STRICT INEQUALITY FOR PHASE TRANSITION BETWEEN FERROMAGNETIC AND FRUSTRATED SYSTEMS 1

We consider deterministic and disordered frustrated systems in which we can show some strict inequalities with respect to related ferromagnetic systems. A case particularly interesting is the Edwards-Anderson spin-glass model in which it is possible to determine a region of uniqueness of the Gibbs measure, which is strictly larger than the region of uniqueness for the related ferromagnetic system. We analyze also deterministic systems with j J b j 2 [ J A ; J B ], where 0 < J A (cid:20) J B < 1 , for which we prove strict inequality for the critical points of the related FK model. The results are obtained for the Ising models but some extensions to Potts models are possible.


Introduction
The problem of proving inequalities in probabilistic models is very common; especially in statistical mechanics, there are models in which many qualitative properties, such as the existence of a phase transition, are only shown by using inequalities. In particular, during the last ten years there was a large effort to prove strict inequalities between critical points (see [AG91,BGK93]). In these works, inequalities are proved between the critical points in percolation and in the Ising model and there is an extension to the Potts model and to many-body interactions in [Gr94].
There are some recent papers that prove this kind of results for disordered systems. These are models in which the interactions are themselves random variables, so that the Gibbs measure becomes a function of these random variables. Important results for disordered models are in [Ca98,Gr99]. In the second work there is a general strategy to study this kind of problems in a wide context.
In this paper, we will prove a strict inequality between phase transitions in the ferromagnetic Ising and in the Edwards-Anderson models (see [Is25,EA75]). We have to stress that it is not rigorously known whether there is a phase transition in the Edwards-Anderson model at some positive temperature, but in any case our result makes sense. In a future paper we will also define a frustrated Potts model and a frustrated many-body model for which it is possible to extended our results for the two-body interactions.
Our results, for the strict inequality between the percolation critical points of the random cluster models, are related in spirit and partly in methodology to the recent work of Grimmett [Gr99] (see also [BGK93]) and to work of Campanino [Ca98]. Then we use a work of Newman [Ne94] (see also [Ne97]) to show, as a consequence of the random cluster percolation result, a strict inequality for the phase transition of the related Gibbs measures. The main differences between our work and [Ca98,Gr99] are: a) we prove a strict inequality for the phase transition, i.e. for uniqueness of the Gibbs distribution, and not only for symmetry breaking; b) we show also a strict inequality for the phase transitions of disordered ferromagnetic Ising models (such a result, but with a different methodology, is also proved by Gandolfi [Ga98]); c) our method can be extended to Potts models and to frustrated many-body models; in a future paper we will provide these extensions (this is not explicit in [Ca98] and we do not know if it is possible to find a related Gibbs measure for the random cluster measure in [Ca98] besides the Ising model). Now we will deal only with the Ising models for the sake of clarity and not to add unnecessary difficulties.
We present here the main result on disordered systems which was suggested as an open problem by Newman (see C. Newman, Topics in Disordered Systems. Lecture in Mathematics, 1997 [Ne97]).
Theorem 1 Let J be an interaction configuration with |J e | = 1. The J e are i.i.d. random variables with the probability Q(J e = −1) =p and 0 <p < 1. Then the Gibbs measure π Jβ is unique in a region strictly larger than the uniqueness region of the ferromagnetic Gibbs measure π F |J|β Q-a.e.: We remark that the standard Edwards-Anderson model has p = 1/2, so we obtain that the region of uniqueness for the E-A model is larger than for the ferromagnetic Ising model. We will prove this theorem at the end of section 4.

General definitions and main results
The Graph. We consider the infinite graph E d = (Z d , E(Z d )) embedded in R d ; the graph has an edge for every pair of vertices having Euclidean distance equal to 1. We will often abbreviate E(Z d ) with E. We will use the following definitions: the distance between two vertices x, y ∈ Z d is d(x, y) = max i=1,...,d |x i − y i | and the distance between two sets A, B ⊂ Z d is d(A, B) = min x∈A,y∈B d(x, y). The same definition of distance applies also to edges thought as a set of two vertices. We say that the vertices i and k belong to an edge e if e = {i, k}. For a given set of edges F ⊂ E we will call Given a finite set of edges γ we say that it is a minimal cut set if there exists a finite connected set of vertices A such that γ = ∂A; in E d , given a minimal cut set γ, there is only one connected set of vertices A verifying the previous condition, so we will call this finite connected set of vertices A the inner part of γ -writing Int(γ) = Aand A c the outer part of γ -writing Out(γ) = A c .
