0,±1 D,c is either HF 0,1 c or HF 0,−1 c , thereby contradicting the invariance of HF 0,±1 D,c under h → −h.In the following remark, we summarise some basic properties of the ATRC marginals µ 1,1 Ω,J,U and µ 1,1 Ω,J,U (defined in Section 3.3) and their infinite-volume limit that we will use in Sections 5.1 and 5.2.
Remark 5.2.Recall that, for domains Ω k L, the measures ATRC 1,1 Ω k ,J,U form a decreasing sequence and converge to ATRC 1,1 J,U .In particular, the same holds for the marginals on ω τ : J,U is invariant under translations and tail-trivial (and hence ergodic).Moreover, ATRC 1,1 Ω k ,J,U (and thus their limit and its marginals) satisfies the finite-energy property.Therefore, the Burton-Keane argument [4] and the non-coexistence theorem [32,11] apply.

Semi-free measures in infinite volume
In this section, we will show weak convergence for some finite-volume spin and height function measures defined in Section 3.2.Lemma 5.3.Let 0 < J < U be on the self-dual line (SD) and take c := coth 2J.Let ω τ ∼ µ 1,1 J,U .Define χ +,± c as the distribution on {±1} L * obtained by assigning ±1 to every cluster of ω * τ uniformly and independently.Then, for any sequence of odd domains D k Z 2 , the marginals of Spin +,± D k ,c on σ • converge weakly to χ +,± c .Moreover, χ +,± c is translation-invariant, positively associated and satisfies the finite-energy property.
Proof.Fix J < U .Let D k be a sequence of odd domains on Z 2 and Ω k the corresponding subgraphs of L such that D k = D (Ω k ) * .Let ω k τ be sampled from µ 1,1 Ω k ,J,U .By Proposition 3.2, assigning ±1 to clusters of (ω k τ ) * uniformly independently gives the marginal of Spin +,± D k ,c on σ • .Since µ 1,1 Ω k ,J,U converges to µ 1,1 J,U that exhibits at most one infinite cluster in ω * τ , the marginal of Spin +,± D k ,c on σ • converges to χ +,± c .Clearly, χ +,± c inherits translation-invariance and the finite-energy property from µ 1,1 J,U .By Proposition 3.8, the marginal of Spin +,± D k ,c on σ • satisfies the FKG inequality.Hence, the same holds for its limit χ +,± c .Working with measures on height functions (rather than spins) is more convenient as they satisfy stochastic ordering in boundary conditions.In the proof of Proposition 5.1, we use an infinite-volume version of HF 0,±1 D,c .We show existence of such subsequential limit in the next lemma by sandwiching HF 0,±1 D,c between HF 0,−1 D,c and HF 0,1 D,c .Lemma 5.4.Let c > 2. For any sequence of domains D k increasing to Z 2 , there exists a subsequence (k ) such that the measures HF 0,±1 D k ,c converge weakly to some HF 0,±1 c as tends to infinity.Remark 5.5.Proposition 5.6 and its proof allow to show that the limiting measure is 1  2 (HF 0,1 c + HF 0,−1 c ) for any (sub)sequence.We do not use this statement and omit the details.
Proof of Lemma 5.4.By [18,Proposition 4.9], it suffices to show that (HF 0,±1 D k ,c ) k≥1 is locally equicontinuous: for any finite V ⊂ Z 2 and any decreasing sequence of local events (A m ) m≥1 supported on V and with ∩ m≥1 A m = ∅, it holds that By Proposition 3.7, finite-volume six-vertex height function measures are stochastically ordered with respect to the boundary conditions, whence Moreover, by Proposition 3.5, HF 0,−1 D k ,c converges to HF 0,−1 c and HF 0,1 D k ,c to HF 0,1 c as k tends to infinity.These statements together easily imply the required local equicontinuity.

