Evolution of the interfaces in a two dimensional Potts model

. We investigate the evolution of the random interfaces in a two dimensional Potts model at zero temperature under Glauber dynamics for some particular initial conditions. We prove that under space-time diﬀusive scaling the shape of the interfaces converges in probability to the solution of a non-linear parabolic equation. This Law of Large Numbers is obtained from the Hydrodynamic limit of a coupling between an exclusion process and an inhomogeneous one dimensional zero range process with asymmetry at the origin.


The Potts model
We consider a 3-state Potts model at zero temperature under Glauber dynamics described by a spin system on Ω sp = {−1, 0, 1} Z 2 , whose generator acting on cylinder functions is given by for all σ ∈ Ω sp , where σ x,j denotes the configuration (σ x,j )(y) = σ(y) for y = x j for y = x , and H is the Hamiltonian defined formally on Ω sp by 1{σ(x) = σ(y)} and ι(−1) > ι(0) = ι(1) > 0 (this last condition will be explained later). This means that at each site a spin is allowed to change at rate 1/2, independently of any other site, if and only if it does not increase the energy of the system. Denote by I = I(Z) the collection of non-decreasing functions on Z and by A the set of configurations σ for which there exists a function f = f σ in I such that, for every x = (x 1 , This gives a one to one correspondence between I and A. Furthermore, A is stable under the dynamics induced by the generator L sp . Indeed, fixed a configuration σ ∈ A, by a direct computation of H(σ x,j ) − H(σ), it is easy to verify that a jump from σ to σ x,j is allowed only in one of the following three cases: (i) j = 1 and x = (x 1 , f σ (x 1 )) with f σ (x 1 ) > f σ (x 1 − 1), (ii) j = 0 and x = (x 1 , f σ (x 1 ) + 1) with x 1 < 0 and f σ (x 1 ) < f σ (x 1 + 1), (iii) j = −1 and x = (x 1 , f σ (x 1 ) + 1) with x 1 > 0 and f σ (x 1 ) < f σ (x 1 + 1). Note that the condition on ι in the definition of H is necessary to prevent a jump from σ to σ (0,fσ(0)),−1 when f σ (0) > f σ (−1). Hence we may investigate the evolution of the Markov process (σ t ) induced by L sp and starting from a configuration σ in A through the process f t = f σt .
In order to establish the hydrodynamical behavior of the described system, we are now going to introduce some notation and impose some restrictions on the initial conditions of the system. Denote by N, Z − and Z * + respectively the sets of non-negative, non-positive and positive integers. It is also clear that a function f in I is determined by its value at a given site y ∈ Z and by its increments, {f (x) − f (x − 1) : x ∈ Z}. Based on this we consider another identification which associates to f ∈ I a configuration η = η f in N Z given by For each α ≥ 0, denote byν α the product measure on N Z whose marginals are given byν The sequence (T m N ) have marginals on N Z− bounded above (resp. below) byν α (resp.ν λ ) for some 0 < λ < α < ∞. (P4) There exists a increasing smooth function λ 0 : R → R with bounded derivative, such that for each continuous function G : R → R with compact support and each δ > 0, Here (P1) allows us to use the correspondence between I and N Z to study the Potts model through the increments process; (P2) says that the initial condition on sites at left of the origin is independent of the initial condition on sites at the right of the origin allowing us to consider the evolution of the system at left of the origin independently, as we shall discuss later; (P3) implies that the left system will be stochastically dominated by a system with reflection at the origin, which is useful to estimate the density of particles over macroscopic boxes; and (P4) is the usual local equilibrium condition in its weakest form for the sequence of initial probability measures (see [5]).
Let D(R + , I) denote the space of right continuous functions with left limits on I endowed with the skorohod topology. For each probability measure m on I, denote by P sp,N m the probability measure on D(R + , I) induced by the Markov process f t = f σt with generator L sp speeded up by N 2 and initial measure m. Our main result is the following: where v t is given by and ρ is the unique weak solution of the nonlinear parabolic equation on with Φ(ρ) = ρ/(1 + ρ). Moreover, for any continuous function G : R → R with compact support and any δ > 0 where λ is the unique weak solution of the nonlinear equation (1.4) In the statement above, the integral in (1.2) is to be understood as the limit as where H n (u) = (1 + u/n) + . Moreover, the precise definitions of weak solutions for equations (1.3) and (1.4) will be given later, on sections 2.2 and 4, respectively.
