CL Oriane Blondel
email: blondel@math.univ-lyon1.fr

Ément Erignoux
email: clement.erignoux@inria.fr

Marielle Simon
email: marielle.simon@inria.fr

STEFAN PROBLEM FOR A NON-ERGODIC FACILITATED EXCLUSION PROCESS

We consider the facilitated exclusion process, which is a nonergodic, kinetically constrained exclusion process. We show that in the hydrodynamic limit, its macroscopic behavior is governed by a free boundary problem. The particles evolve on the one-dimensional lattice according to jump rates which are degenerate, since they can vanish on non-trivial configurations and create distinct phases: indeed, configurations can be totally blocked (they cannot evolve under the dynamics), ergodic (they belong to an irreducible component), or transient (after a transitive period of time they will become either blocked or ergodic). We additionally prove that the microscopic separation into blocked/ergodic phases fully coincides with the moving interface problem given by the hydrodynamic equation.

C.E. would like to thank H. Lacoin and C. Landim for interesting discussions. M.S.

Introduction

In statistical physics, various types of (nonlinear) partial differential equations have been derived from underlying microscopic particle systems which belong to the class of stochastic lattice gases. This mathematical procedure is called hydrodynamic limit: the macroscopic behavior is obtained via a long-time and large-space scaling limit, see for instance [START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF] for a review on the subject. In particular, exclusion processes have attracted a lot of interest due to the variety and complexity of the results which have been obtained in the last decades, despite the simplicity of their description. For these models, the hydrodynamic equations obtained in the limit describe the evolution of the local density, which is conserved by the dynamics. These equations become even more interesting when they involve a phase change in the physical medium: in that case, the process of diffusion is mathematically formulated as a Stefan problem [START_REF] Stefan | Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere[END_REF], or free boundary problem.

Such macroscopic behavior can be naturally expected from kinetically constrained lattice gases, or KCLGs, in which the configuration of particles must satisfy a local constraint in order for a particle to be able to jump. For such models, one may predict distinct behaviors of the system at density ρ, depending on whether the local constraint should typically be satisfied at density ρ. This, however, strongly depends on the specific mixing mechanisms of the models. According to a standard terminology, there are two types of kinetically constrained lattice gases (see e.g. [START_REF] Cancrini | Kinetically constrained lattice gases[END_REF]). In non-cooperative KCLGs, a mobile cluster of particles of a given shape can move autonomously in the system (always respecting the kinetic constraint), and once it reaches a specific neighborhood of a particle, allow the latter to jump. The existence of such mobile clusters gives the system good mixing properties, so that their macroscopic behavior is described by diffusive equations with no phase separation. The model considered here, instead, is cooperative, in the sense that no such mobile cluster exists. This generates intrinsic difficulties, and in our case distorts the equilibrium measures which are no longer product measures 1 .

In the first version of this paper, we asked whether it was possible to build a KCLG which would be at the same time cooperative, gradient (in the sense that the generator is a discrete Laplacian), and reversible with respect to product measures. The motivation to build such a model came from the fact that most of the interesting macroscopic phenomenology of KCLG's comes from their cooperative nature, and also from the effort involved in studying non-gradient or non-reversible models (w.r.t. product measures). We already knew that -the Kob-Andersen model [START_REF] Cancrini | Kinetically constrained lattice gases[END_REF] is cooperative and reversible, -the KCLG whose macroscopic behavior is given by the porous medium equation considered in [START_REF] Gonçalves | Hydrodynamic limit for a particle system with degenerate rates[END_REF][START_REF] Blondel | Convergence of a degenerate microscopic dynamics to the porous medium equation[END_REF] is gradient and reversible, -the facilitated exclusion process [START_REF] Rossi | Universality class of absorbing phase transitions with a conserved field[END_REF] is cooperative and gradient.

The impossibility of combining all three characteristics was proved during the revision process by Shapira in the appendix of [START_REF] Shapira | Hydrodynamic limit of the kob-andersen model[END_REF].

1.1. The facilitated exclusion process. In this paper we consider the last model which has been mentioned above, namely the facilitated exclusion process, introduced in [START_REF] Rossi | Universality class of absorbing phase transitions with a conserved field[END_REF] and further investigated in [START_REF] Baik | Facilitated Exclusion Process: The Abel Symposium[END_REF][START_REF] Basu | Active-absorbing-state phase transition beyond directed percolation: A class of exactly solvable models[END_REF][START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF][START_REF] De Oliveira | Conserved lattice gas model with infinitely many absorbing states in one dimension[END_REF][START_REF] Lübeck | Scaling behavior of the absorbing phase transition in a conserved lattice gas around the upper critical dimension[END_REF]. Its dynamics can be described as follows: on the periodic domain T N , we associate independently with each site a random Poissonian clock ringing at rate 2. When the clock at site x rings, if the site x is occupied, the particle chooses one of its neighbors x ± 1 to jump to, each one with probability 1 2 . However, the jump does not systematically occur, but follows two rules, (i) the exclusion principle: if the target site x ± 1 is already occupied, then the jump is canceled, and (ii) a dynamical constraint: if the other neighbor x ∓ 1 is empty, then the jump is canceled. In other words, a particle, in order to jump, needs to be "pushed" to an empty site by a neighboring particle.

Contrarily to the vast majority of exclusion processes considered in the literature, the grand canonical measures of this process are not products of Bernoulli measures: on the one hand, the strong dynamical constraint creates a phase transition at the critical density 1 2 . Precisely, if the equilibrium density ρ satisfies ρ > 1 2 , then there is a unique invariant measure π ρ , while if ρ 1 2 , all the Dirac measures concentrated on configurations which cannot evolve under the dynamics are invariant. On the other hand, π ρ is not a product measure, but presents non-trivial correlations (which however decay exponentially fast, as proved in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF]Section 6.3]). Another technical issue is that the facilitated exclusion process itself is not attractive, though it can be mapped to an attractive zero-range process (see (3.21)).

At the macroscopic level, one naturally expects the same separation of phases. As conjectured in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF], the macroscopic behavior of this system is described by the free boundary problem in which the same nonlinear diffusion equation as in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF] governs the evolution of the density in the supercritical -active -phase ( 12 , 1], while there is no evolution in the subcritical -frozen -phase [0, 1 2 ]. As the frozen region is progressively filled from the growth of the active region, the latter grows and the frontier (or free boundary) between the two regions moves. More precisely, we show that, in the diffusive space/time scaling, the empirical density of particles is governed in the macroscopic limit by the (weak) solution to the following Stefan problem

2 ∂ t ρ = ∂ 2 u H(ρ) , with H(ρ) = 2ρ-1 ρ 1 {ρ> 1 2 } , (1.1)
where 1 {ρ> 1 2 } is the indicator function which equals 1 on the active phase ( 1 2 , 1] and 0 on the frozen phase [0, 1 2 ]. The solution to (1.1) has very poor regularity properties, since it is generically discontinuous at the free boundary. This hydrodynamic limit result (see Theorem 2.4 below) is the first main outcome of this article. 1.2. Hydrodynamic limit. To derive the Stefan problem as stated in Theorem 2.4, the presence of a phase transition prevents the use of standard methods, as the ones exposed in [START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF]. Indeed, the presence of two phases whose stationary measures have disjoint support prohibits using the entropy method, whose center argument relies on comparing the distribution of the process with a global reference measure. The finer relative entropy method fails as well, because it requires the hydrodynamic limit to be smooth, which is not the case for the Stefan problem. Note that the extension of the relative entropy method to a parabolic differential equation proposed in [START_REF] Blondel | Convergence of a degenerate microscopic dynamics to the porous medium equation[END_REF] would also fail, since we are not able to construct a sufficiently good approximation of the solution to our free boundary problem.

In order to circumvent this difficulty, Funaki [START_REF] Funaki | Free boundary problem from stochastic lattice gas model[END_REF], inspired by [START_REF] Varadhan | Scaling limits for interacting diffusions[END_REF], exploits the concept of Young measures. In his model (originally introduced in [START_REF] Chayes | Hydrodynamic limits for one-dimensional particle systems with moving boundaries[END_REF]), two types of particles are present on the discrete lattice, "ice" particles which never move, and "water" particles which evolve according to a speed-change exclusion process. They form two regions, and they interact only through the interfaces separating both regions. Funaki derives a Stefan problem by adapting Varadhan's idea coming 2 Uniqueness of the weak solution to (1.1) in the sense of Definition 2.1 follows from the monotonicity of H [START_REF] Uchiyama | Scaling limits of interacting diffusions with arbitrary initial distributions[END_REF]Theorem 6,p.10].

from [START_REF] Varadhan | Scaling limits for interacting diffusions[END_REF] to his bi-phased model. One important ingredient to apply his strategy is to give a full characterization of the infinite volume stationary measures. For simple exclusion processes, this characterization follows from De Finetti's Theorem ( [START_REF] Liggett | Interacting particle systems-an introduction[END_REF]Section 4.3], [START_REF] Billingsley | Probability and measure[END_REF]Theorem 35.10,p.473]). In [START_REF] Funaki | Free boundary problem from stochastic lattice gas model[END_REF], the supercritical stationary measures are written as a mixture of canonical Gibbs measures using [START_REF] Georgii | Gibbs measures and phase transitions[END_REF]. In our case, Lemma 3.6 is obtained via the mapping to a zero-range process and [START_REF] Andjel | Invariant measures for the zero range processes[END_REF].

Apart from [START_REF] Funaki | Free boundary problem from stochastic lattice gas model[END_REF], other free boundary problems have been derived from discrete microscopic models. In [START_REF] Tsunoda | Derivation of Stefan problem from a one-dimensional exclusion process with speed change. Markov Process[END_REF], the author considers a generalized exclusion process with positive jump rates, reversible w.r.t. product measures. He then argues that a tagged particle acts as a boundary between two phases and shows that its rescaled velocity converges to the solution of the implicit equation satisfied by the free boundary between two similar phases. [START_REF] Carinci | Free boundary problems in PDEs and particle systems[END_REF] investigates a simple exclusion process with injection and removal of mass at the boundaries, one of which is described as the right-most particle in the system rather than a fixed point in space. In [START_REF] Landim | A microscopic model for Stefan's melting and freezing problem[END_REF], the system of interest is described by two coupled simple exclusion processes with annihilating interaction at the contact point. The facilitated exclusion process stands apart because the two phases arise directly from the dynamics, rather than being implemented from the start in the definition of the model. The resulting Stefan problem is also more complex because it allows for a so-called mushy region, i.e. the frozen phase needs not be flat and featureless.

One can also recall from [START_REF] Gravner | Internal DLA and the Stefan problem[END_REF] that the occupancy set of the so-called internal DLA grows according to a Stefan problem. The microscopic systems considered there are close to the zero-range process (3.24) to which the FEP can be mapped, with the difference that the jump rates grow linearly with the number of particles on a given site.

In [START_REF] De Masi | Fast-reaction limit for Glauber-Kawasaki dynamics with two components[END_REF], the authors derive a two-phase Stefan problem from a system of two exclusion processes (with different rates) in which particles of different type annihilate at a certain rate when they are on the same site. Contrary to ours, the process has product equilibrium measure and is amenable to the relative entropy method. [START_REF] Hayashi | Spatial-segregation limit for exclusion processes with two components under unbalanced reaction[END_REF] generalizes the process to allow different killing rates for the two types of particles, which leads to more complicated behaviors for the limiting PDE.

Let us finally mention the papers [START_REF] Dembo | The criticality of a randomly-driven front[END_REF], [START_REF] Delarue | Global solutions to the supercooled stefan problem with blow-ups: regularity and uniqueness[END_REF]. These consider a somewhat reverse problem: the frozen phase invades a (supercooled) liquid phase. The former grows when a diffusing particle from the liquid phase comes in contact with the frozen region. The parameters in the frozen region are irrelevant, so that in the limit we have a one-phase Stefan problem with no mushy region. On the other hand, since the interface can travel at arbitrary large speeds in the microscopic world, the macroscopic equation can exhibit blow-up in finite time.

1.3. Microscopic phases. As noted in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF], in addition to blocked and ergodic configurations, the facilitated exclusion process also presents transient configurations with mixed features, contrary to [START_REF] Funaki | Free boundary problem from stochastic lattice gas model[END_REF]. It is clear that in finite volume they disappear in finite time, but it would be conceivable that in the hydrodynamic limit the process remains in this undecided state. It turns out that this does not happen.

In [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF] we show that, if the initial density is larger than the critical value 1 2 , after a subdiffusive transition time of order (log N ) α , with high probability the system enters the irreducible component -if the initial configuration belongs to the class of so-called regular configurations , which happens with high probability for reasonable initial conditions (see [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF]Section 4]).

In the present setting with two macroscopic phases, it is clear that this is no longer true. However, we can hope for the next best thing: that after a subdiffusive transition time, there is a way to split the system in two parts, one ergodic and the other blocked, that match the macroscopic super-and subcritical phases. Since our hydrodynamic limit result is obtained in a weak sense, one cannot extract this information directly from Theorem 2.4. Therefore, we formulate this in an additional result, Theorem 2.6, which is the second main outcome of this paper. In order to state the desired property rigorously, we need a good notion of macroscopic interfaces, derived directly from the PDE (1.1), which is given in Proposition 2.5. To prove that result, we use PDE techniques as such exposed in [2,[START_REF] Meȋrmanov | The Stefan Problem[END_REF]. The problems of existence, regularity and uniqueness of solutions to Stefan problems have been investigated for years, and always raise obstacles which are overcome by refined approaches: as the literature is huge, we give here only a partial list of works which treat similar equations as (1.1), see for instance [2,[START_REF] Danielli | On the pointwise jump condition at the free boundary in the 1-phase Stefan problem[END_REF][START_REF] Korten | Nonnegative solutions of ut = ∆(u -1) + : regularity and uniqueness for the Cauchy problem[END_REF][START_REF] Korten | Regularity for solutions of the two-phase Stefan problem[END_REF][START_REF] Meȋrmanov | The Stefan Problem[END_REF][START_REF] Meȋrmanov | One-dimensional Stefan problem with an arbitrary initial enthalpy[END_REF].