Proof of Proposition 5.1
As we argued in Remark 5.2, it suffices to find a dual infinite cluster: Remark 5.7.By the duality relation described in Section 2.1, this is equivalent to saying that ATRC 0,0 J,U [0 The proof of Proposition 5.6 also relies on the non-coexistence theorem -but in the context of site percolation.Following the notation of [21], we let T • be the graph with vertex set L * where a vertex (x, y) ∈ L * is adjacent to (x, y) ± (1, 1), (x, y) ± (1, −1) and (x ± 2, y).
Note that T • is isomorphic to the triangular lattice.
Proof of Proposition 5.6.Fix J < U .Recall that µ 1,1 J,U is the marginal of ATRC 1,1 J,U on ω τ .Assume for contradiction that µ 1,1 J,U does not admit an infinite dual cluster.Set c = coth 2J.Recall that χ +,± c is obtained from ω ∼ µ 1,1 J,U by assigning uniformly independently ±1 to its dual clusters.Since all them are finite by our assumption, χ +,± c inherits ergodicity from µ 1,1 J,U .Also, by Lemma 5.3, χ +,± c is translation-invariant and satisfies the FKG inequality.Thus, by non-coexistence theorem, in T • -connectivity, either χ +,± c admits no infinite cluster of minuses, whence or the same holds for T • -circuits of −.By symmetry, we can assume (24).By Lemma 5.4, there exists a sequence of odd domains D k such that HF 0,±1 D k ,c converge to some infinite-volume height function measure HF 0,±1 c weakly.Recall that Spin +,± D,c is the push-forward of HF 0,±1 D,c under the modulo 4 mapping and, by Lemma 5.3, the marginals of Spin +,± D,c on σ • converge to χ +,± Our goal is to use exponential decay under 0, 1 conditions in Proposition 4.1 to improve non-percolation statement of Proposition 5.1 and get exponential decay in finite volume stated in Proposition 1.1.We use the approach of [6] and [5,Appendix].Additional difficulties in our case come from a weaker domain Markov property of the ATRC measure.Fix J < U and n ≥ 1.For any vertex x ∈ Λ n , define The next lemma provides a lower bound on the size of the boundary cluster of ω τ .Recall that we denote by µ the marginal of the ATRC measure on ω τ .Lemma 6.1.For any δ > 0, there exists α := α(δ, β sd ) > 0 such that Proof.Fix δ > 0. It follows from Proposition 5.1 that This implies that one can find M := M (δ) > 0 such that Fix n M .Without loss of generality, assume that n = (2k + 1)M .One has Denote the expression in the brackets by Y x,M .Then, Define A as the event that N ( ) ≤ δn.Since f (x) ≤ η(x), Lemma 6.1 implies that, up to an error e −cn 2 , event A occurs for some ∈ [4n/5, n], whence where we used that A is measurable with respect to edges of ω τ in Λ n \ Λ and that, conditioned on A , there are maximum 4δn edges that are incident to vertices on ∂Λ that are connected to Λ n and we can disconnect Λ 4n/5 from ∂Λ n by closing all these edges.
On the event {Λ 4n/5 / ωτ ← → ∂Λ n }, there exists a circuit of closed edges in ω τ that surrounds Λ 4n/5 .Denote the exterior-most such circuit by ζ an explore it from the outside: where the sum is over all possible values of ζ and we define Ω C as the subgraph of L bounded by C; the inequality relies on (CBC) and on the 0, 1 boundary conditions being domain Markov for µ.Note that Ω C can be turned into a domain by consecutively removing vertices of degree 1 -denote it by Ω C .Such operations can only increase the measure, whence µ 0,1 Ω C ,β sd ≤ st µ 0,1 Ω C ,β sd , and Λ 4n/5 ⊂ Ω C .Thus, the right-hand side in the last equation is exponentially small by Proposition 4.1.Combining the bounds, we get Taking δ small enough finishes the proof.
7 The case J ≥ U : Proof of Theorem 3 7.1 ATRC for J ≥ U We fix J ≥ U and a finite subgraph Ω of L. The ATRC model is defined via an Edwards-Sokal-type expansion.Since J ≥ U , the leading terms will correspond to interactions in τ and in τ .Thus, the ATRC measure on Ω with boundary conditions η τ , η τ is supported on pairs of percolation configurations (ω τ , ω τ ) ∈ {0, 1} E Ω × {0, 1} E Ω , and is defined by where Z = Z(Ω, J, U, η τ , η τ ) is a normalizing constant and a(0, 0) := e −4J , a(1, 0) = a(0, 1) := e −2(J+U ) − e −4J , a(1, 1) : Similarly to (6), if J > U , we can write the measure as As before, if the parameters J, U are fixed, we write ATRC Coupling of ATRC and AT.As mentioned above, edges in ω τ and in ω τ describe interactions in τ and in τ .In contrast to (8), the correlations of the product τ τ are described by simultaneous connections in both ω τ and ω τ : for any vertex x ∈ V Ω , The statement extends to infinite volume in a standard way, see [22,Proposition 5.11].

Proof of Theorem 3
Fix J ≥ U .By (2), β τ c ≥ β τ τ c .The opposite inequality follows directly from the ATRC representation and the coupling (32).Indeed, To derive part (ii) of Theorem 3, it remains to show that β c = β sd .The inequality β c ≤ β sd follows from a standard argument once the transition is shown to be sharp: for β < β c , there exists c = c(β) > 0 such that, for every n ≥ 1, This can be derived via a general approach [11], see Appendix A. The reverse inequality is a consequence of Zhang's argument provided that β c = β f c , i.e. the transitions for the free and wired measures occur at the same point.This follows from an analogue of Lemma 2.3 (see Appendix D for the proof of both lemmata): Lemma 7.1.There exists D ⊆ {(J, U ) ∈ R 2 : J ≥ U > 0} with Lebesgue measure 0 such that, for any (J, U ) ∈ D c , one has ATRC 0,0 J,U = ATRC 1,1 J,U .
Proof of Theorem 3. Fix J ≥ U .Part (i) follows from Lemma 7.1 and (MON+) in the same way as for J < U , see Section 2.2.