Concerning the proof of Theorem 1.1, as pointed out by Landim,Olla and Volchan [6] the process of the increments of f t evolves as a zero range process on Z with asymmetry at the origin. For this process the evolution of the particles on Z is described by a nearest neighbor, symmetric, space homogeneous zero range process except that a particle is allowed to jump from 0 to 1 but not from 1 to 0. We should think of this as two coupled process: its restriction to Z − , the dissipative system; and its restriction to Z * + , the absorbing system, which acts as a infinite reservoir for the former. For the dissipative system, whose evolution is independent of the absorbing system, the hydrodynamic behavior under diffusive scaling was established in [6] with hydrodynamical equation given by the non-linear parabolic equation (1.3). They also studied the behavior of the total number of particles which leaves the system before a fixed time t > 0. Denoting this number by X t they proved that X −2 t , as → 0, converges in probability to v t defined in (1.2).
Since the rate at which particles leave the dissipative system is equal to the rate at which particles enter the absorbing system, it is then expected for the coupled process a hydrodynamical behavior under diffusive scaling with hydrodynamic equation: (1.5) However the possible accumulation of particles in a neighborhood of the origin appears as a problem in using a direct approach to establish the hydrodynamical behavior of the absorbing system. To avoid this, we are going to consider a clever transform which maps the absorbing system onto a simple one dimensional nearestneighbor exclusion process, see Kipnis [4]. Then, we prove the hydrodynamical behavior of this associated exclusion process. This identification is given by an application that associates to each η ∈ N N the configuration ξ ∈ {0, 1} Z * + given by Therefore if ξ is obtained from η, then η(n) is the number of empty sites between (n-1)th and nth particle of ξ for n > 1 and the number of empty sites before the first particle of ξ for n = 1.
In this way the absorbing system is mapped onto a process in which each particle jumps as in the nearest neighbor symmetric exclusion process on Z * + with reflection at the origin and superposed to this dynamics, when a particle leaves the dissipative system, the whole system is translated to the right and a new empty site is created at the origin. Denoting by (1.6) the macroscopic rate at which mass is transfered from the dissipative to the absorbing system, the hydrodynamic equation associated to the coupled exclusion can be derived from an appropriated transformation of the macroscopic profile of the absorbing system, see section 4, and is given by (1.7) Remark 1.1. From theoretical point of view, the study of the hydrodynamic behavior for such system is important because of the absence of non-trivial equilibrium measures which does not allow a direct application of the usual methods of proof, see [5]. In such cases, the proof becomes particular to each model and few cases have been considered untill now, we refer here to the papers of Chayes-Swindle [2], Landim, Olla and Volchan [6], Landim and Valle [7].
The paper has the following structure: In section 2 we shall consider the coupling between the exclusion process and the dissipative system stating the hydrodynamic limit of the former and section 3 is devoted to its proof. Finally, in section 4 we prove Theorem 1.1.

Hydrodynamics of the coupled exlusion process
This section is divided in four sub-sections. At first, we present the formal description of the system. After that, we introduce some terminology on weak solutions of the parabolic equations (1.3) and (1.7) necessary to state the hydrodynamical behavior of the system. In the third part we consider the hypothesis we need on the initial configurations of the system and in the forth we state of the hydrodynamical behavior.
2.1. The system. The coupled system described informally in the end of the previous section is a Feller process with configuration space Ω = N Z− × {0, 1} Z * + . Denoting by (η, ξ) a configuration in Ω, its generator L may be written as Here L is related to the motion of particles in the exclusion process: where, for every local function F : Ω → R and every x, y ≥ 1, and ξ x,y is the configuration with spins at x, y interchanged: The operator L is related to the motion of particles in the dissipative system: where, for every local function F : Ω → R and every x, y ≤ 0, with g(k) = 1{k > 0} and Finally, L b is the part of the generator related to the coupling between the systems: For every local function F : where x stands for the configuration with no particles but one at x and

Weak solutions of the hydrodynamical equation. Fix a bounded func-
if the following conditions hold: (a) Φ(ρ(t, u)) is absolutely continuous in the space variable and ∂ u Φ(ρ(t, u)) is locally square integrable on (0, (b) ρ(t, 0) = 0 for almost every 0 ≤ t < T .
