Phase transitions for a class of gradient fields

We consider gradient fields on $\mathbb{Z}^d$ for potentials $V$ that can be expressed as $$e^{-V(x)}=pe^{-\frac{qx^2}{2}}+(1-p)e^{-\frac{x^2}{2}}.$$ This representation allows us to associate a random conductance type model to the gradient fields with zero tilt. We investigate this random conductance model and prove correlation inequalities, duality properties, and uniqueness of the Gibbs measure in certain regimes. Moreover, we show that there is a close relation between Gibbs measures of the random conductance model and gradient Gibbs measures with zero tilt for the potential $V$. Based on these results we can give a new proof for the non-uniqueness of gradient Gibbs measures without using reflection positivity. We also show uniqueness of ergodic zero tilt gradient Gibbs measures for almost all values of $p$ and $q$ and, in dimension $d\geq 4$, for $q$ close to one or for $p(1-p)$ sufficiently small.


Introduction
Gradient fields are a statistical mechanics model that can be used to model phase separation or, in the case of vector valued fields, solid materials. Formally they can be defined as a random field (ϕ x ) x∈Z d ∈ R Z d with distribution exp − x∼y V (ϕ(x) − ϕ(y)) Z x∈Z d dϕ(x). (1.1) Here dϕ(x) denotes the Lebesgue measure, V : R → R a measurable symmetric potential, and ∼ indicates the neighbourhood relation for Z d . We can give a meaning to the formal expression (1.1) using the DLR-formalism. The DLR-formalism defines equilibrium distributions usually called Gibbs measure for this type of models as measures µ on R Z d such that the conditional probability of the restriction to any finite set is as above. In the setting of gradient interface models no Gibbs measure exists in dimension d ≤ 2. Therefore one often considers gradient Gibbs measures [19,23]. This means that attention is restricted to the σ-algebra generated by the gradient fields Then infinite volume measures exist if V (s) grows sufficiently fast (linearly is sufficient) as s → ±∞. Gradient Gibbs measures are also useful to model tilted surfaces. For a translation invariant gradient Gibbs measure µ the tilt vector u ∈ R d is defined by where ∇ϕ(x) ∈ R d denotes the discrete derivative, i.e., the vector with entries ∇ i ϕ(x) = ϕ(x+e i )−ϕ(x) with e i denoting the i-th standard unit vector. If the gradient Gibbs measure is ergodic the tilt corresponds to the asymptotic average inclination of almost every realisation of the gradient field. Gradient interface models have been studied frequently in the past years. In particular the discrete Gaussian free field with V (s) = s 2 where the fields are Gaussian caught considerable attention. Many of the results obtained in this case were generalized to the class of strictly convex potentials satisfying c 1 ≤ V (s) ≤ c 2 for some 0 < c 1 < c 2 and all s ∈ R. Let us only mention two results for convex potentials and refer to the literature in particular the reviews [18,30] for all further results and references. Funaki and Spohn showed in [19] that for every tilt vector u there exists a unique translation invariant gradient Gibbs measure. Moreover, the scaling limit of the model is a massless Gaussian field as shown by Naddaf and Spencer [28] for zero tilt and generalised to arbitrary tilt by Giacomin, Olla, and Spohn [21]. In contrast for non-convex potentials far less is known because all the techniques seem to rely on convexity in an essential way. For potentials of the form V = U + g where U is strictly convex and g ∈ L q for some q ≥ 1 with sufficiently small norm the problem can be led back to the convex theory by integrating out some degrees of freedom. This way many results from the convex case can be proved in particular uniqueness and existence of the Gibbs measure for every tilt and that the scaling limit is Gaussian [11,10,14]. This corresponds to a high temperature result. For low temperatures which correspond to non-convexities far away from the minimium of V it was shown that the surface tension is strictly convex and the scaling limit is Gaussian [1,24].
For intermediate temperatures that correspond to very non-convex potentials no robust techniques are known. All results to date are restricted to the special class of potentials introduced by Biskup and Kotecky in [5] that can be represented as where ρ is a non-negative Borel measure on the positive real line. Biskup and Kotecky mostly considered the simplest nontrivial case, denoting the Dirac measure at x ∈ R by δ x , ρ = pδ q + (1 − p)δ 1 (1.5) where p ∈ [0, 1] and q ≥ 1. They show that in dimension d = 2 and for q > 1 sufficiently large there exist two ergodic zero-tilt gradient Gibbs measures. Later, Biskup and Spohn showed in [6] that nevertheless the scaling limit of every zero-tilt gradient Gibbs measure is Gaussian if the measure ρ is compactly supported in (0, ∞). In [33] their result was recently extended by Ye to potentials of the form V (s) = (1 + s 2 ) α with 0 < α < 1 2 . Those potentials can be expressed as in (1.4) but ρ has unbounded support so that the results from [6] do not directly apply.
The main reason to study this class of potentials is that such potentials are much more tractable because the variable κ can be considered as an additional degree of freedom using the representation (1.4). This leads to extended gradient Gibbs measures which are given by the joint law of (η e , κ e ) e∈E(Z d ) . These extended gradient Gibbs measures can be represented as a mixture of nonhomogeneous Gaussian fields with bond potential κ e η 2 /2 for every edge e ∈ E(Z d ) and κ e ∈ R + . This implies that for a given κ the distribution of the random field is Gaussian with covariance given by the inverse of the operator ∆ κ where (1. 6) In all the works mentioned before this structure is frequently used, e.g. in [6] it is proved that the resulting κ-marginal of the extended gradient Gibbs measure is ergodic so that well known homogenization results for random walks in ergodic environments can be applied.
The main purpose of this note is to investigate the properties of the κ-marginal of extended gradient Gibbs measures in a bit more detail. The starting point is the observation that the κ-marginal of an extended gradient Gibbs measures with zero tilt is itself a Gibbs measure for a certain specification. This specification arises as the infinite volume of an infinite range random conductance model defined on finite graphs. On the other hand, we show that starting from a Gibbs measure for the random conductance model we can construct a zero tilt gradient Gibbs measure thus showing a one to one relation between the two notions of Gibbs measures. In particular, we can lift results about the random conductance model to results about gradient Gibbs measures. Note that one major drawback is the restriction to zero tilt that applies here and to all earlier results for this model. Let us mention that massive R-valued random fields have been earlier connected to discrete percolation models to analyse the existence of phase transitions [34]. For gradient models the setting is slightly different because we consider a random conductance model on the bonds with long ranged correlations while for massive models one typically considers some type of site percolation with quickly decaying correlations.
The main motivation for our analysis is that it provides a first step to the completion of the phase diagram for this potential and zero tilt and a better understanding of the two coexisting Gibbs states. Moreover, the random conductance model appears to be interesting in its own right. We could define the random conductance model and prove several of the results for arbitrary ρ but we mostly restrict our analysis to the simplest case where ρ is as in (1.5) and the potential is of the form (1.7) We prove several results about the random conductance model in particular correlation inequalities (that extend to arbitrary ρ). One helpful observation is that the random conductance model is closely related to determinantal processes because its definition involves a determinant weight. This simplifies several of the proofs because all correlation inequalities can be immediately led back to similar results for the weighted spanning tree. Using the correlation inequalities it is possible to show uniqueness of its Gibbs measure in certain regimes. It was already observed in [5] that the gradient interface model with potential V p,q exhibits a duality property when defined on the torus. Moreover, there is a self dual point p sd = p sd (q) ∈ (0, 1) where the model agrees with its own dual. The self dual point satisfies the equation (1.8) In [5] it is shown that the location of the phase transition in d = 2 must be the self dual point. We extend the duality to the random conductance model and arbitrary planar graphs. Using the fact that Z 2 as a graph is self-dual we can use the duality to prove non-uniqueness of the Gibbs measure therefore reproving the result from [5] without the use of reflection positivity. Many of our techniques and results for the random conductance model originated in the study of the random cluster model and we conjecture further similarities. This paper is structured as follows. In Section 2 we give a precise definition of gradient Gibbs measures and state our main results. Then, in Section 3 we introduce and motivate the random conductance model and its relation to extended gradient Gibbs measures. We prove properties of the random conductance model in Sections 4 and 5. Finally, in Section 6 we use the duality of the model to reprove the phase transition result. Two technical proofs and some results about regularity properties of discrete elliptic equations are delegated to appendices.

Model and main results
Specifications. Let us briefly recall the definition of a specification because the concept will be needed in full generality for the random conductance model (see Section 4). We consider a countable set S (mostly Z d or the edges of Z d ) and a measurable state space (F, F) (mostly either |F | = 2 or (F, F) = (R, B(R))). Random fields are probability measures on (F S , F S ) where F S denotes the product σ-algebra. The set of probability measures on a measurable space (X, X ) will be denoted by P(X, X ). For any Λ ⊂ S we denote by π Λ : F S → F Λ the canonical projection. We often consider the σ-algebra F Λ = π −1 Λ (F Λ ) of events depending on the set Λ. Recall that a probability kernel γ from Definition 2.1. A specification is a family of proper probability kernels γ Λ from F Λ c to F S indexed by finite subsets Λ ⊂ S such that γ Λ1 γ Λ2 = γ Λ1 if Λ 2 ⊂ Λ 1 . We define the set of random fields admitted to γ by (2.1) There is a well known equivalent definition of Gibbs measures. A cofinal set I is a subset of subsets of S with the property that for any finite set Λ 0 ⊂ S there is Λ ∈ I such that Λ 0 ⊂ Λ. Then µ ∈ G(γ) if and only if µγ Λ = µ for Λ ∈ I where I is a cofinal subset of subsets of S. See Remark 1.24 in [20] for a proof.