Blocks and cover of Z d . We need to introduce the following subsets of with center v ∈ Z d and size n ∈ N . We will call ∂ E B n (v), defined as follows the outer boundary of B n (v) and The collection of blocks {B n (k)} k∈Cn where C n is the sub-lattice of Z d defined as . Two blocks of size n belonging to the cover are said to be adjacent if the Euclidean distance between their centers is equal to 2n + 1. It is easy to see that the intersection of adjacent blocks is not empty and is equal to the intersection of their inner boundaries. We remark also that ∂ I B n (v) has the important property of being a minimal cut set.   We will also consider a finite set of vertices Λ ⊂ Z d but we will always think of it as a subset of Z d so that the definition of inner and outer part makes sense; we define A cluster C is the maximal connected set of vertices having the property that for all the We say that a potentialφ is a gauge transformation of the potential φ if for each vertex i there exists a 1:1 mapping Random Cluster Measure or FK-measure. We will follow the exposition in [Ne97]. Let Λ ⊂ Z d be a finite set of vertices and τ a configuration in Ω. Define the random cluster model (or the FK-measure) on the finite space H E,Λ with τ boundary conditions as: where k(η) is the number of clusters which do not have open edges in ∂Λ (the clusters not touching the boundary), and Z τ Λ,Jβ is the normalizing factor (or partition function) and I τ η∼J is the indicator function We remark that the random cluster measure can depend on the boundary conditions because the forbidden configurations can change.
We will say that there is a constant boundary condition if τ ≡ i and i = −1, 1. We say that the system is unfrustrated if for η ≡ 1 (we mean that for each e ∈ E: η e = 1) we have I τ ≡1 η∼J = 1. The systems with J ≡ 1 and the systems which are gauge transformations of those are unfrustrated. A plaquette P l is a square of four edges and it is called frustrated if I τ ≡1 η∼J = 0 for all configurations η with η(P l) = 1. In what follows we will show an application of the FK measure to a family of Ising models.
The Hamiltonian and the Gibbs Measure. We define the Hamiltonian on the finite set of vertices Λ with boundary condition τ . Let σ, τ ∈ Ω Λ ; the Hamiltonian is: The related Gibbs measure π τ Λ,Jβ is where Y τ Λ,Jβ is the partition function for this Gibbs measure. If J e > 0 for all e ∈ E we say that the system is ferromagnetic and we add an index F to the Gibbs measures, writing π F ; the ferromagnetic FK measures is µ F . We will use also the free boundary conditions for the Gibbs and the FK measures; these measures with free boundary condition are defined following the same arguments in (6)-(9) just imposing that all the interactions on edges in ∂Λ are identically equal to zero. In this case we simply do not label the boundary condition.
From [Ne97] we know that there is a joint measure of the Gibbs measure and of the FK measure verifying interesting properties (see Proposition 3.3 p. 33 [Ne97]). So we could obtain a spin configuration σ ∈ Ω Λ with Gibbs distribution π just taking η with FK distribution µ and coloring the single clusters independently of each other; for each cluster let us choose a vertex k and with uniform probability a spin value σ k ∈ {−1, 1} and then color the other vertices in the cluster C according to the rule that φ e (σ) > 0 for all e ∈ E(C).
Stochastic Order. Let us define a partial order on the configuration space H E,Λ : η 1 * η 2 if η 1,e ≤ η 2,e for all e ∈ E(Λ), where Λ can be all Z d .
We will introduce a dynamics on the edge configurations. We denote by H T E,Λ = {0, 1} E(Λ)×N the space of the trajectories; the single trajectory is denoted by η T ∈ H T E,Λ , and η e (t) is the value of the configuration at time t and corresponding to the edge e. We have an analogous partial order for the trajectories; we write η T 1 * η T 2 if for all e ∈ E(Λ) and for all t ∈ N we have η 1,e (t) ≤ η 2,e (t).
