Uniform Lipschitz functions on the triangular lattice have logarithmic variations

Uniform integer-valued Lipschitz functions on a domain of size $N$ of the triangular lattice are shown to have variations of order $\sqrt{\log N}$. The level lines of such functions form a loop $O(2)$ model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop O(2) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at the origin; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop $O(2)$ model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model.


Introduction
Height functions occupy a central role in statistical mechanics models on lattices. Indeed, the Ising, six-vertex and dimer models are only some of the lattice models involving height function representations. The predicted conformal invariance of these models is tightly linked to the convergence of their associated height functions to the Gaussian Free Field (GFF) or variations of it. Both statements were proved only in a handful of cases, and remain fascinating conjectures in general.
In this paper we study integer-valued height functions defined on the vertices of the two-dimensional triangular lattice T, or equivalently on the faces of the hexagonal lattice H. It is then natural to impose that height functions are Lipschitz, that is functions whose difference between any two adjacent vertices is at most 1; see Figure 1. More specifically, for any finite domain D of H, we will consider a uniformly chosen Lipschitz function among those with values 0 outside of D. The question of interest is the behaviour of such a function, especially as the domain D increases towards H.
Our goal is to show that the variance of the value at the origin of a uniformly chosen Lipschitz function is of order log N , where N is the radius of the largest ball centred at the origin and contained in D. This result, termed delocalisation (or logarithmic delocalisation to be precise) is in agreement with the conjectural convergence of uniform Lipschitz functions to the GFF.
The essential tool here is a re-interpretation of the uniform Lipschitz functions as the loop O(2) model, which in turn is represented as the superposition of two site percolations on T interacting with each other -below we view these as ±-spin assignments. This double-spin representation is obtained by colouring the loops of the loop O(2) model in two colours (as was done in [7]), then deriving a spin configuration from the families of loops of each colour. This may be viewed as the infinite-coupling-limit of the Ashkin-Teller model on the triangular lattice [26].
The loop O(n) model is defined on collections of non-intersecting simple cycles (loops) on a finite domain D of H and has two real parameters n, x > 0. The probability of each configuration is proportional to n to the number of loops times x to the number of edges in it. The loop O(n) model has a rich conjectural phase diagram [31,4] that remains mostly open; see [35] for an overview of the topic.
For n = 2, the model is expected to exhibit macroscopic loops when x ≥ 1 √ 2 and exponential decay of loop sizes when x < 1 √ 2 . The former is confirmed in this paper for x = 1 and in [13] for x = 1 √ 2 . The latter behaviour is shown to hold for x < 1 √ 3 + for some > 0 in [21]. The correspondence between the loop O(2) model and Lipschitz functions holds for any x > 0, but the corresponding height functions are not uniform: they are weighted by x to the number of pairs of adjacent faces of H having different values. The regime of exponential decay of loop sizes corresponds to localisation for the height function; that of macroscopic loops corresponds to logarithmic delocalisation.
A main difficulty in the study of the loop O(n) model is the lack of monotonicity and positive association. These type of properties are however expected to hold in convenient representations of the model, as illustrated by the present paper and by [13]. Indeed, a core ingredient of our arguments is the FKG inequality, which we show for the marginals of the double-spin representation. We then develop Russo-Seymour-Welsh (or RSW)type results for these marginals, which translate to similar statements for the loop and height function models.
The RSW theory was initially developed for percolation [36,38], and later generalised to other models via more robust arguments (see for instance [3,17,11,42,10]). It has become increasingly clear that for the latter type of arguments to apply the essential feature of the model is an instance of the FKG inequality. Indeed, other restrictions such as independence, symmetries and planarity have been, in some forms, relaxed in recent works. In this paper we do yet another step towards generalising this approach by considering a case where the Spatial Markov property applies only in a limited way.
The FKG inequality mentioned above extends to the case of certain non-uniform distributions on Lipschitz functions (corresponding to the loop O(2) model with x < 1) and more generally to the loop O(n) model with n ≥ 2, x ≤ 1 √ n−1 . Thus, we hope that this instance of the FKG inequality, together with the strategy of our proofs can be useful in other studies of the loop O(n) model.
Finally, we want to emphasise that in this work we do not attempt to prove convergence to the GFF. In general, the RSW theory can be viewed as a robust technique based on geometric constructions, but is not expected to lead to subtle convergence results. Indeed the seminal proofs of convergence of [37,27,40,41,9,8] are all based on some form of exact solvability, which is missing in our case.

Uniform Lipschitz functions
Let H denote the hexagonal lattice, embedded in R 2 with the origin 0 being the center of a face and the distance between the centres of any adjacent faces being 1. • for each u ∈ ∂ in D we have φ(u) = 0.
Since D is finite, only finitely many such functions exist. Write π D for the uniform measure on such functions, and let Φ D denote a random variable with law π D . Theorem 1.1.
(ii) For any increasing sequence of domains (D n ) n≥1 with 0 ∈ D 1 and H = ⋃ n D n , the sequence of variables Φ Dn −Φ Dn (0) converges in law as n → ∞ to a random Lipschitz function Φ H ∶ H → Z that is equal to 0 at 0. Write π H for the law of Φ H .
(iii) There exists c, C > 0 such that, for any distinct x, y ∈ F (H), The same holds for Φ D for any domain D containing the ball of radius 2 x − y around x.
One may wish to study height functions with different values imposed on the boundary via so-called boundary conditions. While we do not attempt to provide the most general form of our result, let us briefly mention some direct generalisations. First, for constant boundary conditions -that is if we study uniform height functions with φ(u) = c for all u ∈ ∂ in D -the law obtained is that of c + Φ D , and the results above adapt readily. Versions of the results above may also be deduced for "flat" boundary conditions, that is boundary conditions whose maximum and minimum differ by at most a constant, independently of D. The results for such boundary conditions may be obtained using the FKG inequality for the height function; we refer the reader to the upcoming paper [15] for formulations and proofs of such results in a slightly different context. In addition to the theorem above, RSW-type statements may be proved for Φ D , see Theorem 5.6. These may be used to prove bounds on the tail of 1 √ log N Φ D (0) in a domain where dist(0, D c ) = N .
To the best of our knowledge this is the first instance when a uniformly distributed Lipschitz function is proven to have logarithmically diverging variance. Previously known results establish that the variance is bounded (referred to as localisation) in high dimensions [32], or when the underlying graph is a tree [33] or an expander [34]. The conjectured convergence of the height function to the GFF indicates that localisation should also hold on lattices in dimensions three and above.
Recently it was established in [13] that the variance is logarithmic in a very similar setup -also on the hexagonal lattice, though the distribution is not uniform but instead the probability of a function φ is proportional to (1 √ 2) #{u∼v∶φ(u)≠φ(v)} . This result follows from [13,Thm. 1] for n = 2.
On the square lattice Z 2 , one may also consider the related model of graph homomorphisms from Z 2 to Z, which are defined as functions on the faces of Z 2 restricted to differ by exactly one between any two adjacent faces. These functions may be viewed as height functions of the six-vertex model that has parameters a, b, c > 0. When a = b = 1 and c > 0 is general, the height functions are weighted by c n 5 +n 6 , where n 5 +n 6 is the number of vertices of Z 2 for which the four adjacent faces contain only two values. For the uniform model c = 1 (termed square ice) a non-quantitative delocalisation result is proved in [6] based on an approach described in [39]. In [14] a dichotomy theorem similar to our Theorem 4.1 is developed and logarithmic delocalisation is shown. In [22] logarithmic delocalisation at c = 2 and localisation for c > 2 are shown, based on the Baxter-Kelland-Wu coupling [2] with the random-cluster model and results of [17] and [12], where the order of the phase transition in the latter model is computed. In the upcoming [15], the logarithmic delocalisation result is generalised to all c ∈ [1,2].
Convergence of the height function of the dimer model to the GFF was proven in a seminal work by Kenyon [27] and was recently extended to the case of a weak interaction [20]. On the square lattice, this corresponds to graph homomorphisms to Z with c ≈ √ 2. Proving convergence of delocalised discrete-valued height functions outside of the free-fermion solution remains a major open problem.
The case of real-valued height functions is better understood. In particular, convergence to the GFF was established for uniformly convex symmetric potentials (under additional regularity assumptions) [29] and the delocalisation was proven for some nonconvex nearest-neighbour potentials [30].

The loop O(2) model
Let D be a domain of H. A loop configuration on D is a subgraph of D in which every vertex has even degree. Thus, a loop configuration is a disjoint union of loops (i.e., subgraphs which are isomorphic to cycles) that are contained entirely in D. In particular, none of these loops contain edges of ∂ where (ω) is the number of loops in ω. The normalising constant Z(D), chosen so that P D is a probability measure, is called the partition function. Write Λ n for the domain defined by a self-avoiding contour going around the set of faces at distance n from 0 (for the graph distance on the dual H * = T of H). A sequence of domains (D n ) n≥1 is said to converge to H if, for all k, all except finitely many domains of (D n ) n≥1 contain Λ k . (i) For any increasing sequence of domains (D n ) n≥1 converging to H, P Dn has a limit denoted by P H .
(ii) The measure P H is supported on even subgraphs of H that contain only finite loops.
(iii) The measure P H is ergodic and invariant under translations and rotations by π 3.
(iv) There exists c > 0 such that, for any even integer n and any finite domain D containing Λ n , or for D = H, P D (there exists a loop in Λ n surrounding Λ n 2 ) ≥ c. where E D denotes the expectation with respect to P D . The same holds if we replace E D (N D ) with E H (N D ). In particular, P H -a.s., there are infinitely many loops surrounding the origin.

6
Any limit of measures of the type P D is supported on even subgraphs of H. Such graphs are in general disjoint unions of loops and infinite paths on H. Thus, point (ii) of the above states that no infinite path exists P H -a.s. Point (iv) of the theorem above resembles a RSW-type statement for the loops of the O(2) model; indeed, it stems from an actual RSW result for a related model (see Corollary 5.2). Due to the imperfect correspondence between the models, the upper bound of (1.2) takes this slightly odd form. We believe that a similar bound should apply to any ρ < 1 (with c depending on ρ), for any n and for a single loop instead of two. This statement is of an independent interest as it would in particular imply, via Aizenman-Burchard [1], tightness of interfaces under Dobrushin 0/1 boundary conditions. Point (v) is a direct consequence of (iv). Moreover, bounds on the deviation of N D from log dist(0, D c ) may be obtained in a straightforward manner.
Finally, we discuss the issue of Gibbs measures for the loop O(2) model. Consider a measure η on {0, 1} H supported on even configurations. Recall that these are disjoint unions of bi-infinite paths and finite loops. Let ω be a configuration in the support of η and D be a finite domain. Then ω ∩ D c induces certain connections between the vertices of ∂ E D. Indeed, each such vertex may be connected to another such vertex, to infinity, or be isolated. These connections constitute a boundary condition on D. Formally we describe boundary conditions as follows.
For ξ 1 and ξ 2 two restrictions to D c of even configurations on H, write ξ 1 ∼ ξ 2 if they induce the same connections on ∂ E D. A measure η on even configurations on H is called a Gibbs measure for the loop O(2) model with edge weight 1 if, for any finite domain D of H and any restriction ξ of an even configuration to D c , for all ω 0 ∈ {0, 1} E(D) . The above equation needs only to hold when the conditioning is not degenerated. Write P ξ D for the measure on {0, 1} E(D) described by the right-hand side above; it is the loop O(2) measure on D with boundary conditions ξ. It is immediate that P ξ D does not depend on the choice of ξ within its equivalency class for ∼. Notice that the infinite paths do not contribute to the right-hand side of (DLR). One may be tempted to add a term of the form (n ′ ) #infinite paths of ω 0 ∪ ξ 0 that intersect D in (DLR) for some n ′ > 0. This would be superfluous, as the number of infinite paths intersecting D is imposed by the boundary conditions. In particular, any Gibbs measure is supported on configurations formed entirely of finite loops. Notice that we do not require that the Gibbs measure be translation invariant or ergodic for it to be equal to P H . However, we do not claim that for any sequence of domains D n converging to H and any sequence of boundary conditions ξ n on these domains, P ξn Dn tends to P H . This is a stronger statement than Theorem 1.3; we believe it to be true, but have no proof. It may appear surprising, but limits of measures P ξn Dn need not be Gibbs in the sense of (DLR). Theorem 1.3 does not imply the uniqueness of the Gibbs measure for height functions. We will discuss more on this point in the following section.

7
The loop model studied here is part of the larger class of loop O(n) models with edge weight x, where n and x are positive parameters. The loop O(n) model with edgeweight x in a domain D is the measure on loop configuration given by where ω is the number of edges in ω and Z(D, n, x) is called the partition function. and n ∈ [1,2], the parafermionic observable is then used to exclude exponential decay of loops, thus proving the equivalent of Theorem 1.2. The uniqueness of the Gibbs measure is shown via the stronger statement which we are unable to prove here: convergence to the unique infinite-volume measure of finite-volume measures on any increasing sequence of domains, with any boundary conditions. The point n = 2, x = 1 is clearly outside of the FKG regime determined in [13], and a more complicated spin representation is required. This representation will involve two spin configurations, and will therefore be sometimes referred to as the double-spin representation (see Section 2 for precise definitions).
Let us also mention that [16] proves that for n large enough and any x > 0, the loops of the loop O(n) model with edge-weight x exhibit exponential decay. Moreover, for n x 6 large enough, it is shown that at least three distinct, linearly independent infinite-volume Gibbs measures exist. For n x 6 small enough (and n large) it was shown in the same paper that at least one Gibbs measure exists, but its uniqueness (though expected) was not proved.