1.4. Outline of the paper. In Section 2 we give a complete description of the microscopic dynamics, together with its main characteristics (presence of distinct phases), and we state the two main results (Theorem 2.4 and Theorem 2.6). Section 3 is devoted to the proof of the hydrodynamic limit, following Funaki's proof based on Young measures. This strategy needs two main ingredients: the ergodic decomposition for the stationary measures (given in Lemma 3.6), and a local law of large numbers reminiscent of the one-block estimate (given in Proposition 3.8). In Section 4 we prove our second main result about the exact correspondence between the microscopic and macroscopic phases, by using ideas coming from [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF] in order to control the transition period of the microscopic system. We prove in the Appendix, for the sake of completeness, several technical results, which do not contain important conceptual novelties, in particular the existence of macroscopic interfaces as stated in Proposition 2.5. 1.5. Notations. We collect here notations and conventions that we use throughout the paper. Since some of the results rely on [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF], we will as often as possible keep the same notations.

First, N := {0, 1, 2, • • • } denotes the set of non-negative integers and N * := N\{0} the set of positive integers. For any finite set Λ we denote by |Λ| its cardinality.

The parameter N ∈ N * is always a scaling parameter and will go to infinity. We let T N := Z/N Z be the discrete torus of size N , which we will also write as {1, . . . , N }. Similarly, T := R/Z = [0, 1) is the one-dimensional continuous torus. For an interval Λ = [a, b] ⊂ T or Λ = [a, b] ⊂ T N of the discrete or continuous torus, we write min Λ = a, max Λ = b, even though the torus is not naturally ordered.

For any ∈ N we set B := {-, . . . , } as the centered symmetric box of size 2 + 1, which can be seen as either a subset of T N (if 2 + 1 N ), or a subset of Z. More generally, we define B (x) := {-+ x, . . . , + x} the box of size 2 + 1 centered at x. Similarly, we set Λ := {0, . . . , } and Λ (x) := {x, . . . , x + }.

We will consider configurations of particles on discrete sets A, with A either Z, the discrete torus T N , or a finite box Λ Z. These configurations are of exclusion type, meaning that no more than one particle can occupy any site of the lattice. They are generically denoted by η ∈ {0, 1} A . In particular, we denote by Σ N := {0, 1} T N the set of periodic configurations and by Σ := {0, 1} Z the set of infinite ones. For any x ∈ A and configuration η ∈ {0, 1} A , we denote by η x ∈ {0, 1} the particle number at site x. For any Λ ⊂ T N (or Λ ⊂ Z) the configuration η ∈ Σ N (or ∈ Σ) restricted to Λ is denoted by η |Λ . We say that a function f : {0, 1} Z → R is local if there exists Λ a finite subset of Z such that f (η) depends only on η |Λ . For any probability measure π on {0, 1} Λ , and f : {0, 1} Λ → R measurable function, π(f ) denotes the expectation of f w.r.t. the measure π. For any f : Σ N → R measurable, and x ∈ T N , we denote by τ x f the function obtained by translation as follows: τ x f (η) := f (τ x η), where (τ x η) y = η x+y , for any y ∈ T N .

More generally, if P is a probability measure on a set E, and f is a measurable function defined on E, we denote by P(f ) the expectation of f with respect to P.

For any sequence (u k ) k∈N , possibly depending on other parameters than the index k, we will denote O k (u k ) (resp. o k (u k )) an arbitrary sequence (v k ) k∈N such that there exists a constant C > 0 (resp. a vanishing sequence (ε k ) k∈N ) -possibly depending on the other parameters -such that

for all k ∈ N, |v k | C|u k | (resp. |v k | |u k |ε k ).
We will omit the subscript k when clear from context.

A function f :

I × T → R, where I ⊂ R + is an interval, is in C α,β (I × T) if it is of class C α in
the first variable, and of class C β in the second variable. If f is defined on a neighborhood of x, we write f (x + ) (resp. f (x -)) for lim y→x,y>x f (y) =: lim y→x + f (y) (resp. lim y→x,y<x f (y) =: lim y→x -f (y)).

Model and results

2.1. The microscopic dynamics. Let us first introduce the facilitated exclusion process described in the introduction, which is a Markov process on the set of periodic configurations η ∈ Σ N = {0, 1} T N .

The infinitesimal generator ruling the evolution in time of this Markov process is given by L N , which acts on functions f : Σ N → R as

L N f (η) := x∈T N c x,x+1 (η) f (η x,x+1 ) -f (η) , (2.1)
where η x,y denotes the configuration obtained from η by swapping the values at sites x and y, namely (η x,y ) x = η y , (η x,y ) y = η x and (η x,y ) z = η z if z = x, y.

Moreover, the jump rates c x,y (η) translate the exclusion rule (no more than one particle at each site) and dynamical constraint (a particle needs to be pushed to an empty site) as follows:

c x,x+1 (η) = η x-1 η x (1 -η x+1 ) + (1 -η x )η x+1 η x+2 . (2.2)
Let us recall the main properties of this model, which have been already detailed in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF]: first, the dynamics conserves the total number of particles x∈T N η x . Elementary computations yield that the following local conservation law holds: for any x ∈ T N ,

L N η x = j x-1,x -j x,x+1 ,
where the instantaneous current

j x,x+1 = -c x,x+1 (η)(η x+1 -η x ) = τ x h -τ x+1 h, is the discrete gradient of the local function h(η) = η -1 η 0 + η 0 η 1 -η -1 η 0 η 1 . (2.3)
Since it satisfies this last property, the facilitated exclusion process considered here is a gradient model. It is also degenerate, since the jump rates can vanish for non trivial configurations. Fix an initial density profile ρ ini : T → [0, 1]. We will consider, as initial condition, a random configuration of particles which is distributed according to a nonhomogeneous Bernoulli product measure on Σ N fitting ρ ini , defined as

µ N (η) := x∈T N ρ ini ( x N )η x + (1 -ρ ini ( x N ))(1 -η x ) . (2.4)
The invariant measures of this process have been deeply investigated in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF]Section 6]. Due to the strong dynamical constraint, they are not independent products of homogeneous Bernoulli measures (as it is often the case for exclusion processes), but they can be made fully explicit. Moreover, there is a critical density ρ (given in the next section) such that, if the density is bigger than ρ , then there is a unique invariant measure, while all the invariant measures are superpositions of atoms if the density is less than ρ . More details will be given in Section 3.3.

Remark 2.1 (On the initial distribution µ N ). Proving the hydrodynamic limit result (Proposition 2.4 below) only requires the convergence in distribution of the empirical density at initial time, namely,

1 N x∈T N ϕ( x N )η x (0) ----→ N →∞ T ϕ(u)ρ ini (u)du,
for any test function ϕ, where the above convergence holds in probability under µ N . However, in the second part, in the investigation of the creation of microscopic fronts (Theorem 2.6 below), one requires some sharp decay of the correlations of the initial distribution. For the sake of clarity, we do not aim at having minimal assumptions on the initial distribution (which is not the main issue here) and choose as initial distribution the product measure (2.4) throughout the paper.

2.2.

Ergodic and frozen phases. The facilitated exclusion process displays a phase transition. Indeed, because of the microscopic jump constraint, pairs of neighboring empty sites cannot be created by the dynamics. In particular, assuming that initially, at least half of the sites are occupied, particles will diffuse in the microscopic system until there are no longer two neighboring empty sites. On the other hand, if initially at least half of the sites are empty, particles will diffuse until the moment when each particle is surrounded by empty sites and can no longer move. For this reason, given Λ ⊂ Z or Λ ⊂ T N , we now introduce the set of ergodic (resp. frozen) configurations as:

E Λ = η ∈ {0, 1} Λ ; η x + η x+1 1, for all x ∈ Λ such that x + 1 ∈ Λ , (2.5)
namely the set of configurations where all empty sites are isolated, resp.

F Λ = η ∈ {0, 1} Λ ; η x + η x+1 1, for all x ∈ Λ such that x + 1 ∈ Λ , (2.6)
namely the set of configurations where all particles are isolated. An example of an element belonging to each set is given in Figure 2.1. At the macroscopic level, this means that there are two distinct regimes for the behavior of this model3 . Either the macroscopic density is larger than the critical value ρ := 1 2 , in which case the system behaves diffusively, or the density is lower than 1 2 , in which case the system remains frozen after a transitive period during which the particles tend to isolate themselves. The interfaces between these two macroscopic phases move as particles from the supercritical phase (ρ > 1 2 ) diffuse towards the subcritical phase (ρ < 1 2 ). As we already noted in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF], there are transitive (or transient) configurations, which are neither ergodic nor frozen (E Λ ∪ F Λ = {0, 1} Λ ). However, they are called transient in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF] because, if Λ Z is finite, then from these transient configurations the process will evolve toward E Λ ∪ F Λ after a number of particle jumps which is a.s. finite. More precisely, in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF] we show that, if the initial configuration of particles is distributed according to µ N (defined in (2.4)), with ρ ini (T) ⊂ ( 1 2 , 1] (therefore, the initial density profile is uniformly larger than the critical density), then the microscopic system of size N needs a subdiffusive time t N = o(N 2 ) in order to reach the ergodic component.

x η ∈ E Λ η ∈ E Λ

2.3.

Free boundary problem. In this section, we turn to the macroscopic point of view, and first give an explicit free boundary problem, for which we explain what we intend by a solution.

Definition 2.1 (Weak solution of the free boundary problem). For any r 0, define the function

H(r) = 2r -1 r 1 {r> 1 2 }
, and denote by f, g the inner product of f and g in L 2 (du) on T.

Fix T 0 and let ρ ini : T → [0, 1] be a measurable initial profile. We call a measurable function ρ : (t, u) → ρ t (u) a weak solution to the free boundary problem

∂ t ρ = ∂ 2 u H(ρ) (2.7)
with initial condition ρ 0 = ρ ini , if:

• for any (t, u) ∈ [0, T ] × T, ρ t (u) ∈ [0, 1],
• and for any test function

ϕ ∈ C 1,2 ([0, T ] × T) ρ T , ϕ T = ρ ini , ϕ 0 + T 0 ρ t , ∂ t ϕ t dt + T 0 H(ρ t ), ∂ 2 u ϕ t dt. (2.8)
Remark 2.2. Let us briefly comment on the function H. As pointed out in Section 2.1, the generator of the process acts as a discrete Laplacian, in the sense that Lη x = τ x+1 h+τ x-1 h-2τ x h. In the frozen phase, the function h vanishes. However, in the ergodic phase, and under the equilibrium measure π ρ at density ρ (defined in Definition 3.2 below), its average equals h(η)dπ ρ (η) = H(ρ).

Intuitively, the configurations evolving according to (2.1) should be separated in two phases: a region of density above 1 2 , where the macroscopic evolution is given by an elliptic equation, and a (frozen) region of density below 1 2 which is gradually filled by the spreading of the supercritical region. Making this picture rigorous is not immediate, given the weakness of the above notion of solution, even for "simple" initial density profiles, as we explain below. Proposition 2.3 (Uniqueness of weak solutions [START_REF] Uchiyama | Scaling limits of interacting diffusions with arbitrary initial distributions[END_REF]). There exists a unique weak solution of (2.7) with initial condition ρ 0 = ρ ini , in the sense of Definition 2.1.

Since H : [0, 1] → [0, 1] is a non-decreasing function, this proposition is already proved by Uchiyama, see [START_REF] Uchiyama | Scaling limits of interacting diffusions with arbitrary initial distributions[END_REF]Theorem 6,p.10], and therefore we do not reproduce the proof here.

2.4. Main results. As already noted in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF], the macroscopic behavior of the facilitated exclusion process appears in the diffusive time scale. Therefore, we define {η(t) ; t 0} as the Markov process driven by the accelerated infinitesimal generator N 2 L N and initially distributed as µ N (given in (2.4)). Fix T > 0 and denote by P µ N the probability measure on the Skorokhod path space D([0, T ], Σ N ) corresponding to this dynamics. We denote by E µ N the corresponding expectation. We emphasize that, even though it is not explicit in the notation, P, E and η(t) strongly depend on N : through the size of the state space, but also through the diffusive time scaling.

Theorem 2.4 (Hydrodynamic limit). Let ρ ini : T → [0, 1] be a measurable initial profile. For any t ∈ [0, T ], any δ > 0, and any smooth test function ϕ : T → R, we have lim

N →∞ P µ N 1 N x∈T N ϕ( x N )η x (t) - T ϕ(u)ρ t (u)du > δ = 0, (2.9)
where ρ is the unique weak solution of (2.7) with initial condition ρ 0 = ρ ini , in the sense of Definition 2.1.

In order to prove Theorem 2.4, we adapt a technique used by Funaki [START_REF] Funaki | Free boundary problem from stochastic lattice gas model[END_REF], and inspired by [START_REF] Varadhan | Scaling limits for interacting diffusions[END_REF], and we exploit the concept of Young measures, as explained in Section 3. One of the drawbacks of this method, however, is that the hydrodynamic limit is proved in a rather indirect way, which says nothing about the separation of the microscopic configuration into two phases matching the macroscopic ones. Therefore, we also prove in Theorem 2.6 below that after a subdiffusive time, the microscopic picture corresponds exactly to the macroscopic one, in two steps: first, with each subcritical or supercritical macroscopic phase can be associated a connected subcritical or supercritical microscopic box, and second, the microscopic interface between those two boxes is indeed close to the macroscopic one coming from the free boundary problem (2.7). We emphasize that this result is not necessary to prove Theorem 2.4, it is an independent outcome.

For our second theorem, we need two extra assumptions on the initial profile

ρ ini . Let us introduce C 0 := (ρ ini ) -1 ({ 1 2 }) ⊂ T the set of critical points of the initial density profile. Assume that C 0 is a finite set with cardinality c(ρ ini ) ∈ N, (H1)
ρ ini ∈ C 2 (T) and ∂ u ρ ini (u) = 0 ∀u ∈ C 0 . ( H2
)
In this setting, unfortunately, formalizing rigorously the notion of macroscopic interfaces for the solutions of (2.7), and the corresponding microscopic features, can be rather cumbersome. We will therefore state and prove all our results in the case

ρ ini < 1, |C 0 | = 2 and (ρ ini ) -1 ([0, 1 2 ]) = [0, u * ]. (T1)
In other words, we assume that there are only two critical points: 0 ≡ 1 and u * ∈ (0, 1), the initial subcritical phase is the segment [0, u * ], the initial supercritical phase is the complementary segment [u * , 1], and at no point in the supercritical phase is density 1 reached. We stress out that this is not necessary for our proof to hold, and that both Definition 2.3 and Theorem 2.6 below can be straightforwardly extended to any even integer c(ρ ini ).