Gradient Gibbs measures. We introduce the relevant notation and the definition of Gibbs and gradient Gibbs measures to state our results. For a broader discussion see [20,30]. In this paragraph we consider real valued random fields indexed by a lattice Λ ⊂ Z d . We will denote the set of nearest neighbour bonds of Z d by E(Z d ). More generally, we will write E(G) and V(G) for the edges and vertices of a graph G. To consider gradient fields it is useful to choose on orientation of the edges. We orient the edges e = {x, y} ∈ E(Z d ) from x to y iff x ≤ y (coordinate-wise), i.e., we can view the graph (Z d , E(Z d )) as a directed graph but mostly we work with the undirected graph.
To any random field ϕ : Z d → R we associate the gradient field η = ∇ϕ ∈ R E(Z d ) given by are nearest neighbours and x ≤ y. We formally write η x,y = η e = ϕ y −ϕ x and η y,x = −η e = ϕ x − ϕ y . The gradient field η satisfies the plaquette condition η x1,x2 + η x2,x3 + η x3,x4 + η x4,x1 = 0 (2 .2) for every plaquette, i.e., nearest neighbours x 1 , x 2 , x 3 , x 4 , x 1 . Vice versa, given a field η ∈ R E(Z d ) that satisfies the plaquette condition there is a up to constant shifts a unique field ϕ such that η = ∇ϕ (the antisymmetry of the gradient field is contained in our definition). We will refer to those fields as gradient fields and denote them by R E(Z d ) g . To simplify the notation we write ϕ Λ for Λ ⊂ Z d and η E for E ⊂ E(Z d ) for the the restriction of fields and gradient fields. We usually identify a subset Λ ⊂ Z d with the graph generated by it and as before we write E(Λ) for the bonds with both endpoints in Λ.
For a subgraph H ⊂ G we write ∂H for the (inner) boundary of H consisting of all points x ∈ V(H) such that there is an edge e = {x, y} ∈ E(G) \ E(H). In the case of a graph generated by Λ ⊂ G we have x ∈ ∂Λ if there is y ∈ Λ c such that {x, y} ∈ E(G). We define and Λ finite and simply connected (i.e., Λ c connected) the following a priori measure on gradient configurations whereφ is a configuration such that ∇φ = ω and ∇ * the push-forward of this measure along the gradient map ∇ : . The shift invariance of the Lebesgue measure implies that this definition is independent of the choice ofφ and it only depends on the restriction ω E(Λ) c since Λ c is connected. For a potential V : R → R satisfying some growth condition we define the specification γ Λ where the constant Z Λ (ω E(Λ) c ) ensures the normalization of the measure. We introduce the notation for the σ-algebra of events depending only on E. Measures that are admitted to the specification γ, i.e., measures µ that satisfy for simply connected Λ ⊂ Z d µ a.s. (2.5) will be called gradient Gibbs measures for the potential V . For a ∈ Z d we consider the shift τ a : Main results. Our first main result is the following almost always uniqueness result for the gradient Gibbs measures for potentials as in (1.7). Theorem 2.3. For every q and d ≥ 2 there is an at most countable set N (q, d) ⊂ [0, 1] such that for any p ∈ [0, 1] \ N (q, d) there is a unique shift invariant ergodic gradient Gibbs measure µ with zero tilt for the potential V p,q .
This theorem is proved in Section 5 below the proof of Theorem 5.1. Moreover, we reprove the non-uniqueness result originally shown in [5] for this type of potential.
Theorem 2.4. There is q 0 ≥ 1 such that for d = 2, q ≥ q 0 , and p = p sd (q) the solution of (1.8), there are at least two shift invariant gradient Gibbs measures with 0 tilt.
The proof of this theorem is given at the end of Section 6. Moreover we prove uniqueness for 'high temperatures' and dimension d ≥ 4. This corresponds to the regime where the Dobrushin condition holds.
Theorem 2.5. Let d ≥ 4. For any q ≥ 1 there exists p 0 = p 0 (q, d) > 0 such that for all p ∈ [0, p 0 ) ∪ (1 − p 0 , 1] there is a unique shift invariant ergodic gradient Gibbs measure with zero tilt for the potential V p,q . Moreover, there exists q 0 = q 0 (d) > 1 such that for any q ∈ [1, q 0 ] and any p ∈ [0, 1] there is a unique shift invariant ergodic gradient Gibbs measure with zero tilt for the potential V p,q .
The proof of this Theorem is given in Section 5 below the proof of Theorem 5.6. The main tool in the proofs of these theorems is the fact that the structure of the potentials V in (1.4) allows us to consider κ as a further degree of freedom and we consider the joint distribution of the gradient field η and κ. We show that the law of the κ-marginal can be related to a random conductance model. The analysis of this model then translates back into the theorems stated before. We will make those statements precise in the next section. Let us end this section with some remarks. Remark 2.6. 1. For spin systems with finite state space and bounded interactions there are general results that show that phase transitions, i.e., non-uniqueness of the Gibbs measure are rare, see, e.g., [20]. Theorem 2.3 establishes a similar result for a specific class of potentials for a unbounded spin space. As discussed in more detail at the end of Section 5 we expect that for every q ≥ 1 the Gibbs measure is unique for all p ∈ [0, 1] except possibly for p = p c for some critical value p c = p c (q). Hence, Theorem 2.3 is far from optimal but we hope that the results provided in this paper prove useful to establish stronger results.
2. Let us compare the results to earlier results in the literature. For p/(1 − p) < 1/q the potential V p,q is strictly convex so that uniqueness of the Gibbs measure is well known and holds for every tilt. The two step integration used by Cotar and Deuschel extends the uniqueness result to the regime p/(1 − p) < C/ √ q (see Section 3.2 in [10]). In particular the case p ∈ [0, p 0 ) in Theorem 2.5 is included in earlier results. However, the potential becomes very non-convex (has a very negative second derivative at some points) for p close to 1 and the uniqueness result for p ∈ (1 − p 0 , 1] and d ≥ 4 appears to be new. In this regime the only known result seems to be convexity of the surface tension as a function of the tilt which was shown in [1] (see in particular Proposition 2.4 there). Their results apply to p very close to one, q − 1 very small, and d ≤ 3.
3. The restriction to dimension d ≥ 4 arises from the fact that the Green's function for inhomogeneous elliptic operators in divergence form decays slower than in the homogeneous case.

Extended gradient Gibbs measures and random conductance model
Extended gradient Gibbs measure. In this work we restrict to potentials of the form introduced in (1.4). As already discussed in more detail in [5] and [6] it is possible to use the special structure of V to raise κ to a degree of freedom. Let µ be a gradient Gibbs measure for V . For a finite set E ⊂ E(Z d ) and Borel sets A ⊂ R E and B ⊂ R E + we define the extended gradient Gibbs measurẽ It can be checked that this is a consistent family of measures and thus we can extendμ to a measure on (R × R + ) E(Z d ) . It was explained in [5] thatμ is itself a Gibbs measure for the specificationγ Λ defined bỹ Note that the distribution (dη, dκ) E(Λ) actually only depends on η E(Λ) c and is independent of κ. Let us add one remark concerning the notation. In this work we essentially consider three strongly related viewpoints of one model. The first viewpoint are gradient Gibbs measures that are measures on R . Thy will be denoted by µ and the corresponding specification is denoted by γ. Then there are extended gradient Gibbs measures for a specificationγ. They are measures on R and will be denoted byμ. The η-marginal ofμ is a gradient Gibbs measure µ. Finally there is also the κ-marginal ofμ which is a measure on R and will be denoted byμ. An important result here is thatμ is a Gibbs measure for a specificationγ if ρ is a measure as in (1.5). In this caseμ is a measure on the discrete space {1, q} E(Z d ) . We expect that this result can be extended to far more general measures ρ but we do not pursue this matter here. To keep the notation consistent we denote objects with single spin space R, e.g., gradient Gibbs measures without symbol modifier, objects with single spin space {1, q}, e.g., the κ-marginal with a bar, and objects with single spin space {1, q} × R, e.g., extended Gibbs with a tilde. Let us also fix a notation for the corresponding relevant σ-algebras. We write as before E E for the σ-algebra on R E(Z d ) generated by (η e ) e∈E and we define E = E E(Z d ) .