(c) For every smooth function with compact support G : R − → R vanishing at the origin and every t > 0, For a uniqueness result for equation (2.1) see [6]. Now, for a fixed bounded function ζ 0 : R + → R. A bounded measurable function ζ : [0, T ) × R + → R is said to be a weak solution of where a : (0, T ) → R + is a bounded measurable function, if (a) ζ(t, u) is absolutely continuous in the space variable and ∂ u ζ(t, u) is a locally square integrable function on (0, T ) × R + such that for all 0 ≤ t ≤ T and for every smooth function G : (b) For every smooth function with compact support G : R + → R and every t > 0, Existence and regularity for equation (2.2) follows from the proof of the hydrodynamical behavior of the system described in Theorem 2.1 on section 2.4. The required uniqueness for the equation follows from diffusion theory, indeed the solution have a stochastic representation in terms of a unique diffusion with generator 1 2 ∆ζ − a t ∂ u ζ reflected at the origin, see Stroock and Varadhan [9].
2.3. Hypothesis on the initial measures. Given any two measures µ, ν on Ω, we denote by H(µ|ν) the relative entropy of µ with respect to ν: where the supremum is carried over all bounded continuous functions. Recall that if µ is absolutely continuous with respect to ν, For a measure µ on Ω, denote by µ − and µ + its marginals on N Z− and {0, 1} Z * + , respectively. Let P ± (Ω) be the space of probability measures µ on Ω that can be written as µ = µ + × µ − . For 0 < α < 1, let ν α denote the Bernoulli product measure of parameter α on {0, 1} Z * + . Fix a sequence of probability measures {µ N : N ≥ 1} on P ± (Ω). To prove the hydrodynamical behavior of the system we shall assume that (E1) The sequence (µ N,− ) is bounded above (resp. below) byν λ1 (resp.ν λ2 ) for , for each δ > 0 and each continuous function G : (E4) The sequence {µ N,+ , N ≥ 1} is associated to a bounded initial profile ζ 0 : R + → R, i.e., for each δ > 0 and each continuous function G : The first three assumptions are used by Landim, Olla and Volchan in [6] to establish the hydrodynamical behavior of the dissipative system. The condition (E4) is the usual law of large numbers imposed on the empirical measure at time 0 for the coupled exclusion. The proof of the hydrodynamical behavior of the dissipative system in [6] is an adaptation of the entropy method introduced by Guo, Papanicolaou and Varadhan in [3], see also [5]. Such an adaptation requires the introduction of appropriate reference measures that plays the role of the missing equilibrium measures. For the dissipative system the reference measures are the product measuresν N γ(·) , N ≥ 1, on N Z− with marginals given bỹ with β > 0 satisfying (E2). For these measures the entropy bound is preserved , which means (see [6]) that Although we do not need an entropy production estimate for the exclusion process we shall adapt some of the estimates found in [6] that requires adequate reference measures for this system. Taking the reference measure for the absorbing system as the canonical measure ν α , the reference measures for the whole system are taken as the product of the reference measures for the dissipative system with ν α , i.e., ν N α,γ :=ν N γ(·) × ν α .
2.4. The hydrodynamical behavior. Let D(R + , Ω) denote the space of right continuous functions with left limits on Ω endowed with the skorohod topology.
For each probability measure µ on Ω, denote by P N µ the probability measure on D(R + , Ω) induced by the Markov process (η t , ξ t ) with generator L speeded up by N 2 and with initial measure µ. The hydrodynamical behavior of the exclusion process is given by the following result: with strictly positive initial profile ζ 0 : R + → R bounded above by 1. Then, for any continuous G : R + → R with compact support, any δ > 0 and where ζ is the unique solution of (2.2), with a t = ∂ u Φ(ρ(t, 0−)) for ρ being the unique solution of (2.1).

The proof of the Hydrodynamic limit
Denote by M = M(R), the space of positive Radon measures on R endowed with the vague topology. Integration of a function G with respect to a measure π in M will be denoted π, G . To each configuration (η, ξ) ∈ Ω and each N ≥ 1 we associate the empirical measure π N +π N in M, where Theorem 2.1 states that the sequence Q N µ N converges weakly, as N → ∞, to the probability measure concentrated on absolutely continuous trajectories π(t, du) = ζ(t, u)du whose density is the solution of (2.2) (see [5]). The proof consists of showing tightness of Q N µ N , that all of its limit points are concentrated on absolutely continuous paths which are weak solutions of (2.2) and uniqueness of solutions of this equation.