Definition 2.2 (Two-phased configurations). We say that a configuration η ∈ Σ N is two-phased if there exists a partition T N = E F, such that E, F ⊂ Σ N are (possibly empty) connected subsets, and

η(t) |E ∈ E E and η(t) |F ∈ F F ,
where the set of ergodic and frozen configurations were defined in (2.5) and (2.6).

We denote by P N the set of non-ergodic two-phased configuration, and for any η ∈ P N , we denote E η and F η (the latter must be nonempty if η is non-ergodic) the corresponding ergodic and frozen sets. To ensure uniqueness, we choose E η to be maximal for inclusion.

Note that a two-phased configuration remains two-phased or ergodic along the dynamics: if η(s) ∈ P N , then for all t s, η(t

) ∈ P N ∪ E T N . (2.10)
Therefore, we can define

Definition 2.3 (Microscopic fronts). Assume that at a time t, η(t) ∈ P N . Then we define u N -(t) = min F η(t) and u N + (t) = max F η(t)
, which correspond to the position of the microscopic fronts, with the convention that

u N -(t) = max s t η(s)∈P N u N -(s) and u N + (t) = min s t η(s)∈P N u N -(s)
if η(t) ∈ E T N has already become ergodic. In other words, once the microscopic fronts have merged and the configuration becomes ergodic, we arbitrarily set the position of the microscopic fronts at the last site where the frozen set was non empty.

Theorem 2.6 below states that the configuration becomes two-phased in a subdiffusive time with high probability. It also states that the boundaries of the frozen set (i.e. the microscopic fronts as defined in Definition 2.3) are never far from the macroscopic interfaces. To state this result, we need to show that the latter are well defined. The following result is proved in Appendix D. Proposition 2.5. Assume Assumptions (H1), (H2) and (T1). For any T > 0, the weak solution ρ of the free boundary problem (2.7) admits continuous macroscopic interfaces u -, u + : [0, T ] → T, respectively non-decreasing and non-increasing, satisfying u -(0) = 0, u + (0) = u * . Moreover, there exists τ ∈ R + ∪ {∞} such that (1) for any t < τ , u -(t) = u + (t), and

ρ t (u) < 1 2 if u ∈ (u -(t), u + (t)) > 1 2 if u ∈ (u + (t), u -(t)); (2) if τ < ∞ then u -(τ ) = u + (τ );
(3) for any t τ , ρ t 1 2 on T, and moreover u + , u -are constant:

u + (t) = u -(t) = u -(τ ).
We are now ready to state our second main result.

Theorem 2.6. Assume (H1), (H2) and (T1).

(1) Creation of fronts.

Letting t N = N -1/4 , lim N →∞ P µ N (η(t N ) ∈ P N ) = 1,
i.e. in a time of order N -1/4 , the microscopic configuration is two-phased with high probability.

(2) Macroscopic match. For any t ∈ (0, τ ] ∩ R + , lim

N →∞ P µ N 1 N u N ± (t) -u ± (t) ε = 0,
where τ, u + , u -are defined in Proposition 2.5 and u N ± in Definition 2.3.

The result actually also holds for t N = N -m for any m < 2 3 , but in order to focus on the important points of the proof, we choose simpler exponents in the required estimates, so that taking t N = N -1/4 is convenient. As will be shown in Section 4.6, point (2) is actually a simple consequence of (1) and the hydrodynamic limit result (Theorem 2.4).

Remark 2.7 (On assumption (H2)). The regularity of the initial profile ρ ini at the critical points is crucial to our proof. However, away from the critical points, the regularity assumption could be weakened. To focus on the important points of the proof, we settle for assumption (H2).

Proof of Theorem 2.4: Young measures and hydrodynamic limit

We prove in this section the hydrodynamic limit result stated in Theorem 2.4, following the strategy given in [START_REF] Funaki | Free boundary problem from stochastic lattice gas model[END_REF].

3.1. Empirical measure. For any t ∈ [0, T ], let us define m N t (du) = m N (η(t), du) := 1 N x∈T N η x (t)δ x/N (du), (3.1)
the empirical measure of the process, where δ a (du) stands for the Dirac measure on T at point a ∈ T. The measure m N t is an element of the set M + (T) of positive measures on the torus T, which we endow with the weak topology. We slightly abuse our notation for the inner product in L 2 (T), and also denote by m, • the integral on T with respect to any measure m(du).

Let us denote by P N the pushforward measure of P µ N by the mapping m N , namely P N := P µ N • m N -1 . Then, P N is a probability measure on the path space D([0, T ], M + (T)), endowed with the Skorokhod topology. In order to prove Theorem 2.4, we are reduced to proving the convergence of the sequence (P N ) towards the Dirac probability measure concentrated on the solution of (2.7).

We already know some properties of the sequence (P N ), which are quite standard in the literature: Proposition 3.1 (Absolute continuity w.r.t. the Lebesgue measure). The sequence (P N ) is weakly relatively compact in D([0, T ], M + (T)), and any of its limit points P * is concentrated on trajectories of measures {m t (du) ; t ∈ [0, T ]} which are (1) continuous in time, i.e.

P * (t → m t is continuous) = 1, (3.2)
(2) and whose marginal at time t is absolutely continuous w.r.t. the Lebesgue measure on T, i.e.

P * ∀ t ∈ [0, T ], ∃ ρ t : T → [0, 1], s.t. m t (du) = ρ t (u)du = 1. (3.3)
In particular, these two assertions prove that

P * ∀ t ∈ [0, T ], ∃ ρ t : T → [0, 1] continuous in t, s.t. m t (du) = ρ t (u)du = 1.
This proposition will be proved in Appendix A for the sake of completeness, but it is standard. In many models, the proof of the hydrodynamic limit can be completed from there by using the entropy method. However, for this model, this standard strategy fails because of the presence of supercritical and subcritical phases on which the time invariant measures are distinct and not absolutely continuous w.r.t. one another. For that reason, we now introduce the concept of Young measures, as given in [START_REF] Funaki | Free boundary problem from stochastic lattice gas model[END_REF].

3.2.

Young measures and sketch of the proof. For that purpose, we need to introduce some notations. Given a configuration η, let us denote by

ρ x = ρ x (η) := 1 2 + 1 y∈B (x) η y (3.4)
the local density in the box B (x) of size 2 + 1 around x (defined in Section 1.5). When x = 0, to simplify notations, we denote ρ = ρ 0 . When η is a time trajectory, and the density is observed at time t, we denote for the sake of clarity ρ x (t) = ρ x (η(t)) and ρ (t) = ρ (η(t)). Similarly as before, let us define, for time trajectories,

π N, t := π N, (η(t)).
Remark 3.2. Observe that, for any smooth function ξ defined on T, and taking ψ(r) = r, an integration by parts shows that there exists a constant

C(ξ) > 0 such that m N t , ξ -π N, t , ξ • r C(ξ) N . (3.5)
We now define P N, as the pushforward measure of P µ N by the mapping

(m N ) -1 , (π N, ) -1 ,
namely for any measurable set B,

P N, {m t , π t } t∈[0,T ] ∈ B = P µ N {m N t , π N, t } t∈[0,T ] ∈ B , which is a probability measure on D [0, T ], M + (T) × M + (T × [0, 1]
) . We first state a technical lemma.

Lemma 3.3. The sequence (P N, ) 1 N is weakly relatively compact, and any of its limit points P * as N → ∞ then → ∞ 4 satisfies

P * ∀ t ∈ [0, T ], ∃ ρ t (•), p t (•, dr), s.t. m t (du) = ρ t (u)du π t (du, dr) = p t (u, dr)du = 1. (3.6)
Proof of Lemma 3.3. Since the first marginal of P N, is P N , the fact that P * -a.s., m t (du) is time continuous and absolutely continuous at every time t w.r.t. the Lebesgue measure is a direct consequence of Proposition 3.1. Moreover, P * -a.s., this is also the case of π t (du, dr), since one can easily check after passing to the limit in π N, t , ξ • 1 that, for any smooth function ξ on T

T [0,1] π t (du, dr)ξ(u) T ξ(u)du,
which proves (3.6). Note that all those estimates are deterministic, in the sense that the only used property is the exclusion rule (at most one particle per site is allowed in the configuration). For this reason, the quantifier "∀ t ∈ [0, T ]" can be inserted inside the probability, thus concluding the proof.

We are now ready to state the main result of this section.

Proposition 3.4. The sequence (P N, ) 1 N is weakly relatively compact, and any of its limit points P * as N → ∞ then → ∞ satisfies

P * ∀ (t, u) ∈ [0, T ] × T, p t u, [0, 1 2 ] = 1, or p t (u, dr) = δ ρt(u) (dr) = 1, (3.7)
where p t and ρ t were defined P * -a.s. by (3.6). 4 More precisely, by this expression we mean that we take limits of convergent subsequences as N → ∞ for fixed , then take a convergent subsequence of these objects as → ∞.

In other words, Young measures in r either only charge the subcritical range of densities, or are trivial and given by a Dirac at ρ t (u). Here, by limit point, we mean that we take any convergent subsequence as N → ∞, and then any convergent subsequence as → ∞.

Note that this proposition does not say anything about the function ρ t (u). However, we prove at the end of this paragraph that it is the weak solution of (2.7).

Proposition 3.4 is a consequence of the following lemma.

Lemma 3.5. Any limit point P * as N → ∞ then → ∞ of the sequence (P N, ) 1

N satisfies P * T 0 T [0,1] H(r) r - [0,1] r p t (u, dr ) p t (u, dr)dudt = 0, (3.8)
where P * (• • • ) denotes the expectation w.r.t. P * .

We postpone the proof of this lemma to the end of the section, see Section 3.5 for the conclusion. Before proving it, we show that Proposition 3.4 follows, and then we prove Theorem 2.4.

Proof of Proposition 3.4. We now show (3.7). Since for any fixed (t, u)

∈ [0, T ] × T, p t (u, •) is a probability measure on [0, 1], and since H(r) = 2r-1 r 1 {r 1 2 } is non- decreasing on [0, 1], we have [0,1] H(r) r - [0,1] r p t (u, dr ) p t (u, dr) [0,1] H(r )p t (u, dr ) [0,1] r - [0,1] r p t (u, dr ) p t (u, dr) = 0.
This follows from the inequality (f (x) -f (y))(g(x) -g(y))dµ(x)dµ(y) 0, valid for any measure µ on R if f, g are non-decreasing. The equality case in the above inequality happens when (f (x) -f (y))(g(x) -g(y)) = 0 a.e. In our case, this means that H should be constant on the support of p t (u, dr).

Therefore, from (3.8), we obtain that almost everywhere w.r.t. the Lebesgue measure in [0, T ] × T:

• either : p t (u, [0, 1 2 ]) = 1 (if p t (u, dr) • H -1 = δ 0 ), • or : p t (u, [0, 1 2 ]) = 0 and there exists b t (u) ∈ ( 1 2 , 1] such that p t (u, dr) = δ bt(u) (dr) (because H is one-to-one on ( 1 2 , 1]
). In the second case, since under P * , for any smooth function ξ on T, we have m t , ξ = π t , ξ • r (recall (3.5) and pass to the limit), one finally obtains that almost everywhere in [0, T ] × T, we must further have b t (u) = ρ t (u), which proves Proposition 3.4.

We now conclude with the proof of the hydrodynamic result, namely (2.9) stated in Theorem 2.4.

Proof of Theorem 2.4. Define the discrete laplacian ∆ N , acting on functions ϕ :

T → R, as ∆ N ϕ(u) = N 2 ϕ(u + 1 N ) + ϕ(u -1 N -2ϕ(u) . (3.9)
Recall from (3.1) the definition of the empirical measure m N . We first write by Dynkin's formula, for any ϕ

∈ C 1,2 ([0, T ] × T) m N T , ϕ T -m N 0 , ϕ 0 - T 0 m N t , ∂ t ϕ t dt - T 0 1 N x∈T N ∆ N ϕ t x N τ x h(η(t)) dt = M ϕ,N T ,
where M ϕ,N T is a martingale whose quadratic variation can be written explicitly (see e.g. [21, Appendix 1.5]) as

M ϕ,N t = N 2 t 0 L N m N s , ϕ s 2 -2 m N s , ϕ s L N m N s , ϕ s ds = 1 2 t 0 x∈T N |y-x|=1 ϕ s x N -ϕ s y N 2 c x,y (η(s))ds.
Since the function ϕ is smooth, [M ϕ,N ] t tC(ϕ)/N and vanishes as N → ∞. Using this, the local ergodicity proved in Proposition 3.8, and the fact that replacing τ x h by 1

+1

y∈B τ x+y h in the integral leads to a term which is bounded by C /N (where C > 0 is a constant), we obtain lim →∞ lim sup

N →∞ E µ N m N T , ϕ T -m N 0 , ϕ 0 - T 0 m N t , ∂ t ϕ t dt - T 0 1 N x∈T N ∆ N ϕ t x N H(ρ t (x))dt = 0. (3.10)
Theorem 2.4 is now a consequence of Proposition 3.4 above. Indeed, the expectation in the left hand side of (3.10) rewrites as

P N, m N T , ϕ T -m N 0 , ϕ 0 - T 0 m N t , ∂ t ϕ t dt - T 0 π N, t , ∆ N ϕ t • H dt ,
where the short notation ∆ N ϕ•H stands for (u, r) → H(r)∆ N ϕ(u) (recall Definition 3.1). In particular, as N → ∞ then → ∞, we obtain according to Proposition 3.4 that for any limit point P * of P N, ,

P * ρ T , ϕ T -ρ 0 , ϕ 0 - T 0 ρ t , ∂ t ϕ t dt - T 0 H(ρ t ), ∂ 2 u ϕ dt = 0,
which yields as wanted that P * is concentrated on trajectories m t (du) = ρ t (u)du such that ρ is a weak solution to (2.7), in the sense of Definition 2.1.

The remainder of the section is dedicated to proving Lemma 3.5. For that purpose, we need to state and demonstrate two important results: first we investigate the grand canonical measures of the process and we prove an ergodic decomposition of any infinite volume stationary measure à la De Finetti (Section 3.3, Lemma 3.6); and second, we obtain a local law of large numbers analogous to the well-known one-block estimate (Section 3.4, Proposition 3.8). The end of the proof is given in the last Section 3.5.