A Sharpness
The proof of sharpness for FK-percolation via the OSSS inequality [11,29] adapts to the ATRC.For completeness, we present a sketch of this argument and give details for the steps that are specific for the ATRC.
Fix J < U .We prove sharpness only for ω τ , since the proof for ω τ τ is the same.Recall that µ 1,1 Ω,β is the marginal of ATRC 1,1 Ω,β on ω τ .The key step in the proof of sharpness in [11] is the extension of the OSSS inequality [29] to dependent measures.The inequality holds for any monotonic (FKG) measure on {0, 1} E , for a finite set of edges E. In particular, it applies also to µ 1,1 Λ 2n ,β , for n ≥ 1.Instead of stating the OSSS inequality, we state its consequence that can be derived in the same way as in [11]: where the covariance is taken with respect to the measure µ 1,1 Λ 2n ,β .We proceed as in [11].Fix β 0 > 0.

A.2 Sharpness for J ≥ U
The proof is the same as for J < U and we only show the analogue of Lemma A.1.

B Proof of Lemma 3.11
Proof of Lemma 3.11.Fix q > 4 and p = p sd (q) and omit them in the notation below.
Let n ≥ 1 and Ω ⊃ Λ 2n be a finite subgraph of L. By Proposition 3.10 and Strassen's theorem, there exists a coupling P of η − ∼ FK w and η + ∼ FK Ω such that P(η − ≤ η + ) = 1.Define C to be the outermost circuit of edges in η − surrounding Λ n and contained in Λ 2n (if there is no such circuit, set C := ∅).By exponential decay of connections for the dual η * − f (Theorem 4), there exists α > 0 such that, for any n ≥ 1, Take any event A depending only on edges in Λ n .We have

Figure 1 :
Figure1: Left: Phase diagram of the Ashkin-Teller model: when J ≥ U , transitions for τ and τ τ occur at the self-dual curve (Theorem 3) and when J < U , the transition occurs at two distinct curves γ τ and γ τ τ dual to each other (Theorems 1 and 2).There are three regimes: disorder in τ and in τ τ (gray), order in τ and in τ (white), disorder in τ and order in τ τ (dashed gray).Right: Domain Ω (in bold black) on L and its dual Ω * (in gray).Notice that Ω * is not a domain on L * .The even domain D Ω (dashed) on Z 2 .

Figure 3 :
Figure 3: Left: Height function with 0, 1 boundary conditions.Right: Its spin representation is given by σ • on L (black circles) and σ • on L * (white circles).

Corollary 3 . 4 .
In the setting of part 1) of Proposition 3.2, take (σ • , σ • ) ∼ Spin +,+ D Ω ,c .Sample a percolation configuration ω on E Ω as follows independently at each edge e: if the endpoints of e * have opposite values in σ • , then ω e = 1; if the endpoints of e have opposite values in σ • , then ω e = 0; if σ • agrees on e * and σ • agrees on e, then P(ω e = 1) = 1 c .

Corollary 3 . 6 .
Fix c > 2, and let λ be the unique positive solution of c = e λ/2 + e −λ/2 .Then, for any sequence of domains D k increasing to Z 2 , the measures Spin +,+ D k ,c and Spin +,+;e λ/2 D k ,c converge weakly to some Spin +,+ c , which is independent of the sequence D k .The height function measures admit useful monotonicity properties and correlation inequalities when c, c b ≥ 1, see [21, Proposition 5.1].Proposition 3.7.Let D be a domain in Z 2 , and let c, c b ≥ 1.Then, for any boundary condition t, the measure HF t;c b D,c satisfies the FKG inequality (9).In particular, if t ≤ t , then HF t;c b D,c is stochastically dominated by HF t ;c b D,c .

Figure 4 :
Figure 4: Left: An edge configuration on the domain Ω ⊆ L from Figure 1 (in black), and its dual on Ω * (in gray).Right: Loops (in red) separating primal and dual clusters within D Ω after opening all edges in E ∂Ω (dashed).
Ω,J,U for the corresponding ATRC measures where 1 refers to the wired boundary condition on ∂Ω.Let η τ , η τ τ be boundary conditions on ∂Ω or ∂Ω.Consider the marginal of ATRC