Relation between the loop O(2) model and random Lipschitz functions
Fix a domain D. For a Lipschitz function ϕ on D, define an edge configuration ω ϕ by ω ϕ (e) = 1 if and only if the two faces separated by e have different values of ϕ. It is straightforward to check that ω ϕ is indeed a loop configuration.
(ii) Given some loop configuration ω, the law of Φ conditionally on ω Φ = ω is obtained as follows: define → ω by choosing a clockwise or a counter-clockwise orientation uniformly and independently for each loop of ω. Then, for each face u of D, set where ↻ ( → ω ; u) and ↺ ( → ω ; u) stand for the number of clockwise (resp. counterclockwise) oriented loops of → ω surrounding u.
Proof The correspondence between oriented loop configurations and Lipschitz functions defined by (1.4) is in fact a bijection. Indeed, the reverse mapping can be defined as follows: given a Lipschitz function ϕ ∶ D → Z, each loop of the corresponding (unoriented) loop configuration ω ϕ is oriented clockwise if the values of ϕ inside of the loop are higher than those outside, and is oriented counter-clockwise otherwise. The push-forward of π D under this bijection is a uniform measure on all oriented loop configurations on D. Considering the projection on the set of unoriented loop configurations we obtain P D , since each loop has two possible orientations. This proves (i), and (ii) follows readily. ◻ Using the correspondence between Lipschitz functions and loop configurations described above, Theorem 1.1 follows easily from Theorem 1.2.
Proof of Theorem 1.1 (assuming Theorem 1.2) (i) By Proposition 1.4 (ii), a random Lipschitz function Φ D distributed according to π D can be sampled from a random loop configuration ω distributed according to P D by orienting each loop of ω uniformly and independently. Then Φ D (0) has the distribution of a simple random walk on Z with N D steps, where N D is the number of loops in ω surrounding 0. Thus, Var(Φ D (0)) = E D (N D ). The conclusion follows from (1.3).
(ii) Using the coupling from Proposition 1.4, we get that for any u ∈ F (D), the value of Φ D (u) − Φ D (0) is a function of number of loops separating u from 0 and their orientations. By items (i) and (ii) of Theorem 1.2 the infinite-volume limit of P D exists and consists only of finite loops. Thus, the infinite-volume limit of Φ D − Φ D (0) also exists.
(iii) We will prove the statement for Φ H ; that for Φ D is proved in the same way. Similarly to the previous items, we have 5) where N x∖y stands for the number of loops surrounding x but not y and N y∖x for those surrounding y but not x.
For the lower bound, notice that N x∖y is larger than the number of loops surrounding x and contained in Λ x−y . Thus, by (1.3), E H (N x∖y ) ≥ c log x − y for some universal constant c > 0. The desired lower bound on Var(Φ H (x) − Φ H (y)) follows.
For the upper bound, define Γ to be the outermost loop surrounding x but not y, provided such a loop exists. Let γ be a possible realisation of Γ and let Int(γ) be the interior of the domain delimited by γ. Notice that the event {Γ = γ} is measurable in terms of the configuration on and outside γ. Therefore, conditionally on Γ = γ, the restriction of P H to Int(γ) is the uniform measure among all loop configuration in Int(γ), which is to say it is equal to P Int(γ) . Thus where the sum is over all possible realisations γ of Γ and N Int(γ) (x) in the right hand side stands for the number of loops surrounding x and contained in Int(γ). Now, since y ∉ Int(Γ), dist(x, Int(γ) c ) ≤ x − y for any path γ appearing in the sum. Thus, (1.3) proves that E H (N Int(γ) (x)) ≤ 1 + C log x − y for some universal constant C.
The same holds for E H (N y∖x ). Using this and (1.5), we obtain the desired upper bound on Var(Φ H (x) − Φ H (y)). ◻ Let us briefly comment on the uniqueness of infinite-volume measures for Lipschitz functions. One may think that, due to Theorem 1.3, π H should be the only infinite-volume measure with the property that its restriction to any finite domain is uniform among Lipschitz functions that take the value 0 at the origin. This is not the case. Indeed, the correspondence between the loop and Lipschitz functions models is not perfect, and does not allow us to deduce this.
Moreover the claim is false, as an infinite family of infinite-volume measures for uniform Lipschitz functions is expected to exist, one for each global "slope". The loop representation of any of these contains infinite paths and is not Gibbs in the sense of (DLR).
Structure of the paper The rest of the paper is entirely dedicated to the loop O(2) model with x = 1. In Section 2 we derive a representation of the loop model in terms of two loop O(1) configurations conditioned not to intersect. These are in turn represented in terms of spin configurations that are shown to satisfy the FKG inequality and a certain form of Spatial Markov property.
In Section 3 this spin representation is used to construct an infinite-volume, ergodic loop O(2) measure. The infinite-volume measure is then shown to be unique (in some sense that will be made precise later). In doing so, we show that 0 is surrounded by infinitely many loops. For height functions, this corresponds to the delocalisation of Φ(0) or equivalently to the divergence of covariances. At this stage, the delocalisation/divergence is not quantitative.
Section 4 contains a dichotomy theorem. In the language of uniform Lipschitz functions, the dichotomy theorem roughly states that the covariance between two points either is bounded or diverges logarithmically in the distance between the points.
Finally, in Section 5, the non-quantitative delocalisation result and the dichotomy theorem are used to prove Theorem 1.2. Theorem 1.3 is also proved here. Moreover, we provide an RSW result for height functions in Section 5.4.
The paper is structured so as to isolate the different ingredients of our argument; some of them may be useful for the analysis of the loop O(n) model with other values of n and x, or other similar models. We further discuss in Section 2.1 the various properties of the loop O(2) model that are necessary for our proof.
Notation Below is a list of notation used throughout the paper. Some of it was already mentioned, some is new.
Recall that H denotes the hexagonal lattice; its dual is the triangular lattice, written H * = T. We will call edge-path any finite or infinite sequence of adjacent edges of H with no repetitions. A face-path is a sequence of adjacent faces of H with no repetitions, or equivalently it is a path on T that does not visit the same vertex twice.
Domains D = (V (D), E(D)) are interior of edge-polygons of H. The edges of the polygon form the edge-boundary of D, written ∂ E D. The faces of H adjacent to ∂ E D and inside ∂ E D (outside, respectively) form the inner face boundary of D, written ∂ in D (and the outer face-boundary written ∂ out D, respectively). The set of faces of H inside ∂ E D is written F (D); those not adjacent to ∂ E D form the interior of D, denoted by An edge configuration on D is an element ω ∈ {0, 1} E(D) ; it is identified to the graph with vertex set V (D) and edge-set {e ∈ E(D) ∶ ω(e) = 1}. Write u ω ← → v to indicate that two vertices u, v of V (D) are connected in ω. The same notation applies to H and D * .
A spin configuration on D is an element σ ∈ {−, +} F (D) ; the notation extends to H. Below we will use two superposing spin configurations. We identify one as red, the other as blue and denote the relevant spins by { , } and { , } for legibility.
For a red-spin configuration σ ∈ { , } F (D) and two faces u, v ∈ F (D), write u ← → v (or u in D ← → v when the choice of D is unclear) to indicate that there exists a face-path in D starting at u and ending at v, formed entirely of faces with σ-spin . Such a path will be called a -path or simple-path. Connected components for this notion of connectivity are called -clusters.
A double-path will be an edge-path for which all adjacent faces have spin ; connections by double-paths will be denoted by ← →. The same applies to spins , and .
Write ← → for the negation of ↔.
Acknowledgements The authors would like to thank Hugo Duminil-Copin for numerous discussions and tips, especially concerning the dichotomy theorem of Section 4, and Ron Peled for suggesting to develop the loop-weight 2 = 1 + 1 following Chayes and Machta. Our conversations with Matan Harel, Marcelo Hilario, and Nick Crawford also were very helpful. We acknowledge the hospitality of IMPA (Rio de Janeiro), where this project started. The first author is supported by the Swiss NSF grant P300P2_177848, and partially supported by the European Research Council starting grant 678520 (LocalOrder). The second author is a member of the NCCR SwissMAP.

1+1 = 2
Fix a domain D. Choose a loop configuration ω according to P D and colour each loop of ω in either red or blue, with equal probability, independently for each loop. Extend P D to include this additional randomness. Write ω r and ω b for the configurations of blue and red loops. Then, for any two disjoint loop configurations ω r , ω b , .
In other words, P D is the uniform distribution on pairs of loop configurations (ω r , ω b ) that do not to intersect. In the context of Lipschitz functions, one may think of ω r as the level lines with higher value on the inside and ω b as those with higher value on the outside (that is the clockwise and counter-clockwise, respectively, oriented loops in the language of (1.4)). While accurate, this interpretation is not relevant below.
Keeping the idea of colouring loops as the intuition, in the next section we introduce a measure on pairs of red and blue ±1 spin configurations. Though this measure is tightly linked to the loop O(2) measure on pairs of red and blue loops and under certain boundary conditions these two measures will be shown to coincide, we emphasise that this is not always the case.
To shorten notation, we will use the symbols , to denote the values of red spins and , for blue spins.

Spin representation
Define µ D to be the uniform measure on all pairs of spin configurations σ r ∈ { , } F (D) and σ b ∈ { , } F (D) such that for every two adjacent faces u, v ∈ F (D) at least one of the equalities σ r (u) = σ r (v) and σ b (u) = σ b (v) holds. We call such configurations σ r , σ b coherent and denote this relation by σ r ⊥ σ b .
Given a spin configuration σ ∈ {±1} D , define ω(σ) to be set of edges of D separating adjacent faces bearing different spin in σ. Then ω(σ) consists of disjoint loops and paths linking boundary vertices in D.
The correspondence σ ↦ ω(σ) is a classical tool in the study of the Ising model, called the high temperature representation (see for instance [19,Sec. 3.10.1]). If σ is chosen according to a Ising distribution, then ω(σ) has the law of a loop O(1) model, with parameter x depending on the temperature of the Ising measure. For the loop O(n) model with general values of n, this correspondence was used in [13] with the name cluster representation.
The following proposition describes the relation between µ D and P D . Define where ≡ should be understood as "equal everywhere to". The notation µ D comes from Theorem 2.3, where these boundary conditions are shown to be equivalent to setting on the interior boundary of D and on its exterior boundary.
Proof The map σ r ↦ ω(σ r ) is a bijection between spin configurations σ r ∈ { , } F (D) that are equal to on ∂ in D and all loop configurations on D. Indeed, due to the constant spin of σ r on ∂ in D, ω(σ r ) is indeed a loop configuration. Moreover, the reverse mapping is the following: a loop configuration ω on D is mapped to the spin configuration σ r ∈ { , } F (D) that is equal to (resp. ) at all faces of D that are surrounded by an even (resp. odd) number of loops of ω. Similarly, the map σ b ↦ ω(σ b ) defined on the set of spin configurations σ b ∈ { , } F (D) that are constant on ∂ in D and taking values in L (D) is two to one, due to its invariance under global spin flip.
The condition σ r ⊥ σ b corresponds to ω(σ r ) ∩ ω(σ b ) = ∅. Thus, µ D induces a uniform measure on all pairs (ω(σ r ), ω(σ b )) of non-intersecting red and blue loop configurations on D, that is P D . As described above, the marginal of this measure on the non-coloured loop configuration ω(σ r ) ∪ ω(σ b ) is the loop O(2) measure on D. We will show below that, under µ D , the marginals σ r and σ b satisfy the FKG inequality. Moreover the spin measures of the type µ D satisfy the Spatial Markov property in the following sense. If D ′ is a domain contained in some larger domain D, then the restriction of µ D to D ′ , conditionally on the spins σ r , σ b outside D ′ , is entirely determined by the values of σ r and σ b on ∂ out D.
It may be tempting to think that these two observations suffice to apply the techniques developed for the random-cluster model to our setting (such as those of [17,18]). Unfortunately this is easier said than done. Indeed, many of these techniques use a form of monotonicity of boundary conditions. In our case, it is unclear how to compare boundary conditions consisting of pairs of spins, as the FKG inequality applies only individually to the single-spin marginals of µ D .
To circumvent this difficulty, we will focus our study on one of the single-spin marginals of µ D ; we arbitrarily choose the red-spin marginal, and call it ν D . As already stated, this measure satisfies the FKG inequality, but fails to have a general spatial Markov property. However, we show in Theorem 2.3 and Corollary 2.4 that a limited version of the spatial Markov property applies to ν D , under certain restrictions.
One may attempt to apply the same strategy to other values of n and x. Our argument is quite intricate, and different parts of it use different properties of the double spin representation described above. The paper is organised to separate the different arguments, so as to facilitate the identification of blocks that may be applied to other models. Below is brief list of the essential properties of the double spin representation and their uses.
• The FKG inequality for the red-spin marginal is crucial and is used extensively throughout the proof. As mentioned in Remark 2.11 (iii), the FKG inequality extends to the red-spin marginal of a certain double spin representation of the loop O(n) model with parameters n ≥ 2 and x ≤ 1 √ n−1 . • That x = 1 is essentially only used for the spatial Markov property. In its current form, the property does not apply to x ≠ 1.
• The symmetry between the red and blue spin marginals (which, in light of Remark 2.11 (iii) boils down to n = 2) is akin to a self-duality property, and is used to prove RSW type estimates (see Lemma 3.8).
Finally, let us mention that it is expected that the loop O(2) model for n = 2 and x ≥ 1 √ 2 has a similar behaviour to the case x = 1, that is macroscopic loops exist at every scale. However, for all n > 2 and any x > 0 or n = 2 and x < 1 √ 2, loops are expected to exhibit exponential decay. Thus, parts of our proof need to fail for more general values of n and x. The dichotomy theorem of Section 4 (or similar statements) may be expected to hold for all values of n and x, but no proof is generally available.

Spatial Markov property
In general, the measures ν D , that is the red-spin marginals of µ D , do not have the spatial Markov property. However, a version of this property holds in certain cases. Recall the definition (2.1) of µ D and set Let ν D and ν D , respectively, be the marginals on σ r of the above two measures. Define the measures µ D , µ D , µ D etc. in a similar ways, and write ν etc. their red-spin marginals.
where symbol D = means that the two measures are equal when σ r and σ b are restricted to D.
Proof All measures under consideration are uniform over sets of coherent pairs (σ r , σ b ) that agree with the corresponding boundary conditions. Thus, it is enough to show that the two sets corresponding to the two sides of (2.2), and of (2.3), respectively, are equal.
(i) Consider a pair of coherent configurations σ r ∈ { , } F (D) and σ b ∈ { , } F (D) contributing to the RHS of (2.2); let us show that they also contribute to the LHS. By definition, σ r ≡ on ∂ in D, which is to say that σ r = τ r on ∂ in D.
. Then the following statements hold: where by symbol D = we mean that the two measures are equal when σ r is restricted to D.
Remark 2.5. It is tempting to think that the above Spatial Markov property holds for any boundary conditions on ∂ in D ∪ ∂ out D. This is not the case. One significant example is that of the boundary conditions consisting of four arc of alternating spins , , , . Indeed, these boundary conditions are coherent with non-intersecting loop configurations 1 (ω r , ω b ) where ω b contains • paths between the arcs , • paths between the arcs or • none of the above.
The three cases above are mutually exclusive. Depending on the red configuration outside D one or both of the first two cases may be excluded.

FKG inequality
In this section we show that the red-spin marginals ν of the measures µ satisfiy the FKG inequality. This property is crucial to all our proofs. Similar properties were found in [13] for the single-spin representation of the loop O(n) for a certain range of parameters and in [22] for a spin representation of height functions on Z 2 arising from the six-vertex model. Fix some domain D. We start by introducing a partial order on A probability measure P on { , } F (D) is said to satisfy the FKG inequality (or called positively associated) if for any two increasing events A, B ⊂ { , } F (D) , we have (2.4) Recall that the marginal of µ D on the red spin configurations is denoted by ν D .
Before proving the FKG inequality, let us compute ν D . For a spin configuration σ on D, let θ(σ) ∈ {0, 1} E(D * ) be the set of all edges e = uv ∈ E(D * ) such that σ(u) ≠ σ(v). If σ is associated to a loop configuration ω, then e * ∈ θ(σ) if and only if e is present in ω. For readers familiar with the notion of duality in percolation (where the dual configuration is written ω * ), we mention that θ(σ) = (ω * ) c . See Figure 3 for an example. Denote by k(θ(σ)) the number of connected components of θ(σ); note that isolated vertices of D * (that is faces of D) are also counted as connected components.
Proposition 2.7. (i) The law of σ r under µ D is given by