An event C ∈ F E (Λ) is increasing -write C ↑ -if η ∈ C and η * η imply that η ∈ C; an analogous definition applies to the trajectory space H T E,Λ . Associated with the partial order we define the stochastic order between two measures µ 1 and µ 2 : µ 1 µ 2 if for all increasing events A ∈ F E we have µ 1 (A) ≤ µ 2 (A); in the following we will refer to this definition as Definition A.
then by means of Strassen's theorem (see [St65], [Li85], [Lin92]) we have the stochastic order µ 1 µ 2 if and only if there exists a probability measure ( We callμ the joint representation of the stochastic order µ 1 µ 2 or the coupling measure. We will refer to this equivalent characterization of the stochastic order as Definition B.
Related to the same partial order we have this definition: for transition kernels K and K of two Markov chains in H E,Λ , we say that K is dominated by K if If (13) is true for K = K , we say that K is attractive.
Notice that the stochastic order is a partial order but the relation of domination is not. We will use both K(η, η ) and K η,η for the kernel.
We quote standard results in coupling theory (see [Lin92]): if µ 1 µ 2 and there are two sequences of kernels {K A2) the induced processes µ T 1 and µ T 2 on the trajectory space H T E,Λ are stochastically ordered: µ T 1 µ T 2 ; Using Strassen's theorem it is easy to see that there exists a joint representation of the stochastic order µ T 1 µ T 2 . We stress that the relations A1) and A2) are also verified using a single kernel if it is attractive. We will call a Markov chain attractive if its kernel is attractive.
Given two measures ν 1 and ν 2 on the dyadic space {0, 1} we will indicate with standard coupling this joint measure P : In the same way we will use the expression standard coupling also for the coupling of the trajectories of two Markov chains with only two states. In this case we will think the coupling, as usual, to be driven by the extraction of a uniform random variable in [0, 1] that updates at the same time the coupled Markov chains (see [Li85]). We try to explain the construction of the coupling; by hypothesis we have two independent Markov chains {X n } n∈N and {Y n } n∈N with values in {0, 1} and transition probabilities Obviously, the transition probability in 1 are also known. We consider i.i.d. r.v.'s {θ n } n∈N uniform in [0, 1] that are also independent of {X n } n∈N and {Y n } n∈N . We couple the Markov chains using the variables θ n ; if θ n < p 1 (X n , 0) then X n+1 = 0 otherwise X n+1 = 1 and also if θ We quote without a proof some properties of the FK measures (see for more details [Ne97]). If for each e ∈ E(Λ), p e = 1 − e −β |J e | ≤ p e = 1 − e −β|J e | , then the ferromagnetic FK measures are stochastically ordered: The same stochastic order relation is true in the case of free boundary conditions: Moreover a FK measure with the same absolute value of the interactions |J e | = J e for all e ∈ E and the same β and with any boundary conditions τ verifies the relation: these relations will play an important role in the paper. Also, the ferromagnetic FK measures with free boundary condition or with τ ≡ i satisfy the FKG inequalities (see [FKG71,Gr95,Ne97]).
For the ferromagnetic Ising model we recall Proposition 3.8 p. 35 [Ne97] (see also the original proof [LM72]) that prove the equivalence of the transition phase in the Ising model with percolation in the related FK model.

Proposition 2.1 For the ferromagnetic Ising model there is a unique infinite Gibbs measure at inverse temperature β if and only if
We will not prove Proposition 2.1. A proof can be found in [Ne97] pp. 35-37. In any case we will use some related ideas in the proof of Theorem 3 that is presented in the appendix.
Ferromagnetic and frustrated measures. Let us define two Gibbs measures and their related FK measures. We define these measures on a finite set of vertices Λ ⊂ Z dfinite volume-fixing the configurations J and the boundary conditions τ . We take a ferromagnetic and a frustrated system, and we will use the label i = 1, 2 to indicate respectively the ferromagnetic and the frustrated system. Take Let J 1,e > 0 for all edges e -so that system 1 is ferromagnetic-and J 2 is such that in each block B n (v) ⊂ E(Λ) there is a frustrated plaquette. We remind that we have covered the edges E(Z d ) with blocks of a fixed size n.