For the κ-marginal we similarly consider the σ-algebra F E on {1, q} E(Z d ) generated by (κ e ) e∈E and we write again . It was already remarked in [5] that this setting resembles the situation for the Potts model that can be coupled to the random cluster model via the Edwards-Sokal coupling measure. The random conductance model. As explained before our strategy is to analyse the κ-marginal of extended gradient Gibbs measures and then use the results to deduce properties of the gradient Gibbs measures for V p,q . The key observation is that the κ-marginal of extended gradient Gibbs measures is given by the infinite volume limit of a strongly coupled random conductance model. To motivate the definition of the random conductance model we consider the κ-marginal of the extended specificationγ defined in (3.2). For zero boundary value0 ∈ R We write Λ w =Λ/∂Λ for the graph where the entire boundary is collapsed to a single point (this is called wired boundary conditions and we will discuss this below in more detail). We denote the lattice Laplacian with conductances λ and zero boundary condition outside of where we set f (y) = 0 for y / ∈ • Λ. The definition (3.2) and an integration by parts followed by Gaussian calculus imply theñ It simplifies the presentation to introduce the random conductance model of interest in a slightly more general setting. We consider a finite and connected graph G = (V, E). The combinatorial graph Laplacian ∆ c associated to set of conductances c : E → R + is defined by for any function f : V → R. Note that we defined the graph Laplacian as a non-negative operator which is convenient for our purposes and common in the context of graph theory. In the following we view the Laplacian ∆ c as a linear map on the space H 0 = {f : V → R : x∈V f (x) = 0} of functions with vanishing average. We define det ∆ c as the determinant of this linear map. By the maximum principle the Laplacian is injective on H 0 , hence det ∆ c > 0. Sometimes we clarify the underlying graph by writing ∆ G c . Remark 3.1. In the general setting it is more natural to let the Laplacian act on H 0 instead of fixing a point to 0 as in the definition of∆ Λ w λ above where this corresponds to Dirichlet boundary conditions. It would also be possible to fix a point x ∈ V(G) and consider∆ G c acting on functions It is easy to see using, e.g., Gaussian calculus and a change of measure that the determinant of∆ G c is independent of x 0 and Motivated by (3.4) we fix a real number q ≥ 1 and consider the following probability measure on where Z = Z G,p denotes a normalisation constant such that P G,p is a probability measure. In the following we will often drop G and p from the notation and we will always suppress q. We restrict our attention to q ≥ 1 because by scaling the model with conductances {1, q} has the same distribution as a model with conductances {α, αq} for α > 0 so that we can set the smaller conductance to 1. Let us state a remark concerning the relation to the random cluster model.
Remark 3.2. 1. We chose the notation such that the similarity to the random cluster model is apparent. Both models have an a priori distribution given by independent Bernoulli distribution with parameter p on the bonds that is then correlated by a complicated infinite range interaction depending on q. They reduce to Bernoulli percolation for q = 1. At the end of Section 5 we state a couple of conjectures about the behaviour of this model that show that we expect similarities with the random cluster model in many more aspects.
2. While there are several close similarities to the random cluster model there is also one important difference that seems to pose additional difficulties in the analysis of this model. The conditional distribution in a finite set depends on the entire configuration of the conductances outside the finite set (not just a partition of the boundary as in the random cluster model). In particular the often used argument that the conditional distribution of a random cluster model in a set given that all boundary edges are closed is the free boundary random cluster distribution has no analogue in our setting.

3.
We refer to the model as a random conductance model since we will (not very surprisingly) use tools from the theory of electrical networks. Note that in the definition of the potential V the parameters correspond to different (random) stiffness of the bonds.

Basic properties of the random conductance model
Preliminaries. As before we consider a connected graph G = (V, E). To simplify the notation we introduce for E ⊂ E and κ ∈ {1, q} E the notation for the number of hard and soft edges respectively and we define h(κ) = h(κ, E) and s(κ) = s(κ, E). Let us introduce the weight of a subset of edges t ⊂ E by defining We will denote the set of all spanning trees of a graph by ST(G). We identify spanning trees with their edge sets. In the following, we will frequently use the Kirchhoff formula for the determinant of a weighted graph Laplacian (cf. [32] for a proof). Let us remark that the Kirchhoff formula is frequently used in statistical mechanics and has also been used in the context of gradient interface models for some potentials as in (1.4) in [8].
Remark 4.1. Note that equation (4.4) remains true for graphs with multi-edges and loops. Indeed, loops have no contribution on both sides and multi-edges can be replaced by a single edge with the sum of the conductances as conductance.
Correlation inequalities We will now show correlation inequalities for the measures P = P G,p . We start by recalling several of the well known correlation inequalities. To state our results we introduce some notation. Let E be a finite or countable infinite set. Let Ω = {1, q} E and F the σ-algebra generated by cylinder events. We consider the usual partial order on Ω given by ⊂ Ω is increasing if its indicator function is increasing. We writē µ 1 μ 2 ifμ 1 stochastically dominatesμ 2 which is by Strassen's Theorem equivalent to the existence of a coupling (ω 1 , ω 2 ) such that ω 1 ∼μ 1 and ω 2 ∼μ 2 and ω 1 ≥ ω 2 (see [31]). We introduce the minimum ω 1 ∧ ω 2 and the maximum ω 1 ∨ ω 2 of two configurations given by (ω 1 ∧ ω 2 ) e = min(ω 1 e , ω 2 e ) and (ω 1 ∨ ω 2 ) e = max(ω 1 e , ω 2 e ) for any e ∈ E. We call a measureμ on Ω strictly positive ifμ(ω) > 0 for all ω ∈ Ω. Finally we introduce for f, g ∈ E and ω ∈ Ω the notation ω ±± f g ∈ Ω for the configuration given by We define ω ± f similarly. We sometimes drop the edges f , g from the notation. We writeμ(ω) =μ({ω}) for ω ∈ Ω andμ(X) = Ω X dμ for X : Ω → R.
A strictly positive measure is called strongly positively associated if it satisfies the FKG lattice conditionμ  The next theorem provides a simple way to verify the assumptions of Theorem 4.2 and Theorem 4.3. Basically it states that it is sufficient to check the conditions when varying at most two edges.
We state one simple corollary of the previous results. , q} E such that one of the measurē µ 1 ,μ 2 is strongly positively associated. Then Proof. Assuming thatμ 1 is strongly positively associated we find using first the assumption (4.10) and then (4.6)μ Now Theorem 4.4 implies the claim. The proof ifμ 2 is strictly positively associated is similar.
It is convenient to derive the following correlation results for the measures P G,p from corresponding results for the weighted spanning tree measure. The weighted spanning tree measure on a connected weighted graph .
This model has been studied extensively, see [3] for a survey. An important special case is the uniform spanning tree corresponding to constant conductances κ that assigns equal probability to every spanning tree.
The following lemma provides the basic estimate to check the condition (4.9) for the measures P G,p . Recall the notation κ ±± f g introduced before Theorem 4.2 and also the shorthand κ ±± .
Lemma 4.6. For a finite graph G and κ ∈ {1, q} E as above The lemma can be derived from the fact that the weighted spanning tree has negative correlations. It is well known (see, e.g., [3]) that for all positive weights κ on a finite graph G the measure Q G κ has negative edge correlations (4.14) Simple algebraic manipulations show that this is equivalent to We introduce the following sums (4.16) With this notation multiplication by It remains to show that the statement in the lemma can be deduced from (4.17) (actually the statements are equivalent). Clearly we can assume κ = κ −− , i.e., κ f = κ g = 1. Using the Kirchhoff formula (4.4) we find the following expression Hence we obtain , Subtracting those two identities we find that only the cross-terms between A f , A g and between A f g , A do not cancel and we get (4.20) We can conclude using (4.17).
The previous lemma directly implies that the measures P G,p are strongly positively associated.
Corollary 4.8. The measure P G,p satisfies the FKG lattice condition for any and the FKG inequality Proof. Lemma 4.6 and the trivial observation that , q} E and f, g ∈ E the lattice inequality Then Theorem 4.4 applied toμ 1 =μ 2 = P G,p implies that the FKG lattice condition (4.21) holds and therefore by Theorem 4.3 also the FKG-inequality (4.22).
Let us first state a trivial consequence of this corollary.
Lemma 4.9. The measures P G,p and P G,p satisfy for p ≤ p Proof. Using Corollary 4.8 and Corollary 4.5 we only need to check whether (4.10) holds forμ 1 = P G,p andμ 2 = P G,p . This is clearly the case if p ≤ p .
The next step is to show correlation inequalities with respect to the size of the graph. More specifically we show statements for subgraphs and contracted graphs. This will later easily imply the existence of infinite volume limits. Moreover, we can bound infinite volume states by finite volume measures in the sense of stochastic domination. Let F ⊂ E be a set of edges. We define the contracted graph G/F by identifying for every edge f ∈ F the endpoints of f . Similarly for a set W ⊂ V of vertices we define the contracted graph G/W by identifying all vertices in W . The resulting graphs may have multi-edges. We also consider connected subgraphs G = (V , E ) of G. Recall the notation κ ± = κ ± f for f ∈ E. We use the notation ∆ G κ for the graph Laplacian on G where we restrict the conductances κ to E and we denote by ∆ G/F κ the graph Laplacian on G/F . The following lemma relates the determinants of the different graph Laplacians.
Remark 4.11. The lemma again extends to κ ∈ R E The proof is similar to the proof of Lemma 4.6. We derive the statement from a property of the weighted spanning tree model. For graphs as above and e ∈ E the estimate holds (see Corollary 4.3 in [3] for a proof). We can rewrite (assuming again .
and therefore (using Similar statements hold for the graphs G/F and G . Hence (4.26) implies (4.25).
Let us remark that the probability Q G κ (f ∈ t) can also be expressed as a current in a certain electrical network. In order to avoid unnecessary notation at this point we kept the weighted spanning tree measure and we will only exploit this connection when necessary below.
Again, the previous estimates implies correlation inequalities for the measures P G,p . In the following we consider a fixed value of p but different graphs so that we drop only p from the notation but we keep the graph G. We introduce the distribution under boundary conditions for a connected where (λ, κ) ∈ {1, q} E denotes the conductances given by κ on E and by λ on E \ E . This definition implies that we have the following domain Markov property for ω ∈ {1, q} E Since the measure P G is strongly positively associated, (4.31) and Theorem 2.24 in [22] implies that the measure P G,E ,λ is strongly positively associated. We now state the consequences of Lemma 4.10 on stochastic ordering. , q} E such that λ 1 ≤ λ 2 the following holds More generally, we have for λ ∈ {1, q} E and E ⊂ E or E ∩ F = ∅ respectively Proof. From Lemma 4.10 we obtain for f ∈ E and any κ ∈ {1, q} E Similarly, Lemma 4.10 implies for f ∈ E \ F and κ ∈ {1, q} E\F Then the the strong positive association of P G and Corollary 4.5 imply the first and the last stochastic ordering claimed in (4.32). The stochastic domination result in the middle of (4.32) follows from (4.31) and a general result for strictly positive associated measures (see [22,Theorem 2.24]). The proof of (4.33) is similar.