5)
where Z D is a normalising constant.
(ii) The law of σ b on D under the conditional measure µ D (. σ r ) is obtained by colouring independently and uniformly the clusters of θ(σ r ) in either or . . Thus, there are exactly 2 k(θ(σr)) blue spin configurations coherent with σ r , and (i) follows readily. In addition, when conditioned on σ r , the measure on these blue spin configurations is uniform, thus asserting (ii). ◻ Remark 2.8. A straightforward adaptation of the proof above shows that, for A ⊂ F (D), the law of σ r under µ D (. σ b ≡ on A ) is given by 1 Z 2 k A (θ(σr)) , where k A (θ(σ r )) is the number of connected components of θ(σ r ) when all components intersecting A are counted as a single one. When A is connected, k A (θ(σ r )) may be viewed as the number of connected components of the configuration obtained by adding to θ(σ r ) all edges between pairs of adjacent faces of A .
As a consequence where k ∂D (θ(σ r )) is the number of connected components of θ(σ r ), where all components intersecting ∂ in D are counted as a single one.
We are in a position to prove Theorem 2.6.
Proof of Theorem 2.6 By [25,Thm. 4.11], it is enough to show the FKG lattice condition, which states that, for any two spin configurations σ andσ, where σ ∨σ, σ ∧σ ∈ { , } T are defined by σ ∨σ(u) = max(σ(u),σ(u)) and σ ∧σ(u) = min(σ(u),σ(u)) for every u ∈ T. Moreover, by [24,Thm. (2.22)], it is enough to show (2.6) for any two configurations which differ for exactly two faces. That is, that for any σ ∈ { , } F (D) and u, v ∈ F (D) two distinct faces, where σ ab is the configuration coinciding with σ except (possibly) at u and v, and such that σ ab (u) = a and σ ab (v) = b. By Proposition 2.7, the ratio of the LHS and RHS of (2.7) is written Our goal is thus to show that First we will treat the simple case where σ is such that u ← → v in σ and u ← → v in σ . Then u and v are not adjacent and there exist two paths or circuits, one of the other of , that separate u from v in D. Hence, there exists a path or loop γ in ω(σ) that separates u from v and does not contain any edges of the faces u or v. For any choice of a, b ∈ { , }, edges in T that cross γ belong to θ(σ ab ), thus forming a path or a circuit of edges in θ(σ ab ) that separates u from v. The effect on k(θ(σ)) of switching the spin at v from to is then independent of the value of the spin at u: As a consequence, the LHS of (2.9) is zero.
We move on to the case where u and v are connected by a path of or by a path of . Before diving into the core of the proof, we need to eliminate a degenerate case: when u and v are neighbouring faces and no face of D is adjacent to both u and v. Then D may be split into two domains D u and D v containing all faces connected to u in D ∖ {v} and those connected to v in D ∖ {u}, respectively. It is then immediate to see that the number of connected components of θ(σ ) intersecting D u is the same as that for θ(σ ). The same statement applies to θ(σ ) and θ(σ ). A similar statement may be formulated for D v , by pairing θ(σ ) with θ(σ ) and θ(σ ) with θ(σ ). Finally, in θ(σ ) and θ(σ ), faces u and v are in the same connected component, while in θ(σ ) and θ(σ ) they are in different components. Thus, we find Henceforth we may assume that, if u and v are neighbours, then there exists at least one face of D adjacent to both u and v. Moreover, we will suppose that u and v are connected by a path of in σ or by a path of in σ . By symmetry, we may limit our study to the case where u is connected to v in σ by a -path; when u and v are neighbours, we may choose the path to contains at least one vertex other than u and v.
Denote by P the -cluster of u (and implicitly of v as well) in σ ; denote by M the union of all -clusters in σ that are adjacent to u or v. Both P and M are fixed sets of faces of D. Then all the connected components of θ(σ ), θ(σ ), θ(σ ), and θ(σ ) that do not intersect P ∪ M are the same in these four configurations, and thus cancel out in (2.9). It remains to study the contribution of connected components of θ(.) that do intersect P ∪ M .
For a spanning subgraph Θ of D * , define k P (Θ) to be the number of connected components of Θ that intersect P , and k M (Θ) as number of connected components that intersect M and do not intersect P . Clearly, k P (Θ) + k M (Θ) is equal to the number of connected components in Θ that intersect P ∪ M . Thus, is suffices to prove the following two inequalities: We start by proving the easier inequality (2.11). Four types of components contribute to k M (θ(σ )): those who contain faces adjacent to both u and v, those who contain faces adjacent to u but not v, those who contain faces adjacent to v but not u, and those containing no faces adjacent to u or v. Write K {u,v} , K {u} , K {v} and K ∅ for the number of components in each category above. By the definition of k M and the fact that u, v ∈ P , any connected component contributing to k M (θ(σ )) is such that all its faces that are adjacent to u or v have spin in σ . When turning the spin of u from to , all components of the type K {u,v} , K {u} become connected to u, and thus cease to contribute to k M . The same holds for v, and we find: Let us now prove (2.10). Denote by E u , E v ⊂ E(D * ) the sets of all edges linking u (resp. v) to adjacent vertices in V (D * ). The next claim constitutes the core of the proof and, as we will see below, implies readily (2.10).
Then γ is a non-trivial simple cycle on T. Since the domain D is simply connected, γ delimits a simply connected domain which we denote by D γ . The boundary of P ∖ {u} intersects γ at two places: the midpoint of the edge γ m−1 γ m and the midpoint of the Figure 4: The constructions used in the proofs of (2.12) (left) and (2.14) (centre and right). Spins + are pink and − are gray; only spins of interest are depicted. The path γ (black bold) uses faces of alternating spins until it enters P , then it continues on P , whose faces (except for v in the central and right diagrams) are of spin +. The paths χ, χ 1 and χ 2 (in red) are part of the boundary of P and separate faces of distinct spins. Their dual edges contain paths linking u to γ n , u to v and v to γ n , respectively. edge γ n−1 γ n . Thus the boundary of P ∖ {u} contains a path χ that is contained in D γ and that connects these two midpoints of edges.
Finally notice that, for any two adjacent faces a, b with a ∈ P ∖ {u} and b ∉ P ∖ {u}, we have σ (a) = and σ (b) = , hence ab ∈ θ(σ ). Applying this to faces on either side of χ, we find that all edges of T crossing χ are contained in θ(σ ). In particular, we deduce that u is connected in θ(σ ) to γ n , hence also to w. This completes the proof of (2.12). The same argument proves (2.13).
We turn to the proof of (2.14). We will prove this in two steps: The second equality above is implied by (2.13). Indeed, we have proved that no edge of {vw ∈ E(D * ) ∶ σ (w) = } may connect two distinct clusters contributing to k P (θ(σ )).
That is also true for clusters contributing to k P (θ(σ )∪E u ), since the latter configuration dominates the former. The first equality of (2.16) is similar to (2.12), with the only difference that it applies to σ rather than σ . This apparent detail complicates the proof slightly as u is not necessarily connected to all points of P by paths of in σ . The middle and right diagram of Figure 4 helps illustrate the argument below.
As for (2.12), the proof goes through the equivalent of (2.15). Fix a face w neighbouring u with σ (w) = and which belongs to a connected component of θ(σ ) that intersects P . Our goal is to prove that w is connected to u in θ(σ ) In a first instance let us suppose that w ≠ v. Then, as in the proof of (2.12), we may produce a path w = γ 0 , . . . , γ n , . . . , γ m = u such that γ 0 , . . . , γ n−1 ∉ P , γ n , . . . , γ m ∈ P and γ 0 , . . . , γ n uses only edges of θ(σ ). If such a path may be constructed to not include v, then we choose γ such, and the same reasoning as in (2.12) (applied with P ∖{v} instead of P ) allows us to conclude that u Suppose now that no path γ with the properties above and which avoids v exists. Then pick γ to visit v at some index k ≥ n and with k < m − 1 (see Figure 4, center). We have k ≥ n since v ∈ P ; we may pick k < m − 1 since, even when u and v are adjacent, u is connected to v by a non-trivial path of , and we include this path in γ. It is also true that k > n, since σ (γ k−1 ) = necessarily. Let D γ be the domain delimited by γ.
Consider the boundary of P ∖ {u, v} inside the domain D γ ; it intersects the boundary of D γ at four points: the midpoint of the edges γ n−1 γ n , γ k−1 γ k , γ k γ k+1 and γ m−1 γ m . Since no path γ avoiding v exists, the boundary of P ∖ {u, v} contains two non-empty segments χ 1 and χ 2 which connect γ n−1 γ n to γ k−1 γ k and γ k γ k+1 to γ m−1 γ m , respectively. By the choice of χ 1 and χ 2 as parts of the boundary of P ∖ {u, v}, all edges of T that intersect χ 1 and χ 2 are present in θ(σ ). In particular, we find u Finally let us study the case when w = v and hence u and v are adjacent (see Figure 4, right). Then, due to our assumption that u and v are connected by a non-trivial path of in σ , we may choose a face-path γ = γ 0 , . . . , γ m with m ≥ 2, γ 0 = v, γ m = u and σ (γ k ) = for all 1 ≤ k < m. The cycle γ ∪ {uv} delimits a simply connected domain D γ . By considering the interface between P and the cluster of u in σ , we deduce the existence of an edge-path χ on E(D γ ) with on one side and on the other, that starts on an edge adjacent to v and ends on one adjacent to u. This implies that u θ(σ ) ← → v, and the proof is complete. ◻ Using Claim 2.9, (2.10) becomes The LHS above is the number of distinct connected components in θ(σ ) that contain at least one endpoint of an edge of E u minus one. The RHS is the same number for θ(σ )∪E v instead of θ(σ ). Clearly, the former is greater or equal than the latter, and the proof of (2.10) is finished. ◻ Below we formulate several corollaries about the FKG inequality under various boundary conditions that we are going to use in the proofs.
Corollary 2.10. The FKG inequality (2.4) holds also in the following cases: (i) for the red-spin marginal of µ D , when the red spins are conditioned to take given values on a set of faces of D and the blue spins are conditioned to be on a connected set of faces of D. More precisely, for σ r chosen according to µ D (. σ r = σ 0 on A and σ b ≡ on B), where A is any set of faces of D, σ 0 is any red spin configuration on A , and B is a connected set of faces of D; (ii) for the measures ν D , ν D , ν D and ν D .
Proof (i) First let us show the FKG inequality when A is empty. As described in Remark 2.8, conditioning the blue spins to be on B boils down to counting all connected components of θ(σ r ) that intersect B as a single one. When B is connected, this may be achieved by adding to θ(σ r ) all edges linking pairs of neighbouring vertices in B. The proofs of (2.12), (2.13) and (2.14) adapt directly to this situation. Indeed, as already discussed in the proof above, adding edges to θ(σ r ) only helps in proving (2.12), (2.13) and (2.14). The rest of the proof of Theorem 2.6 applies directly.
Next assume that A is non-empty and σ 0 is given. The FKG lattice condition for µ D (. σ r = σ 0 on A and σ b ≡ on B) is a subset of the inequalities that constitute the FKG lattice condition for µ D (. σ b ≡ on B). Since the latter were proved to hold, so do the former.
(ii) Let D ′ be a domain containing F (D) ∪ ∂ out D. Then ν D ′ satisfies the FKG inequality. By point (i) and the Spatial Markov property (Corollary 2.4), the FKG inequality also applies to ν D , ν D , ν D and ν D . ◻ when B is not connected. A counter-example is provided by a domain formed of six faces in a line, with B being formed of the first and last face, u and v being the second and fifth face, respectively, and σ being the red spin configuration formed of alternating and spins.
Nor does the FKG inequality apply to the red spin marginal of µ D (.
where B + and B − are disjoint sets of faces of D.
(ii) The proof of the FKG inequality only uses limited features of the hexagonal lattice. Indeed, it adapts to any planar trivalent graph whose set of faces forms a simply connected domain. It is however worth mentioning that the condition of simply connectedness is essential. Indeed, counter-examples may be given for sets of faces of H which are not simply connected: the counter-example of point (i) above may easily be adapted.
(iii) A similar instance of the FKG inequality extends to the loop O(n) model with n ≥ 2 and x ≤ 1 √ n − 1, when the red spin configuration is obtained by colouring loops in red with probability 1 n and in blue otherwise, independently. The only difference in the proof is that the term 2 k(θ(σ)) in (2.5) should be replaced by the partition function of the Ising model on the graph obtained by collapsing each cluster of θ(σ r ) into a single vertex. The FKG property of the FK-Ising representation then leads to the analogue of (2.17). We do not give further details of this generalisation as it is irrelevant here; the reader is referred to [22], where similar ideas are used to prove a FKG statement for the spin representation of a six-vertex model.

Comparison between boundary conditions
Above he have introduced a number of boundary conditions for the positively associated measure ν D . As for the random cluster model or other positively associated models, the boundary conditions may have an increasing or decreasing effect on the measure.
For two measures

(i) Let D be a domain and let
(ii) For any domain D the following comparison inequalities hold: where A = is the set of faces where σ 1 and σ 2 agree and A ≠ that where they disagree. By the ordering σ 1 ≤ σ 2 , we deduce that σ 1 is constantly on A ≠ while σ 2 is constantly on this set. Due to the positive association of ν D (. σ r = σ 1 on A = ) shown in Corollary 2.10, (ii) Let us begin with the first and last inequalities of (2.18). Let D ′ be a domain containing F (D) ∪ ∂ out D. By point (i) above, The Spatial Markov property (Corollary 2.4) translates the above to ν D ≤ st ν D . The first inequality of (2.18) is proved in the same way.
We move on to the middle inequality of (2.18). Considering (2.3) and the symmetry of blue spins, this inequality may be written as where the stochastic ordering refers only to the red-spin marginal. Clearly, the set ∂ in D is connected in H * , thus the inequality follows from Corollary 2.10 (i). ◻ Let D be a domain with vertices a, b, c, d on its boundary ∂ E D, arranged in counterclockwise order, and such that the edges incident to a, b, c and d all belong to ∂ E D or to E(D). Call (ab) the segment of ∂ E D between a and b, when going around D in the counter-clockwise direction. Define (bc), (cd) and (da) similarly.
Let µ a b c d a D be the uniform measure on pairs of coherent red and blue spin configurations on D with the property that σ r is equal to on all faces of ∂ in D adjacent to (ab) or (cd) and on all other faces of ∂ in D. The condition above also imposes that the blue spins of the two faces of ∂ in D that are adjacent to a are equal, and the same for the pairs of faces adjacent to b, c and d. Other than this, there is no restriction for the blue spins on ∂ in D. The marginal on red spins of the above is denoted by ν a b c d a . We say that a configuration τ r on D ′ ∖ Int(D) imposes boundary conditions a b c d a on D if all the faces adjacent to (ab) ∪ (cd) but not to (bc) ∪ (da) have spin in τ r and all those adjacent to (bc) ∪ (da) but not to (ab) ∪ (cd) have spin . The faces of ∂ in D ∪ ∂ out D that are adjacent to both (bc) ∪ (da) and (ab) ∪ (cd) may have spins or in τ r 2 (see Figure 5).
As already discussed in Remark 2.5, the spatial Markov property does not apply to the boundary conditions a b c d a. Indeed, the connectivity of the edges of θ(τ r ) that are adjacent to a, b, c and d influences the measure in D and is not determined by the spins on ∂ in D ∪ ∂ out D. It may be that certain configuration σ b are awarded positive probability in µ a b c d a D , but null probability in µ D ′ (. σ r = τ r on D ′ ∖ Int(D)). However, if we limit ourselves to the red-spin marginal, the Radon-Nikodim derivative of the second measure with respect to the first may be shown to be uniformly bounded.
, by which we mean that the former stochastically dominates the restriction to D of the latter.
(iii) Finally, if a ′ , b ′ , c ′ , d ′ ∈ ∂ E D are another set of four points with the same properties as a, b, c, d and such that (bc) Proof (i) Both measures in the statement are supported on configuration ς r ∈ { , } F (D) that agree with τ r on ∂ in D. By Proposition 2.7, for any such configuration ς r , ,Z are normalising constants and k D (θ(ς r ∪τ r )) is the number of connected components of θ(ς r ∪ τ r ) that intersect D. Indeed, the number of connected components that do not intersect D does not depend on ς r , hence cancel out.
Observe that θ(ς r ∪ τ r ) contains more connections than θ(ς r ), hence fewer connected components. However, in θ(ς r ), there are at most four distinct connected components that may be connected in θ(ς r ∪ τ r ). Thus, By summing the above over all configuration σ r in the support of ν a b c d a Inserting the last two inequalities in (2.19) provides the desired bound.
The FKG inequality implies the desired stochastic domination.
Corollary 2.12 (i) implies that the first measure dominates the second. ◻ 3 Infinite-volume measure: existence and uniqueness In this section we construct an infinite-volume Gibbs measure for the loop O(2) model. As for the Ising, Potts or FK models, the infinite-volume limit will be created as a limit of finite-volume measures. The existence of the limit rests on the monotonicity of the measures ν D in their boundary conditions. The same argument may be used to construct measures µ H , µ H and µ H as limits of finite-volume measures with the proper boundary conditions. Below we prove that these measures are all equal to a single measure µ H . We also show that the measures µ D converge to µ H as D increases to H.
For the double-spin representation, the theorem below may be understood as a partial uniqueness theorem; see Remark 3.3 for more on why it is not a complete uniqueness theorem. For Lipschitz functions, the theorem below amounts to non-quantitative delocalisation: it proves that the value at 0 is not tight as the domain increases to H, but does not offer the speed at which its variance increases.
For n ≥ 1, let Circ (n) be the event that there exists a simple closed path of edges of H surrounding Λ n with the property that the red spin of all faces adjacent to any of its edges is . The events Circ (n), Circ (n) and Circ (n) are defined similarly.
In particular µ H = µ H = µ H = µ H , and we will simply write µ H . Also, for any sequence of finite domains D n that increases to H, the measures µ Dn , µ Dn , µ Dn and µ Dn all converge to µ H . Remark 3.3. The theorem above states that any finite-volume measure with any red boundary condition converges to µ H . However, we do not claim this for mixed red and blue boundary conditions. It may be tempting to believe that, for any assignment of red and blue spins ξ r n , ξ b n on ∂ in D n , the measure µ Dn conditioned to have spins ξ r n and ξ b n on ∂ in D n also converge to µ H . Unfortunately this is not the case: counter examples may be created where the boundary conditions ξ r n , ξ b n force one single configuration inside the domain.
The rest of the section is dedicated to the proofs of the two theorems above. The RSW theorem developed in Section 3.2 will be of great use also in Section 4.