We are ready to write the finite volume Gibbs measures π τ ≡i 1,Λ,J 1 β , π τ 2,Λ,J 2 β and the related FK measures µ τ ≡i 1,Λ,J 1 β , µ τ 2,Λ,J 2 β . We denote with π 1,J 1 β any infinite volume Gibbs measure obtained as a weak limit along any subsequence {Λ L } L and any boundary condition τ , i.e. π τ 1,Λ L ,Jβ → L→∞ π 1,Jβ ; we follow the same notation for the other measures taking a weak limit of finite volume measures. The above construction for the FK and Gibbs measures shall be called hypothesis (H1). Sometimes we use the label τ of the boundary condition also for the infinite volume measure if it is relevant. The critical point is i.e. the weak limit of the measures is independent of the boundary conditions and of the sequence {Λ L } L ; the percolation critical point of the FK measure is The measure µ Jβ need not be translation invariant. In any case, using a lemma of [DG99] it is known that if the probability to percolate is larger than zero at a vertex, then it is larger than zero for all the vertices if all the interactions are different from zero; so, in this case, we could indicate only the cluster at the origin in (20). For ferromagnetic systems, using Proposition 2.1, one has that β F K c (J) = β c (J) (see [Ne97]). When the Gibbs measure is not unique one says that there is a phase transition; for the Ising models one can also define the region of parameter β in which there is broken symmetry, that is E (σ x ) > 0 for some boundary conditions. It is known that broken symmetry implies a phase transition but in general for the Ising models it is not known if the converse is true; only for ferromagnetic systems, by Proposition 2.1, it is known that phase transition and broken symmetry are equivalent. For notational convenience we will if the two systems satisfy the hypothesis (H1), and analogously for the critical point β c . We have this first strict inequality for the FK measures (see for the same result with a different proof [Ca98]).
Theorem 2 Let us consider the graph E d with d ≥ 2, let the FK measures µ τ ≡i 1,|J|β and µ τ 2,Jβ verify the hypothesis (H1), and let and We will give the proof of the stochastic order in Theorem 2 in the next section. It is known that for d ≥ 2, 0 < β F K c (J) < ∞, for the ferromagnetic systems with J e > > 0 for all the edges (see [Ne97]). So the second part of the Theorem is nontrivial.
is a consequence of (21). By the stochastic order, formula (21), we have the inequality β F K c (2, J) > β F K c (1, |J|) because percolation is an increasing event and the following relation is obviously true: So J). We know also that β F K (2, J) < ∞, this follows from a modification of [DG99] in which only FK measures without boundary conditions are analyzed. The idea is that all the boundary conditions can be thought of as a particular choice of J interactions and the configuration η on some edges connecting the vertices of the boundary. So we can do the same kind of proof as with free boundary conditions [DG99]. We have also Theorem 3, which is similar to Proposition 2.1 in the results and it is also related to [Ne94] in the idea of the proof and in the kind of the result. and then for all finite sets of vertices A ⊂ Z d : therefore the related infinite volume Gibbs measure π Jβ is unique.
We will give the proof of Theorem 3 in the appendix. The next result is an immediate consequence of Theorem 2 and Theorem 3, but we want to point it out because it is relevant for Statistical Mechanics (for the non strict inequality see [Ne94])).

Non disordered systems
In the previous section we have defined a ferromagnetic and a frustrated measure on the space H E,Λ for a particular choice of interactions J = {J e } e∈E . Now we want to construct non homogeneous Markov chains having the finite volume FK measures µ τ 2,Λ,Jβ and µ τ ≡i 1,Λ,Jβ as stationary measures. The Markov chains that we will construct will be irreducible and so the stationary measure will be unique. We will study the coupling between these Markov chains to deduce stochastic order between their stationary measures. In what follows Λ will indicate a set of vertices of shape Z d ∩ [−L, L] d for any L > 2.
We consider Markov chains that update the configurations on the edges or on the frustrated plaquettes. Before giving the kernels of the Markov chains we will fix the order in which the configurations are updated. Let's give a lexicographic order to the blocks B n (v) ⊂ E(Λ) that are used to cover E(Z d ); we write {B 1 , B 2 , . . . , B N }. In every block B n we mark a frustrated plaquette that we will call pl n . We update the configuration in the given lexicographic order, and for each block we use this procedure (a) update the configuration on the edges e ∈ ∂ E B n in a lexicographic order; (b) update the configuration on the edges e ∈ B n in a lexicographic order; (c) update the configuration on the marked frustrated plaquette pl n ; (d) update the configuration on every plaquette belonging to B n and then mark it. The first marked plaquette is pl n updated in point (c). Update the edges on the unmarked plaquettes that are adjacent to any marked plaquette and that are in the block which we are considering. Let's update the edges of this plaquette in the order e 1 , e 2 , e 3 , e 1 , e 2 (see figure 2) leaving unchanged the edges belonging to marked edges; (e) update the configuration on the edges e ∈ ∂ I B n in this order: e 1 , e 2 , e 1 , where e 1 and e 2 are two edges touching the same side of ∂ E B and then let update in a lexicographic order all the other edges touching the same side. Then repeat this procedure for the four sides of ∂ I B (see fig. 2).