Infinite volume measures. The definition of the measure P shows that it is a finite volume Gibbs measure for the energy E(κ) = ln(det ∆ κ )/2 and a homogeneous Bernoulli a priori measure. We would like to define infinite volume limits for the measures P G and define a notion of Gibbs measures in infinite volume. This requires some additional definitions. Recall the definition of the σ-algebras F E for E ⊂ E(Z d ) and note that there is a similar definition for general graphs which will be used in the following. An event A ⊂ F is called local if it measurable with respect to F E for some finite set E, i.e., A depends only on finitely many edges. Similarly we define a local function as a function that is measurable with respect to F E for a finite set E. We say that a sequence of measures µ n on {1, q} E(Z d ) converges in the topology of local convergence to a measure µ if µ n (A) → µ(A) for all local events A. For a background on the choice of topologies in the context of Gibbs measures we refer to [20]. The construction of the infinite volume states proceeds similarly to the construction for the random cluster model by defining a specification and introducing the notion of free and wired boundary conditions. For simplicity we restrict the analysis to Z d but the generalisation to more general graphs is straightforward. First, we define infinite volume limits of the finite volume distributions with wired and free boundary conditions. Let us denote by Λ n = [−n, n] ∩ Z d the ball with radius n in the maximum norm around the origin and we denote by E n = E(Λ n ) the edges in Λ n . We introduce the shorthand Λ w n = Λ n /∂Λ n for the box with wired boundary conditions. We definē µ 0 n,p = P Λn,p ,μ 1 n,p = P Λ w n ,p (4.36) for the measure P on Λ n with free and wired boundary conditions respectively. From Corollary 4.12 and equation (4.31) we conclude that for any increasing event A depending only on edges in E n We conclude that for any increasing event A depending only on finitely many edges the limits lim n→∞μ 0 n,p (A) and similarly lim n→∞μ 1 n,p (A) exist. Using standard arguments we can write every local event A as a union and difference of increasing local events and we conclude that lim n→∞μ 0 n,p (A) and lim n→∞μ 1 n,p (A) exist. It is well known (see [4]) that this implies convergence ofμ 0 n,p andμ 1 n,p to a measure on {1, q} E(Z d ) in the topology of local convergence. We denote the infinite volume measures byμ 0 p andμ 1 p . Lemma 4.13. The measureμ 0 p andμ 1 p satisfy the FKG-inequality and for 0 ≤ p ≤ p ≤ 1 the relations µ 0 Moreover they are invariant under symmetries of the lattice and ergodic with respect to translations.
Proof. This is a consequence of Corollary 4.8 and Corollary 4.12 and a limiting argument. See the proof of Theorem 4.17 and Corollary 4.23 in [22] for a detailed proof for the random cluster model which also applies to the model considered here. Ergodicity is proved by showing that the measures are even mixing.
Infinite volume specifications. We now introduce the concept of infinite volume Gibbs measures for this model. We first consider the case of a finite connected graph G.
where the normalisation Z λ ensures thatγ G E (·, λ) is a probability measure. A simple calculation shows thatγ G is indeed a specification, i.e.,γ G E are proper probability kernels that satisfy for The measure P G is a finite volume Gibbs measure, i.e., it satisfies We would like to call µ a Gibbs measure on {1, q} Z d for the random conductance model if holds for all E ⊂ E(Z d ) finite. However,γ G E is a priori only well defined for finite graphs so that we use an approximation procedure for infinite graphs. Let G be an connected infinite graph. We are a bit sloppy with the notation and do not distinguish betweenγ H E for a subgraph H of G and its proper extension to F × {1, q} E(G) , i.e., we define for κ, λ ∈ {1, q} E(G) (4.44) We denote for f ∈ E(G) and κ ∈ {1, q} E(G) by κ + and κ − as before the configurations such that κ + e = κ − e for e = f and κ − f = 1, κ + f = q. In the following we assume p ∈ (0, 1). For p ∈ {0, 1} the measures P G,p agree with the Dirac measure on the constant 1 or constant q configuration. Since we assume that E is finite the specification γ H E is uniquely characterized by the fact that it is proper and it satisfies for κ, where we used (4.29) in the second step. We show that we can give meaning to this expression in infinite volume. For this we sketch the definition of spanning trees in infinite volume but we refer to the literature for details (see [3]). A monotone exhaustion of an infinite graph G is a sequence of subgraphs G n such that G n ⊂ G n+1 and G = n≥1 G n . It can be shown that for any finite sets In fact this is a consequence of (4.26) and the arguments we used forμ 0 n above. Hence it is possible to define a measure Q G,0 κ on 2 E(G) , the power set of E(G) which will be called the weighted free spanning forest on G (as the name suggest the measure is supported on forests but not necessarily on trees, i.e., on connected subsets of edges). Similarly, we can define the wired spanning forest Q G,1 κ replacing the subgraphs G n by the contracted graphs G n /∂G n . By definition those measures satisfy for any f ∈ E. Then it is possible to define two families of proper probability kernelsγ G,0 E andγ G,1 E for E ⊂ (E(G)) finite by the property that for f ∈ E and κ, λ  Note that the concatenation forγ G,0 for E , E ⊂ E(Z d ) is given bȳ in particular it only involves a finite sum in the case of a finite spin space. We conclude using (4.50) and (4.51) thatγ G,0 E andγ G,1 E define two specifications on G. Suppose the wired and the free uniform spanning forest on G agree. This implies that also the weighted wired and free spanning forest Q G,0 κ and Q G,1 κ on G agree if the conductances κ e are contained in a compact subset of (0, ∞) (see Theorem 7.3 and Theorem 7.7 in [3]). Thusγ G,1 E =γ G,0 E in this case. In particular we obtain thatγ Z d ,0 E =γ Z d ,1 E because the free and the wired uniform spanning forest on Z d agree (Corollary 6.3 in [3]). In the following we will denote this specification byγ E . To ensure consistency with the earlier definition ofγ we define for a connected subset Λ ⊂ Z d thatγ Λ =γ E(Λ) . We can now give a formal definition of Gibbs measures for the random conductance model.  We show that both sides converge in the topology of local convergence as n → ∞. Let A be an increasing event depending on a finite number of edges. We have seen in (4.37) that µ 0 n (A) is an increasing sequence and converges by definition to µ 0 (A). We derive the convergence of the left hand side of equation (4.53) from the following three observations. First, we conclude from (4.32) and (4.42) thatγ Λn E (A, ·) is an increasing function. Second, using (4.33) and (4.42) we obtainγ The third observation is that (4.37) can also be applied to an increasing function instead of an increasing event. These three facts implȳ µ 0 n (γ Λn E (A, ·)) ≤μ 0 (γ Λn E (A, ·)) ≤μ 0 (γ E (A, ·)). Let us briefly introduce the class of quasilocal specifications which is a natural and useful condition for a specification. For an extensive discussion we refer to the literature [20]. A quasilocal function on a general state space is a bounded function X : F S → R that can be approximated arbitrarily well by local functions, i.e., (4.59) A specification γ is called quasilocal if γ Λ X is a quasilocal function for every local function X. We will show that the specificationγ E is quasilocal. This will be a direct consequence of the following result that shows uniform convergence ofγ Λ w n E toγ E . This convergence will be of independent use later.  Proof. First, we claim that it is sufficient to show that Indeed, using (4.61) in (4.45) we obtain Since E n is finite this implies the claim. It remains to prove (4.61). This is a consequence of the transfer current theorem (see Theorem 4.1 in [3]) that states in the special case of the occupation property that for f = {x, y} ∈ E(G) where the expression I f (f ) denotes the current through the edge f when 1 unit of current is induced respectively removed at the two ends of f . In the last step we used that I f (f ) can be calculated by applying the inverse Laplacian to the sources to obtain the potential which can be used to calculate the current through f . Now (4.61) follows from the display (4.63) and Lemma B.3.
Proof. Let X be a local functions. We need to show thatγ E X is quasilocal. Lemma 4.16 implies that the local functionsγ (4.64) Relation to extended gradient Gibbs measures In this paragraph we state the results that relate the random conductance model to extended gradient Gibbs measure. This is finally the justification to consider this model. The proofs of the results in this paragraph are deferred to Section A. The first Proposition establishes that the κ-marginal of extended gradient Gibbs measures are Gibbs states for the random conductance model. The second main result in this paragraph is a reverse of Proposition 4.18, namely that it is possible to obtain an extended Gibbs measure with zero tilt for the potential V p,q , given a Gibbs measureμ for the random conductance model with parameters p, q.
Proposition 4.19. Letμ be a Gibbs measure in the sense of Definition 4.14 for parameters p and q and κ ∼μ. Let ϕ κ be the random field that for given κ is a Gaussian field with zero average, ϕ κ (0) = 0, and covariance (∆ κ ) −1 , i.e., ϕ κ satisfies for f : Z d → R with finite support and x f (x) = 0 (4.65) Letμ be the joint law of (κ, ∇ϕ κ ). Thenμ is an extended Gibbs measure for the potential V p,q with zero tilt, in particular its η-marginal is a gradient Gibbs measure with zero tilt.
As a last result in this direction we state a very useful result from [6] that characterizes the law of ϕ given κ for extended gradient Gibbs measures if ϕ is distributed according to a gradient Gibbs measure.