Infinite-volume measure for red marginal
We will work here only with the red-spin marginals ν of the measures µ. , where the above only refers to the restrictions of the measures to D. Thus, the sequence of measures (ν Dn ) n≥0 is decreasing, hence converges to a measure on { , } F (H) , which we denote by ν H . Since the limit exists for any sequence of domains, it necessarily is the same for any sequence of domains (D n ) n≥0 . In particular, the same limit is obtained for any sequence (Λ n +z) n≥1 with z ∈ V (H), which implies that ν H is invariant under translations.
That ν H is positively associated for increasing events depending only on the state of finitely many faces follows by passing to the limit. The property extends to arbitrary increasing events by the monotone class theorem (see [24,Prop. 4.10]).
In order to prove that ν H is ergodic, we will show that it has the following mixing property. The measure ν H is said to be mixing if, for any events A, B, if τ x (B) denotes the translation of B by some x ∈ V (H), The above implies that ν H is ergodic with respect to translations, as explained in [24,Cor. 4.23]. By the monotone class theorem, it suffices to prove (3.2) for events A and B that are increasing and only depend on finitely many faces. We do this below. Let A, B be increasing events depending only on the states of faces in some finite Conversely, positive association implies that These two inequalities and the fact that > 0 is arbitrary imply (3.2), and hence the ergodicity of ν H . ◻

Crossing estimates for double-plus percolation (weak version)
We will work in the rest of the paper with two percolation models derived from spin configurations. Let us describe them for a red spin configuration σ r on H; the definitions adapt readily to blue spins, to − instead of +, and to domains of H.
The first corresponds to connections via face-paths of spins . This percolation, along with its paths, clusters etc. will be referred to as simple-; connections between two sets of faces A and B are denoted by A ← → B. This notion was implicitly used in the proof of Theorem 2.6.
The second is termed double-percolation. The double-plus configuration dp(σ r ) ∈ {0, 1} E(H) associated to σ r is formed of the edges of H whose two adjacent faces have spins . We regard dp(σ r ) as a bond percolation on H and use the ensuing notion of connectivity. In particular, for sets A and B of vertices, we write A ← → B for the event that there exists an edge-path in dp(σ r ) with one endpoint in A and the other in B.
More generally, we call a double-path, or a double-path of spin , a path of edges in dp(σ r ). All notions related to this percolation (clusters, crossings, circuits etc.) will be referred to as double-. Thus, the event Circ (n) of Theorem 3.2 may be described as the existence of a double-circuit surrounding Λ n . The appeal of this second, more restrictive percolation model is that double-circuits isolate the inside from the outside in the sense of the Spatial Markov property (2.2). Since σ r is positively correlated under µ D , so is dp(σ r ). Indeed any event A which is increasing for dp(σ r ) is also increasing for σ r . A double-minus configuration dm(σ r ) ∈ {0, 1} E(H) is defined in a similar way and is also positively correlated under µ D . We want to stress however that the union dp(σ r ) ∪ dm(σ r ) is not necessarily positively correlated.
Write Par m,n for the set of faces of H with centres at k + e iπ 3 with 0 ≤ k ≤ m and 0 ≤ ≤ n (see Figure 6). They form a domain approximately shaped as a parallelogram. Its boundary ∂ E Par m,n may be partitioned into four sides called Bottom(m, n), Right(m, n), Top(m, n) and Left(m, n), defined as their name indicates. To be precise, Right(m, n) and Left(m, n) start and end with vertical edges. Below we will also use the notation Bottom(m, n), Right(m, n), Top(m, n) and Left(m, n) to refer to the faces of ∂ in Par m,n ∪ ∂ out Par m,n that are adjacent to these sections of ∂ E Par m,n . Faces in the corners of Par m,n belong to two such sets.
Write C h (m, n) (and C v (m, n)) for the event that there exits a face-path in Par m,n formed only of faces with spin , with the first face adjacent to the left side of Par m,n and the last face adjacent to the right side (and top and bottom sides, respectively). Call such face-paths horizontal (respectively, vertical) simple-crossings of Par m,n .
Write C h (m, n) (and C v (m, n)) for the event that there exits a path of edges of dp(σ r ) contained in Par m,n with one endpoint on the left side of Par m,n and the other on the right side (and top and bottom sides, respectively); for technical reasons, we ask that the endpoints of the paths not be corners of Par m,n . We call such paths horizontal (and vertical, respectively) double-crossings of Par m,n .
The Russo-Seymour-Welsh (or RSW for short) theory first appeared in the simultaneous works of Russo and Seymour and Welsh for Bernoulli percolation [36,38]. Its ultimate conclusion is that rectangles are crossed with probability bounded by constants that only depend on the rectangles' aspect-ratios, not their sizes. Such crossing probability bounds were obtained for Bernoulli percolation using two separate arguments: • a self-duality argument proves that the probability of crossing a square of any size is 1 2 (or more generally bounded uniformly away from 0); • the so-called RSW lemma proves that crossing a rectangle of aspect ratio 2 in the long direction is bounded by a function of the probability of crossing a square of (roughly) the same size.
The same two step procedure will be used below for the double-percolation. While for bond percolation on Z 2 with parameter 1 2 the first point is immediate due to self-duality, in our context a more complex argument is needed. The second point also requires special attention, due to the lack of independence and even of a general Spatial Markov property. A weak version of the RSW lemma is obtained easily using a general argument due to Tassion [42] (see Proposition 3.10 below). A more elaborate statement is proved later on (see Proposition 4.7); it requires considerable work.

Crossings of symmetric domains
This part contains results on crossing of symmetric domains; they are akin to the consequences of self-duality for site percolation on the triangular lattice or bond percolation on Z 2 . Two type of crossings will be treated: simple-crossings and double-crossings. We start with the former, where self-duality applies as for percolation. Proof Fix D, ζ and n. Drop n from the notation Par, Bottom and Top. First observe that by the monotonicity in boundary conditions (Corollary 2.12 (i)), the LHS of (3.3) is minimal when ζ ≡ on D ∖ (Bottom(n, n) ∪ Top(n, n)). We will assume this to be the case. All faces of Bottom(n, n)∪Top(n, n) have spin in ζ as required by the proposition; we will switch the sign of the two left-most faces of Top(n, n) to -this only decreases further the LHS of (3.3). For σ r a red spin configuration on Int(Par), write σ r ∪ ζ for the configuration on D obtained by completing σ r with ζ on D ∖ Int(Par). Write τ (σ r ) for the configuration obtained by applying the symmetry with respect to the line e iπ 6 R to −σ r . It is a known fact (see duality of site-percolation on T [28, Sec. 1.2]) that either σ r ∈ C v (n, n) or τ (σ r ) ∈ C v (n, n).
Recall from Proposition 2.7 that µ D (σ r ∪ ζ) is proportional to 2 k(θ(σr∪ζ)) . Also notice that θ(τ (σ r ) ∪ ζ) is easily determined in function of θ(σ r ∪ ζ). Indeed, the configuration ζ restricted to ∂ in Par ∪ ∂ out Par is (almost) invariant under τ . Thus, θ(τ (σ r ) ∪ ζ) restricted to Par ∪ ∂ out Par is the reflection with respect to the line e iπ 6 R of θ(σ r ∪ ζ). See Figure 6. with the configuration ζ on ∂ in Par ∪ ∂ out Par. In the left image no simple-+ crossing exists between the red arcs Top and Bottom. The right image is τ (σ r )∪ζ; it contains a simple-+ crossing between these arcs. Observe that θ(τ (σ r ) ∪ ζ) is the reflection of θ(σ r ∪ ζ) with respect to the diagonal of the rhombus. In this concrete example k[θ(σ r ∪ζ)]−k[θ(τ (σ r )∪ ζ)] = 1 because the clusters of Top and Bottom are linked together after the reflection but not before.
All other edges have same state in θ(τ (σ r ) ∪ ζ) and θ(σ r ∪ ζ): they are determined by ζ and are quite simple. Indeed, in θ(ζ), all faces except those in the corners of ∂ out Par are isolated points. Moreover, the top corners of ∂ out Par are connected, as are the bottom ones.
It follows that, for any By summing the above over σ r ∈ C v (n, n) we find . This proves the desired bound. ◻ Next we turn to double-crossings. The absence of such a crossing does not induce the existence of a double-crossing, and we may not apply the same argument as above. We do however have a similar statement. Lemma 3.6. For any m, n ≥ 1, and any pair of coherent configurations σ r ∈ { , } F (Parm,n) and σ b ∈ { , } F (Parm,n) , either Par m,n is crossed horizontally by a double path of constant red spin, or it is crossed vertically be a double path of constant blue spin. That is Proof Recall that dp(σ r ), dm(σ r ), dp(σ b ), dm(σ b ) ⊂ E(Par m,n ) denote the sets of double plus and double minus edges in σ r and σ b , respectively. Also, recall that to each edge e ∈ E(H) we associate its dual e * ∈ E(T) that is defined as the unique edge on T that intersects e. For a set S ⊂ E(H) we denote by S * ⊂ T the set of edges dual to the edges in S. By duality between H and T, either dp(σ r ) ∪ dm(σ r ) contains a left-right crossing of Par m,n , or [E(Par m,n ) ∖ (dp(σ r ) ∪ dm(σ r ))] * contains a top-bottom crossing of Par m,n .

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First consider the case when dp(σ r ) ∪ dm(σ r ) contains a left-right crossing of Par m,n . Any such crossing consists either entirely of edges of dp(σ r ) or entirely of edges of dm(σ r ). Indeed, edges of dp(σ r ) and dm(σ r ) can never share a vertex. In conclusion, in this case at least one of C h (m, n) and C h (m, n) occurs.
It remains to consider the case when [E(Par m,n ) ∖ (dp(σ r ) ∪ dm(σ r ))] * contains a top-bottom crossing of Par m,n . Let γ * be such a crossing.
For each edge e ∈ E(D), let N (e) ⊂ E(Par m,n ) denote the set of edges consisting of e and all edges in E(Par m,n ) that share a vertex with e. Then, if e * ∈ γ * , we claim that N (e) ⊂ dp(σ b )∪dm(σ b ). Indeed, the two faces of Par m,n separated by e have opposite red spin, hence same blue spin. Moreover, the blue spins of the two faces adjacent the endpoints of e but not containing e in their boundary must also coincide with the spins on either side of e.
It remains to observe that the union of N (e) taken over all e such that e * ∈ γ * contains a top-bottom crossing of Par m,n . Thus dp(σ b ) ∪ dm(σ b ) contains a top-bottom crossing of Par m,n , and thus either C v (m, n) or C v (m, n) occurs. ◻ Remark 3.7. It is obvious from the proof that Lemma 3.6 may be generalised to other domains with four arcs marked on the boundary. Later we will also use the fact that, if an annulus Λ N ∖ Λ n does not contain a circuit around Λ n of either double-or double-, then Λ n is connected to Λ c N by a double-path of constant blue spin. Proof By Lemma 3.6 we have In the equality we used the fact that the blue spin marginal of µ D is 1 2 (ν D + ν D ) and that µ D [C h (n, n)] = µ D [C h (n, n)] ; in the last line we used that ν D ≥ st ν D . This provides the desired result. ◻ Corollary 3.9. For any n ≥ 1 Proof For any domain D as in Lemma 3.8, by the monotonicity of boundary conditions, ν D C h (n, n) ≥ 1 4 .
By taking the limit of the above as D grows to H, we obtain the desired bound. As a consequence lim sup n→∞ ν H Circ (n, 2n) > 0.
The proof of Proposition 3.10 uses a technique introduced by Tassion in [42]. Indeed, the main argument of [42,Thm. 1] shows that lim inf n→∞ ν H (C h (n, n)) > 0 (which is the result of Corollary 3.9) implies (3.6). This technique applies to general percolation measures with the FKG property and sufficient symmetry. Our model fits in this framework and the relevant part of the proof of [42, Thm. 1] applies readily. We simply point out that, in order to harness the symmetries of the hexagonal lattice, one should apply the argument using crossings of hexagonal domains between opposite sides, rather than crossings of squares or lozenges.
Note that [42, Thm. 1] actually claims a stronger result than (3.6), where lim sup is replaced by lim inf. This improvement requires an additional ingredient which is lacking here. For now we are content with the above sub-sequential form of RSW.
A stronger statement (with the lower bound valid for all n) will be proved in Section 4 -see Proposition 4.7. All the ingredients for it are already available, however the proof is tedious and is not necessary at this point. The argument of [42] is elegant, short and quite robust, and suffices to prove Theorem 3.2; we prefer it for now.
Proof The argument of [42, Thm. 1] requires minor modifications because the hexagonal lattice in invariant under rotations of π 3, unlike the square one, which is invariant under rotations of π 2. We briefly sketch the adapted argument below.
Write T and B for the top and bottom horizontal sections of ∂Λ n , and let C v (Λ n ) be the event that T and B are connected to each other by a double-path contained in Λ n . From Corollary 3.9, using standard applications of the FKG inequality and the invariance of Λ n under rotations by multiples of π 3, we deduce that ν H C h (2n, n) is bounded away from 0 uniformly in n.
Following [42], define 2α n as the maximal width of a centred interval I on B such that Finally, since 0 ≤ α n ≤ n for all n, there exist infinitely many values of n such that α n ≤ 2α 3n 4 , and the proof is complete. ◻   Figure 7: The construction that proves that if α n ≤ 1 2 α 3n 4 , then Λ n ∪Λ n is crossed vertically with uniformly positive probability.
Proof Suppose the opposite, that is that with positive ν H -probability, 0 is surrounded by a finite number of disjoint double-circuits. Since ν H is ergodic and the above event is translation invariant, it occurs with probability 1.
Set N = min{n ≥ 1 ∶ Circ (n) does not occur}; observe that N is a random variable that is, by our assumption, ν H -a.s. finite. Then there exists n 0 such that since the right-hand side is strictly positive by Proposition 3.10. Using that, for all n > n 0 , we obtain a contradiction. ◻ Corollary 3.12. The graph θ(σ r ) contains ν H -a.s. no infinite cluster.
Proof Observe that a double--circuit in σ r blocks connections in θ(σ r ). More precisely, if σ r ∈ Circ (n), then all clusters of θ(σ r ) that intersect Λ n are finite. Now Corollary 3.11 states that ν H (Circ (n)) = 1 for all n, which implies by the observation above that θ(σ r ) contains no infinite cluster a.s. ◻