(f) update the configuration on the edges e ∈ ∂ E B n in a lexicographic order; (h) let's take the next block and repeat all the procedure. Now we will write the kernels of three Markov chains which are P Λ,β , P Λ,β and P Λ,β ; the first and second Markov chains have respectively µ τ ≡i 1,Λ,|J|β and µ τ 2,Λ,Jβ as stationary measures. We will drop the labels Λ and β in the kernels for simplicity.
In general we will indicate with P the Markov kernel that is the result of the composition where P D i updates the configuration on the set of edges D i . On each set of edges D -it will be an edge or a plaquette-we will use as kernel the conditional probability (Gibbs sampler); for every η ini , η fin ∈ {0, 1} Λ and every region D ⊂ Λ we use this kernel to update the configuration:

kernel (or transition probability) is independent of the initial value on D but it depends on η ini (E(Λ) \ D); we also point out that η ini (E(Λ) \ D)) = η fin (E(Λ) \ D)).
Analogously we proceed for the Markov chain P (2) substituting µ τ 2,Λ,Jβ to the µ τ ≡i 1,Λ,|J|β in (26). Finally we define the kernel P that has for the marked frustrated plaquette pl and P e = P (1) e for all the edges that are not in a marked frustrated plaquette. The technique, that we will explain later, is applicable also to obtain strict inequality for two ferromagnetic systems (see [BGK93,Gr95,Gr99]) but in that case we do not need to introduce the measure P . For the proofs we could also choose a continuous time Markov process as in [BGK93] but it seems to us that with Markov chains there are more explicit bounds for the differences of the percolation critical points β F K c (2, J)−β F K c (1, |J|). Recall also that the measures µ 1 and µ 2 are reversible with respect to the defined Markov chains P (1) and P (2) , so they are stationary measures for them. The stationary measure is unique for each chain because the chain is irreducible.
In the next lemma we prove, adding an hypothesis, the Strassen's theorem in a stronger form which is useful to prove strict inequalities.

Lemma 3.1 Let µ 1 and µ 2 be measures on the finite space
The following assertions are equivalent: (a) There exists ε > 0 such that for each increasing event E ⊂ X with E = ∅, X µ 1 (E) + ε ≤ µ 2 (E).

Proof. (b) ⇒(a)
is a consequence of this direct computation; let E be as in the hypothesis of the theorem µ 1 (E) = P 12 (E, X) = P 12 (E, E) (28) the equalities in (28)-(30) and the inequality (29) are true because of the hypothesis, and the inequality in (30) because (a) ⇒(b). There exists P 12 which respect the Strassen's theorem with µ 1 (E) = P 12 (E, X) and P 12 (X, E) = µ 2 (E) because µ 1 µ 2 ; so we have only to prove that there is such a coupling measure verifying P 12 (η 1 ≡ 0, η 2 ≡ 1) ≥ ε. Let's define for each event A ∈ X the function where I is the indicator function and ε is given by point (a) of the theorem. It is easy to see that ν ε is a probability measure coinciding with µ 2 on the configurations that are not constant (η ≡ 0, 1). Moreover, Then ν ε is a probability measure and by construction: in fact for every increasing event E = ∅, X by hypothesis µ 1 (E) ≤ µ 2 (E) − ε = ν ε (E) whereas for the events ∅, X we have the equality.
This ends the proof.
It is known that P (1) is attractive (see [Gr95]), we want now to prove that also P has this relevant property.

Lemma 3.2 The kernel P is attractive.