3.3.

Canonical and grand canonical measures. Let us define the infinite volume generator associated with our dynamics (recall (2.1)), which acts on local functions f : {0, 1} Z → R, as

L ∞ f (η) := x∈Z c x,x+1 (η) f (η x,x+1 ) -f (η) . (3.11)
In this section, we investigate the measures on {0, 1} Z which are stationary for L ∞ . One of the main ingredients needed to apply the same arguments as in Funaki's proof [START_REF] Funaki | Free boundary problem from stochastic lattice gas model[END_REF] is to prove that any stationary measure for the generator L ∞ , once restricted to the active phase {ρ > ρ = 1 2 }, admits a decomposition along spatially ergodic measures.

Let us first introduce the grand canonical measures π ρ for the facilitated exclusion process, which have been studied in detail in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF]: Definition 3.2 (Grand canonical measures).

• For any ρ ∈ ( 1 2 , 1), and any local configuration σ = (σ 0 , . . . , σ ) on Λ , we define

π ρ η |Λ = σ = 1 {σ∈EΛ } (1 -ρ) 1-ρ ρ -p 2ρ-1 ρ 2p--σ0-σ , (3.12)
where p = p(σ) := y∈Λ σ y is the number of particles in σ, and E Λ was defined in (2.5) as the set of local ergodic configurations. • For any ρ ∈ [0, 1 2 ], we define

π ρ = 1 2 δ •• + 1 2 δ •• , (3.13)
where •• (resp. ••) is the configuration in which there is a particle at x iff x is odd (resp. even), and δ η is the Dirac measure concentrated on the configuration η. • For ρ = 1, let π 1 = δ 1 , where 1 denotes the configuration identically equal to 1.

We know from [7, Section 6] that the measures π ρ are invariant for the generator L ∞ . Here we prove important additional properties of theses measures. The main result of this section is the following: Lemma 3.6 (Ergodic decomposition of stationary measures). Let µ be a translation invariant, infinite volume, measure on {0, 1} Z , which is stationary for L ∞ , i.e. such that for any local function f , µ(L ∞ f ) = 0.

Then, there exist λ ∈ [0, 1], a probability measure µ F with support included in F Z (the set of frozen configurations, cf (2.6)), and a probability measure (dρ) on

[ 1 2 , 1], such that, µ(•) = λµ F (•) + (1 -λ) [ 1 2 ,1] (dρ)π ρ (•). (3.14)
Proof of Lemma 3.6. We first discard the degenerate case, where the translation invariant measure µ satisfies µ(η 0 = 1) = 1: in this case, by translation invariance, µ = π 1 = δ 1 . Then the result is trivially true. Fix now a translation invariant measure µ on {0, 1} Z which is stationary w.r.t. the generator L ∞ , and such that µ(η 0 = 1) < 1 (i.e. µ(η 0 = 0) > 0). Recall from (2.5) and (2.6) the definition of the sets of infinite ergodic and frozen configurations E Z and F Z . We first claim that, since µ is stationary, we must have

µ {0, 1} Z \ (E Z ∪ F Z ) = 0, (3.15)
i.e. µ charges configurations which are either completely ergodic, or completely frozen. To expose the argument as clearly as possible, let us indicate the occupied sites by •, and the empty sites by •, and any local configuration η by a finite sequence of • and •. Since µ is translation invariant, there will be no need to specify the support of the configurations in the following argument. We further use the notation

[••] k := • • • • • • • 2k sites and [••] k := • • • • • • • 2k sites
.

We are going to show that for any k 0,

µ(• • [••] k • •) = µ(• • [••] k • •) = 0, (3.16)
the box where the configuration is observed being arbitrary, but fixed. Since any configuration which is not in

E Z nor in F Z must contain either • • [••] k • • or • • [••] k • •
for some k, this will prove (3.15). For k = 0, we write by definition and using the translation invariance of

µ µ L ∞ 1 {••} = -µ(• • • •) -µ(• • • •) = 0,
since µ is stationary. Therefore both probabilities on the right hand side, which are non-negative, must be equal to 0. This proves (3.16) for k = 0. Assume now that (3.16) holds for any < k, then write (more explanations on the following identity are given right below):

0 = µ L ∞ 1 {•• [••] k-1 ••} = -µ • • • • [••] k-1 • • -µ • • [••] k-1 • • (3.17) -µ • • [••] k-1 • • • + µ • • [••] k • • (3.18) + k-3 =0 µ(• • [••] • • • • [••] k--3 • •) (3.19) + k-2 =0 µ(• • [••] • • [••] k--2 • •). (3.20)
Let us comment briefly on the identity above: the only terms that give a non-zero contribution to

L ∞ 1 {•• [••] k-1 ••} are:
(1) the jumps that can happen in the configuration

• • [••] k-1
• • (giving contributions with the minus sign). There are three possibilities: first, if there are two extra particles to the left, then the first pair of empty sites may be broken by a particle coming from the left,

• • • • [••] k-1 • • → • • • • [••] k-1 • •,
and this gives the first term in (3.17). Similarly, the transitions corresponding to the second term in (3.17) and first term in (3.18) are given respectively by

• • [••] k-2 • • • • → • • [••] k-2 • • • • and • • [••] k-1 • • • → • • [••] k-1 • • •.
(2) Then, there are the jumps that, starting from another configuration, lead to

• • [••] k-1
• • (giving the three contributions with the plus sign). The corresponding transitions are depicted below:

• • [••] k-1 • • • • → • • [••] k-1 • • • •, • • [••] • • • • [••] k--3 • • → • • [••] • • • • [••] k--3 • •, • • [••] • • [••] k--2 • • → • • [••] • • [••] k--2 • •.
Note that all terms in (3.

(• • [••] k • •).
Therefore the latter must vanish as well. An analogous computation for µ

L ∞ 1 {••[••] k-1 ••}
proves the second identity, so that (3.16) holds for any k. Now, let λ = µ(F Z ) ∈ [0, 1] be the total mass of frozen configurations. Note that any translation invariant measure with support included in F Z is necessarily stationary for L ∞ . In particular, in order to prove Lemma 3.6, we only need to treat the decomposition of µ restricted to the ergodic component. Without loss of generality, we can therefore assume that λ = 0, i.e. µ(E Z ) = 1. Let us also put aside the case where µ gives positive weight to 1 and assume µ(1) = 0. Let us define the set of configurations with infinitely many zeros both right and left of the origin:

Σ ∞ := η ∈ {0, 1} Z ; x 0 (1 -η x ) = x 0 (1 -η x ) = ∞ .
We claim that, since we assumed µ(η 0 = 0) > 0 and µ(1) = 0, we must have µ(Σ c ∞ ) = 0. To prove this claim, fix a semi infinite configuration η + on N with a finite number of empty sites (see Figure 3.1), denote c = µ(η |N = η + ). If η + = 1, denote z its rightmost empty site, and define the set

E k := η ∈ {0, 1} Z ; η |{k(z+1),... } = τ -k(z+1) η + .
x 0 z etc.

E k ⊂ η ∈ {0, 1} Z ; x k(z+1) (1 -η x ) > 0 and x (k+1)(z+1) (1 -η x ) = 0 ,
therefore in particular, we must have c = 0. Since there are countably many configurations with a finite number of empty sites to the right of the origin, and since we just proved that their probabilities vanish, making the same statement for configurations to the left of the origin yields as wanted µ(Σ c ∞ ) = 0.

To prove the ergodic decomposition (3.14), we use a classical mapping between the facilitated exclusion process and a zero-range process, introduced in [START_REF] Basu | Active-absorbing-state phase transition beyond directed percolation: A class of exactly solvable models[END_REF] and already exploited in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF]. For simplicity, we define this mapping on the set Σ 0 ∞ := {η ∈ Σ ∞ ; η 0 = 0} of configurations with an empty site at the origin. Then, given η ∈ Σ 0 ∞ , and for any integer k > 0 (resp. -k < 0) we denote x k (η) (resp. x -k (η)) the position of the k-th empty site to the right (resp. to the left) of the origin, and let x 0 = 0. We then define, for any k ∈ Z and η ∈ Σ 0 ∞ ω η k := x k+1 (η) -x k (η) -1. In other words, ω η ∈ N Z is the zero range configuration such that the number of particles on site k > 0 (resp. -k < 0) is the number of particles between the k-th and (k+1)-th empty site to the right (resp. to the left) of the origin in η (see Figure 3 We first note that for any ergodic configuration η ∈ Σ 0

∞ ∩ E Z , we must have ω η ∈ N Z * . Let us denote by Π the one-to-one mapping

Π : Σ 0 ∞ ∩ E Z → N Z * η → ω η . (3.21)
Recalling that we assume µ(E Z ) = 1 and µ(η 0 = 0) > 0, we now define a measure ν on the set N Z * of infinite zero-range configurations, ν(F

) := µ(ω η ∈ F | η 0 = 0), F ⊂ N Z * measurable. (3.22)
In particular, for any E ⊂ E Z measurable,

µ(η ∈ E | η 0 = 0) = ν Π(E ∩ Σ 0 ∞ ) . (3.23)
Define the infinite volume zero-range generator L ZR ∞ which acts on local functions f :

N Z → R L ZR ∞ f (ω) := x∈Z δ=±1 1 {ωx 2} f (ω x,x+δ ) -f (ω) , (3.24)
with ω x,x+δ representing the zero-range configuration where one particle in ω was moved from x to x + δ, i.e.

ω x,x+δ y =        ω x -1 if y = x ω x+δ + 1 if y = x + δ ω y else .
One easily checks that for any α 1 the geometric product homogeneous measures ν α with marginals

ν α (ω 0 = p) = 1 {p∈N,p 1} 1 α 1 - 1 α p-1 (3.25)
are reversible for L ZR ∞ , and that α = E να (ω 0 ) then represents the average particle density per site. We claim the following.

Lemma 3.7. The measure ν defined by (3.22) on N Z * is translation invariant, and stationary w.r.t. the zero-range generator L ZR ∞ . In particular from [START_REF] Andjel | Invariant measures for the zero range processes[END_REF], there exists a probability measure ZR on [1, +∞), such that

ν(•) = [1,+∞) ZR (dα)ν α (•) . (3.26)
Before proving this result, we show that Lemma 3.6 follows. For any event E ⊂ Σ 0 ∞ ∩ E Z , we can now write according to Lemma 3.7 and using (3.23),

µ(E | η 0 = 0) = [1,+∞) ZR (dα)ν α Π(E) .
for some measure ZR (dα) on [1, +∞). Define G(α) = α/(1 + α), which is an increasing bijection from [1, +∞) to [ 1 2 , 1). Given the explicit expressions (3.25) and (3.12) for ν α and π ρ , one easily checks that

π G(α) (E | η 0 = 0) = ν α Π(E) .
We now define the measure on [ 1 2 , 1) as the pushforward of ZR by G

= ϕ ZR • G -1 ,
which yields after a change of variables

µ(E | η 0 = 0) = [ 1 2 ,1) (dρ)π ρ (E | η 0 = 0). Finally, let (dρ) = µ(η 0 = 0) π ρ (η 0 = 0) (dρ),
and we obtain

µ(E ∩ {η 0 = 0}) = [ 1 2 ,1) (dρ)π ρ (E ∩ {η 0 = 0}).
Since by assumption µ(Σ ∞ ) = 1, for any event E ⊂ E Z we can write µ(E) = ∞ k=0 µ(E ∩ {η 0 = . . . = η k-1 = 1, η k = 0}) and similarly with π ρ for any ρ ∈ [ 1 2 , 1). Using the translation invariance of both µ and π ρ in these identities, we obtain

µ(E) = [ 1 2 ,1)
(dρ)π ρ (E) as wanted, which concludes the proof of Lemma 3.6.

We now prove Lemma 3.7.

Proof of Lemma 3.7. It was proved by Andjel in [1, Theorem 1.9] that any translation invariant, stationary measure for the zero-range process with constant jump rate can be decomposed as

[1,+∞) (dα) ν α (•), (3.27)
where ν α is the product measure on N Z with marginals ν α (ω 0 = p) = 1 α (1-1 α ) p . We can couple this zero-range process (which lives on N Z ) with the process generated by L ZR ∞ and restricted to configurations in N Z * by simply adding a particle at every site. The decomposition (3.27) then yields (3.26) for any translation invariant measure on N Z * which is stationary w.r.t. the zero-range generator L ZR ∞ . We therefore only need to prove the first two claims, namely the translation invariance and stationary properties.

(i) Translation invariance. Recall that we denote Λ = {0, . . . , }. To prove that ν is translation invariant, fix 0 and consider a local zero-range configuration σ = (σ 0 , . . . , σ ) ∈ N +1 * . We are going to prove that for any x ∈ Z,

ν(ω |x+Λ = σ ) = ν(ω |Λ = σ ),
where we shortened x+E = {x+y, y ∈ E}. To prove it, first note that by definition ν(ω |x+Λ = σ ) = µ(η 0 = 0) -1 µ ω η |x+Λ = σ and η 0 = 0 .

Assume that x = 0 (otherwise the statement is trivial), and first consider the case where 0 ∈ x + Λ , so that in particular we must have x -1. In this case, denote = µ ω η |Λ = σ and η 0 = 0 . To establish the second identity, we use the translation invariance of µ and make the change of variable η = τ k η . To establish the third (resp. fourth) identity, we simply lie down what the configuration ω τ k η |Λ +x (resp. ω η |Λ ) means for η. This proves the translation invariance, assuming that 0 ∈ x + Λ .

k := k(σ ) = -x-1 y=0 (1 + σ y ),
We now consider the case x > 0. The case x < -being strictly analogous, we will not detail it here. For two zero-range configurations σ k and σ defined respectively on Λ k and Λ , denote σ k σ the concatenated configuration (defined on Λ k+ +1 ) σ k σ = (σ k 0 , . . . , σ k k , σ 0 , . . . , σ ). Fix x > 0, and write

µ ω η |x+Λ = σ and η 0 = 0 = σ x-1 ∈(N * ) x µ ω η |Λ x+ = σ x-1 σ and η 0 = 0 .
As in the first case, define k = where the third identity is derived as above. Summing over all σ x-1 ∈ (N * ) x , one finally obtains as wanted µ ω η |x+Λ = σ and η 0 = 0 = µ ω η |Λ = σ and η 0 = 0 . This proves that the measure ν is translation invariant.