Joint infinite-volume measure
We now turn to the existence of limiting measures for the joint law of the red and blue spins, that is Theorem 3.1.
The crucial property here is that by Lemma 2.7, conditionally on σ r , σ b is obtained by colouring independently and uniformly the clusters of θ(σ r ) in either or . This procedure may also be applied in infinite-volume for red-spin configurations sampled according to ν H . The absence of infinite clusters in θ(σ r ) is used to show that the result of this procedure in a finite but large volume is close to that in infinite-volume.
Proof of Theorem 3.1 Let (D n ) n≥0 be an increasing sequence of domains with ⋃ n≥0 D n = H. Recall from Theorem 3.4 that the red-spin marginals of µ Dn converge to an ergodic translation-invariant limiting measure denoted by ν H . Let µ H be the measure obtained by sampling σ r according to ν H , then awarding to all faces of each cluster of θ(σ r ) a blue spin uniformly chosen in { , }, independently for each cluster. Let us prove that µ Dn converges to µ H .
Fix k ∈ N and > 0. We will show that the total-variation distance between the restrictions of µ Dn and µ H to Λ k is smaller than 2 , provided that n is large enough.
Let K ≥ k be such that ν H (Λ k θ(σr) ← → Λ c K ) < . Due to Corollary 3.12, it is always possible to choose K with this property. Now, let N = N ( , K) be such that, for any n ≥ N , the distance in total variation between the restrictions of ν Dn and ν H to Λ K is smaller than . Thus, one may couple ν Dn and ν H to produce configurations σ r , σ ′ r in such a way σ r = σ ′ r on Λ K with probability at least 1 − . Moreover, by choice of K, with probability at least 1 − 2 , σ r = σ ′ r on Λ K and there is no connected component of θ(σ r ) that intersects both Λ k and Λ c K . On this event, the connected components of θ(σ r ) and θ(σ ′ r ) that intersect Λ k are identical. Using the same blue spin assignment for these components, we have produced a coupling of µ Dn and µ H that is equal inside Λ k with probability at least 1 − 2 , which was our goal.
Since > 0 and k are arbitrary, we conclude that µ Dn converges to µ H . The translation invariance of µ H follows from that of ν H . Since µ Λn is invariant under rotations by multiples of π 3 and converges to µ H , the latter is also invariant under such rotations. The ergodicity of µ H follows from that of ν H and from the absence of infinite clusters in θ(σ r ). ◻ Proposition 3.13. Under µ H , σ b contains a.s. no infinite -cluster and no infinitecluster. As a consequence, ω b is formed entirely of finite loops µ H -a.s.
The proof below is a straightforward application of the uniqueness argument of Burton and Keane [5] and of Zhang's argument for non-coexistence of clusters (see [23,Lem 11.12] for an illustration of this argument which was never published by Zhang himself).
Proof To start, observe that under µ H the number N of infinite -clusters is a.s. constant. This is a direct consequence of the ergodicity of σ b under this measure. The same applies to infinite -clusters.
The technique introduced by Burton-Keane in [5] applies readily to the blue-spin marginal under µ H . Indeed, this marginal satisfies the finite-energy property required by [5]. As a consequence we obtain that either N = 0 µ H -a.s. or N = 1 µ H -a.s.
Finally, let us prove that N = 0 µ H -a.s. by contradiction. Assume that N = 1 µ Ha.s. Then, by the symmetry of the blue-spin marginal, the number of -infinite clusters is also equal to 1 a.s. Thus, there exists some n ≥ 1 such that µ H (Λ n ← → ∞) > 1 − 1 4 6 . Write ∂ 1 Λ n , . . . , ∂ 6 Λ n for the six sides of ∂Λ n in counter-clockwise order. Then The inequality is due to the FKG property for σ b and to the invariance of the measure under rotations by π 3. Thus we find

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The same holds for any side of Λ n and also for -connections instead of ones.

Define the event
Using the union bound, we find Now notice that when the above event occurs, then necessarily either there exist two infinite -clusters or two infinite -clusters. This contradicts the uniqueness of the infinite cluster proved above.
Finally, the existence of an infinite path in ω b implies the existence of both infinite and clusters, which was excluded above. ◻

Uniqueness of infinite-volume measure: proof of Theorem 3.2
Proof of Theorem 3.2 Let us first prove that µ H -a.s., there exist infinitely many loops surrounding the origin in the loop representation of (σ r , σ b ). To that end, it is enough to show that for any n, µ H -a.s. there exists at least one loop surrounding Λ n . Fix n and consider the union of all -and -clusters that intersect Λ n . Due to Proposition 3.13, all these clusters are finite. The outer boundary of their union is then a finite blue loop surrounding Λ n . Hence, 0 is surrounded a.s. by infinitely many loops. Let us now prove (3.1). Fix n. By the above, µ H -a.s. there exist infinitely many loops surrounding Λ n which may be ordered starting from the inner most. Since each loop is blue or red with probability 1 2 independently, there exist a.s. four consecutive loops γ 1 , . . . , γ 4 surrounding Λ n that have colours red, blue, red, blue, in this order, from inside out. Then both γ 1 and γ 3 have constant blue spins on all faces adjacent to them, but that for γ 1 is opposite to that for γ 3 . That is, either γ 1 is double-and γ 3 is doubleor γ 1 is double-and γ 3 is double-. Similarly, of γ 2 and γ 4 , one is double-and the other is double-. This proves (3.1).
A direct consequence of µ H (Circ (n)) = 1 is that the restriction of ν H to Λ n is dominated by ν H . Thus ν H = ν H . Moreover, due to the monotonicity of boundary conditions, for any sequence of finite domains D n that increases to H, the measures ν Dn and ν Dn , as well as the red-spin marginals of µ Dn , µ Dn , µ Dn and µ Dn all converge to ν H .
Finally, due to the procedure that selects blue spins knowing the red spins, we conclude that µ ξ

A dichotomy theorem
Below we state a dichotomy result similar to those of [17] and [18]. The result states that the model is in one of two states: co-existence of phases (see case (i) of Corollary 4.2) or (stretched)-exponential decay of diameters for clusters of one phase inside the other (see case (ii)). In Section 5, we show that the latter case contradicts Theorem 3.2.
Compared to the setting of [17] and [18], the present model exhibits considerable additional difficulties due to the lack of a general Spatial Markov property and to the absence of monotonicity in the boundary conditions . These difficulties appear in several places, most notably in proving a crossing estimate inside mixed boundary conditions (Corollary 4.14) and in eliminating case (ii). Theorem 4.1. There exist constants ρ > 2 and C > 1 such that, if for n ≥ 1 we set α n = µ Λρn [Circ (n, 2n)], we have α (ρ+2)n ≤ Cα 2 n for all n ≥ 1. (4.1) Corollary 4.2. For ρ > 2 given by the above, one of the two following statements holds (i) inf n α n > 0 or (ii) there exist constants c, C > 0 and n 0 ≥ 1 such that α n ≤ Ce −n c for all n = (ρ + 2) k n 0 with k ∈ N.
The constant ρ in Theorem 4.1 will be chosen large enough to accommodate certain geometric constructions used in the proof. Its choice only affects scenario (ii) of Corollary 4.2, which we will see is contradictory. Thus, any value of ρ suffices for our purposes. It is immediate that the Spatial Markov property applies to µ Cyl m,n in the same way as for planar domains. In particular, µ Rectm,n is related to µ Cyl m,n by the following. While Cyl m,n is not a planar domain, the FKG inequality applies to it.  Proof We will not give a full proof of this, only a sketch. Notice that Cyl m,n ∪∂ out Cyl m,n , may be embedded in the plane and rendered simply connected by adding a face of degree 4m below the bottom of ∂ out Cyl m,n , as drawn in Figure 8, right diagram. Write D for the planar graph obtained by this procedure. Then a straightforward adaptation of Theorem 2.6 (see also remark 2.11 (ii)) shows that the FKG lattice condition also holds for µ D .

Preparation: measure in cylinder
As explained in Corollary 2.10, conditioning on the value of red spins on a given set conserves the FKG lattice condition for the red spin marginal. In particular, the red-spin marginal of µ D conditioned on the event that all faces adjacent to the top of Cyl m,n have σ r ≡ while all those adjacent to the bottom have σ r ≡ also satisfies the FKG lattice condition. Finally, the Spatial Markov property states that the conditional measure above is identical to µ Cyl m,n (when restricted to Cyl m,n ). ◻

Strong RSW theory
As promised in Section 3.2.2, we will now prove a stronger RSW result. Variations of it may be envisioned; we will state it in the form most useful to us. We start with a general lemma that allows to lengthen crossings of long rectangles. The main result of the section (Proposition 4.7) is given afterwards. It will be stated and proved for the cylinder, but may also be deduced in other settings.

Lengthening crossings
Recall the definition of Par m,n and its boundary segments Top and Bottom from Section 3.2.  The previous lemma may be used to glue crossings of long rectangles as described below. where (n, 0) + Par 4n,n is the translate of Par 4n,n by n units to the right.
Due to the Spatial Markov property, the statements of Lemma 4.5 and Corollary 4.6 also apply to measures with boundary conditions such as µ D .
Observe that Par 3n,n may be partitioned into three translations of Par n,n ; call them R 1 , R 2 and R 3 ordered from left to right (see Figure 9). Due to Lemma 3.5 and the FKG inequality, If C v (R 1 ) occurs, let Γ L be the left-most simple-vertical crossing of R 1 . Formally, let Γ L be the edge-path running along the left side of the spin-crossing. Also, let Γ R be the right-most simple-vertical crossing of R 3 when C v (R 3 ) occurs.
Fix two possible realisations γ L , γ R of Γ L , Γ R . Let D 0 be the domain formed of the faces of Rect 3n,n between γ L and γ R . The events Γ L = γ L and Γ R = γ R are both measurable in terms of the configuration outside D 0 .
Consider the line running through the bottom left and upper right corner of R 2 and let ρ be the orthogonal symmetry with respect to ; note that H is invariant under ρ. Let D be the domain formed of the faces of D 0 and those of ρ(D 0 ).
Let a, b, c, d be the corners of R 2 ordered in counter-clockwise order, starting from the top-left corner. Write (bc) and (da) for the arcs of ∂D in counter-clockwise order. Then, due to the monotonicity in boundary conditions and the FKG inequality, The equality is due to the specific to the boundary conditions induced by Γ L , Γ R and ζ on D 0 (a brief analysis is needed to ensure that the is no multiplicative constant appearing between the two sides). The first inequality is due to the FKG inequality and the inclusion of events; the last one is due to Lemma 3.8. Averaging the above over all possible values of Γ L and Γ R and using (4.4), we obtain the desired bound. ◻ Proof of Corollary 4.6 Let Γ + L and Γ − L be the top and bottom most, respectively, double-horizontal crossings of Par 4n,n . Define Γ + R and Γ − R similarly for the rectangle (n, 0) + Par 4n,n . When both Par 4n,n and (n, 0) + Par 4n,n are crossed horizontally by double-paths, then either Γ + L intersects or is higher than Γ − R inside the middle parallelogram (0, n) + Par 3n,n , or Γ − L intersects or is lower than Γ + R . As a consequence, In the last inequality we used the FKG property. We focus next on the first term in the LHS above. For any realisation of Γ + L and Γ − R with the former intersecting or higher than the latter, if Γ + L and Γ − R intersect or if they are connected to each other by a double-path, then µ D (C h (5n, n)) occurs. Below we will show that, conditionally on Γ + L and Γ − R , the two paths intersect or are connected by a double-path with positive probability. The case where the paths intersect is trivial; we assume henceforth that Γ + L and Γ − R are disjoint. Notice that Γ + L is measurable in terms of the spins of the faces of Par 4n,n above it, and Γ − R is measurable in terms of the spins of the faces of (n, 0) + Par 4n,n below it. Let U be the set of all faces of Par 5n,n which are in neither of the two categories above. Let a be the right endpoint of Γ + L and c be the left endpoint of Γ − R . Orient Γ + L from left to right and Γ − R from right to left. Let b be the last point of intersection of Γ + L with the left side of (n, 0) + Par 3n,n , and d be the last point of intersection of Γ − R with the right side of (n, 0) + Par 3n,n . Write D for the domain contained in (n, 0) + Par 3n,n , delimited at the top by the section of Γ + L between b and a, and at the bottom by the section of Γ − R between d and c.
Let H be the event that there exists an edge-path γ contained in D, connecting the arcs (ab) and (cd) of ∂ E D, such that all faces of U ∩ D adjacent to it are of red spin . Then, by the FKG inequality and the properties (i) and (ii) of Lemma 2.13, The second inequality is due to the inclusion between the two events. Notice also that when H occurs, Γ + L , Γ − R are necessarily connected by a double-path. Figure 10: Left: the paths Γ + L and Γ − R are measurable in terms of the hashed regions; its complement is U. Right: The domain D is delimited by parts of Γ + L and Γ − R . Any crossing between the arcs (ab) and (cd) in D induces a connection between Γ + L and Γ − R .
Now, applying again Lemma 2.13, we conclude that where a ′ , b ′ , c ′ and d ′ are the corners of the parallelogram, ordered in counter-clockwise order, starting from the top left. The RHS of the above is bounded below by 1 36, as proved in Lemma 4.5. Combining the last two displayed equations, we find Averaging over all values of Γ + L and Γ − R as above, we find µ D (C h (5n, n)) ≥ 1 288 µ D (Γ + L intersects or higher than Γ − R ). The same bound holds for the second term in (4.5), and (4.3) follows.
In other words, the above tells us that if wide rectangles are crossed vertically with positive probability (that is in the easy direction), then they are also crossed horizontally (i.e. in the hard direction) with positive probability. This is a typical RSW result in that it relates probabilities of crossings in the easy direction to those of crossings in the hard direction. What is remarkable is that the measure to which it applies, namely µ Cyl M,N , is not rotationally invariant. Thus, vertical crossings are "orthogonal" to horizontal ones. The exact aspect ratio of the two rectangles (4 and 6, respectively) is not essential; they have been chosen to simplify statements later on.
The proof of Proposition 4.7 is quite intricate. It is based on similar results from [18], but with additional difficulties due to the two layers of spins necessary for the Spatial Markov property (see Theorem 2.3). A very similar version appears in [14]. The next section is dedicated to proving Proposition 4.7.