Proof. We shall show that for all configurations η ξ, P (η, ·) P (ξ, ·). It is enough to consider only configurations which are different on a single edge e, i.e. η e = 0 and ξ e = 1, showing on these configurations the monotone property. At time n if an edge e is updated then P is equal to the ferromagnetic kernel ( P e = P (1) e for all the edges e). But P (1) e is attractive (see [Gr95,Ne97]) so also P e is attractive. If at time n the frustrated plaquette pl i is updated there are two case: 1) η(B i ) = 0; again P pl is equal to the ferromagnetic kernel, so it is attractive. 2) η(B i ) ≡ 0; if the edge e in which η e = ξ e is not in B i then it does not influence P pl because the plaquette is inside a minimal cut set (in two dimensions a dual circuit) γ with all the edges closed and the kernels P (η, ·) and P (ξ, ·) are equal; this is equivalent to free boundary conditions on the plaquette. If e ∈ B i : the first stochastic order follows from µ F 2 µ F 1 with free boundary condition (16), the second stochastic order follows from the attractiveness of the ferromagnetic kernel P (1) Now we can proof the domination between the kernels P (2) , P and P (1) . In the next lemma we denote with δ η the probability measure supported on the configuration η.

Lemma 3.3 For all the initial conditions η ∈ {0, 1} E(Λ) and for all
Proof. δ η (P (2) ) t δ η ( P ) t . Using the condition A1) we have only to prove the result on the update of a single set D i . So we should prove that for each D i , η and η we have: ·) as a consequence of the stochastic order (17); in fact the conditional probabilities are FK measure on the particular set of edges D i (plaquette or the single edge). The kernel P D i is attractive, so we have ·), which shows the right stochastic order. An analogous argument works also for the second domination.
We remaind that in the rest of the section the configuration J verifies Lemma 3.4 For every inverse temperature β ∈ (0, ∞) there exists ε 1,J (β) > 0 such that for each frustrated plaquette pl and for each increasing event E ∈ H pl = {0, 1} pl and E = ∅, H pl we have: Proof. The conditional probabilities in (33) are independent of the boundary conditions and the size of the box Λ because the plaquette is isolated being inside a minimal cut set (in two dimension a dual circuit) γ in which η(γ) = 0. There is only a forbidden configuration on H pl , that is the configuration η(pl) ≡ 1. Let's write µ 1 = µ τ ≡i 1,Λ,|J|β and µ 2 = µ τ 2,Λ,Jβ . Let µ 1 (E|η(B\pl) ≡ 0) = A(E) Z and being µ 1 (η(pl) ≡ 0|η(B\pl) ≡ 0) = δ > 0 we have Z > A(E) for all E that respect the hypothesis; in fact {η(pl) = 0} is not in any increasing event E = ∅, H pl . Then µ 2 (E|η(B \ pl) ≡ 0) = A(E)−ε Z−ε and ε = b∈pl p b > 0 that is the weight of the configuration η(pl) ≡ 1 for the measure µ 1 . Instead, for the probability measure µ 2 the configuration η(pl) ≡ 1 has a null weight because such a configuration is forbidden. For all E respecting the hypothesis So there exists a constant ε 1,J (β) > 0 verifying (33) because the number of events E verifying the inequality (34) is finite.
In the next lemma η is a configuration with η(pl) = 0 and η is a configuration with η (pl) = 1. Let's define P (1) Proof. We will use in the proof a dynamic argument. We will find a lower bound for µ τ ≡i 1,Λ,|J|β (E|η , η (pl) = 1) − µ τ ≡i 1,Λ,|J|β (E|η, η(pl) = 0) (37) using the same kernel P (1) e to update the configuration for two systems with a different initial configuration on the plaquette pl and eventually out of the considered block B. Being P (1) B\pl irreducible, aperiodic and with finite number of states we know that, for every configuration on pl and on B c , its stationary measure is unique and in the limit t → ∞ the measure at time t of the Markov chain converges to this stationary measure. We use the standard coupling for the kernel P (1) e ; this kernel is attractive so at every time t we will have η(t) * η (t).
There is a positive probability that η ≡ 0 on all the edges e ∈ B \ pl and so, using the coupling, also η ≡ 0 on B \ pl; there is a positive probability that also η = η ≡ 1 on the set ∂ E B, in fact all the configurations on a finite set of edges has a positive probability to be realized (finite energy). We start at such a configuration, i.e. η (B \ pl) = η(B \ pl) = 0 and η (∂ E B) = η(∂ E B) = 1. Let's update the configuration on plaquettes having at least one edge belonging to a plaquette already updated (see point (d) for the description of the dynamics). If in all the previous steps the updated plaquettes in the two systems verify η = 0 and η = 1, there is a different probability to have η e = 1 and η e = 1 with e belonging to the plaquette that we are updating (see [BGK93]). This is so because updating the first two edges there is a positive probability to have η e(1) = η e(2) = 1 (see figure 2). So on the third edge, conditioning on the event η e(1) = η e(2) = 1 and on all the updated plaquettes in the block η = 0 and η = 1, we obtain that the probability to have η(e(3)) = 0 and η (e(3)) = 1 is Formula (38) is a consequence of the fact that the vertices in i, k belonging to e(3) are connected in the configuration η independently of η (e 3 ), whereas they are not in the configuration η (see [BGK93] and [Gr95] p. 1471).