We have already discussed uniqueness of weak solutions of (2.2) in section 2.2. Note that all limit points of the sequence Q N µ are concentrated on absolutely continuous measures since the total mass on compact intervals of the empirical measure π N is bounded by the size of the interval plus 1/N .
In order to show that all limit points of the sequence Q N µ N are concentrated on weak solutions of (2.2) we will need the following result: We shall divide the proof of Theorem 2.1 in four parts: We start proving tightness in section 3.1. The section 3.2 is devoted to prove of Lemma 3.1 under a condition on the entropy of the system with respect to an equilibrium measure for the system with reflexion at the origin. From Lemma 3.1, to conclude the proof that all limit points of the sequence Q N µ N are concentrated on weak solutions of (2.2), we have to justify an integration by parts to obtain conditions (a) and (b) in the definition of weak solutions of (2.2). This is consequence of an energy estimate which is the content of section 3.3. We finish the proof of Theorem 2.1 removing the imposed condition on the entropy at section 3.4.
3.1. Tightness. The sequence Q N µ is tight in the space of probability measures on D([0, T ], M), if for each smooth function with compact support G : R + → R, π N t , G is tight as a random sequence on D(R + , R). Now fix such a function, denote by F t = σ((π s , π s ), s ≤ t), t ≥ 0, the natural filtration on D([0, T ], M), and by T T the family of stopping times bounded by T . According to Aldous [1], to prove tightness for π N t , G we have to verify the following two conditions: Condition (i) is a trivial consequence of the fact that the empirical measure has finite total mass on any compact interval. In order to prove condition (ii), consider the (F t )-martingale vanishing at the origin From the previous expression and Chebyshev inequality, (ii) follows from In order to show (3.3) and (3.4) and complete the proof of tightness, let us first derive some formulas and fix some notation. An elementary computation shows that N 2 (L + L b ) π N , G is given explicitly by where ∆ N and ∇ N denote respectively the discrete Laplacian and gradient: We also derive an explicit formula for the quadratic variation of M G,N t , see Lemma 5.1 in Appendix 1 of [5]. It is given by Proof of (3.3): From the optional stopping theorem and the martingale property Hence, applying formula (3.6), by the Taylor expansion for G, we have that Recall from the proof of Lemma 3.8 in [6] that for every 0 < t ≤ T , Proof of (3.4): From formula (3.5) and the Taylor expansion for G we obtain that We postpone the proof (3.10), assuming it we have (3.4).
It remains to prove (3.10) to complete the proof of (3.1). However we first show that in fact the expectation in The reason is that (3.11) is required in the next sections and its proof is similar to that of (3.10).
Proof of 3.11: We introduce a second class of martingales. Let G : R − → R be a smooth function with compact support and denote byM G,N the (F t )-martingalẽ By a straightforward computation we have that Moreover, the quadratic variation M G,N t is given by Therefore, for each configuration η ∈ N Z− , By Fatou's Lemma, we have that the expectation of the right hand side term in the previous equality is bounded above by Note that In particular, by (3.8), Using the martingaleM G l ,N and its quadratic variation, we verify by usual computations that the expectation of the middle term at the right hand side of this equation is uniformly bounded in both N and l. The other two terms are bounded respectively by whose expectation is also uniformly bounded. To prove this last statement, use the fact that µ N,− ≤ν α to construct a coupling, between the dissipative system and the nearest-neighbor, symmetric, space-homogeneous zero-range process with reflection at the origin, which preserves the stochastic order and recall that for this last systemν α , defined in (1.1), is an equilibrium state. Therefore (3.11) holds.
Proof of (3.10): As in (3.12), we have that On the one hand, as in the proof of (3.3), obtain by quadratic variation ofM On the other hand, | π N τ +θ , H l − π N τ , H l | is bounded above by with G l taken as in the proof of (3.11). Using again the explicit formulas forM G l ,N and for its quadratic variation, we show that Therefore, we just have to prove that converges to 0 as C → 0. The previous expression is dominated by which, by the coupling also described in the end of the proof of (3.11), is bounded by whereĒ Ñ να denotes the expectation with respect to the distribution of the nearestneighbor, symmetric, space-homogeneous zero-range process on Z − with reflection at the origin speeded up by N 2 and with initial measureν α . Therefore, we conclude the proof of (3.10) with the following result: whereπ is the empirical measure associated to the zero-range with reflection. Thus an upper bound for (3.15) is given by Now, choosing β > 0 sufficiently small such that Eν α [exp{βη(0)}] is finite, we have that the expectation in the first term of (3.16) is dominated by since exp{max 1≤i≤K a i } ≤ 1≤i≤K exp a i andν α is a product measure invariant for the zero-range reflected at the origin. Therefore the first term of (3.16) is of order O(C). On the other hand, the expectation in the second term of (3.16) is proved to be of order O(C −1 K −1 ) as N → ∞ by standard techniques, using the martingale associated to the zero-range process in the semi-infinite space with reflexion at the origin (see chapter 5 in [5]), which means that the second term of (3.16) is zero. This proves (3.15).