Proof of Proposition 4.7
The structure is reminiscent of a proof by contradiction: assuming that (4.6) fails (more precisely that C h ([−2n, 2n] × [k, k + n]) has probability below a certain threshold), we prove that double-vertical crossings of [−3n, 3n] × [k, k + n] have some particular behaviour. This is done in a series of lemmas (Lemmas 4.8, 4.9, 4.10 and 4.11) -the title of each lemma indicates a constraint that vertical crossings need to satisfy for (4.6) to fail. Then we prove that two typical vertical crossings of [−3n, 3n] × [k, k + n] with starting points sufficiently close to each other are connected with positive probability by a doublepath (see Lemma 4.12). Proposition 4.7 follows from this last statement. Lemma 4.12 is the heart of the proof; it relies on the construction of a symmetric domain, similarly to what was done for Lemma 4.5.
Fix n, k, M, N as in the proposition. We will work with n large; the function ψ in the proposition may be adjusted to incorporate all small values of n. For ease of writing, translate the cylinder Cyl M,N vertically by −k; write µ for the measure with boundary conditions on this translated cylinder. Using this notation, our goal is to prove that, for any C > 0 there exists ∆ > 0 depending only on C, not on n, k, M or N , such that Below we will talk about double-paths contained in Strip h with endpoints in Mid 0 (m) and Mid h (m), respectively. While not explicitly stating it each time, we will always ask that such a path have trivial winding around the cylinder. For j ∈ Z, let M j = (2j m, 0) + Mid 0 (m ). Using again the invariance of µ under horizontal shift, we find that Finally, if all the events above occur simultaneously, then C h (Rect 2m,m ) also occurs. By the FKG inequality, we find Then, for any C wig > 0 there exists ∆ wig > 0 such that, for any m ≤ n, Rect 6ρ in m,m adaptations to the proof need to be made; we leave this to the reader. See also Figure 11 -right diagram.) Using this and the lower bound on the probability of A , we find The above is akin to (4.8). We conclude as in the proof of Lemma 4.8 that for some sufficiently small constant ∆ wig > 0 depending only on C wig and ρ in . ◻ The last two lemmas will be used for different scales m ≤ n. The following will only be used at scale n. To simplify notation, we only state it at this scale.
Then, for any constant C loc > 0 there exists ∆ loc = ∆ loc ( , C loc ) > 0 such that Notice that G loc ( , k) contains all configurations with no double-path connecting Mid 0 ( n) to Mid n ( n) inside Strip n . Indeed, the condition is trivially satisfied.
Proof Fix ≤ ρ in and C loc > 0; to simplify notation we will consider ρ in to be an integer. Assume that µ[G loc ( , k)] < 1 − C loc for all k ≥ ρ in .
For k ≥ 1, let E k be the event that there exists a double-path in Strip n with endpoints in Mid 0 ( n) and Mid n ( n) and which intersects L v (k n). Also write E 0 for the event that Mid 0 ( n) and Mid n ( n) are connected by a double-path contained in Strip n , with no other restriction. The events E k are increasing each, but form a decreasing sequence.
Additionally, define A k as the event that there exists a double-path in Strip n that connects Mid 0 ( n) to Mid n ( n) and which does not intersect L v (k n). Then each A k is an increasing event and the sequence A k is increasing in k. Moreover G loc ( , k) = (A k ∪E k+5 ) c .
Notice that E c 0 is contained in all events G loc ( , k), hence µ(E 0 ) ≥ C loc . Set k to be the smallest index such that µ(A k ) ≥ C loc 2. The existence of k is guaranteed by the fact that lim j µ(A j ) = µ(E 0 ) ≥ C loc .
Suppose first that k ≤ ρ in . Then, H in wig (n) ≥ µ(A k ) ≥ C loc 2, and Lemma 4.9 shows that µ[C h (Rect 2n,n )] is bounded below by some constant that only depends on C loc .
Henceforth we assume that k > ρ in . Then, due to our initial assumption, which implies µ(E k+4 ) > C loc 2. WriteÃ k for the horizontal shift of the event A k by 4 n. By the choice of k, we have µ(Ã k ) = µ(A k ) ≥ C loc 2. Using the FKG inequality, we find Notice now that, if bothÃ k and E k+4 occur, then the paths in the definition of these two events necessarily intersect. In conclusion As in the proof of Lemma 4.8, this implies that µ[C h (Rect 2n,n )] is larger than some threshold depending only on and C loc , and the lemma is proved. ◻ In the proof of Proposition 4.7 we will work with two scales: the scale n and a lower scale m chosen below. Moreover, the endpoints of the vertical paths will be fixed in some segment of length 2 n where > 0 is also chosen below.
Fix m = ρ in 2ρout n. Then, fix some > 0 so that n < 1 2 ρ in m and ρ out m < ρ in n − c loc n. (4.11) In conclusion, the scales n, m and n are fixed so that n is much smaller than m, which in turn is much smaller than n. All constants below depend on the ratios between these scales. Write Γ L and Γ R for the left-and right-most, respectively, double-paths contained in Strip n , with lower endpoint on Mid 0 ( n) and top endpoint in Mid n ( n) (recall that these are paths formed of edges of the hexagonal lattice). If no such crossings exists, set Γ L = Γ R = ∅. We will always orient such paths from their endpoint on L h (0) towards that on L h (n).
When Γ L and Γ R exist and are disjoint, write Int(Γ L , Γ R ) for the domain with boundary formed of the concatenation of Γ R , the segment of L h (n) between the top endpoints of Γ R and Γ L (from right to left), Γ L (in reverse), and the segment of L h (0) between the bottom endpoints of Γ L and Γ R (from left to right). Also let Ext(Γ L , Γ R ) be the set of faces of Strip n which are not strictly inside Int(Γ L , Γ R ); precisely, Ext(Γ L , Γ R ) contains all faces of Strip n ∖ Int(Γ L , Γ R ) as well as all faces adjacent to Γ L or Γ R .
It is standard that Γ L and Γ R may be explored from their left and right, respectively. That is, for γ L and γ R two possible realisations of Γ L and Γ R , respectively, the event {Γ L = γ L and Γ R = γ R } is measurable with respect to the state of faces in Ext(γ L , γ R ).

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Our next goal is to show that, whenever Γ L and Γ R exist and behave reasonably well, they have a positive probability to be connected inside Int(Γ L , Γ R ) by a path of double-. The notion of well-behaved vertical paths is defined below.
For an edge-path Γ contained in Strip n , with starting point in Mid 0 ( n) and endpoint in Mid n ( n), let Γ b be the segment of Γ contained between its starting-point and its first visit of L h (m). Let Γ t be the segment of Γ between its last visit of L h (n − m) and its endpoint.
Define G wb (k) as the event that any double-path Γ contained in Strip n , with starting point in Mid 0 ( n) and endpoint in Mid n ( n) is well-behaved (for this value of k), that is (i) Γ b has one endpoint in Mid m (2 n) and Γ t has one endpoint in Mid n−m (2 n); (ii) Γ b and Γ t are both contained in Rect(ρ out m, n) but each crosses L v (ρ in m), In order to apply our reasoning, we will ask that Γ L and Γ R are well-behaved, that is, we will ask that G wb (k) occurs for some k. This is guaranteed by the following result.
Proof Fix C wb > 0 and assume µ[G wb (k)] < 1 − C wb for all k ≥ ρ in . Then, one out of the three conditions defining G wb (k) fails with probability at least C wb 3 for every k.
Thus, at least one of the following cases occurs: • (i) fails with probability at least C wb 3 for some k. Then Lemma 4.8 states that µ[C h (Rect 2m,m )] > ∆ mid for some ∆ mid > 0 depending only on C wb . Using the horizontal translation invariance of µ, the same bound applies to any horizontal translate Rect 2m,m + j(m, 0) of Rect 2m,m , with j ∈ Z. Using this and Corollary 4.6, we find for some universal constant c > 0.
• (iii) fails with probability at least C wb 3 for all k. Then, by Lemma 4.10 applied with C loc = C wb 3, we deduce µ[C h (Rect 2n,n )] > ∆ loc .
We conclude that in all cases, µ[C h (Rect 2n,n )] is bounded below by a constant depending only on C wb , as required. ◻ Now that we proved that some G wb (k) occurs with high probability, we will show that, when it does occur, Γ R and Γ L connect to each other. Since Γ R and Γ L are measurable in terms of the spins in Ext(Γ L , Γ R ), the same applies to G wb (k). Indeed, if Γ L and Γ R satisfy the conditions of G wb (k), then so do all double-paths contained in Strip n , from Mid 0 ( n) to Mid n ( n). Lemma 4.12. There exists some universal constant C cnt > 0 such that, for any possible realisations γ L , γ R of Γ L , Γ R with the property that G wb (k) occurs for some k ≥ ρ in and any red spin configuration ζ such that Γ L = γ L and The conditioning in (4.13) may be reduced simply to {σ r = ζ on Ext(γ L , γ R )}, since this determines Γ L = γ L , Γ R = γ R , which in turn implies that G wb (k) occurs. We included the latter conditions in (4.13) to emphasise their importance.
The lemma above is the heart of the proof of Theorem 4.7.
Proof Fix γ L , γ R and ζ as in the statement. Let k ≥ ρ in be some value for which G wb (k) occurs. We may assume that γ L and γ R are disjoint, otherwise the conclusion is trivially attained. We will proceed in two steps, first we will create simple-connections between γ L and γ R , close to the top and bottom of Strip n , respectively. In a second stage, we connect γ L and γ R by a double-path contained between the two simple-paths shown to exist in the previous step.
Recall that γ L and γ R are oriented from bottom to top. Let R 1 = Rect(ρ out m, m) and R 2 = R 1 + (0, n − m) be the vertical translation of R 1 contained between L h (n − m) and L h (n). Due to G wb (k) occuring, γ b L and γ b R are contained in R 1 , while γ t L and γ t R are contained in R 2 .
Step 1: Simple-crossings. Let I be the event that γ L and γ R are connected by two simple-paths contained in R 1 and R 2 , respectively. We will now prove that I has positive probability, uniformly in m, n, γ L , γ R and ζ. We do this for the connection in R 1 ; the same argument applies in R 2 . The argument used in this step is exactly that of [18]. Figure 12 contains an illustration of the construction below.
Recall that both γ b L and γ b R intersect L v (ρ in m) but that their endpoints are in Mid 0 ( n) and Mid m (2 n), hence to the left of L v (ρ in m). Let A be the first point where γ L intersects L v (ρ in m) and write γ ′ L for the subpath of γ L from its starting point up to A. Then γ b R contains at least one subpath contained in the part of R 1 to the right of L v (ρ in m), which has both endpoints on L v (ρ in m), one below A and one above A (this is because γ b R has both its endpoints to the left of L v (ρ in m)). Write γ ′ R for the left-most such path and let C be the endpoint of γ ′ R above A. Write τ for the reflection with respect to L v (ρ in m) (actually, with respect to the vertical axis {⌊ρ in m⌋} × R). Then τ leaves the lattice invariant.
Observe now that τ (γ ′ L ) intersects γ ′ R . Indeed, τ (γ ′ L ) runs from A to L h (0) and is contained in the region to the right of L v (ρ in m), while γ ′ R separates A from L h (0) in this same region. Let B be the first intersection point of τ (γ ′ L ) with γ ′ R when starting from A and let γ A be the subpath of τ (γ ′ L ) between A and B. Let γ B be the subpath of γ ′ R between B and C. Finally set γ C = τ (γ B ), γ D = τ (γ A ) and D = τ (B). The paths γ A , γ B , γ C and γ D only intersect at their endpoints and their concatenation bounds a domain which we call D.
Let us derive a bound on the crossing probability of D, independently of how D was formed. Consider the red spin configuration ξ on Cyl consisting only of with the exception of the faces adjacent to γ B and those adjacent to γ D , which have spin . By the same reasoning as in Lemma 3.5 and due to the invariance of D under τ , we obtain Figure 12: The paths γ b L and γ b R are drawn in solid lines; the thicker parts are γ ′ L and γ ′ R . The reflections of parts of γ ′ L and γ ′ R are in dashed lines. The domain D is shaded. Observe that any crossing from γ B to γ D in D induces a connection between γ b L and γ b R Write µ ξ D for the conditional measure above. By Corollary 2.12 and due to the condition Γ L = γ L and Γ R = γ R , the measure µ[. σ r = ζ on Ext(γ L , γ R )] restricted to D ∩Int(γ L , γ R ) dominates the restriction of µ ξ D to this same set of faces.
Set A to be the event that there exists a face-path χ in D, with the first and last faces adjacent to γ B and γ D , respectively, and such that all faces of χ that are contained in Int(γ L , γ R ) have spin . Then Now observe that a path χ as in the definition of A necessarily contains a subpath contained in Int(γ L , γ R ) with the first and last faces adjacent to γ L and γ R , respectively. We conclude that Using the same argument in R 2 and the FKG inequality, we obtain µ I σ r = ζ on Ext(γ L , γ R ) ≥ 1 9 . (4.14) Step 2: Double-crossing. We will now prove that The procedure is similar to that of Step 1, but at scale n rather than m and with some additional difficulties. We recommend that the reader inspects Figure 13, which contains the strategy of the proof as well as the relevant notation. When I occurs, we will denote by Ξ 1 and Ξ 2 be the lowest and highest, respectively, paths of simple-from γ L to γ R , contained in Int(γ L , γ R ). More precisely, define Ξ 1 to be the lowest edge-path contained in Int(γ L , γ R ), with endpoints on γ L and γ R , respectively, with the property that all faces above it have spin . Define Ξ 2 similarly, only that it is highest and that all faces below it are required to have spin . By the definition of I . Bottom right: a deformation of D allows to embed it in the plane; it contains Int, whose deformation is shaded. Any crossing from (AB) to (CD) contains a path from γ L to γ R in Int. and the first condition of G (k), Ξ 1 and Ξ 2 are contained in R 1 and R 2 , respectively, whenever I occurs.
Let χ 1 and χ 2 be possible realisations of Ξ 1 and Ξ 2 , respectively, such that I occurs. Define the domain Int = Int(γ L , γ R , χ 1 , χ 2 ) as the set of faces delimited by these four paths. Also let Ext = Ext(γ L , γ R , χ 1 , χ 2 ) be the set of faces outside Int along with those of ∂ in Int. By a standard exploration argument, the event {Ξ 1 = χ 1 , Ξ 2 = χ 2 } is measurable with respect to the spins on Ext. Fix a red spin configuration ξ on Ext(γ L , γ R , χ 1 , χ 2 ) with ξ = ζ on Ext(γ L , γ R ) and such that Ξ 1 = χ 1 , Ξ 2 = χ 2 . This implies in particular that all faces of ∂ in Int have spin in ξ.
The line L v (k n) separates χ 1 from χ 2 inside the simply connected domain Int. It follows that there exists at least one segment of L v (k n) that is fully contained in Int and that separates χ 1 from χ 2 inside this domain. Indeed, L v (k n) needs to intersect both γ L to γ R in order to separate χ 1 from χ 2 . Consider the intersections of L v (k n) with γ L and γ R in increasing vertical order; there necessarily exists one intersection with γ R followed by one with γ L . The segment of L v (k n) between these two intersections has the desired property.
Let [A, C] be the first such segment when going from χ 1 to χ 2 , where A denotes its higher endpoint (the segments with this property are naturally ordered, for instance by their end-points on Γ L ). Then A is a point of γ L while C is a point of γ R . Write γ 1 L and γ 2 L for the subpaths of γ L from the intersection with χ 1 to A and from A to the intersection with χ 2 , respectively. The same notation applies to γ R . Then [A, C] separates Int into two sub-domains. The first, which we call Int 1 , has boundary formed of χ 1 , γ 1 L , [A, C] and γ 1 R . The boundary of the second, called Int 2 , is the concatenation of χ 2 , γ 2 R , [A, C] and γ 2 L . Let τ be the reflection with respect to the vertical axis L v (k n). Now define D 1 as the union of the sets of faces of Int 1 and τ (Int 2 ). Then D 1 is itself a domain, whose boundary consists of χ 1 , τ (χ 2 ), [A, C] and pieces of γ 1 L , γ 1 R , τ (γ 2 L ) and τ (γ 2 R ). It is particularly important that χ 1 is fully part of the boundary of D 1 . This is because τ (Int 2 ) lies entirely to the right of L v ((k − c loc ) n), and thus does not intersect R 1 (property (iii) of wellbehaved paths, see also (4.11)). For similar reasons, τ (χ 2 ) is also fully contained in the boundary of D 1 .
Let Observe now that D contains Int. Moreover, any path crossing from (AB) to (CD) in D contains a subpath which is contained in Int and which has endpoints on γ L and γ R , respectively. Indeed, the segment (AB) is above γ 1 L , while (CD) is below γ 2 R . Finally, we claim that the restriction of µ(. σ r = ξ on Ext) to Int dominates that of µ D . We start off with a heuristic explanation. The key to this argument is to observe that D may be obtained from Int by "pushing away" parts of the boundary of D, but that these only belong to γ L and γ R , not to χ 1 or χ 2 . Since these are double-paths in ξ, the monotonicity of boundary conditions applies, and we may conclude.
Let us now present a rigorous proof of this domination with a slightly weaker conclusion. As already explained, D is part of two copies of H glued along the segment [A, C]. Let D ′ be a planar domain of this graph that contains D along with all faces adjacent to it. Then, due to the Spatial Markov property that also applies in this slightly different setting, µ D is the restriction to D of µ D ′ (. σ r ≡ on D ′ ∖ D and σ r ≡ on ∂ in D).
Since D ′ is planar, the FKG inequality holds for µ D ′ . Let A − be the set of faces of ∂ out Int adjacent to χ 1 or χ 2 . Let A + be all the other faces of D ′ ∖ Int along with ∂ in Int. Then, by the monotonicity of boundary conditions (Corollary 2.12 (i)) and the consideration above, the restriction of µ D to Int is dominated by that of µ D ′ (. σ r ≡ on A + and σ r ≡ on A − ). Moreover, the latter is equal to the restriction of µ(. σ r ≡ on Ext ∖ A − and σ r ≡ on A − ) to Int. (Here A − is also viewed as a subset of Cyl.) In conclusion Conclusion. Equations (4.14) and (4.15) imply that We are finally ready to prove the main result of the section, namely Proposition 4.7.
Proof of Proposition 4.7 Recall from (4.11) that m and are fixed.
The bottom boundary of Rect 3n,n may be partitioned into 18 segments of length n 3. At least one of these segments is connected inside Strip n by a double-path to L v (n) with probability at least C v 18. Since the measure µ is translation invariant, Let ∆ mid be given by Lemma 4.8 with C mid = Cv 36 and 6 instead of . If µ[C h (Rect 2n,n )] > ∆ mid the proof is complete. We will therefore assume that µ[H mid (m)] < C mid , which along with (4.17) implies For j ∈ Z, write M b j = (j 2 3 n, 0) + Mid 0 ( n 3) and M t j = (j 2 3 n, 0) + Mid n ( n 3). Let J be the event that M b j is connected to M t j by a double-path inside Strip n for both j = −1 and j = 1. Using again the translation invariance of µ, the FKG inequality and (4.18), we find Applying now Lemma 4.12, we find When J occurs, the endpoints of Γ L are contained in M b −1 and M t −1 , respectively, while those of Γ R are in M b 1 and M t 1 . Thus, when all three events above occur simultaneously, M b −1 and M b 1 are connected inside Strip n by a path of double-. We conclude that We conclude in the same way as in the proof of Lemma 4.8: the lower bound above Using the FKG inequality, the intersection of all these translations occurs with probability at least (C cnt ⋅ C wb ) 3 +1 . When all the events above occur, Rect 2n,n contains a double-horizontal crossing. Thus Since is a universal constant and C cnt and C wb only depend on C v , the above provides the desired bound. ◻