(ii) Stationarity. We now prove that ν is stationary for the zero-range generator L ZR ∞ as well. To do so, it is sufficient to prove that for any 0 and any local configuration

σ ∈ N +1 * , ν L ZR ∞ 1 {ω |Λ =σ } = 0. (3.28)
Proving this identity is a matter of elementary, though lengthy, computations; to facilitate reading, we will only write it for = 1 and let the reader check that this identity also holds for = 0 and > 1. In order not to burden the notations, we now omit the exponent = 1 in the configuration σ. Fix a pair of integers (σ 0 , σ 1 ) ∈ N 2 * , we can write

L ZR ∞ 1 {ω0=σ0, ω1=σ1} = 1 {ω-1 2} 1 {ω0=σ0-1, ω1=σ1} -1 {ω0=σ0, ω1=σ1} + 1 {ω0 2} 1 {ω0=σ0+1, ω1=σ1} + 1 {ω0=σ0+1, ω1=σ1-1} -21 {ω0=σ0, ω1=σ1} + 1 {ω1 2} 1 {ω0=σ0, ω1=σ1+1} + 1 {ω0=σ0-1, ω1=σ1+1} -21 {ω0=σ0, ω1=σ1} + 1 {ω2 2} 1 {ω0=σ0, ω1=σ1-1} -1 {ω0=σ0, ω1=σ1} . (3.29)
Analogously, for η ∈ E Z ,

L ∞ 1 {η0=0, ω η 0 =σ0, ω η 1 =σ1} = 1 {η0=0} -1 {ω η -1 2, ω η 0 =σ0, ω η 1 =σ1} + 1 {ω η 0 2} 1 {ω η 0 =σ0+1, ω η 1 =σ1-1} -21 {ω η 0 =σ0, ω η 1 =σ1} + 1 {ω η 1 2} 1 {ω η 0 =σ0, ω η 1 =σ1+1} + 1 {ω η 0 =σ0-1, ω η 1 =σ1+1} -21 {ω η 0 =σ0, ω η 1 =σ1} + 1 {ω η 2 2} 1 {ω η 0 =σ0, ω η 1 =σ1-1} -1 {ω η 0 =σ0, ω η 1 =σ1} + 1 {η1=0, ω τ 1 η -1 2, ω τ 1 η 0 =σ0-1, ω τ 1 η 1 =σ1} + 1 {η-1=0, ω τ -1 η 0 =σ0+1, ω τ -1 η 1 =σ1} . (3.30)
The last two terms rewrite as

τ 1 1 {η0=0, ω η -1 2, ω η 0 =σ0-1, ω η 1 =σ1} + τ -1 1 {η0=0, ω η 0 =σ0+1, ω η 1 =σ1} . (3.31)
In particular, since µ was assumed to be translation invariant, the expectation of (3.31) with respect to µ is also equal to the expectation of

1 {η0=0, ω η -1 2, ω η 0 =σ0-1, ω η 1 =σ1} + 1 {η0=0, ω η 0 =σ0+1, ω η 1 =σ1} .
Using this, and taking the µ-expectation in both identities (3.29) and (3.30) we obtain as wanted:

µ L ∞ 1 {η0=0, ω η 0 =σ0, ω η 1 =σ1} = µ 1 {η0=0} L ZR ∞ 1 {ω0=σ0, ω1=σ1} (ω η ) = µ(η 0 = 0) ν L ZR ∞ 1 {ω0=σ0, ω1=σ1} .
Since µ is stationary for L ∞ , the left hand side above vanishes, therefore so does the right hand side. As seen previously, we assumed that µ(η 0 = 0) > 0, which proves ν L ZR ∞ 1 {ω0=σ0, ω1=σ1} = 0, and in turn (3.28). This concludes the proof of Lemma 3.7.

Local ergodicity.

We now turn to the second brick necessary to prove Lemma 3.5. Let µ N t denote the distribution on Σ N of η(t), and define the space-time average

µ N T := 1 T N T 0 x∈T N µ N t • τ -1 x dt. (3.32)
Recall definition (3.4) of ρ x and recall that we set ρ = ρ 0 . We are now ready to state the following result, which, although proved differently, is analogous to the so-called one-block estimate (see [START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF]Section 5.4]). Proof of Proposition 3.8. Thanks to the work of the previous section, and the correlation decay for the grand canonical measures π ρ proved in [7, Section 6.3], we are able to prove this proposition using the same arguments as in [START_REF] Funaki | Free boundary problem from stochastic lattice gas model[END_REF]. Therefore, we simply sketch out the proof here. First, note that for any ρ ∈ [0, 1],

π ρ (h) = 2ρ -1 ρ 1 {ρ> 1 2 } = H(ρ). (3.34)
Recall the infinite volume generator L ∞ introduced in (3.11). By periodically extending the configurations, one can see µ N T as a measure on the set of infinite exclusion configurations, namely {0, 1} Z . For any local function f defined on this set, any x ∈ T N , and any t 0, we can then write

dµ N t (τ x f ) dt = µ N t (N 2 L N τ x f ) = N 2 µ N t (τ x L ∞ f )
for any N large enough (depending on the support of f ). In particular, averaging over x ∈ T N and over the segment [0, T ], the identity above rewrites

1 N 3 T x∈T N µ N T (τ x f ) -µ N 0 (τ x f ) = µ N T (L ∞ f ).
Since f is a local function, it is in particular bounded, therefore the left hand side above vanishes as N → ∞. As a consequence, any limit point µ T of µ N T is stationary for the infinite volume generator L ∞ , and to obtain (3.33) it is sufficient to prove that lim

→∞ µ T 1 2 + 1 y∈B τ y h -π ρ (h) = 0, (3.35)
for every measure µ T which is stationary and translation invariant. According to Lemma 3.6, we can decompose µ T as

µ T = λµ T F + (1 -λ) [ 1 2 ,1] T (dρ)π ρ ,
where λ ∈ [0, 1], where the support of µ T F is included in F Z (recall (2.6)), and T (dρ) is a probability measure on [ 1 2 , 1]. If η ∈ F Z is a frozen configuration, then it cannot contain two neighboring particles, therefore ρ (η) 1 2 , and both h(η) and π ρ (η) (h) vanish a.s. Hence (3.35) trivially holds with µ T F instead of µ T . Similarly, π 1 -a.s. we have

1 2 +1 y∈B τ y h -π ρ (h) = 0. We now prove lim →∞ [ 1 2 ,1) T (dρ)π ρ 1 2 + 1 y∈B τ y h -π ρ (h) = 0.
To do so, fix ε > 0, and split the integral in ρ over [ 1 2 , 1) as a first contribution over [ 1 2 , 1 2 + ε) and a second over [ 1 2 + ε, 1):

• If ρ ∈ [ 1 2 + ε, 1)
, one can straightforwardly show using the same proof as in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF]Corollary 6.6], that the correlations under the measures π ρ between two boxes at distance decay exponentially as e -C , uniformly in ρ ∈ [ 1 2 + ε, 1). Using the Lipschitz continuity of ρ → π ρ (h), one easily obtains that there exists a constant C = C(ε) > 0 such that

π ρ 1 2 + 1 y∈B τ y h -π ρ (h) π ρ 1 2 + 1 y∈B τ y h -π ρ (h) + π ρ π ρ (h) -π ρ (h) (3.36) = O (e -C ), which vanishes as → ∞ uniformly in ρ ∈ [ 1 2 + ε, 1). • If ρ ∈ [ 1 2 , 1 2 + ε),
we use the exact same bound, namely (3.36), and the fact that π ρ (h) = 2ρ-1 ρ . Moreover, ρ 1 2 a.s. under π ρ , since π ρ only charges configurations without consecutive empty sites. In particular, for any ρ 1 2 + ε, the second term in (3.36) can be estimated for any K > 0 by π ρ π ρ (h) -π ρ (h) 8Kε + π ρ ρ -1 2 > Kε . By Markov inequality, the second term in the right hand side is less than 1/K. Therefore, letting → ∞, then ε → 0 and then K → ∞, proves that the second term in (3.36) vanishes uniformly as → ∞ and ε → 0. The correlations, however, no longer decay uniformly. But we can write, recalling the expression (2.3) for the function h,

π ρ 1 2 + 1 y∈B τ y h -π ρ (h) 2π ρ (|h|) 6π ρ (η 0 η 1 ) 12ε, uniformly in ρ ∈ [ 1 2 , 1 2 + ε
). This proves Proposition 3.8.

3.5.

Proof of Lemma 3.5. We now closely follow Funaki's proof [START_REF] Funaki | Free boundary problem from stochastic lattice gas model[END_REF]. Recall that to prove Lemma 3.5, one needs to show the following: any limit point P * as N → ∞ then → ∞ of the sequence (P N, ) 1 N satisfies

P * T 0 T [0,1] H(r) r - [0,1]
r p t (u, dr ) p t (u, dr)dudt = 0, (3.37) with H(r) = 2r-1 r 1 {r 1 2 } and p t defined by (3.6).

For u ∈ 1 N T N , t ∈ [0, T ],
we let H N : t, u → H N t (u) be the solution of the discrete heat equation on 1 N T N with N particles initially at the origin, that is

∂ t H N t (u) = ∆ N H N t (u), u ∈ 1 N T N , t ∈ [0, T ] H N 0 (u) = N 1 {u=0} , u ∈ 1 N T N .
, (3.38) where ∆ N is the discrete laplacian defined in (3.9). One obtains straightforwardly (cf. [16, p. 589]) an explicit expression for

H N t ( x N ): for any x ∈ T N H N t x N = 1 -1 {N is even} e -4tN 2 cos(πx) + 2 N/2 k=1 e -tλ N k cos 2kπx N , (3.39)
where λ N k := 4N 2 sin 2 kπ N . For any time τ 0, let us now introduce

V N, τ : = E µ N T 0 1 N 2 y∈T N H N τ y N x∈T N ρ x+y (t) + ρ x-y (t) H(ρ x (t))dt (3.40) = T µ N T 1 N y∈T N H N τ y N ρ y + ρ -y H ρ 0 , (3.41)
where in the last identity we used the definition of µ N T given in (3.32). For convenience sake, we assume that the sequence P N, converges to P * as N → ∞ then → ∞ (in particular the intermediate limits as N → ∞ for fixed exist). At any moment this assumption can be dropped by taking an arbitrary convergent subsequence instead. We prove two important results about V N, τ :

Lemma 3.9.

lim Θ→∞ lim →∞ lim N →∞ V N, Θ/N 2 = 2P * T 0 T [0,1]
rH(r)p t (u, dr)dudt .

(3.42) Lemma 3.10.

lim θ→0 lim →∞ lim N →∞ V N, θ = 2P * T 0 T [0,1] H(r)p t (u, dr) [0,1]
r p t (u, dr )dudt .

(3.43)

The proofs of Lemmas 3.9 and 3.10 are straightforward adaptations of Lemmas 5.5 and 5.6 in [START_REF] Funaki | Free boundary problem from stochastic lattice gas model[END_REF].

Proof of Lemma 3.9. We first consider H N Θ/N 2 ( y N ) sampled at a large microscopic time Θ/N 2 . Roughly speaking, H N Θ/N 2 converges to a macroscopic Dirac measure at 0, so that convoluted with H N Θ/N 2 , we get: as N → ∞, ρ t (x ± y) ρ t (x) for large enough, which yields the identity in the lemma. More precisely, we use the second expression of V N, Θ/N 2 given in (3.41), and we split the sum in y in two parts, depending on whether |y| Θ or |y| > Θ:

• in the second case |y| > Θ, H N Θ/N 2 ( y N
) is small: more precisely, consider a continuous time random walk X t initially at site 0, and jumping at rate N 2 on each of its neighbors on T N , by Feynman Kac's formula, we can write 2rH(r) π N, t (du, dr)dt.

Thanks to Lemma 3.3 we get the result (3.42).

Proof of Lemma 3.10. We now consider a sample of H N θ at a small macroscopic time θ. Denote h s (u) the heat kernel on T, namely andt T , we obtain from Lemma 3.3

h s (u) = 1 + 2 ∞ k=1 e -sk cos(2πku). Since E µ N [H(ρ x (t))] is bounded uniformly in N , x ∈ T N ,
lim θ→0 lim →∞ lim N →∞ E µ N T 0 1 N 2 y∈T N H N θ y N x∈T N ρ x+y (t) + ρ x-y (t) H(ρ x (t))dt = lim θ→0 lim →∞ lim N →∞ 2E µ N T 0 1 N 2 x,y∈T N H N θ x-y N η y (t)H(ρ x (t))dt = lim θ→0 2P * T 0 dt T du T dv h θ (u -v)ρ t (v) [0,1]
H(r)p t (u, dr) , which converges as θ → 0 to the wanted quantity since h θ (v)dv converges to a Dirac at the origin, and since ρ t (v) = [0,1] r p t (v, dr ).

To end the proof of Lemma 3.5, we now need to show that both limits (3.42) and (3.43) are equal. For a configuration η ∈ Σ N , we define the averaged empirical measure on T N , where the density at each point is averaged out over a large microscopic box of size 1, namely

m N, (du) = m N, (η, du) := 1 N x∈T N δ x/N (du)ρ x (η),
where ρ x (η) was defined in (3.4) as the density in a box of size around x. Once again, when η depends on time, we shorten m N, t = m N, (η(t)). Note in particular that for any function ξ on T, we have m N, , ξ = π N, , ξ • r , where the Young measure π N, was introduced in Definition 3.1, and r is the short notation for the identity application on [0, 1]. For any 1

N and any time s 0, we introduce

R N, s := E µ N m N, T , m N, T * H N s -m N, 0 , m N, 0 * H N s , (3.44)
where the convolution * between a measure m and a function ξ

N on 1 N T N is defined as the function (m * ξ N )(u) = m, ξ N (u -•) for any u ∈ 1 N T N .
In particular, for any times t, s we have (recall Definition 3.1)

m N, t , m N, t * H N s = 1 N 2 x,y∈T N H N s y N ρ x-y (t)ρ x (t).
We need the following two results.