Now updating the configurations on the edges e(1) and e(2) we will find the same bound for the probability of the event η e(1) < η e(1) .
So we can bound the the probability of the event η (e(i)) < η (e(i)) for i = 1, 2, 3 with This bound is uniform in the time t so, for the unique stationary measures, we have that there is a probability larger than zero to have η(B) ≡ 0, η (B) ≡ 1 and in each case the configurations are ordered η * η ; in fact for every updated region we can obtain only ordered configurations (doing a comparison on all the terms), so the coupling has all the measure supported on the space M. Now using the Lemma 3.1 we have (36). It is clear that all the bounds are also uniform in Λ and in the configurations η, η ∈ {0, Let's remark that the lower bound (39) does not hold if β = 0 or β = ∞ (means zero temperature) because the inequality (38) becomes equal to zero (remind that p e = 1 − e −|Je|β ). Also, (36) falls for β = 0, ∞, so the condition β ∈ (0, ∞) is necessary for Theorem 3.5.
The proof of Theorem 2 will be similar to [BGK93,Gr99]; as a matter of fact we want to prove the strict inequality using only the attractive Markov chains P and P (1) . Now we are ready to prove the main result of this section.
Proof of Theorem 2. Let's put back the labels β, J and Λ on the measures µ 1 and µ 2 , and therefore on the kernels P and P (1) . The kernel P where P (1) Let's remark that not all the boxes [−L, L] d ∩ Z d can be divided into blocks of fixed size n, but in any case we can take boxes of different sizes k with n ≤ k ≤ 2n and, using all these types of blocks, we can cover every box [−L, L] d ∩ Z d if L is larger than n. On each block B, we will find a different value of ε 3,J(B) (β), but in any case the different sizes are a finite number (between n and 2n), so the minimum exists and is larger than zero. We redefine α := min B α(B).
On each block we can replace the kernel P B,Jβ with P for all boundary conditions τ , where the first stochastic order is a consequence of Lemma 3.3. Remember that α in (44) is defined uniformly in Λ, so we can take the weak limit on every subsequence of boxes Λ k , on which the limit exists, to have This ends the proof.
Notice that it is possible to use the stochastic order (45) for every increasing event, not only for percolation. For the increasing events in the tail σ-algebra one can define a critical point analogous to (20) and one has a strict inequality in the critical point for the frustrated and the ferromagnetic measure if the ferromagnetic critical point is different from zero and infinite. For the Edwards-Anderson model, if one wants only to show a strict inequality between the regions in which there is broken symmetry, one could use the result in [Ca98].

Strict inequality for disordered systems
In this section we will find a strict inequality in the phase transition for disordered systems.
In all the section we will consider the configuration of the interactions J = {J e } e∈E as random variables. We will denote with (Θ, A, Q) the abstract probability measure on which the random variables {J e } e∈E are defined. The random variables {J e } e∈E are i.i.d. and the support of the distribution is The second claim of Proposition 3.9 in [Ne97] is given for the Ising model but the same proof works for the frustrated Ising model. So we will write the proposition without proof. The second point of the next Theorem could be obtained using the paper [OPR83] in which a strict inequality for the magnetization of some ferromagnetic systems is proved. We need a similar estimate for the FK measure. We will give the proof of the theorem because it is different from [OPR83] and it is directly applicable to the FK measures that we will use at the end of this section. We will write J 1 for the configuration with all the interactions equal to J 1 . Analogously for J 2 .
Theorem 4 Let J be the absolute value of the random variable defined before, then and Proof. We will abbreviate the notation writing η \b = η(E \b), J \b = J(E \b) and omitting the labels β and Λ in the measures. We will prove (46) for the FK measures with free boundary conditions; then we will finish the proof using the fact that the percolation is an increasing event, the equality between µ F Jβ and µ τ ≡i,F Jβ for almost all the β, and the fact that µ τ ≡i,F Jβ respects the relation (15).