Proposition 4.20. Let µ be a translation invariant, ergodic gradient Gibbs measure with zero tilt andμ the corresponding extended gradient Gibbs measure. Then the conditional law of ϕ given κ is µ-almost surely Gaussian. It is determined by its expectation E ϕ x | F (κ) = 0 (4.66) and the covariance given by (∆ κ ) −1 , i.e., for f : Z d → R with finite support and x f (x) = 0 Proof. This is Lemma 3.4 in [6].
In particular those results establish the following. Assume that µ is an ergodic zero tilt gradient Gibbs measure. Letμ be the κ-marginal of the corresponding extended gradient Gibbs measureμ (which by Proposition 4.18 is Gibbs for the random conductance model). We can use Proposition 4.19 to construct an extended gradient Gibbs measureμ . Using the definition ofμ in Proposition 4. 19 and Proposition 4.20 we conclude that we get back the extended gradient Gibbs measure we started from, i.e.,μ =μ .

Further properties of the random conductance model
In this section we state and prove more results about the random conductance model considered in this work and use the results from the previous section to derive corresponding results for the associated gradient interface model. We end this section with some conjectures and open questions. We start by provingμ 0 p =μ 1 p for d ≥ 2 and almost all values of p which will in particular implies uniqueness of the Gibbs measure for those p. Proof. It is a standard consequence of the invariance under lattice symmetries andμ 0 p μ 1 p that µ 1 p =μ 0 p is equivalent toμ 1 p (κ e = q) =μ 0 p (κ e = q) for one and therefore any e ∈ E(Z d ) (see, e.g, Proposition 4.6 in [22]). Lemma 5.3 below implies for e ∈ E(Z d ) for any p > p. In particular, we can conclude thatμ 0 p =μ 1 p holds for all points of continuity of the map p →μ 0 p (κ e = q). Since this map is increasing by Lemma 4.9 it has only countably many points of discontinuity.
We are now in the position to prove Theorem 2.3.
Proof of Theorem 2.3. We note that a translation invariant zero tilt Gibbs measure exists for any p and q, e.g., as a limit of torus Gibbs states (see the proof of Theorem 2.2 in [5]). It remains to show uniqueness. Consider p such thatμ 1 p =μ 0 p which is true for all but a countable number of p ∈ [0, 1] by Theorem 5.1 above. Let µ 1 and µ 2 be ergodic zero tilt gradient Gibbs measures for V = V p,q . By Proposition 4.18 the corresponding κ-marginalsμ 1 andμ 2 of the extended Gibbs measuresμ 1 and µ 2 are Gibbs measures in the sense of Definition 4.14 and therefore equal. Using Proposition 4.20 we conclude that since µ 1 and µ 2 are ergodic zero tilt gradient Gibbs measures their laws are determined byμ 1 andμ 2 , hence µ 1 = µ 2 .
Remark 5.2. Similar arguments for this model appeared already in the proof of Theorem 2.4 in [5] where they use the convexity of the pressure to show that the number of q-bonds on the torus is concentrated around its expectation in the thermodynamic limit. However, this is not sufficient to conclude uniqueness.
The key ingredient in the proof of Theorem 5.1 is the following lemma that comparesμ 1 p (κ e = q) withμ 0 p (κ e = q) for p < p . Intuitively the reason for this result is that a change of p is a bulk effect of order |Λ| while the effect of the boundary conditions is of order |∂Λ|.

Lemma 5.3. For any p < p we have
Proof. The proof follows the proof of Theorem 1.12 in [16] where a similar result for the random cluster model is shown. The only difference is that the comparison between free and wired boundary conditions is slightly less direct. We define a =μ 0 p (κ e = q) and b =μ 1 p (κ e = q). Comparison between boundary condition impliesμ 0 n,p (κ e = q) ≤μ 0 p (κ e = q) = a for any e ∈ E n . Recall that h(κ) = |{e ∈ E(G) : κ e = q}| denotes the number of q-bonds and s(κ) similarly the number of 1-bonds. The definition of a and b implies for 0 < ε < 1 − ā µ 0 n,p h(κ) ≤ a|E n | ⇒μ 0 n,p h(κ) ≤ (a + ε)|E n | ≥ ε.
Our goal is to show that b − ε ≤ a + ε. We denote by ∆ 0 and ∆ 1 the graph Laplacian on Λ n with free and wired boundary conditions respectively. To compare the boundary conditions we denote by T 1 = ST(Λ w n ) the set of wired spanning trees on Λ n and by T 0 = ST(Λ n ) the set of spanning trees on Λ n with free boundary conditions. There is a map Φ : T 0 → T 1 such that Φ(t) Λn−1 = t Λn−1 . Indeed, removing all edges in E n ⊂ E n−1 from t we obtain an acyclic subtree of Λ w n , hence we can find a tree Φ(t) such that t Λn−1 ⊂ Φ(t) ⊂ t. The observation |t \ Φ(t)| = |∂Λ n | − 1 implies that w(κ, t) ≤ w(κ, Φ(t))q |∂Λn|−1 . Since Φ does not change the edges in E n−1 each tree t ∈ T 1 has at most 2 |En\En−1| preimages. We obtain that Similarly, there is an injective mapping Ψ : Inserting the bound |E n \ E n−1 | ≤ 2d|∂Λ n | we infer from the definition (3.7) for any κ ∈ {1, q} En .
The next result is a non-uniqueness result for the random conductance model.
Theorem 5.4. In dimension d = 2 and for q > 1 sufficiently large there are two distinct Gibbs measuresμ 1 p sd =μ 0 p sd at the self-dual point defined by equation (1.8). The proof uses duality of the random conductance model and can be found in Section 6. This result easily implies Theorem 2.4.
Proof of Theorem 2.4. Using Proposition 4.19 we infer from Theorem 5.4 the existence of two translation invariant extended gradient Gibbs measuresμ 0 andμ 1 constructed fromμ 0 p sd =μ 1 p sd . Their η-marginals µ 0 and µ 1 are not equal since then the κ-marginalsμ 1 andμ 2 would agree. They both have zero tilt by Proposition 4.19 and the definition ofμ shows thatμ is translation invariant ifμ is translation invariant.
Remark 5.5. A proof similar to Lemma 3.2 in [6] shows that ergodicity ofμ 1 andμ 2 implies that µ 0 and µ 1 are itself ergodic. The only difference is that η given κ is not independent (which κ given η is). Instead one has to rely on the decay of correlations for Gaussian fields stated in Appendix B.
Theorem 5.6. For d ≥ 4 there is q 0 > 1 such that for p ∈ [0, 1] and q ∈ [1, q 0 ) the Gibbs measure for the random conductance model is unique. Similarly, for d ≥ 4 and q ≥ 1 there is a p 0 = p 0 (q, d) > 0 such that the Gibbs measure is unique for Proof. We are going to apply Dobrushin's criterion (see, e.g., [20,Theorem 8.7]. The necessary estimate is basically a refined version of the proof of Lemma 4.6. Fix two edges f, g ∈ E(Z d ). Recall the notation λ ±± = λ ±± f g and λ ± = λ ± f introduced above Theorem 4.2. We will writeγ f =γ {f } in the following. Note that (4.45) andγ f (λ + , λ) +γ f (λ − , λ) = 1 imply that where Q λ − denotes the weighted spanning forest measure on Z d with conductances λ − . We need to bound the entries of the Dobrushin interdependence matrix given by (5.12) Since the derivative of the map x → p/(p To simplify the notation we assume λ = λ −− . We can express Q λ −+ (f ∈ t) through the measure Q λ −− = Q λ as follows (5.14) A sequence of manipulations then shows that The numerator can be rewritten using the transfer-current Theorem for two edges (see [3,Page 10] and equation below 4.3 in [27]) where I κ f (g) denotes the current through g in a resistor network with conductances κ when 1 unit of current is inserted (respectively removed) at the ends of f (using a fixed orientation of the edges here, e.g., lexicographic). All together we have shown that Using electrical network theory we can express for f = (x, x + e i ) and g = (y, y + e j ) where G κ denotes the inverse of the operator ∆ κ which exists in dimension d ≥ 3 and whose derivative exists in dimension d ≥ 2. Combining the bound (B.6) in Lemma B.2, (5.17), and (5.18) we conclude In dimension d ≥ 4 the sum is finite. Now, for fixed q, the sum becomes smaller 1 for p sufficiently close to 0 or 1. Therefore there is p 0 = p 0 (q, d) such that the Gibbs measure is unique for On the other hand, the constant C(q, d) from Lemma B.1 is decreasing in q. Therefore we can estimate uniformly for p ∈ [0, 1] and for q ≤ 2 Hence the Dobrushin criterion is satisfied for q sufficiently close to 1 and all p ∈ [0, 1].
Remark 5.7. 1. Note that the gradient-gradient correlations in gradient models at best only decay critically with |x| −d (which is the decay rate for the discrete Gaussian free field). In particular, the sum of the covariance g∈E(Z d ) Cov(η f , η g ) diverges in this type of model. We use crucially in the previous theorem that the decay of correlations is better for the discrete model: They decay with the square of the gradient-gradient correlations.
2. The averaged (annealed) second order derivative of the Greens functions decays with the optimal decay rate |x| −d as shown in [13]. For the application of the Dobrushin criterion we, however need deterministic bounds which are weaker.