Lemma 3.11.

lim θ→0 lim Θ→∞ lim →∞ lim N →∞ θ Θ/N 2 R N, s ds = 0 .
Lemma 3.12. For any θ > 0,

lim Θ→∞ lim →∞ lim N →∞ θ Θ/N 2 R N, s -T N, s ds = 0,
where

T N, s := E µ N T 0 1 N 2 y∈T N ∆ N H N s y N x∈T N ρ x+y (t) + ρ x-y (t) H(ρ x (t))dt .
Since we now have all the ingredients to do so, before turning to the proof of Lemma 3.11 and Lemma 3.12, we conclude the proof of Lemma 3.5.

Proof of Lemma 3.5. Since by definition ∆

N H N s = ∂ s H N s , we have θ Θ/N 2 T N, s ds = V N, θ -V N, Θ/N 2 ,
where V N, τ was defined in (3.40). As a consequence of Lemmas 3.11 and 3.12, we have

lim θ→0 lim Θ→∞ lim →∞ lim N →∞ θ Θ/N 2 T N, s ds = 0. which proves lim Θ→∞ lim →∞ lim N →∞ V N, Θ/N 2 = lim θ→0 lim →∞ lim N →∞ V N, θ . (3.45)
In particular, Lemma 3.5 follows from Lemmas 3.9 and 3.10.

It remains to prove Lemma 3.11 and Lemma 3.12.

Proof of Lemma 3.11. This is immediate: since H N s is non-negative, for any t we have

E µ N m N, t , m N, t * H N s 1 N y∈T N H N s y N E µ N sup x∈T N ρ x (t) 2 1.
In particular, R N, s is uniformly bounded. Since we integrate it over a time segment s ∈ [Θ/N 2 , θ] whose length vanishes in the limit, this concludes the proof.

We now turn to the proof of Lemma 3.12, for which we need the following two technical lemmas. Lemma 3.13. For any function ξ N on 1 N T N ⊂ T, we have the identity

N 2 L N m N , m N * ξ N = 1 N 2 x,y∈T N ∆ N ξ N y N (η x+y + η x-y ) τ x h(η) + ∆ N ξ N (0) N 2 x∈T N (η x+1 + η x-1 -2η x ) τ x h(η).
Lemma 3.13 follows from rather elementary computation, we give its proof for the sake of completeness in Appendix B. Lemma 3.14 (Equivalent formula for R N, s ). Let us introduce, for any 1 N and x ∈ T N , and any function ξ N defined on 1 N T N , the average function

ξ N N, x N = 1 (2 + 1) 2 y1,y2∈B ξ N x+y1+y2 N , (3.46)
and recall from (3.1) the definition of the empirical measure m N t . Then, seeing m N t as a measure on 1 N T N , we have

R N, s = E µ N m N T , m N T * H N s N, -m N 0 , m N 0 * H N s N, (3.47)
= E µ N T 0 1 N 2 y∈T N ∆ N H N s N, y N x∈T N (η x+y + η x-y ) (t)τ x h(η(t)) + ∆ N H N s N, (0) N 2 x∈T N (η x+1 + η x-1 -2η x ) (t)τ x h(η(t)) dt . (3.48)
Proof of Lemma 3.14. The first identity (3.47) is an easy integration by parts. The second one is obtained by, first, writing Dynkin's formula, and then using Lemma 3.13.

We are now ready to prove Lemma 3.12.

Proof of Lemma 3.12. From (3.39) one can easily obtain

∆ N H N s N, ∞ ∆ N H N s ∞ 2 N -1 k=0 λ N k e -sλ N k .
One first estimates the contribution to θ Θ/N 2 R N, s ds of the second term in the right hand side of (3.48): this contribution can be crudely bounded from above for any N by

θ Θ/N 2 E µ N T 0 ∆ N H N s N, (0)
N 2 x∈T N (η x+1 + η x-1 -2η x ) (t)τ x h(η(t)) dt ds 4T N N -1 k=0 e -Θλ N k N -2 ----→ N →∞ 4T 1 0 e -4Θ sin 2 (uπ) du,
and therefore vanishes as

N → ∞ then Θ → ∞.
Furthermore, by integrations by parts, one can rewrite the quantity appearing in the first term of (3.48), namely

y∈T N ∆ N H N s N, y N x∈T N (η x+y + η x-y ) (t)τ x h(η(t)), as y∈T N ∆ N H N s y N x∈T N ρ x+y (t) + ρ x-y (t) 1 2 + 1 z∈B τ x+z h(η(t)).
In order to prove Lemma 3.12, it is therefore enough to show that

1 N 2 E µ N T 0 x∈T N y∈T N θ Θ/N 2 ∆ N H N s y N ds ρ x+y (t) + ρ x-y (t) × 1 2 + 1 z∈B τ x+z h(η(t)) -H(ρ t (x)) dt (3.49)
vanishes in the limit of the statement. As before, we first rewrite the integral in s as

θ Θ/N 2 ∆ N H N s y N ds = H N θ y N -H N Θ/N 2 y N .
The absolute value of (3.49) is therefore bounded from above by the sum of four terms

A + θ + A - θ + A + Θ/N 2 + A - Θ/N 2 ,
where A ± τ is given by the following expression

E µ N 1 N 2 T 0 x∈T N y∈T N H N τ y N ρ x±y (t) 1 2 + 1 z∈B τ x+z h(η(t)) -H(ρ x (t)) dt ,
which is bounded uniformly in τ from above by

T µ N T 1 2 + 1 z∈B τ z h(η) -H(ρ 0 ) , since ρ x±y ∈ [0, 1] and N -1 y∈T N H N τ ( y N ) = 1. Proposition 3.
8, together with (3.34), then conclude the proof.

4. Proof of Theorem 2.6: Creation of the microscopic interfaces 4.1. Creation of the microscopic fronts. By definition of the initial measure for our process, the supercritical macroscopic phase {ρ ini 1 2 } can contain neighboring empty sites at the microscopic level, whereas the subcritical macroscopic phase {ρ ini 1 2 } can contain non-frozen particles (i.e. neighboring particles). Because of the indirect way it proves the hydrodynamic limit using Young's measures, Funaki's scheme (which we have adapted here) does not provide any information on the microscopic structure of the free boundary problem.

In this section we prove Theorem 2.6, i.e. that under reasonable assumptions on the initial profile ρ ini , after a macroscopic time of order t N = o(1), the microscopic structure of the configuration matches the macroscopic one.

Recall that we now make the following assumptions on ρ ini :

(ρ ini ) -1 ([0, 1 2 ]) = [0, u * ], ρ ini < 1, (H1-T1) ρ ini ∈ C 2 (T) and ∂ u ρ ini (0), ∂ u ρ ini (u * ) = 0. ( H2
)
We emphasize once again that the assumptions on the number of critical points (only two), and on the initial density which never hits 1, are purely for the simplicity of the presentation, and are not required for the proof. Indeed, most of the work of this section concerns the study of the critical interfaces, since the supercritical region (in which the density is larger than, and bounded away from the critical value ρ c = 1/2) has already been thoroughly studied in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF]. In particular, the technical issues specific to the case where the density hits 1 are solved therein. In order not to burden this section with analogous results we assume (H1-T1).

4.2.

Mapping with the zero-range process. To prove Theorem 2.6, we will once again exploit the mapping with the zero-range process. Given a configuration η ∈ Σ N , define K(η) = N -x∈T N η x the number of empty sites in the exclusion configuration η, and, for K < N , define Σ 0 N,K = η ∈ Σ N ; η 0 = 0 and K(η) = K . We define on Σ 0 N,K the finite volume counterpart Π N,K of (3.21), namely

Π N,K : Σ 0 N,K → N T K η → ω η , (4.1)
where, as before, for any k ∈ T K(η) , ω η k is the number of particles between the k-th and (k + 1)-th empty site (to the right of 0) in η.

Given a trajectory η(t) of the exclusion process, define K 0 = K(η(0)) the initial number of empty sites in the configuration. Mark the first empty site in η(0) to the right of site 0 if it exists. We keep track of the motion of this empty site and denote X(t) its position5 at time t. We then denote by

η(t) = τ X(t) η(t) ∈ Σ 0 N,K0
the exclusion configuration seen from the marked empty site.

We now denote by ω(t) = ω η(t) = Π N,K0 ( η(t)) the associated zero-range configuration. If η(0) ≡ 1 (the constant configuration with particles at each site), which happens with vanishing probability, then we let by convention ω(0) = N ∈ N T1 , the zero-range configuration with only one site and N particles on this site.

Then, as detailed in [7, Section 3], { ω(t)} t 0 is a Markov process, initially in the state ω(0) = ω η(0) , and driven by the generator N 2 L ZR K0 (recall also (3.24) for the infinite volume version), where

L ZR K f (ω) := x∈T K δ=±1 1 {ωx 2} f (ω x,x+δ ) -f (ω) . (4.2)
As already noted, for any α 1 one can define an equilibrium (grand canonical) distribution ν K α of the zero-range generator L ZR K on T K , as the geometric product homogeneous measure

ν K α (ω 0 = p) = 1 {p∈N,p 1} 1 α 1 - 1 α p-1 , (4.3)
which satisfies the detailed balance condition w.r.t. L ZR K . We then denote by ν * α the product measure on the set N N of semi infinite zero-range configurations with marginals given by (4.3). Given an integer K and an initial zero-range configuration ω ∈ N T K , we denote by Q K,ω the probability distribution on the path space D([0, T ], N T K ) of the zerorange process started from a fixed configuration ω, and driven by the non-accelerated zero-range generator L ZR K .

Remark 4.1. If η(0) is distributed according to the initial measure µ N fitting ρ ini , then the distribution of ω(0) = ω η(0) can also be associated with a profile in the following way. For u ∈ T, define

v(u) = u 0 (1 -ρ ini (u ))du , v = v(1) > 0, (4.4)
and

α ini : [0, v) → R + such that, for any v = v(u) ∈ [0, v), α ini (v) = ρ ini 1 -ρ ini (u). (4.5)
Then one could prove that, for all δ > 0 and smooth test function φ,

P µ N v K 0 K0 x=1 φ xv K 0 ω x (0) - v 0 φ(v)α ini (v)dv > δ ----→ N →∞ 0. (4.6)
We will not need this result, but a weaker version can be found in Appendix C.1.

Note that by assumption (T1), we have ρ ini < 1, therefore α ini is well defined. Under this mapping, if (T1) holds, the two critical points 0 and u * are mapped respectively to 0 and v

* := v(u * ) ∈ (0, v], which satisfy (α ini ) -1 ([0, 1]) = [0, v * ].
The main advantage of working with the zero-range process is the following monotonicity property (see e.g. [21, Chapter 2, Section 5]). Consider two trajectories {ω(t)} t∈[0,T ] and {ω (t)} t∈[0,T ] driven by the generator L ZR K , respectively started from two configurations ω ω . Then, one can couple both processes ω and ω in such a way that at any positive time t, ω(t) ω (t). In particular, given an event E ⊂ N T K increasing in the configuration, and if ω ω , for any t 0,

Q K,ω (ω(t) ∈ E) Q K,ω (ω (t) ∈ E). (4.7)
4.3. Typical zero-range configurations. In this section we define a set T K of typical zero-range configurations. Define K = K 3 4 , and denote

B K := { K , . . . , k * -K } , and A K = T K \ B K , (4.8)
where k * := Kv * v is the microscopic site corresponding to the macroscopic critical point v * . The set B K is the set of sites in the subcritical phase at distance at least K of the macroscopic critical points {0, v * }. Note that for any fixed K, the sets A K and B K only depend on the initial macroscopic profile ρ ini .

Given a zero-range configuration ω and a set Λ, we denote by

α Λ (ω) = 1 |Λ| x∈Λ ω x (4.9)
the empirical density of ω in the set Λ. Define

c * = 4v min -∂ u ρ ini (0) ; ∂ u ρ ini (u * ) (4.10) = v min -∂ v α ini (0) ; ∂ v α ini (v * ) > 0,
and introduce

α K = 1 + c * K K > 1 and Λ + K = {1, . . . , 10 K }.
Throughout, we will not burden the notations and write for example α K K instead of α K K . We further define

c K (ω) = x∈Λ + K ω x (x -5 K ), (4.11)
which sums the arithmetic distances between particles in Λ + K and the center of Λ + K . We now introduce the subset Ω + K ⊂ N T K given by

Ω + K = ω ∈ N T K ; α Λ + K (ω) = α K , c K (ω) 0, and ∀ x / ∈ Λ + K , ω x = 0 .
Note that in the last definition we slightly abused our notation, and by

α Λ + K (ω) = α K , we actually mean that Λ + K ω x = α K |Λ + K | .
We denote by Ω - K the set of configurations such that the configuration ω x = ω -x (obtained by symmetry w.r.t. the origin) is in Ω + K . In other words, configurations in Ω ± K have slightly more than one particle per site in a box of size 10 K to the left/right of the origin, and those particles, on average, are closer to the origin than to the other extremity of the box. (ii) For any x ∈ A K , there exists ω ∈ Ω + K ∪ Ω - K (depending on x) such that ω τ -x ω .

We denote by T K ⊂ N T K the set of typical configurations.

The first condition states that no large subcritical box has an abnormally large density. The second one states that for any site x close enough to the supercritical phase, one can always find a neighboring large box x + Λ K , containing at least α K > 1 particle per site on average. In ω , we keep only the particles closest to x, which will ensure that c K (ω ) 0 w.h.p. Then, w.h.p, at least one of those excess particles will eventually exit the box through site x.

Lemma 4.2. Recall that K 0 (η) is the number of empty sites in the exclusion configuration η(0), which is distributed according to µ N . We have lim

N →∞ P µ N K 0 / ∈ I N or ω(0) ∈ T K0 = 0,
where I N = vN -log 2 N, . . . , vN + log 2 N .

The proof of this lemma requires Assumption (H2

Q eq K,α * (G c K ) = ∞ n=0 P(N T K = n)P ∃ i n, ∃ x ∈ T K s.t. ω d x (i) log 2 K K ∞ n=0 nP(N T K = n)ν * α * (ω 0 log 2 K) KE[N T K ] 1 - 1 α * log 2 K-1 = K 11/4 1 - 1 α * log 2 K-1
.

Since K vN -log 2 N for any K ∈ I N , we obtain lim

N →∞ sup K∈I N Q eq K,α * (G c K ) = 0.
Combining this with (4.15) and Lemma 4.2, we conclude the proof of the lemma.

4.5.