3.2.
The proof of Lemma (3.1). Let G : R + → R be a smooth function with compact support. By Doob inequality, for every δ > 0, which, by the explicit formula for the quadratic variation of M G,N t , is bounded by Thus, by (3.11), for every δ > 0, In the previous expression we claim that we can obtain the integral term as a function of the empirical measure replacing N g(η s (0)) by a s , given in (1.6), and ξ s (1) by a mean of ξ over small boxes around 1. At first we are going to justify the replacement of N g(η s (0)) by a s . This is the content of Lemma 3.2 just below. Let us note before that for all δ > 0, and 0 ≤ t ≤ T , which indicates that the right candidate to replace N g(η s (0)) is a s . Actually, this follows from (3.12), (3.13) and Proposition 5.1 in [6], which states that for all δ > 0, and 0 ≤ t ≤ T , Here, are to be understood respectively as where H l is defined in section 3.1.

Lemma 3.2.
For every smooth function G : R + → R with compact support and Proof: The supremum in the statement is bounded above by for every K > 0. The third term clearly goes to 0 as K goes to ∞. To deal with the second term we are going to show that lim sup Considering the supremum out of the expectation in the last two expression, both of them were proved in the last section. The same proof carried out there can be applied for the case where the supremum is inside the expectation, since we have (3.15) and the expectation of the supremum of a martingale is dominated by the expectation of its quadratic variation. So it remains to consider the first term in (3.19). Since we first make N → ∞ and then K → ∞, we only have to show that lim sup for all 0 < t ≤ T . Fix C > 0 and let P : 0 = t 0 < t 1 < . . . < t n = t be a partition on [0, t] such that |P| := max(t i+1 − t i ) ≤ C min(t i+1 − t i ). Bound (3.21) by In order to prove (3.21), we start estimating the Q N µ N probability of the first term in (3.22) to be greater than δ. Using (3.2) and (3.5), we obtain an upper bound of Therefore, since (a + b) Applying Schwarz inequality to the first term, we have that (3.25) is dominated by Together with (3.11) this proves Claim 1.
Proof of Claim 2: This expectation is bounded by Together with (3.11) this proves Claim 2.
Analogously, considering the second term in Now we consider the replacement of ξ s (1). At this point we impose one more condition on the sequence of initial measures which is the following (E5) There exists 0 < α < 1 such that H(µ N,+ |ν α ) ≤ CN for some constant C > 0. At section 3.4 we justify how this condition can be removed from the hypotheses to prove Theorem 2.1.
Proof: As in the proof of Lemma 3.2, since is a family of functions indexed by N which is almost surely uniformly equicontinuous in the variable t on [0, T ], we may omit the supremum in the statement. It is also easily seen that we can replace ξ s (0) by [ N ] N ξ s (0) in the statement and it is enough to show that By the entropy inequality we have that the expectation in (3.26) is bounded above by for every A N , where ν N α,γ is defined in section 2.3.. We are going to estimate the second term in (3.27). Since e |x| ≤ e x + e −x and lim sup N N −1 log{a N + b N } ≤ max{lim sup N N −1 log a N , lim sup N N −1 log b N }, we may suppress the absolute value in the exponent. Define for every bounded function f on {0, 1} Z * + . We have that In order to obtain an upper bound for the right hand term in this inequality we will show below, finishing this section, that for every B > 0, where the supremum in f is taken over all densities with respect to ν α , · ∞ denote the supremum norm and D(f ) denote the Dirichlet form In particular, since by (2.3) and (E5) we have that H(µ N |ν N α,γ ) ≤ CN , for some C > 0, it follows from (3.28) and the Gronwall inequality that (3.27) is bounded by which goes to 0 as → ∞, proving the Lemma.