3. To extend the uniqueness result for q close to 1 to dimensions d = 3 and d = 2 one would need estimates for the optimal Hölder exponent α depending on the ellipticity contrast of discrete elliptic operators. Here the ellipticity contrast can be bounded by q. There do not seem to be any results in this direction in the discrete setting. In the continuum setting the problem is open for d ≥ 3, but has been solved for d = 2 in [29]. In this case α → 1 as the ellipticity contrast converges to 1. A similar result in the discrete setting would imply uniqueness of the Gibbs measure for small q in dimension 2.
Note that we can again lift the uniqueness result for the Gibbs measure of the random conductance model to a uniqueness result for the ergodic gradient Gibbs measures with zero tilt.
Proof of Theorem 2.5. The proof follows from the uniqueness of the discrete Gibbs measure proven in Theorem 5.6 in the same way as the proof of Theorem 2.3 which can be found above Remark 5.2.
Open questions Let us end this section by stating one further result and two conjectures regarding the phase transitions of this model. They are most easily expressed in terms of percolation properties of the model even though the interpretation as open and closed bonds is somehow misleading in this context. We write x ↔ y for x, y ∈ Z d and κ if there is a path of q-bonds in κ connecting x and y and similarly for sets. Observe that the results of [17] can be applied to the model introduced here and we obtain the existence of a sharp phase transition.
Theorem 5.8. For every q the model undergoes a sharp phase transition in p, i.e., there is p c (q, d) such that the following two properties hold. On the one hand there is a constant c 1 > 0 such that for p > p c sufficiently close to p cμ On the other hand, for p < p c there is a constant c p such that Proof. The proof of Theorem 1.2 in [17] for the random cluster model applies to this model. Indeed, it only relies on µ 1 n,p being strongly positively associated and a certain relation for the p derivative of events stated in Theorem 3.12 in [22] which is still true since the p-dependence is the same as for the random cluster model.  [17] for the random cluster model .
In the random cluster model the most interesting phenomena happen for p = p c and the subcritical and supercritical phase are much simpler to understand (in particular in d = 2). Due to the differences explained in Remark 3.2 those questions seem to be harder for our random conductance model. Nevertheless we conjecture the following stronger version of Theorem 5.1 and Theorem 5.6 Conjecture 5.10. For p = p c there is a unique Gibbs measure.
Note that the sharpness result Theorem 5.8 shows that the probability of subcritical q-clusters to be large is exponentially small. Nevertheless it is not clear how this can be used to show uniqueness of the Gibbs measure in our setting.
The behaviour at p c is also very interesting. A phase transition is called continuous if µ 1 pc (0 ↔ ∞) = 0 and otherwise it is discontinuous. For the random cluster model in dimensions d = 2 the phase transition is continuous for q ≤ 4 and otherwise discontinuous. Moreover, the uniqueness of the Gibbs measure at p c is equivalent to a continuous phase transition. We do not know whether the same is true for the random conductance model considered here. But we expect the general picture to be true also for the random conductance and we make this precise in a second conjecture.
Conjecture 5.11. There is a q 0 = q 0 (d) such that for q > q 0 there is non-uniqueness of Gibbs measuresμ 1 pc,q =μ 0 pc,q at the critical point while for q < q 0 the Gibbs measures agree, i.e.,μ 1 pc,q =μ 0 pc,q .
A partial result in the direction of this conjecture is Theorem 5.4 that states non-uniqueness for large q in dimension d = 2 and Theorem 5.6 that shows uniqueness for q close to 1 and d ≥ 4.

Duality and coexistence of Gibbs measures
In this section we are going to prove that µ 0 p sd = µ 1 p sd for large q which implies the non-uniqueness of gradient Gibbs measures stated in Theorem 2.4. This is a new proof for the result in [5]. They consider conductances q 1 , q 2 with q 1 q 2 = 1 which makes the presentation slightly more symmetric.
In contrast to their work we do not rely on reflection positivity but instead we exploit the planar duality that is already used in [5] to find the location of the phase transition. Therefore it is not possible to extend the argument given here to d ≥ 3 while the proof using reflection positivity is in principle independent of the dimension (note that the spin wave calculations in [5] can be simplified substantially and generalised to d ≥ 3 using the Kirchhoff formula cf. [9,Section 5.7]). In addition to planar duality we rely on the properties proved in Section 4, in particular on the Kirchhoff formula. Similar arguments were developed in the context of the random cluster model and we refer to [22,Section 6 and 7].
We proceed now by stating the duality property in our setting. For a planar graph G = (V, E) we denote its dual graph by G * = (V * , E * ). The dual graph has the faces of G as vertices and the vertices of G as faces and each edge has a corresponding dual edge. For a formal definition of the dual of a graph and the necessary background we refer to the literature, e.g., [32].
For any configuration κ : E → {1, q} we define its dual configuration κ * ∈ {1, q} E * by κ * e * = 1 + q − κ e where e * ∈ E * denotes the dual edge of an edge e ∈ E. More generally we denote for E 1 ⊂ E by E * 1 = {e * : e ∈ E 1 } the dual edges of the edges E 1 . We also introduce the notation E d 1 = {e * ∈ E * : e / ∈ E 1 } = (E c 1 ) * for E 1 ⊂ E for the dual set of an edge subset. Note that E 1 is acyclic if and only if E d 1 is spanning, i.e., every two points x * , y * ∈ V * are connected by a path in E d 1 . In particular, t ⊂ E is a spanning tree in G if and only if t d is a spanning tree in G * and the map t → t d is an involution and in particular bijective from ST(G) to ST(G * ).
Recall that h(κ, t) = |{e ∈ t : κ e = q}| denotes the number of q-bonds in the set t ⊂ E(G) of κ and the similar definition of s(κ, t) for the number of soft 1-bonds in t. The definitions imply that The last two identities follow from the observation that s(κ * , t d ) = h(κ, E \ t) and similarly for s and h interchanged. We calculate the distribution of κ * if κ is distributed according to P G,p This implies that if κ is distributed according to P G,p the dual configuration κ * is distributed according to P G * ,p * where q * = q and Note that the self dual point p sd defined by p * sd = p sd is given by the solution of (6.5) We will now restrict our attention to Z 2 . Let us mention that detailed proofs of the topological statements we use can be found in [26]. We can identify the dual of the graph (Z 2 , E(Z 2 )), which will be denoted by ((Z 2 ) * , E(Z 2 ) * ), with Z 2 shifted by the vector w = ( 1 2 , 1 2 ). We also consider the set of directed bonds E(Z 2 ) and E(Z 2 ) * . For a directed bond e = (x, y) ∈ E(Z 2 ) we define its dual bond as the directed bond e * = ( 1 2 (x + y + (x − y) ⊥ ), 1 2 (x + y + (y − x) ⊥ ) where ⊥ denotes counter-clockwise rotation by 90 • , i.e., the linear map that satisfies e ⊥ 1 = e 2 , e ⊥ 2 = −e 1 . In other words, the dual of a directed bond e is the bond whose orientation is rotated by 90 • counter-clockwise and crosses e. Every point x ∈ Z 2 determines a plaquette with corners z 1 , z 2 , z 3 , z 4 ∈ (Z 2 ) * where z i are the four nearest neighbours of x in (Z 2 ) * and the plaquette has faces e * 1 , e * 2 , e * 3 , e * 4 ∈ E(Z 2 ) * where e * i are the dual bonds of the four bonds e i that are incident to x. Vice versa every point z ∈ (Z 2 ) * determines a plaquette in Z 2 . We write P(Z 2 ) for the set of plaquettes of Z 2 .
For a bond e = {x, y} we define the shifted dual bond e + w = {x + w, y + w}. Similarly, we define . For a subgraph G ⊂ Z 2 we denote by P(G) = {P ∈ P(Z 2 ) : all faces of P are in E(G)} the plaquettes of G. A subgraph G ⊂ Z 2 is called simply connected if the union of all vertices v ∈ V(G), all edges {x, y} ∈ E(G) which are identified with the line segment from x to y in R 2 and all plaquettes P(G) is a simply connected subset of R 2 .
An important tool in the analysis of planar models from statistical mechanics is the use of contours. Let us provide a notion of contours that is useful for our purposes. Our definition is slightly more complicated than the definition of contours for the random cluster model. We consider closed paths γ = (x * 1 , . . . , x * n , x * 1 ) with x * i ∈ (Z 2 ) * (not necessarily all distinct) along pairwise distinct directed dual bonds b * 1 = (x * 1 , x * 2 ), . . . , b * n = (x * n , x * 1 ). We denote the vertices in the contour by V(γ) * = {x * i : 1 ≤ i ≤ n} and the bonds by E(γ) Similarly we write E(γ) = { b i : 1 ≤ i ≤ n} for the corresponding primal bonds. We also consider the underlying sets of undirected bonds E(γ) and E(γ) * . Finally, we denote the heads and tails of b i by y i and z i , i.e., b i = (z i , y i ).
Definition 6.1. A contour γ is a closed path in the dual lattice without self-crossings in the sense that there is a bounded connected component int(γ) of the graph (Z 2 , E(Z 2 ) \ E(γ)) such that ∂(int(γ)) = {z i : 1 ≤ i ≤ n}. We denote the union of the remaining connected components by ext(γ) and we define the length |γ| of the contour as the number of (directed) bonds it contains, i.e., |γ| = | E(γ)| = n.
Contours are a suitable notion to define interfaces between hard and soft bonds.
Definition 6.2. A contour γ is a q-contour for κ if the following two conditions hold. First, the primal bonds b ∈ E(γ) are soft, i.e., κ b = 1. Moreover, for every plaquette with center x * ∈ V(γ) * all its faces b such that b ∈ E(int(γ)) are hard, i.e., satisfy κ b = q.