Front creation for the zero-range process. We denote by P ZR K ⊂ N T K the set of two-phased zero-range configurations. Note that a zero-range configuration ω η is two-phased iff an associated exclusion configuration η also is (regardless of the marked empty site chosen in the exclusion configuration).

The main ingredient to prove Theorem 2.6 is an analogous result for the zerorange process started from a typical configuration. Recall that Q K,ω denotes the distribution of the non-accelerated zero-range process with initial configuration ω and infinitesimal generator L ZR K .

Proposition 4.5 (Hitting time of

P ZR K ). Recall T K = K 7 4 , lim K→∞ sup ω∈T K Q K,ω G K ∩ ω(T K ) ∈ P ZR K = 0,
where G K has been defined in (4.14).

lim K→∞ max ω∈T K Q K,ω G K ∩ {∃ x ∈ A K , ω x (T K ) = 0} = 0. ( 4
lim K→∞ max ω∈T K Q K,ω G K ∩ ∃ x < y < z ∈ B K , ω x (T K ) = ω z (T K ) = 0 and ω y (T K ) > 1 = 0. (4.17)
First, we prove that these two lemmas imply the result stated in Proposition 4.5.

Proof of Proposition 4.5. One can choose

B = max {x, . . . , z} ⊂ B K , ω x (T K ) = ω z (T K ) = 0 ,
where the max is taken for the inclusion. The configuration ω(T K ) is subcritical on B according to Lemma 4.7, and supercritical on A := T K \ B with high probability, according to Lemma 4.6. In what follows, we will couple ω with an auxiliary process χ where the particles in Λ copy exactly the jumps performed by ω, but any jump occurring from a site y / ∈ Λ is canceled. Under this coupling, particles in χ behave as those in ω up to the time when they leave Λ, where they get stuck. Defined in this way, the process {χ(t)} t is a Markov process, driven by the generator L st Λ , defined as

L st Λ f (χ) := x∈Λ |x-y|=1 1 {χ(x) 2} f (χ x,y ) -f (χ) .
We denote by Q st Λ,ω the distribution of the process {χ(t)} t started from ω and driven by the generator L st Λ above, and we denote by

T Λ χ = inf t 0 : sup x∈T K χ x (t) > log 2 |Λ| or χ y (t) 1, ∀ y ∈ Λ
the time at which either the number of particles became too high at some site, or all the particles got stuck (either by leaving Λ or by remaining alone on a site).

The following result is analogous to Lemma 4.4 of [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF], and is proved in the same way: Lemma 4.8. For any θ > 0, there exists λ 0 (θ) such that for any sequence of sets Λ(K) ⊂ T K satisfying log 2 K|Λ(K)| 2+θ K 2 , and |Λ(K)| λ 0 (θ) for all K large enough,

Q st Λ(K),ω T Λ(K) χ log 2 K|Λ(K)| 2+θ log 2 Ke -|Λ(K)| θ/2 .
Proof of Lemma 4.8. The proof is based on a coupling argument and can be obtained with small modifications from the proof of [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF]Lemma 4.4]. We sketch here its more salient points. Let us fix K and write Λ := Λ(K). The first step consists in coupling the process χ with another process σ, namely a system of independent symmetric random walks that jump at rate 1/ log 2 K inside Λ and get stuck when they exit it. Letting

T Λ σ = inf t 0 : sup x∈T K σ x (t) > log 2 |Λ| or σ y (t) 1, ∀ y ∈ Λ ,
standard arguments allow to show that T Λ χ T Λ σ . It then remains to prove Lemma 4.8 with T Λ σ in place of T Λ χ , which follows from standard estimates on random walks (see e.g. [START_REF] Varadhan | Probability theory[END_REF] p. 173). Note that before T Λ σ , there are at most log 2 K|Λ| particles to consider. 4.5.3. Supercritical phase; proof of Lemma 4.6. We will simply sketch the proof of Lemma 4.6, since given the definition of typical configurations it is analogous to Proposition 4.1 in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF]. To prove Lemma 4.6 it is enough to show sup

ω∈T K x∈A K Q K,ω G K ∩ {ω x (T K ) = 0} = o(K -1 ). ( 4.18)
Fix x ∈ A K , any typical configuration ω satisfies condition (ii) in Definition 4.1.

Assume for example that there exists ω ∈ Ω + K such that ω τ -x ω ; the other case is treated in the exact same way. Then, since the event G K ∩ {ω x (T K ) = 0} is decreasing in the configuration, translating the problem back to the origin, to prove (4.18) it is sufficient to prove that lim sup

K→∞ K sup ω∈Ω + K Q K,ω G K ∩ {ω 0 (T K ) = 0} = 0.
As outlined in Section 4.5.2, we now couple ω with the auxiliary zero-range process χ with generator L st Λ * K , where Λ * K = {1, . . . , 10 K }. Let us introduce the event

G χ K = sup x∈Λ * K sup t T K χ t (x) < log 2 K , which is G K 's counterpart for χ. Fix ω ∈ Ω + K . We can write Q K,ω G K ∩ {ω 0 (T K ) = 0} Q st Λ * K ,ω G χ K ∩ {χ 0 (T K ) = 0} . (4.19) We shorten T χ = T Λ * K χ . Choose θ = 1 4 and recall that K = K 3/4 , T K = K 7/4
log 2 K 2+θ K . Therefore, according to Lemma 4.8, for any K large enough,

Q st Λ * K ,ω T χ T K e -K 1/16 . ( 4.20)
In particular, in order to prove Lemma 4.6, it is enough to prove lim sup

K→∞ K sup ω∈Ω + K Q st Λ * K ,ω ({χ 0 (T χ ) = 0} ∩ G χ K ∩ {T χ T K }) = 0. (4.21)
On the event G χ K ∩ {T χ T K }, we have χ(T χ ) 1 on Λ * K . In particular, T χ = T χ := inf{t 0 : χ y (t) 1, ∀y ∈ Λ * K }. Therefore, we only have to prove lim sup

K→∞ K sup ω∈Ω + K Q st Λ * K ,ω {χ 0 (T χ ) = 0} = 0. (4.22)
First recall that any configuration ω ∈ Ω + K has 10 K (1 + c * K K ) particles. Denote by t j the jump times 0 = t 0 < t 1 < • • • < t L = T χ of the process χ before time T χ , and define for any j L

Z j = 10 K +1 x=0 χ x (t j )(x -5 K ),
Since ω ∈ Ω + K , recalling (4.11), we have Z 0 0. Furthermore, at time T χ , at least 10c * 2 K /K particles have exited Λ * K , and if none is at site 0, all those particles got stuck at site 10 K + 1 and it is therefore straightforward to show that

χ 0 (T χ ) = 0 ⇒ Z L 50c * 3 K /K,
+ K Q st K,ω L 3+θ K = O(e -θ/2 K ).
Moreover, the process {Z j } j is distributed as a discrete time, nearest-neighbor, symmetric random walk up to time L, so that sup

ω∈Ω + K Q st K,ω χ 0 (T χ ) = 0 and L < 3+θ K P sup 0 j 3+θ K X j 50c * 3 K /K = O e -C 3-θ K /K 2 ,
for some positive constant C depending on c * where P is the distribution of a discrete time random walk X initially at the origin. Since K = K 3 4 , the last two bounds together finally yield for θ = 1 4 that

K sup ω∈Ω + K Q st K,ω χ 0 (T χ ) = 0 = O Ke -K 1/16 .
The right hand side vanishes as wanted as K → ∞, which proves (4.21) and then Lemma 4.6.

4.5.4. Subcritical phase; proof of Lemma 4.7. Denote by E the event inside brackets in (4.17), and define

E x,y,z = ω x (T K ) = ω z (T K ) = 0, ω y (T K ) > 1
and ω j (T K ) 1, ∀ j ∈ {x + 1, . . . , z -1} , which yields straightforwardly

E = x<y<z∈B K E x,y,z . Since |B K | K for K large enough, it is enough to show that, uniformly in x < y < z ∈ B K , K 3 Q K,ω (G K ∩ E x,y,z ) vanishes.
Assume first that z -x > K and set Λ = {x + 1, . . . , z -1}. If ω is a regular configuration, it satisfies condition (i) in Definition 4.1, and in particular, we must have α Λ (ω) 1. However, no particle can cross an empty site, so that on the event E x,y,z we also have α

Λ (ω(T K )) = α Λ (ω) 1.
By definition, on the event E x,y,z we have α Λ (ω) > 1, because an extra particle is at site y, so that, finally for any z -x > K , and any regular configuration ω, Q K,ω (E x,y,z ) = 0.

We can therefore assume that z -x K . Now set Λ = {x + 1, . . . , x + K }. On E x,y,z , there is at least one free (still able to move) particle in Λ. However, on E x,y,z this particle must have remained in Λ from time 0 to T K because sites x and z are still empty at time T K . In particular, couple as in Section 4.5.2 on Λ the process ω with the stuck zero-range in Λ. Then, by Lemma 4.8

Q K,ω G K ∩ E x,y,z Q st Λ,ω T Λ χ T K = O e -1/8 K .
Finally, Note that for N large enough, for any K ∈ I N , T K = K 7/4 N 2 t N , therefore the probability in the right hand side above is less than Q K,ω G K ∩ ω(T K ) ∈ P ZR K . This yields P µ N (η(t N ) ∈ P N ) max

Q K,ω G K ∩ E K 3 sup x<y<z∈B K Q K,ω G K ∩ E x
K∈I N ω∈T K Q K,ω G K ∩ ω(T K ) ∈ P ZR K + o N (1).
For K ∈ I N , K → ∞ as soon as N → ∞. Therefore, letting N → ∞, the right hand side vanishes according to Proposition 4.5.

It remains to show point (2) of Theorem 2.6. Fix t ∈ R + ∩ (0, τ ]; we give to u N ± (t) the arbitrary value 0 if the configuration never became two-phased before time t (which, according to point (1), occurs with vanishing probability). Recall Definition 2.3 and (2.10), which ensures that u N ± (t) are well defined. Let us start with the leftmost interface at position u -(t). Note that, by Proposition 2.5, u - can be identified without any ambiguity with a continuous non-decreasing function from R + to [0, 1). We first show that, for any ε > 0, and any t < τ ,

P µ N 1 N u N -(t) -u -(t) ε ----→ N →∞ 0.
The other cases can be treated in the exact same way and are left to the reader.

In the following we denote I ε (t) := [u -(t), u -(t) + ε] and we take ε small enough such that I ε (t) ⊂ [u -(t), u + (t)). We also introduce its microscopic counterpart We conclude by using Theorem 2.4.

I N ε (t) = N u -(
where I and II respectively correspond to the cases where y / ∈ {1, N -1, N } and y ∈ {1, N -1}. Note that the contribution of the terms for y = N vanishes because To prove (ii) it is enough to show that with probability going to 1, one has We are on the event E c N , therefore each empty site is at most at a distance log 2 N of its expected position in the initial exclusion configuration. Since ω k (0) = y k+1y k -1, Fix T > 0 and ε > 0, and a bounded non-negative function g : T → [0, +∞), we define ψ ε as the classical solution to the elliptic equation

∂ t ψ ε t = χ T -t + ε ∂ 2 u ψ ε t ψ ε 0 = g . (D.6)
Since the initial profile g is non-negative, by maximum principle so is ψ ε t for any t T , so that (D.5) yields, choosing

ϕ t = ψ ε T -t , ρ T , g -ε T 0 ρ t , ∂

Figure 2 . 1 .

 21 Figure 2.1. Example of configurations belonging to the ergodic and frozen sets, with |Λ| = 10.

Definition 3 . 1 (

 31 Young measure). Let us fix an integer . The Young measure π N, on T × [0, 1] is given for any configuration η of particles byπ N, (du, dr) = π N, (η, du, dr) := 1 N x∈T N δ x/N (du) δ ρ x (dr).For any measure π on T × [0, 1], any function ξ defined on T, and any function ψ defined on [0, 1], we denote by π, ξ • ψ the integral of the function (ξ • ψ)(u, r) := ξ(u)ψ(r) w.r.t. the measure π.

Figure 3 . 1 .

 31 Figure 3.1. An example of configuration η + with support N.

Figure 3 . 2 .

 32 Figure 3.2. An exclusion configuration η with an empty site at the origin and its corresponding zero-range configuration ω η .

( 1 +( 1 +

 11 and note the following: for any configuration η ∈ Σ 0 ∞ (with an empty site at the origin), if ω η |x+Λ = σ then η -k = 0, since -k is the position of the (exclusion) x-th empty site in η, corresponding to the (zero-range) site x in ω η . In particular, we can write µ ω η |x+Λ = σ and η 0 = 0 = µ ω η |x+Λ = σ and η -k = 0 and η 0 = 0 = µ ω τ k η |x+Λ = σ and η 0 = 0 and η k = 0 = µ η 0 = 0 and ∀ y ∈ 1, . . . , z=0 σ z ) , η y = 0 ⇔ ∃ i ∈ Λ , y = i z=0 σ z )

x- 1 y=0 ( 1 + σ x- 1 y)

 111 , and rewrite for any fixed σ x-1 ∈ (N * ) x : µ ω η |Λ x+ = σ x-1 σ and η 0 = 0 = µ ω η |Λ x+ = σ x-1 σ and η 0 = 0 and η k = 0 = µ ω τ -k η |Λ x+ = σ x-1 σ and η -k = 0 and η 0 = 0 = µ ω η |-x+Λ = σ x-1 σ and η 0 = 0 ,

Proposition 3 . 8 (

 38 Local law of large numbers in the supercritical phase). Recall that π ρ was defined for any ρ ∈ [0, 1] in Definition 3.2 and that B = {-, . . . , }. Recall also from (2.3) the definition of the function h. Then we have lim h -π ρ (h) = 0.(3.33)

2 yN

 2 = P(|X Θ/N 2 | > Θ) = O(e -Θ ), where the second identity comes from a standard estimate on symmetric random walks. In particular, since µ N T (ρ y ) is uniformly bounded in y, the contribution of the sum |y| > Θ vanishes as N → ∞, then → ∞, and then Θ → ∞ ; • in the first case |y| Θ, |ρ y -ρ 0 | 2Θ 2 +1 . Therefore, the left hand side in (3.42) rewrites as lim →∞ lim N →∞ T µ N T 2ρ 0 H ρ 0 , which is also equal to (recalling Definition 3.1, and also (3.32) and (3

Definition 4 . 1 .