Proof of (3.28): By Feynman-Kac formula, {P s,t : 0 ≤ s ≤ t} is a semigroup of operators associated to the non-homogeneous generator L s = N 2 L + A N U N (s, ·). Moreover, the first Chapman-Kolmogorov equation holds: ∂ s P s,t = −L s P s,t . Hence We shall estimate separately each term in this expression.
Proof of Claim 1: Denote P N s,t 1 by h. We have that Here, after some changes of variable, In the previous summation we can neglect the first two terms, which are negative, and add the last two terms in x, for x ≤ −1, obtaining that where ∆γ(x) = γ x+1 + γ x−1 − 2γ x . Observing that ∆γ(x) is zero except at x = −N + 1, when it is negative, and that γ −1 /γ 0 = 2, we have shown Claim 1.
Claim 2: Proof of Claim 2: Denote P N s,t 1 by h. We are considering an integral of the form where 0 and τ are defined in section 2.1. Add and subtract to this the term We end up with three terms, the first is which is negative and may be neglected, and the others are Since γ 0 = β/N , we have Claim 2.
Claim 3: Proof of Claim 3: We have the bound where Γ N s is the greatest eigenvalue of the generator N 2 L + A N U N (s, ·). By the variational formula (see Appendix 3 in [5]) Γ N s is equal to If we can show that for any B > 0, then replacing B by B/H(s) we conclude the proof of the Claim from (3.29). The left hand side in (3.30) is equal to which can be rewritten as and applying the elementary inequality 2ab ≤ Ba 2 + B −1 b 2 , that holds for every a, b in R and B > 0, we have that previous expression is bounded by This show (3.30). Now, is easy to see that (3.28) is a consequence of Claim 1-3. Therefore we have proved the Lemma. Theorem 3.4. Every limit point of the sequence Q N is concentrated on paths ζ(t, u)du with the property that ζ(t, u) is absolutely continuous whose derivative ∂ u ζ(t, u) is in L 2 ([0, T ] × R + ). Moreover To prove the previous theorem we make use of the following energy estimate: Lemma 3.5. There exists K > 0 such that if Q * is a limit point of the sequence Q N then where the supremum is taken over all functions H in C 0,1 K ([0, T ] × R + ). Proof: For every > 0, δ > 0, H : R + → R smooth function with compact support and ξ ∈ {0, 1} Z * + , denote by W N ( , δ, H, ξ) the following expression where ξ δ (x) = δ −1 x+δ y=x ξ(y). We claim that there exists K > 0 such that for any dense subset for every k ≥ 1 and every > 0. We postpone the proof of (3.32), using it, since Q * is a weak limit point of the sequence Q N , it follows that for every k ≥ 1. Since, letting δ → 0 and then → 0, it follows from (3.33) that To conclude the proof we apply the monotone convergence theorem, noting that is continuous as a real function on C 0,1 K ([0, T ] × R + ). Proof of (3.32). Since H is a continuous function, an integration by parts justify the replacement of W N ( , δ, H, ξ) as δ → ∞ in (3.32) by which we denote by W N ( , H, ξ). By the entropy inequality Now hypothesis (E5) and the elementary inequality e max ai ≤ e ai imply that this last expression is bounded by Here, since max{lim sup N N −1 log a N , lim sup N N −1 log b N } is greater or equal to lim sup N N −1 log{a N + b N }, the second term is dominated by Analogously to the proof of (3.28) in Lemma 3.3, we have that the previous expression is bounded by where the supremum is taken over all densities f with respect to ν N α . By (3.11) we just have to estimate the first term in the previous formula, which we are now going to show that is in fact non-positive. Since, from (3.31), we have, for each B > 0, that Hence Summing up in x and taking B = N we obtain from (3.34) that 3.4. Removal of the entropy condition. We have proved Theorem 2.1 under condition (E5). We apply a coupling argument to obtain Theorem 2.1 without such condition. The idea is to couple the dissipative system not to one but to two exclution processes in a way that the exclution processes are themselves coupled in a appropriated way. This basic coupling for the system described is section 2.1 is a Feller process on Ω 0 = N Z− × {0, 1} Z * + × {0, 1} Z * + whose generator acting on local functions F : Ω 0 → R is given by [F (η, ξ x,y ,ξ x,y ) − F (η, ξ,ξ)] [F (η, ξ x,y ,ξ) − F (η, ξ,ξ)] [F (η, ξ,ξ x,y ) − F (η, ξ,ξ)] where the notation in the above expression is taken from section 2.1. For existence results and properties of such a coupling see chapter 8 of [8]. One important property is that this coupling preserves stochastic order in the sense that: if we consider two inital conditions for the system µ 1 = µ − × µ + 1 and µ 2 = µ − × µ + 2 such that µ + 1 is stochastically dominated by µ + 2 then there exists a coupling measure on D(R + , Ω 0 ) concentrated on {(η t , ξ t ,ξ t ) ∈ Ω 0 : ξ t ≤ξ t }, for every 0 ≤ t ≤ T , and with marginals P µ1 with respect to (η, ξ) and P µ2 with respect to (η,ξ).