Our goal is to show that q-contours are unlikely for large values of q and p ≤ p sd . We now fix a contour γ and introduce some useful notation and helpful observations for the proof of the following theorem. We use the shorthand G i = int(γ) and E i = E(int(γ)). We observe that G i is simply connected because γ is connected and without self-crossings. Therefore the faces of G i consist of plaquettes in Z 2 and one infinite face. We also consider the graph G with edges E = E i ∪ E(γ) and endpoints of edges as vertices. Let1 ∈ {1, q} E denote the configuration given by1 e = 1 for all e ∈ E. We write G w = G/∂G = G/ext(γ) for the graph G with wired boundary conditions. Moreover we introduce the graph H * with edges E * i and their endpoints as vertices. We claim that H * /∂H * agrees with the graph theoretic dual of G i . To show this we need to prove that we identify all vertices that lie in the same face of G i . First we note that every point in • H * = H * \ ∂H * determines a plaquette in P(G i ) and this is a bijection. Then it remains to show that all vertices in ∂H * lie in the infinite face of G i . This follows from the observation To show the observation we note that if x * ∈ ∂H * then there are edges e * 1 / ∈ E(H * ) and e * 2 ∈ E(H * ) incident to x * . This implies that there is a face e = {z 1 , z 2 } of the plaquette with center x * such that z 1 ∈ V(G i ) but e / ∈ E(G i ). Then e ∈ E(γ) and therefore x * ∈ V(γ) * because x * is an endpoint of e * ∈ E(γ) * . This ends the proof of the inclusion '⊂'. Now we note that if x * ∈ V(H * ) ∩ V(γ) * there is an edge e * ∈ E(γ) * incident to x * which is not contained in E(H * ) and therefore x * ∈ ∂H * .
Finally we remark that if γ is a q-contour for κ then Indeed, we argued above that if e * ∈ E(H * ) is incident to ∂H * then x * ∈ V(γ) * . Thus e ∈ E i is a face of the plaquette with center x * so that the definition of q-contours implies that κ e = q. Theorem 6.3. Let γ be a contour. The probability that γ is a q-contour under the measure P G w ,Ei,1 for p = p sd is bounded by Remark 6.4. The general idea of the proof is the same as when proving similar estimates for the Ising model. One tries to find a map from configurations where the contour is present to configurations where this is not the case and then estimates the corresponding probabilities. The more similar argument for the random cluster model can be found, e.g., in Theorem 6.35 in [22]. For an illustrated version see [16]. Proof. We denote the set of all κ ∈ {1, q} E such that γ is a q-contour for κ by Ω γ .
Step 1. We define a map Φ : Ω γ → {1, q} E with Φ(κ) = κ # as follows. Recall the definition of the dual configuration κ * on E * ⊂ E(Z 2 ) * and define for e ∈ E We claim that By definition of κ # , we only need to consider the case e − w ∈ E i * = E(H * ). We will show a slightly more general statement. Let us introduce the setẼ = E(H * ) + w = E * i + w ⊂ E and the graph G consisting of the edgesẼ and their endpoints as vertices. See Figure 2 for an illustration of this construction. We remark thatG agrees with H * shifted by w, which we denote byG = H * + w. Equation (6.7) implies that κ # e = κ * e−w = 1 for e ∈G incident to ∂G (6.11) because then e − w ∈ E(H * ) is incident to ∂H * . It remains to show that all edges e ∈ E ∩Ẽ \ E i are incident to ∂G. From e ∈ E \ E i we conclude that e ∈ E(γ). The edge e − w has a common endpoint with e * ∈ E(γ) * and is therefore incident to V(γ) * in this case. Using the observation (6.6) this implies that e − w ∈ E(H * ) is incident to ∂H * . Our goal is to compare the probabilities of P G w ,Ei,1 (κ Ei ) and P G w ,Ei,1 (κ # Ei ). To achieve this we use a strategy similar to the proof of Lemma 5.3.
Step 2. We define a map Ψ : ST(G w ) → ST(G w ) with Ψ(t # ) = t in the following steps 1. We choose deterministically a subsett ⊂ t # Ẽ such thatt is a spanning tree onG/∂G and all edges in t # Ẽ \t are incident to ∂G.

We set
3. We consider a fixed b ∈ E(γ) that is incident to int(γ) and ext(γ) and we define t = Ψ(t # ) = Ψ(t # ) Ei ∪ b See Figure 3 for an illustration of the construction.
We have to show that this construction is possible, in particular that t ∈ ST(G w ). We start with the first step. The relationG ⊂ G implies ∂G ∩G ⊂ ∂G. (6.12) HenceG/∂G agrees with (G/∂G)/(G c ∪ ∂G) up to self loops. This implies that t # Ẽ is spanning iñ G/∂G if t # ∈ ST(G w ). We consider the subset t ⊂ t # Ẽ consisting of all edges e ∈ t # Ẽ that are not incident to ∂G. The set t contains no cycles because t # ∈ ST(G w ) and no edge in t is incident to ∂G by (6.12). Therefore we can select a spanning treet inG/∂G with t ⊂t ⊂ t # Ẽ deterministically, e.g., using Kruskal's algorithm.
We now argue that the second and third step yield a spanning tree in G w . Clearly it is sufficient to show that Ψ(t # ) Ei ∈ ST(G i ). We note that the relation betweenG and H * implies thatt − w is a spanning tree on H * /∂H * . As shown before the theorem H * /∂H * agrees with the dual of G i and thus (t − w) d ∈ ST(G i ).
Step 3. The next step is to consider κ # = Φ(κ) and t = Ψ(t # ) and compare the weights w(κ # , t # ) and w(κ, t). First we argue that w(κ # , t # ) = w(κ # ,t). (6.13) Sincet ⊂ t # it is sufficient to show that t # \t contains only edges e such that κ # e = 1. Indeed, let e be an edge in t # \t. For e / ∈Ẽ we have κ # e = 1 by definition. Let us now consider e ∈Ẽ ∩ (t # \t). (6.14) By construction oft the edge e is incident to a vertex v ∈ ∂G. This implies that e − w ∈ E(H * ) is incident to v − w ∈ ∂H * ⊂ V(γ) * . Using (6.7) we conclude that For the treest and Ψ(t # ) E we can apply the usual duality relations stated before. Using (6.2) and as before κ # = Φ(κ) and t = Ψ(t # ) we obtain (6.16) We compute In the last step we used that t ∩ E i is a free spanning tree on G i and therefore has |V(G i )| − 1 edges.
Step 4. We bound the number of preimages of a tree t under Ψ. Note that Ψ factorizes into two maps t # →t → t. The second map is injective since we only pass to the dual tree which is an injective map and we add one additional edge. For the first map we observe that we only delete edges e incident to ∂G. However, for x ∈ ∂G the point x − w ∈ ∂H * is contained in the contour by (6.6). Therefore there are at most 4|γ| such edges. We conclude that for every t ∈ ST(G w ). The displays (6.17) and (6.18) imply w(κ, t).

(6.19)
Step 5. We can now estimate the probabilities of the patterns κ and κ # = Ψ(κ) under P G w ,Ei,1 using (6.19) and κ # e = 1 =1 e for e ∈ E \ E i where we used equation (6.5) of p sd in the last step. The definition of κ # implies that h(κ # , E i ) = h(κ # ,Ẽ) = s(κ, E i ) and we get We end up with the estimate Conclusion. Note that the map Φ is injective, hence Using correlation inequalities we can derive the following stronger version of the previous theorem. For a simply connected subgraph H ⊂ Z 2 we say that γ is contained in H if all faces of plaquettes with center x * for x * ∈ V(γ) * are contained in E(H). Corollary 6.5. For any p ≤ p sd , any simply connected subgraph H ⊂ Z 2 , and a contour γ that is contained in H the probability that γ is a q-contour can be estimated by Proof. We estimate For the measure P H,E(H)\V(γ),1 the bonds crossing the contour are fixed to the correct value. Hence the event that γ is a q-contour for κ is increasing under this event, such that the stochastic domination results proved in Lemma 4.9 and Corollary 4.12 imply that where G denotes the graph corresponding to γ as introduced above Theorem 6.3. Theorem 6.3 implies the claim.
We can now give a new proof for the coexistence result stated in Theorem 2.4.
Proof of Theorem 5.4. First we note that the duality between free and wired boundary conditions in finite volume implies that µ 0 p sd and µ 1 p sd are dual to each other in the sense that if κ ∼ µ 0 p sd then κ * ∼ µ 1 p sd (on (Z 2 ) * )). The proof is the same as for the random cluster model, see, e.g., [22,Chapter 6]. Hence, it is sufficient to show that µ 0 p sd (κ e = q) < 1/2 because then we can conclude that µ 1 p sd (κ e = q) =μ 0 p sd (κ e = 1) > 1/2 (6.27) whenceμ 1 p sd =μ 0 p sd . Note that if κ e = q and there is any contour γ such that e ∈ E(int(γ)) and κ b = 1 for b ∈ E(γ) then there is a q-contour surrounding e. We can thus estimate for e ∈ E n P Λn+1,En,1 (κ e = q) ≤ P Λn+1,En,1 (there is a q-contour around e) (6.28) where as before1 e = 1 for all e. The shortest contour γ that surrounds the edge e has length 6 so the bound in Corollary 6.5 implies that P Λn+1,En,1 (γ is a q-contour) ≤ C/q 1 4 for any γ surrounding e. Using Corollary 4.12 we can compare boundary conditions to obtain the relationμ 0 n P Λn+1,En,1 . This and a standard Peierls argument imply for q sufficiently largē Taking the limit n → ∞ we obtainμ 0 p sd (κ e = q) ≤ 1 4 .