Crossing rectangles in mixed boundary conditions
We give two statements that are crucial in the proof of Theorem 4.1. They are crossing probability estimates similar to those of Proposition 3.10. What is essential here is that they are in finite domains with mixed boundary conditions. Proposition 4.13. For C ≥ 3 there exists δ = δ(C) > 0 such that, for all n ≥ 1, Corollary 4.14. For all C h ≥ 3 and C v ≥ 1 there exists δ = δ(C h , C v ) > 0 such that, for all n ≥ 1, Corollary 4.14 is referred to in [18] as the "pushing" lemma; it is an essential result in establishing the dichotomy of Corollary 4.2.
The results stated above mimic the structure of the original RSW theory: the first result serves as an input (such as self-duality in critical bond percolation on Z 2 or as Corollary 3.9 for the weaker RSW statement of Proposition 3.10), the second states that horizontal crossings may be extended to longer rectangles. The latter follows from the former in a fairly standard way using Proposition 4.7. For clarity, we will avoid using Proposition 4.7 in the proof of Proposition 4.13. We start with the proof of the proposition; the proof of the corollary may be found at the end of the section. The probability that R is crossed (horizontally or vertically) by a double red-plus path is higher in the right image than in the middle.
Proof of Proposition 4.13 Fix C ≥ 3. We will proceed by contradiction and will assume that for some constant δ > 0 that we will choose later. It will be obvious that the choice of δ only depends on C.
Write Cyl for Cyl Cn,5n and µ = µ Cyl . The cylinder is split into five strips of height n: The proof of the proposition is based on two claims that we state and prove below. The whole argument is summarised in Figure 14. Proof The same argument as in the proof of Lemma 3.8 shows that either Strip 2 is crossed vertically by double-path of constant red spins, or it contains a horizontal doublecircuit (winding around the cylinder) of constant blue spins. Thus where Circ is defined similarly to Circ and C v (Strip 2 ) is the event that Strip 2 contains a path of double-with one endpoint on its bottom and one on its top. If C v (Strip 2 ) occurs, then at least one of the rectangles [kn, (k + 6)n] × [2n, 3n] with −C ≤ k < C is crossed vertically by a double-path, or one of the rectangles [kn, (k + 4)n] × [2n, 3n] with −C ≤ k < C is crossed horizontally by a double-path. Due to our assumptions (4.21) and to the monotonicity with respect to boundary conditions, all of the crossing events above occur with probabilities at most δ. Thus,

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The same argument applies to double-crossings and we find For this second inequality, the monotonicity of boundary conditions was not used, but rather the invariance of µ under vertical reflection composed with red spin flip.
Assume now that δ ≤ 1 16C . Then the first two terms of (4.22) sum up to at least 1 2. Moreover, µ is invariant under blue spin flip, hence these two terms are equal. In conclusion, each is larger than 1 4. ◻ Proof If Circ (Strip 2 ) occurs, let Γ be the lowest circuit as in its definition. Let γ be a possible realisation for Γ and let γ be the reflection of γ with respect to the horizontal line R × {7n 2}. Then γ lies entirely above R × {5n}, hence above the top of Cyl. It will be useful to view γ and γ as drawn on the infinite vertical cylinder Cyl ∞ ∶= (R (2Cn + 4)Z) × R, of whom Cyl is a subset. Let Cyl be the cylinder contained between γ and γ and let Top be the top boundary of Cyl (it may be seen as a horizontal circuit in Cyl). Let µ Cyl be the measure on Cyl with boundary conditions on γ and on γ. Precisely, µ Cyl is the uniform measure on pairs of coherent spin configuration (σ r , σ b ) on Cyl with the property that all faces adjacent to γ have blue spin σ b = and all faces adjacent to γ have σ r = .
Then, both the red-spin and blue-spin marginal of µ Cyl have the FKG property. We sketch the proof of this fact next. Embed Cyl in the plane in the same way that Cyl m,n was embedded in Figure 8; call D the planar graph thus obtained. The measure µ Cyl is equal to that on D with the faces adjacent to γ conditioned to have blue spin and those adjacent to γ to have red spin . By Corollary 2.10, this conditioned measure does satisfy the FKG inequality for both the blue and red-spin marginals. Moreover, the boundary conditions of µ Cyl satisfy the following Spatial Markov property for measures on Cyl ∞ . Let A r be the set of faces that are either below γ, above γ, or below γ but adjacent to it. Similarly, let A b be the set of faces that are above γ, below γ, or above γ but adjacent to it. Then, for any red spin configuration ξ r with the property that all faces adjacent to γ have spin and any blue spin configuration ξ b with the property that all faces adjacent to γ have spin , we have where the equality refers only to the restriction on Cyl. This fact may be proved exactly as Theorem 2.3 and we do not give further details.
In addition, the Spatial Markov property for boundary conditions holds under µ Cyl . Thus, using the FKG property for µ Cyl , we find where Top ≡ stands for the event that all faces adjacent to the top of Cyl have red spin . The same holds for C v (R).
Finally, let us mention that µ Cyl is invariant under reflection with respect to the horizontal line R × {7n 2} composed with colour inversion. This transformation maps C v (R) onto C v (R) and C v (R) onto C v (R). Now, due to Lemma 3.8, R contains either a horizontal double-crossing of constant red spin, or a vertical one of constant blue spin. Thus, Define the boundary conditions on Cyl in the same way as . Then µ Cyl is obtained from µ Cyl by flipping the sign of all red spins. Using the same argument as in Corollary 2.12 (ii), it may be shown that the red spin marginal of µ Cyl dominates that of µ Cyl . In particular, The same holds for vertical crossings. Insert the above in (4.24) and use (4.23), to find In conclusion where the sum is over all possible realisations γ of Γ. ◻ Finally, let us finish the proof of Proposition 4.13. For δ > 0 small enough for Claim 4.15 to hold, using Claims 4.15 and 4.16, we find However, by our assumption (4.21), the left-hand side is bounded above by 2δ. This leads to a contradiction if δ is chosen smaller than 1 16. ◻ Proof of Corollary 4.14 Fix C h , C v and n as in the statement. We may assume n larger than some constant depending on C v and C h ; the inequality for smaller values may be satisfied by altering the value of δ(C v , C h ). Apply Proposition 4.13 to N = Cv 5 n and C = C h 5Cv to obtain that Thus, up to replacing δ by ψ(δ), we may suppose that (4.25) holds always. Then, by repeated applications of Corollary 4.6 we deduce that for some δ 0 > 0 depending only on C h and C v . A consequence of Lemma 4.3 and of the FKG property is that the red-spin marginal of µ Cyl C h n,Cv n is dominated by that of µ Rect C h n,Cv n . Using this, and the fact that N = Cv 5 n, we find Define the rectangles R j = Rect C h n,(4 5) j Cvn and . Then (4.28) applies to any of the rectangles R j with j ≥ 0, and we find for some δ j > 0 that depends on C h , C v and j, but not on n 3 . When C h (S j ) occurs for some j ≥ 0, there exists a double-path contained in S j ⊂ R j+1 , connecting the left and right side of R 0 . Any such path separates the top of R 0 from its bottom. Let Γ be the highest such path and Under(Γ) be the set of faces of R 0 that are separated from the top of R 0 by Γ. Then Γ is measurable with respect to the spins above and adjacent to Γ. For any possible realisation γ of Γ, due to the Spatial Markov property, the red-spin marginal of µ R 0 [. Γ = γ] restricted to Under(γ) stochastically dominates that of µ R j+1 . It follows that This may appear surprising, as it is not always the case that S j+1 ⊂ Under(γ). Let us explain briefly why (4.30) is nevertheless true. Couple the red-spin marginals of µ R j+1 and µ R 0 [. Γ = γ] in an increasing fashion (this is possible do to the stochastic domination of the former by the latter). Then, if (σ r ,σ r ) is a sample of this coupling,σ r is equal to for all faces adjacent to γ and is greater of equal to σ r for the faces of Under(γ). If σ r is such that C h (S j+1 ) occurs, thenσ r ∈ C h (S j+1 ) as well. See Figure 15 for an illustration. Summing over all possible values of γ, we find R 0 Figure 15: The rectangle R 0 = Rect C h n,Cvn with + + boundary conditions on the top and lateral sides and − − on the bottom. If C h + + (S j ) occurs, the measure under Γ dominates µ + + − − R j+1 and C h + + (S j+1 ) is more likely to occur than in R j+1 .
Iterating this for j < J ∶= ⌊log 5 4 C v ⌋, we find Notice that S J is included in Rect C h n,n , hence the above implies that The right-hand side of the above is a positive constant depending only on C h and C v , and the proof is complete. ◻

Proof of dichotomy theorem (Theorem 4.1 and Corollary 4.2)
Proof of Theorem 4.1 Fix n and let ρ be some large constant (we will see below how to choose it). We will work in the domain B ∶= Λ ρ(ρ+2)n , under the measure µ B . The steps of the proof are described in Figure 16. Let x L = (−ρn, 0) and x R = (ρn, 0). Write Λ k (x L ) for the ball of radius k centred at x L , and use the same notation for x R . Let Circ (x L ) and Circ (x R ) be the events that there exists a double-circuit in Λ 2n (x L ) ∖ Λ n (x L ) and Λ 2n (x R ) ∖ Λ n (x R ), respectively. Notice that both Circ (x L ) and Circ (x R ) depend only on the spins inside Λ (ρ+2)n . Recall that Circ (k, ) is the event that there exists a double-circuit in Λ that surrounds Λ k .
By the monotonicity of boundary conditions In the second inequality we used the monotonicity of boundary conditions (Corollary 2.12) and the definition of α (ρ+2)n ; the last inequality is due to the positive association of σ r Figure 16: Left: Create Circ + + (x L ) and Circ + + (x R ) by first creating one double-− circuit surrounding the whole of Λ (ρ+2)n (at a cost α (ρ+2)n ), then creating two smaller circuits inside Λ (ρ+2)n which come at constant cost. Right: when Circ + + (x L ) ∩ Circ + + (x R ) occurs, Corollary 4.14 allows us to create two long double-− crossings in the strips above and below Λ 2n (x L ).
under µ Λ 2(ρ+2)n . It is a standard consequence of the comparison of boundary conditions and Corollary 4.14 that µ Λ 2(ρ+2)n [Circ (x L )] > c 0 for some constant c 0 > 0 that does not depend on n. In conclusion (4.31) We will now condition µ B on the event Circ (x L ) ∩ Circ (x R ), and will construct double-circuits around Λ 2n (x L ) and Λ 2n (x R ). Using the Spatial Markov property, these will allow to bound the probability in (4.31) as a product of two probabilities α n .
When Circ (x L ) occurs, write Ξ L for the innermost double-circuit as in the definition of Circ (x L ). Then Ξ L is measurable in terms of the spins of the faces inside and adjacent to it. Define Ξ R in the same way. Let χ L and χ R be two possible realisations of Ξ L and Ξ R , respectively. A straightforward variant of the Spatial Markov property (Theorem 2.3) states that the restriction of µ B [. Ξ L = χ L and Ξ R = χ R ] to the faces of B outside of χ L and χ R is independent of the values of the spins strictly inside χ L and χ R . In particular, the restricted measure above is equal to µ B [. χ L ≡ and χ R ≡ ], and its red-spin marginal satisfies the FKG inequality (see Corollary 2.10).
Consider the horizontal strip Strip T = R×[ √ 3n, 2 √ 3n]; it sits above Λ 2n (x L ) and Λ 2n (x R ). Write C h (Strip T ∩ B) for the event that Strip T ∩ B is crossed horizontally by a doublepath (Strip T ∩ B is not technically a rectangle, but we use the same notation). Then Corollary 4.14 (or rather its variant with and inverted) implies the existence of a constant c 1 > 0 independent of n, χ L and χ R such that The estimate (4.32) also holds for Strip B , the symmetric of Strip T with respect to the horizontal axis R × {0}. Thus, by the FKG inequality, Summing over all possible values χ L and χ R of Ξ L and Ξ R and using (4.31), we find Write D for the domain that is the intersection of B with the strip R×[−2 √ 3n, 2 √ 3n]. Then, by conditioning on the highest and lowest double-crossings of Strip T and Strip B , respectively, using the spatial Markov property and the monotonicity of boundary conditions, we find 1 c 2 0 α (ρ+2)n . Now consider the parallelogram Par formed of the faces with centres at k + e iπ 3 with 0 ≤ k ≤ 24n and −4n ≤ ≤ 4n. Define its horizontal translates Par LL = Par − ((ρ + 26)n, 0), Par LR = Par − ((ρ − 2)n, 0), Par RL = Par + ((ρ − 26)n, 0) and Par RR = Par + ((ρ + 2)n, 0). These are all contained in D, touch its top and bottom and are left of Λ 2n (x L ), right of Λ 2n (x L ), left of Λ 2n (x R ) and right of Λ 2n (x r ), respectively. Let us assume that ρ is large enough so that Par LL and Par LR are included in Λ ρn (x L ) (ρ ≥ 30 suffices). Then Par RL and Par RR are included in Λ ρn (x R ) and, in particular, are disjoint from the first two parallelograms. Now observe that, due to Lemma 4.5 (applied with and exchanged) for some universal constant c 2 > 0. Then, using Bayes rule Finally, by conditioning on the left-most vertical double-crossing of Par LL and the right-most of Par LR , and using the monotonicity of boundary conditions, it may be shown that the restriction of µ D [. C v (Par LL ) ∩ C v (Par LR )] to Λ 2n (x L ) is dominated by that of µ Λρn(x L ) . Moreover, due to the Spatial Markov property, this is true even when conditioning on the spins to the right of Par LR . The same procedure may be applied to µ D [. C v (Par RL )∩C v (Par RR )] for the measure in Λ 2n (x R ). Notice that the areas that determine the restriction of is dominated by the independent product of µ Λρn(x L ) and µ Λρn(x R ) . In conclusion, The last two displayed equations yield the desired conclusion. ◻ Proof of Corollary 4.2 Let ρ, C be the constants of Theorem 4.1. Suppose that inf n α n = 0. Let n 0 be such that α n 0 ≤ 1 2C . Then a simple induction involving (4.1) implies that α (ρ+2) k n 0 ≤ 1 C 2 −2 k for all k ≥ 0. This implies the stated inequality for c < log 2 log(ρ + 2) and C chosen accordingly. ◻

Conclusions
In this section we prove Theorems 1.2 and 1.3. To this end, we first resolve the dichotomy stated in Corollary 4.2 and then transfer the results from the spin representation to the loop O(2) model.