The order preserving property may even be described in a less restrictive sense if we improve a bit our coupling considering that the particles are all distinct and that once a ξ particle is attached to aξ particle, which means that from the moment they share the same site, they remain attached from this moment on. This coupling will be called the Stirring coupling for the system. Then the property we have mentioned is the following: consider a set Λ ∈ Z * + and for a measure µ on Ω let µ Λ be the marginal of µ on Λ, i.e., µ Λ (ς) = µ{(η, ξ) : ξ(x) = ς(x) for all x ∈ Λ}, for all x ∈ {0, 1} Λ .
Fix Λ and consider two inital conditions for the system such that µ Λ 1 is stochastically dominated by µ Λ 2 . Denote by K t s (ς, Λ, Γ) = number of ς particles at sites of Γ at time t that were at sites of Λ at time s for every trajectory (ς t ) t≥0 on {0, 1} Z * + and s < t. Thus for the Stirring coupled process with marginals P µ1 with respect to (η, ξ) and P µ2 with respect to (η,ξ), we have that for any other subset Γ of Z * + , we have that K t 0 (ξ, Λ, Γ) ≤ K t 0 (ξ, Λ, Γ), for almost all trajectories (η, ξ,ξ) on Ω 0 with respect to the coupling measure.
Let {µ N : N ≥ 1} be a sequence in P ± (Ω) associated to a initial profile ζ 0 : R + → R. Now, let ξ be an exclusion process with arbitrary initial condition. Then K t 0 (ξ, Γ, Λ) is dominated by the same quantity related to the process starting at configurations with a particle on each site in Λ. Based on this, Let ν N,M = µ N,− × ν N,M,+ be the measure on Ω such that ν N,M,+ is the Bernoulli product measure on Z * This number is clearly bounded by the number of particles at sites in (−∞, CN ] at time t for the simple symmetric excusion process (construct a coupling similar to the previous one on Z), which divided by N converges to the integral on (−∞, C] of the solution of ∂ t ζ(t, u) = 1 2 ∆ζ(t, u), t ∈ R + , u ∈ R, ζ(0, u) = 1{(M, +∞)}(u), u ∈ R .
By Diffusion Theory, the unique solution of the previous equation has a stochastic representation given by E u [h(B t )] , u ∈ R, t ∈ R + , where (B t ) is a standard Brownian Motion and h = 1{(M, +∞)}. It is then a straightforward computation, using the Gaussian kernel, to show that the integral on (−∞, C] of this last expression converges to 0 exponentially fast as M → ∞. Thus (3.39) holds.

From the exclusion process to the Potts model
We prove in this section Theorem 1.1. As a first identification we associate to each configuration f in I such that f (0) = 0 a configuration in N Z representing the increments of the former: η(x) = f (x + 1) − f (x) for every x ∈ Z. This allows us to associate the Potts model dynamics to a zero range dynamics as described in section 1.
For technical reasons we consider the zero range as two coupled processes: the dissipative and the absorbing systems. A configuration η in N Z is associated to a configuration (η, ξ) in Ω = N Z− × {0, 1} Z * + in such a way that, for x ≥ 1, η(x) represents the number of consecutive holes that precede the x th particle in configuration ξ. Since for the exclusion process the total number of sites in a given finite box equals the total number of holes plus the total number of particles, we obtain the following relation: This function ρ : R + × R + → R is the unique solution of (1.5) with initial condition ρ 0 (u) = 1 ζ 0 (M −1 (0, u)) − 1, u > 0.
Thus we have proved: Therefore, λ is the unique solution of 1.4. This concludes the proof of Theorem 1.1.