A Proofs of Proposition 4.18 and Proposition 4.19
In this section we pay the last remaining debt of proving two propositions from Section 2.
Proof of Proposition 4.18. For λ ∈ {1, q} E(Z d ) and E ⊂ E(Z d ) finite we define the cylinder event With a slight abuse of notation we drop the pullback from the notation when we consider the set Since all local cylinder events in F can be written as a union of events of the form A(λ E L ) it is by Remark 2.2 sufficient to show for all L, n ≥ 0 and all λ ∈ {1, q} E(Z d ) . Using the quasilocality ofγ stated in Corollary 4.17 and Remark 4.21 in [20] it is sufficient to consider L = n and we will do this in the following. We are going to show the claim in a series of steps.
Step 1. We investigate the distribution of the κ-marginal conditioned on ω E c N . Sinceμ is a gradient Gibbs measure we know by (3.2) that for ω ∈ R where Z is the normalisation and denotes the partition function corresponding to the configuration κ. Let ϕ ∈ R Z d be the configuration such that ∇ϕ = ω and ϕ(0) = 0. We denote by χ κ the corrector of κ, i.e., the solution of ∇ * κ∇χ κ = 0 with boundary values ϕ Λ c . A shift of the integration variables and Gaussian calculus implies (see also (3.4)) where0 is the configuration with vanishing gradients, i.e.,0 e = 0 for e ∈ E(Z d ). The necessary calculation to obtain (A.5) basically agrees with the calculation that shows that the discrete Gaussian free field can be decomposed in a zero boundary discrete Gaussian free field and a harmonic extension. We now restrict our attention to We introduce the law of the κ-marginal for wired non-constant boundary conditions for κ ∈ {1, q} E N bȳ Note thatμ 1,0 N =μ 1 N whereμ 1 N was defined in (4.36).
Step 2. In this step we are going to show that there is N 0 ∈ N depending on n such that for N ≥ N 0 and uniformly in λ ∈ {1, q} E(Z d ) i.e., the boundary effect is negligible. We start by showing that typically the difference between the corrector energies for configurations κ andκ that only differ in E n will be small. This will allow us to estimate the difference betweenμ 1 N andμ 1,ω N conditioned to agree close to the boundary. Recall that we consider the case that Λ = Λ N is a box. The Nash-Moser estimate stated in Lemma B.1 combined with the maximum principle for the equation ∇ * κ∇χ κ = 0 imply for b ∈ E n and some α = α(q) > 0 We introduce the event M (N ) = {ω : max x∈∂Λ N ϕ(x) − min y∈∂Λ N ϕ(y) ≤ (ln N ) 3 }. Consider configurations κ,κ ∈ {1, q} E(Z d ) such that κ e =κ e for e / ∈ E n . Using the fact that the corrector is the minimizer of the quadratic form (∇χ κ , κ∇χ κ ) E N with given boundary condition we can estimate (A.9) From (A.8) we infer that for N ≥ 2n and ϕ ∈ M(N ) By choosing N 1 ≥ 2n sufficiently large we can ensure that for N ≥ N 1 , ϕ ∈ M(N ), and uniformly in κ,κ as before Using this in (A.5) we conclude that for N ≥ N 1 ∨ 2n, ω ∈ M(N ), ε < 1/3, and λ ∈ {1, q} E(Z d ) From Lemma A.1 below and Proposition 4.20 we infer that for an extended gradient Gibbs measurẽ µ associated to an ergodic zero tilt Gibbs measure µ and any λ ∈ {1, q} E(Z d ) 14) for all N ≥ N 2 and N 2 sufficiently large. We conclude that for N ≥ N 0 : Step 3. Using the previous results we can now finish the proof. We rewritē (A. 16) The identity above and the fact thatγ En is proper imply adding and subtracting the same term We continue to estimate the right hand side of this expression. We start with the first term. Sincē µ 1 Λ N is a finite volume Gibbs measure (see (4.53)) we have for Using this and the bound (A.7) we obtain for N ≥ N 0 We now address the second term on the right hand side of (A.17). By Lemma 4.16 there is N 3 such that for N ≥ N 3 and any λ, This ends the proof.
The following simple Lemma was used in the proof of Proposition 4.18.
Lemma A.1. Let λ ∈ {1, q} E(Z d ) and denote by ϕ λ the centred Gaussian field on Z d with ϕ(0) = 0 and covariance ∆ −1 λ . Then ϕ λ satisfies P max Proof. We use the notation1 ∈ {1, q} E(Z d ) for the configuration given by1 e = 1 for e ∈ E(Z d ). The Brascamp-Lieb inequality (see [7,Theorem 5.1]) implies for the centred Gaussian fields ϕ λ and ϕ1 that It is well known that for a centred Gaussian random vector X ∈ P(R m ) with E(X 2 i ) ≤ σ 2 the expectation of the maximum is bounded by It remains to provide a proof of Proposition 4.19. We will only sketch the argument.
Proof of Proposition 4.19. First we remark that the law of (κ, ∇ϕ κ ) is a Borel-measure on . This follows from Carathéodory's extension theorem and the observation that for a local event A ∈ E E with E ⊂ E(Z d ) finite the function κ → µ ϕ κ (A) is continuous (this can be shown using Lemma B.3). By Remark 2.2 it is sufficient to prove thatμγ Λn =μ for all n. To prove this we use an approximation procedure. We fix n and define for N > n a measureμ N on R E(Z d ) g × {1, q} E(Z d ) as follows. The κ-marginal ofμ N is given byμ N =μγ Λ w N En where as before we extendedγ Λ w N En to a proper probability kernel on {1, q} E(Z d ) . For given κ, let ϕ κ be the centred Gaussian field with zero boundary data outside of • Λ N and covariance (∆ Λ w N κ E N ) −1 where∆ Λ w N κ E N was defined in Section 3. The measureμ N is the joint law of (κ, ϕ κ ) where κ has lawμ N . We claim that for N > ñ µ NγΛn =μ N .
(A. 28) We prove this by showing the statement for the measuresμ N · |A(λ E N \En ) for every configuration λ ∈ {1, q} E(Z d ) . To shorten the notation we writeμ λ N =μ N · |A(λ E N \En ) . By definition ofμ N the ϕ-field conditioned on κ has density exp(− 1 2 (ϕ,∆ where dϕ Λ = x∈Λ dϕ x denotes the Lebesgue measure. This implies for B ∈ B(R Λ N ) and σ ∈ {1, q} E(Z d ) such that We use the definition ofμ N and the fact that specifications are proper to rewritẽ . (A.31) The last three displays, a summation by parts, and (3.6) lead us tõ Combining this with the definition (3.2) we conclude that for σ ∈ {1, q} E(Z d ) such that σ E N \En = λ E N \En and ω ∈ R This impliesμ λ NγΛn =μ λ N and (A.28) follows directly. It remains to pass to the limit in equation (A.28), i.e., we show that the right hand side converges in the topology of local convergence toμ and the left hand side toμγ Λn thus finishing the proof. We only sketch the argument. Sinceγ(A, ·) is a measurable, local, and bounded function if A is a local event it is sufficient to show thatμ N converges toμ locally in total variation, that is for every Λ ⊂⊂ Z d Where we used the σ-algebra A E on R E(Z d ) g × {1, q} E(Z d ) defined in Section 3 as the product of the pullbacks of E E and F E . We first consider the κ-marginals ofμ N andμ. They are given byμ N =μγ In the second step we used that the specifications are proper thus we can assume A ⊂ A(κ E c n ) and use that |A(κ E c n )| ≤ 2 |En| . Using Lemma 4.16 we conclude We address the η-marginals of the measuresμ andμ N . We writeμ(· | κ),μ N (· | κ) ∈ P(R E(Z d ) g ) for the conditional distribution of the η-field for a given κ ∈ {1, q} E(Z d ) . From the construction this is well defined for every κ. We define the centred Gaussian field ϕ κ by ϕ κ (0) = 0 and its covariance (∆ κ ) −1 and the centred fields ϕ κ N pinned to 0 outside of • Λ N with covariance (∆ Λ w N κ ) −1 and we denote their gradients by η κ = ∇ϕ κ and η κ N = ∇ϕ κ N . Note that by definition ofμ andμ N the law of η κ and η κ N coincides withμ(· | κ) andμ N (· | κ). Fix an integer L. We introduce the Gaussian vectors X κ = (ϕ κ (x) − ϕ κ (0)) x∈Λ L and X κ N = (ϕ κ N (x) − ϕ κ N (0)) x∈Λ L . Note that given X κ , X κ N the gradient field η κ E(Λ L ) respectively η κ N E(Λ L ) can be expressed as a function of X κ and X κ N respectively. This implies that (A.37) Theorem 1.1 in [15] states that the total variation distance between two centred Gaussian vectors Z 1 , Z 2 with covariance matrices Σ 1 and Σ 2 can be bounded by 3 2 |Σ −1 1 Σ 2 − 1| F where | · | F denotes the Frobenius norm. Using this theorem and the uniform convergence of the covariance of η κ N to the covariance of η κ stated in Lemma B.3 we conclude that

B Estimates for discrete elliptic equations
In this appendix we collect some regularity estimates for discrete elliptic equations. We consider as before uniformly elliptic κ : E(Z d ) → R + with 0 < c − ≤ κ e ≤ c + < ∞ for all e ∈ E(Z d ). We denote corresponding set of conductances by M (c − , c + ) = [c − , c + ] E(Z d ) .
Next we state a discrete version of the well known Nash-Moser estimates for scalar elliptic partial differential equations with L ∞ coefficients. (B.2)