Excluding stretched-exponential decay
The goal of this section is to show that the case (ii) of Corollary 4.2 is incoherent with Theorem 3.2. Once it is established that case (i) holds, it is fairly standard to deduce Theorem 1.2; this is done in Section 5.2.
The constant ρ and the ratio between the inner and outer radii of the annulus above may actually be chosen arbitrarily, as we prove below. Other variants referring to rectangle crossings may also be formulated.
Corollary 5.2. For any a > 1, inf µ Λan Circ (n, an) ∶ n ≥ 3 a−1 > 0. The lower bound on n in the infimum above is to ensure that the annulus is thick enough to allow the existence of a double-circuit. We start by proving the corollary, based on Proposition 5.1. The remainder of the section is then dedicated to proving Proposition 5.1. Λ n Λ bn Λ an Figure 18: The small annuli Ann 1 , . . . , Ann K placed inside Λ bn ∖ Λ n have inner radius m and outer radius 2m. They are such that the balls of radius ρm around each of their centres are contained in Λ an . If they all contain double-+ circuits, then these form a circuit around Λ n , contained in Λ bn ⊂ Λ an .
Proof This is a standard application of Proposition 5.1, the FKG property and the monotonicity of boundary conditions. Fix a > 1 and let b = (1 + a) 2. We may limit ourselves to values of n larger than some threshold depending on a; smaller values of n only add strictly positive numbers to the set whose infimum we are considering.
Recall that ρ is fixed by Theorem 4.1. Let m = ⌊min{ a−b ρ ; b−1 4 } ⋅ n⌋, and suppose that n is large enough so that m ≥ 2. Then there exists a number K = K(a, ρ), not depending on m or n such that one may place K translates Ann 1 , . . . , Ann K of the annulus Λ 2m ∖ Λ m inside Λ bn ∖ Λ n in such a way that, if all of them contain a circuit of double-, then Circ (n, an) occurs. See Figure 18 for an example.
Since mρ < (a − b)n, all faces at distance ρm from each Ann j are contained in Λ an . It follows from the FKG inequality and the monotonicity of boundary conditions that µ Λan Circ (n, an) ≥ K j=1 µ Λan Circ (Ann j ) ≥ µ Λρm Circ (m, 2m) where c = inf n µ Λρn [Circ (n, 2n)] is a strictly positive constant due to Proposition 5.1. Since the ultimate lower bound above does not depend on n, the proof is complete. ◻ We now turn to proving Proposition 5.1. We will proceed by contradiction. Fix ρ > 2 given by Theorem 4.1 and recall that α n = µ Λρn Circ (n, 2n)). Will assume that case (ii) of Corollary 4.2 occurs, namely that there exist constants c, C > 0 and n 0 ≥ 1 such that We start by proving a series of results based on (ExpDec). All constants below depend implicitly on the values of n 0 , ρ, c and C of (ExpDec).
τ (x T ) x B x T R τ (R) Figure 19: Left: One may place six copies of R around Λ n so that, if {Λ n + + ←→ Λ c 2n } occurs, then at least one of them is crossed in the short direction (here the crossed copy is shaded). Right: When x T is connected to x B in R and τ (x T ) is connected to τ (x B ) in τ (R), then τ (x T ) is connected to x T in τ (R) ∪ R.
Proof Fix κ ≥ 2 and n arbitrary. Let R ∶= Rect 2n,n 2 . The annulus Λ 2n ∖ Λ n may be covered by six translations and rotation R 1 , . . . , R 6 of R in such a way that, if {Λ n ← → Λ c 2n } occurs, then at least one of R 1 , . . . , R 6 is crossed in the short direction by a double-path (see Figure 19). For 1 ≤ i ≤ 6, write C v (R i ) for the appropriate rotation and translation of C v (R). Then, using the union bound and the monotonicity of boundary conditions, we deduce that Henceforth we aim to prove a stretched-exponential upper bound for µ Λ (κ+2)n [C v (R)]. Let x B and x T to be the points of the bottom and top, respectively, of R that are most probable under µ Λ (κ+2)n to be connected by a double-path contained in R. Then since there are 16n 2 potential pairs of points (x L , x R ). Let τ be the reflection with respect to the horizontal line R × {0}. Then we also have If the events of (5.3) and (5.4) occur simultaneously, then x T and τ (x T ) are connected inside R ∪ τ (R) = [−2n, 2n] × [−n 2, n 2] (see Figure 19). Thus, by the FKG inequality, Using the above, the FKG inequality again and the monotonicity of boundary conditions, we find, ← → x T }. The box has been increased to Λ (κ+10)n so that all of these events occur in rectangles with distance to the boundary greater than (κ + 2)n.
Recall the fixed values ρ > 2 and n 0 of (ExpDec). Let k be minimal such that, for N ∶= (ρ + 2) k n 0 , one has N n ≥ max 16, κ+10 ρ−2 . By the minimality of k, we have N ≤ c 0 n for some constant c 0 depending on ρ, n 0 and κ only. Then, there exists a constant K = K(ρ, n 0 , κ), that depends on ρ, n 0 and κ but not on n or on the resulting value of N , such that one may construct a circuit in Λ 2N ∖ Λ N by combining at most K vertical crossings of translates of [−2n, 2n]×[−8n, 8n] and rotations by 2π 3 and 4π 3 of this rectange, all contained in Λ 2N (see Figure 20). Due to the choice of N , the faces at distance (κ+10)n from any of these rectangles are all contained in Λ ρN . Thus, by the monotonicity of boundary conditions and the FKG inequality, Due to (ExpDec), this implies for constants c 1 , c 2 > 0 depending only on κ, ρ and n 0 . Finally, from (5.2) we deduce that µ Λκn Λ n ← → Λ c 2n ≤ 6c 2 (4n) 2 e −c 1 n c . This implies (5.1) with C 1 chosen small enough to absorb the multiplicative factor. ◻ Lemma 5.4. Under assumption (ExpDec), for any κ ≥ 2 there exists C 2 = C 2 (κ) > 0 such that µ Λκn Λ n ← → Λ c 2n < e −C 2 n c ∀n ≥ 1.

(5.6)
Proof Fix κ ≥ 2 and let C 2 be some small constant to be fixed below (it will be obvious that the bound on C 2 depends only on κ). It suffices to prove the statement for n large enough; small values may be incorporated by adjusting C 2 . Suppose by contradiction that there exists n ≥ 1 (large) such that µ Λκn [Λ n ← → Λ c 2n ] < 1 − e −C 2 n c . From now on n is fixed and it is crucial that we use the assumption above only for this particular value of n.
Due to the monotonicity of boundary conditions, we deduce that µ D Λ n ← → Λ c 2n < 1 − e −C 2 n c for any domain D containing Λ κn . Suppose now that C 2 is chosen smaller than C 1 (4κ) 2, with C 1 (4κ) given by Lemma 5.3. Then, assuming n is above some threshold (which we will do from now on), we have Due to the two displays above, and to the monotonicity of boundary conditions, for any domain D with Λ κn ⊂ D ⊂ Λ 4κn . As in Lemma 3.6 , the absence of a double-or double-connection between Λ n and Λ c 2n implies that at least one of Circ (n, 2n) and Circ (n, 2n) occurs (see also Remark 3.7). Under µ D , the blue spins are interchangeable, and we deduce that µ D Circ (n, 2n) = µ D Circ (n, 2n) ≥ 1 4 e −C 2 n c , (5.8) for any domain D with Λ κn ⊂ D ⊂ Λ 4κn . Next we work in the domain Λ 3κn . Place translations Ann 1 , . . . , Ann K of the annulus Λ 2n ∖ Λ n around the outside of Λ κn as in Figure 18 so that, if all of them contain double-circuits, then there exists a double-circuit in Λ (κ+2)n surrounding Λ κn . As discussed in the proof of Corollary 5.2, this procedure employs a number K of translates that only depends on κ, not on n. Thus µ Λ 3κn Circ (κn, (κ + 2)n) ≥ µ Λ 3κn The second inequality is due to the FKG property of blue spins under µ Λ 3κn (see Remark 2.11 (i) with reversed colours). The last inequality is a consequence of (5.8).
This contradicts (5.1) provided that C 2 is small enough (any C 2 < C 1 (3κ) (K +1) suffices) and n is large enough. ◻ Lemma 5.5. Under assumption (ExpDec), for any κ ≥ 2 there exists C 3 = C 3 (κ) > 0 such that µ Λκn [Circ (n, 2n)] > 1 − e −C 3 n c , ∀n ≥ 1. (5.10) As a consequence, for any > 0, there exists n 1 ≥ 1 such that Circ (2 j n 1 , 2 j+1 n 1 ) > 1 − . for all n and some fixed constant C 4 > 0. Notice that we are aiming to show that a thin rectangle is crossed in the long (vertical) direction with very high probability. Heuristically, Lemma 5.4 says that such rectangles are crossed with high probability in the short (i.e. horizontal) direction. To pass from crossing in the short direction to crossings in the long direction, we will use the same argument as in the proof of (5.5). However, since this argument applies to small probabilities rather than large ones, we will use it for the model dual to double-connections. Recall the notation dm(σ r ) for the set of edges of H with spin on either side. Let dm(σ r ) * be the dual of dm(σ r ); it is a percolation configuration on T, with edges open if at least one of their endpoints is a face of spin . Then C v (Rect n 16,n 2 ) fails if and only if Rect n 16,n 2 is crossed horizontally by a path in dm(σ r ) * . The same holds with Rect 2n,n 2 instead of Rect n 16,n 2 . Moreover, dm(σ r ) * is increasing in σ r , hence satisfies the FKG inequality under µ Λ (κ+2)n .
The same strategy as in the proof of Lemma 5.3 applies here, namely choosing the points on the left and right sides of Rect n 16,n 2 that are most likely to be connected in dm(σ r ) * , using horizontal reflection, the FKG inequality and monotonicity of boundary conditions, we find that 1 − µ Λ (κ+4)n [C v (Rect 2n,n 2 )] ≥ 4 n 2 1 − µ Λ (κ+2)n [C v (Rect n 16,n 2 )] 32 .
by a random variable 2G j , where G j has a geometric distribution of parameter c > 0. Then E D (N D ) ≤ ∑ K j=1 2E(G j ) = 2K c , as required. Finally, (5.16) also applies to P H instead of P D , and directly implies that Thus, there are indeed infinitely many loops surrounding the origin P H -a.s.. ◻

Proof of Theorem 1.3
Finally we prove Theorem 1.3. It may be worth mentioning that the proof below may be adapted to circumvent the use of the results of Section 4. Indeed, the non-quantitative delocalisation result of Theorem 3.2 suffices.
Proof of Theorem 1.3 From the construction of P H as limit of finite-volume measures, it is immediate that it is a Gibbs measure. The rest of the proof is dedicated to showing it is the only one. Let η be a Gibbs measure. For any configuration ω chosen according to η, colour each loop of ω independently in red or blue; colour each infinite path in red. Write ω r and ω b for the obtained red and blue configurations, respectively. Extend η to incorporate this additional randomness.
Additionally, associate to (ω r , ω b ) a pair of spin configurations (σ r , σ b ) obtained by choosing the spins at 0 (σ r (0), σ b (0)) ∈ { , } × { , } uniformly, then assign spins to all other faces with the constraint that two faces have distinct red spin (and blue spin, respectively) if and only if they are separated by an edge of ω r , and ω b , respectively. Thus η is both a law on pairs of red and blue loop configurations, as well as a law on pairs of red and blue spin configurations. We call the latter the double-spin representation of η.
Let us show that the red-spin marginal of η is equal to ν H . Notice that (DLR) implies that the double-spin representation of η has the spatial Markov property in that, for any domain D, the restriction of η to Int(D) conditionally on the double-spin configuration outside Int(D) is measurable in terms of the double-spin configuration on ∂ in D.
Fix > 0 and n ≥ 1. Let N > n be chosen so that for any event A that depends only on the spins in Λ n . Write ω ∖ Λ N for the subgraph containing only the edges of ω that lie outside of Λ N . Then, the connected components of ω ∖ Λ N may be loops, bi-infinite paths, as well as semi-infinite or finite paths with endpoints on ∂Λ N .

RSW theorem for height functions
We finish the paper with a RSW result for the uniform Lipschitz functions model. Recall the notation of Section 1.1. When considering Lipschitz functions on a domain containing Λ 2n , write Circ ≥k (n) for the event that there exists a closed face-path contained in Λ 2n , surrounding Λ n and formed entirely of faces for which the function is larger than k.
Theorem 5.6. For any k ≥ 1 there exists c(k) > 0 such that for all n large enough and any domain D containing Λ 2n .
Proof Fix k ≥ 1. Let n ≥ 1 be a large integer and D a domain containing Λ 2n . Recall from Propositions 1.4 and 2.1 that the loop representation of a height function chosen according to π D has law P D , and its spin representation has law µ D .
Write H for the event that there exist k + 1 closed edge-paths γ 1 , . . . , γ k+1 in Λ 2n that surround Λ n , that are numbered from outer-most to inner-most, and such that γ j is a double-path if j is odd and a double-path if j is even. By repeated applications of Corollary 5.2, there exists a constant c(k) > 0 independent of n or D such that µ D (H) ≥ c(k).
When H occurs, there exist at least k loops in the loop representation that are contained in Λ 2n ∖ Λ n and surround Λ n . WriteH for set of loop configurations which contain at least k such loops, and denote by Γ 1 , . . . , Γ k the outermost k loops as above. Recall that, in order to obtain the height function Φ from the loop configuration ω chosen according to P D , loops need to be oriented uniformly, and that the orientation of each loop dictates whether the height inside the loop is larger or smaller than the one outside. By symmetry, conditionally on any loop configuration ω ∈H, with probability at least 1 2 the height of the faces outside and adjacent to Γ 1 is at least 0. Moreover, independently of the above, all paths Γ 1 , . . . , Γ k are oriented clockwise with probability 2 −k . When both of the above occur, the height of the faces inside and adjacent to Γ k is at least k. Thus π D (Circ ≥k (n)) ≥ 2 −(k+1) P D (H) ≥ 2 −(k+1) µ D (H) ≥ 2 −(k+1) c(k), which is the desired conclusion. ◻