We want to show that 0 < p a < 1. We will first prove that p a > 0. Let It is known that We apply the inequality (54) to (51) to obtain, with some algebraic calculations, where the first part in the max function corresponds to a configuration η \e in which the connection of {x, y} = e is independent of the value of η e ; the number 1+e −βJ 1 1+e −βJ 2 , instead, depends on the configurations η \e connecting the vertices x and y -with {x, y} = eif and only if η e = 1. Applying again the inequality (54) we have for all β > 0 and 0 < J 2 < J 1 . Then In the same way we can show that 1 − p a (η \e ) > 0.
As in the last section, we will define two processes P 1 and P , where the elements of the kernels are conditional probabilities. Remember that [η η] e is the configuration which coincides with η on the edge e and is equal to η on all the other edges. Let's define for the kernel (transition matrix) that updates the configuration on the edge e; analogously for the integrated measure P e,β (η, [η η] e ) = µ β (η e |η \e ).
We can choose β A and β B such that β F K c (J 1 ), β F K c (J 2 ) ∈ [β A , β B ] with 0 < β A < β B < ∞. In fact β F K c (J 1 ) and β F K c (J 2 ) are finite positive numbers. So due to the stochastic order of the FK measures (15), also all the ferromagnetic measures with J e ∈ [J 1 , J 2 ] have β F K c (J) ∈ [β A , β B ]. The Markov chain P 1,αJ 1 β is attractive and dominates P e,β on every edge because of (59)-(62); here the edges play the role of the blocks in the previous section. Following the exposition of the previous section, we have that the stationary measures of the processes P 1,αJ 1 β and P e,β are stochastically ordered and, rewriting the parameters Λ and β, one has µ Λ,β µ Λ,αJ 1 β ∀β ∈ [β A , β B ]. We remark that the inequalities (59)-(62) are independent of Λ, so we can take the limit of Λ → Z d preserving the stochastic order between the measures. So, for all the weak limits of the integrated measure µ β , one has of a fixed size, we can choose n = 6. In each block there is a positive probability to find a frustrated plaquette because 0 < Q(J e = −1) < 1. In all the blocks that do not have frustrated plaquettes we set the interaction equal to the constant 1; on all the remaining edges we set J e = J 2 < 1. We will indicate this configuration withJ. If J 2 is chosen close to 1, we know from the previous section that the kernel P B,J is dominated by P (1) B,J 2 and the kernel P B,J 1 dominates the kernel P B,J≡J 2 if there is at least one frustrated plaquette, we have a kernel that dominates the previous one. We can calculate, using the reversibility, the stationary measure of this new kernel which turn out to be a ferromagnetic FK measure. So we find that this last FK measure and the original systems are stochastically ordered by taking the limit for N → ∞ in A1). Again, the limit exist and is unique because the Markov chain is irreducible and aperiodic. So we have We want to compare the ferromagnetic measure µ F Jβ with µ F Jβ ; we know by Theorem 4 that ifJ are dyadic i.i.d. random variables then we have the strict inequality for the critical point. From the construction one can see that on each block, independently from the others, there is a positive probability to find a frustrated plaquette. Therefore, in the coupling on each block B there is a positive probability to have J b = J 2 < 1. The considered procedure to construct the configurationJ on the ferromagnetic system induces a measure P 1 on the interactions. It is simple to see that the measure P 1 is stochastically ordered with a dyadic product measure B p 1 (independent on all the edges) where B p 1 (J b = 1) = p 1 and B p 1 (J b = J 2 ) = 1 − p 1 . Obviously the stochastic order is with respect to the partial order on the configurations of interactions. To have P 1 B p 1 it is enough that where |B| is the number of edges in a block. So there is a ferromagnetic FK measure, called µ F (J 1 ,J 2 )β , with distribution of the interactions B p 1 satisfying the following stochastic order Using Theorem 4 one obtains that for almost all the realizations of J Finally from the formula (66) using Theorem 3 one has β c (J) ≥ β c ((J 1 , J 2 )) > β c (J ≡ 1).
This ends the proof.
The strategy of the proofs is quite general and can be applied to other models. For example with this method it is possible to study strict inequalities in the phase transition of disordered systems having also a random magnetic field (see [BK88] for the definition of the model).