 41 We call a configuration ω ∈ N T K typical if it meets the following two conditions :(i) For any x ∈ B K , and any connected set Λ ⊂ B K containing x such that |Λ| K , we have α Λ (ω) 1.

4. 5 . 1 .

 51 Typical ZR configurations become two-phased in subdiffusive time.Definition 4.2 (Two-phased zero-range configurations). A zero-range configuration ω ∈ N T K is called two-phased if there exists a partition T K = A B, where A and B are both connected subsets of T K , and ω |A 1 and ω |B 1.

Proposition 4 .

 4 5 is a consequence of Lemmas 4.6 and 4.7 below, which are proved respectively in Sections 4.5.3 and 4.5.4. Recall the definition (4.8) of the sets A K and B K .

Lemma 4 . 6 .

 46 With high probability the set A K becomes supercritical before time T K , precisely:

4. 5 . 2 .

 52 Stuck zero-range. In this paragraph we introduce an auxiliary process which will be used to prove both Lemma 4.6 and Lemma 4.7. Fix a box Λ ⊂ T K , and define its exterior boundary ∂Λ = {x ∈ T K , d(x, Λ) = 1} and set Λ = Λ ∪ ∂Λ.

7 . 4 . 6 . 4 P

 7464 ,y,z = O e -K 1/16 , which vanishes as wanted as K → ∞ and proves Lemma 4.Proof of Theorem 2.6. Let us show point (1) of Theorem 2.6. Choose t N = N -1/4 , we first write, using Lemmas 4.2 and 4.µ N (η(t N ) ∈ P N ) = P µ N ω(t N ) ∈ P ZR K0 = P µ N {K 0 ∈ I N } ∩ { ω(0) ∈ T K0 } ∩ G N ∩ ω(t N ) ∈ P ZR K0 + o N (1) max K∈I N ω∈T K P µ N G N ∩ ω(t N ) ∈ P ZR K0 K 0 = K and ω(0) = ω + o N (1).

  x∈T N L N η 2x = L N x∈T N η x = 0, since the dynamics is conservative.More precisely, shortening F (x) = τ x h(η), and defining its discrete Laplacian asδ N F (x) := F (x + 1) + F (x -1) -2F (x), y δ N F (x) + η x δ N F (x -y) N (η x+y + η x-y ) δ N F (x)andII = ξ N ( 1 N ) + ξ N ( N -1 N ) N 2 x∈T N (η x+1 + η x-1 + η x ) δ N F (x).

2 C. 1 .

 21 x+y + η x-y ) δ N F (x) + ∆ N ξ N (0) N 4 x∈T N η x δ N F (x).Successive summations by parts in x and y in the first sum, and in x in the second one, prove the lemma.Appendix C. Proof of Lemma 4.Law of large numbers for the positions of zeros. We first check that the microscopic and macroscopic mappings defined respectively in (4.1) and (4.4)-(4.5) match in the limit.For any k N v, denote u k the solution of N u k 0 (1 -ρ ini (u))du = k, and for k > N v, we let u k = 1. Note in particular thatu k = v -1 (k/N ), (C.1)where the function v(u) was introduced in (4.4). From the law of large numbers, we expect that the k-th empty site y k ∈ T N in the initial configuration η(0) should be close to site N u k .Lemma C.1. limN →∞ P µ N (E N ) = 0, (C.2)whereE N := |K 0 -vN | log 2 N or max k K0 |y k -N u k | > log 2 N .

ω

  k (0) (k -5 K0 -x 0 ) 0. (C.5)

-v * 2 + 2 Nk 2 -

 222 -y k -1) n k k=x0 (y k+1 -y k -1), which rewrites, on E c N , 0 n -(y k -y x0 -( k -x 0 )) y k+1 -y k 2 log 2 N + N sup k (u k+1 -u k ). (C.6)Recall that K0 N = N 3/4 , using (C.1) and (C.4) yields that for any k ∈ Λ + ∪ {x 0 },u k = u * + 2 k N -v * + 4∂ u ρ ini (u * ) k N O(N -3/4 ).For any integer j,shorten j = j -k * = j -N v * + O(log 2 N ), on E c N , the identity above yields y k -y k * = 2k + c 0 k + O(N 1/4 ), (C.7)where c 0 = 4∂ u ρ ini (0) > 0 by Assumption (H2). Using (C.7), we now rewrite the left hand side of (C.5),k k=x0 ω k (0) (k -5 K0 -x) = k k=x0 (y k+1 -y k -1) (k -5 K0 -x 0 ) = k k=x0 k (y k+1 -y k -2) -(5 K0 + x 0 ) n + ( k -x 0 )( k + x 0 + 1) -(5 K0 + x 0 ) n + ( k -x 0 )( k + x 0 (5 K0 + x 0 ) n + ( k -x 0 )( k + x 0 ) 2 + O( N ). (C.8)Note that by definition, x 0 = -K0 . Using equations (C.6) and (C.7), one can easily check that n = ( k -x 0 )(1 + c 0 ( k + x 0 )/N ) + O(N 1/4 ), so that by Taylor expansionk = x 0 + n -c 0 n N (2x 0 + n) + O(N 1/4 ) = 9 K0 -c 1 2 K0 K 0 + O(N 1/4 ),where we denoted c 1 = 80c 0 v -10c * > 70c 0 v by definition (4.10) of c * . After elementary computations, the dominant terms of order O(

  19) contain • • ••, and all terms in (3.20) contain • • •• and therefore vanish. Since we assumed that (3.16) holds for any k -1, all terms in the right hand side (3.17)-(3.18) vanish, except µ

  ). It is fairly technical but poses no significant difficulty, we give it in Appendix C. Let us denote by {ω d (n)} n∈N this discrete-time Markov chain with initial distribu-= ω d (N t ) t , where {N t } t 0 is a standard Poisson process independent of ω d . Moreover, ω d is reversible w.r.t. ν * α * . Therefore, writing P for the joint distribution of ω d and N ,

	tion ν * α * . Then
	ω(t) t	(d)

  because then the minimal value for Z L is the case whereχ(T χ ) |Λ * K ≡ 1,andχ 10 K +1 (T χ ) = 10c *

	2 K /K.
	Recall that there are less than 20 K particles initially in Λ * K , and each of those par-ticles either gets stuck or exits Λ * K in O( 2+θ K ) jumps with probability 1 -O(e -θ K ).
	Elementary computations yield that
	sup
	ω∈Ω

  t) , . . . , N (u -(t) + ε) . As before, for the sake of clarity we omit integer parts • in all that follows. By definition (recall Definition 2.3),|I N ε (t) ∈ E I N ε (t) + ε N ,where the error term ε N comes from the (very unlikely) scenario where1 N u N -(t) u -(t) + ε 1 N u N + (t) u -(t). Since according to Lemma C.1, u N + (0) is at distance at most log 2 N of N u * , and given the respective monotonicities of u N ± , u ± , the error term ε N vanishes as N → ∞.We now estimateP µ N η(t) |I N ε (t) ∈ E I N ε (t) .Let us choose ϕ : T → [0, 1] as a smooth test function which satisfies1 [u-(t)+δ,u-(t)+ε-δ] ϕ 1 Iε(t) ,where δ satisfies (recall Proposition 2.5)

	P µ N	1 N	u N
			T	ρ t (u)ϕ(u)du		Iε(t)	ρ t (u)du <	ε 2	-2δ.	(4.23)
	Moreover, on the event {η(t) |I N ε (t) ∈ E I N ε (t) }, we have (recall (2.5))
		1 N	x∈T N	ϕ x N η x (t)	1 N	N (u-(t)+ε-δ) x=N (u-(t)+δ)	η x (t)	ε 2	-δ.	(4.24)

-(t) -u -(t) ε P µ N η(t)

  t (u) ρ t (u)du M max u --u -∞ , u + -u + ∞ , (u)ρ t (u)du is continuous in (u -, u + ) w.r.t. |||•|||.Since ∂ t g is also continuous w.r.t. |||•|||, one obtains straightforwardly that ) ρ s (u)∂ t g s (u)duds also is. This, together with (D.3), proves thatt 0 ∂ u ρ s (u -(s) -)ds is continuous in (u -, u +). An analogous argument withg t (u) = u -(t) -u u -(t) -u + (t) 1 {u∈[u+(t),u-(t)]} , (D.4)proves that t 0 ∂ u ρ s (u + (s) + )ds also is, and concludes the proof.Proof of Lemma D.2. This is a simpler case of [28, Theorem 10, p.30], we give it for the sake of completeness. Define ρ ini = ρ ini,> -ρ ini,< , as well as ρ t = ρ > t -ρ < t , and χ t =

		u-(t)	u-(t)
		g t (u)ρ t (u)du -
	u+(t)	u+(t)
	so that	u-u+ g t t	u-(s)
			0 u+(sH(ρ > t ) -H(ρ < t ) ρ > t -ρ < t 1 {ρ > t =ρ <

2 K0 ) in (C.8) cancel out, g t } .

Since classical solutions of (2.7) are also weak solutions, for any smooth function

ϕ ∈ C 1,2 ([0, T ] × R), we have ρ T , ϕ T = ρ ini , ϕ 0 + T 0 ρ t , ∂ t ϕ t + χ t ∂ 2 u ϕ t dt. (D.5)

  2 u ψ T -t dt. (D.7)where we used that ρ ini 0. Assume now that g ∈ C 2 (T), multiplying the first line of (D.6) by ∂ 2

		u ψ ε t , and integrating over [0, T ] × T, yields
	1 2 T	(∂ u g) 2 du =	1 2 T	(∂ u ψ ε T ) 2 du +	0	T	T	χ T -t + ε (∂ 2 u ψ ε t ) 2 dudt,
	so that in particular
					ε ∂ 2 u ψ ε T , ∂ 2 u ψ ε T	1 2		∂ u g, ∂ u g ,
	and by Hölder's inequality, (D.7) yields
	ρ T , g	-	√	2ε

T 0 ρ t , ρ t 1/2 ∂ u g, ∂ u g 1/2 dt -T √ 2ε ∂ u g, ∂ u g 1/2 .

Note however that E Λ ∩ F Λ is non-empty since it contains alternated particle/empty site configurations.

If X(t -) = x and a particle jumps from x ± 1 to x at time t, then X(t) = x ± 1.

thanks C. Cancès and M. Herda for very enlightening exchanges about the solutions to free boundary problems. C.E. gratefully acknowledges funding from the European Research Council (ERC) under the European Unions Horizon 2020 Program, ERC Consolidator Grant UniCoSM (grant agreement n o 724939). M.S. thanks Labex CEMPI (ANR-11-LABX-0007-01). O.B. thanks the ANR projects LSD (ANR-15-CE40-0020) and MALIN (ANR-16-CE93-0003) and acknowledges support from INSMI (CNRS) through a PEPS JCJC grant. This project is partially supported by the ANR grant MICMOV (ANR-19-CE40-0012) of the French National Research Agency (ANR), and it has also received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovative program (grant agreement n o 715734). the physics litterature as reversible dynamics w.r.t.

Remark 4.3. Theorem 2.6 holds for initial measures different from µ N , as long as they satisfy the analog of Lemma 4.2.

4.4.

Bound on the maximum of the zero-range process. Because the jump rate per site is always 1 (provided that the constraint ω x 2 is satisfied), the facilitated zero-range process can be interpreted as a family of random walks, where each random walker jumps "independently" at a rate 1/k, where k is the number of other random walkers on the same site, assuming the random walker is not alone on the site, in which case it remains there. With this in mind, we prove a technical lemma, giving a uniform bound on the number of particles at any site in ω(t), which will be useful to bound from below the jump rate of each individual particle. Lemma 4.4. Let T K = K 7/4 . Then, the following limit holds:

where

Proof of Lemma 4.4. Let us denote by Q eq K,α the distribution of the zero-range process generated by L ZR K , started from its equilibrium distribution ν K α on T K (recall (4.3)). We first claim that, letting ρ * = sup T ρ ini < 1 and α * = ρ * 1-ρ * , we have

where G K is defined as

Indeed, one can prove by standard arguments that there exists a coupling between ω(0) and a semi-infinite zero-range configuration ω * with distribution ν * α * such that ω(0) ω * |T K 0 (identifying T K0 with {1, . . . , K 0 }). In particular, by monotonicity of the zero-range process (4.7),

where we used to establish the second bound both the coupling above and the fact that the event G c K is increasing in the initial configuration. We now estimate the equilibrium probability Q eq K,α * (G c K ). The process {ω(t)} t 0 can be constructed as a time-change of a discrete-time Markov chain on (N * ) T K , where N * = {1, 2, . . . } is the set of positive integers. Consider the transition matrix given by p(ω, ω ) =

2K

if there exist x ∈ T K and δ ∈ {±1} s.t. ω = ω x,x+δ and ω x 2; 0 else.

Appendix A. Proof of Proposition 3.1

We start by proving tightness of (P N ) and assertion (1). To do so, first note that since only one particle per site is permitted,

therefore we only need to show (cf. Theorem 1.3 and Proposition 1.6, p. 51 in [START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF]) that for any limit point P * , the following is satisfied: for any function ξ ∈ C 2 (T) and for any positive ε,

To prove (A.1), we can rewrite for any fixed N , by Dynkin's formula,

where M N,ξ t is a martingale w.r.t. the filtration σ η(τ ), τ t . Since the model is gradient, and recalling (2.3), the first term in the right hand side can be rewritten as

where

where the last estimate comes from elementary and classical computations, using the fact that the function ξ is smooth. In particular, the martingale terms in (A.2) vanish as well, which proves (A.1) and assertion [START_REF] Andjel | Invariant measures for the zero range processes[END_REF].

We now prove assertion (2), which is immediate because only one particle is allowed per site. This yields in particular that any limit point P * of (P N ) satisfies

which proves the assertion.

Appendix B. Proof of Lemma 3.13

Let us compute explicitly

Proof. This estimate is a simple consequence of the facts that

together with standard large deviation estimates for sums of independent variables.

Throughout the rest of the proof, we now assume that E c N holds, we are going to show that condition (i) and (ii) of Definition 4.1 hold as well for any N large enough, which will prove Lemma 4.2. C.2. Subcritical phase. We first deal with condition (i). By Assumption (H2), there exists c 0 = c 0 (ρ ini ) > 0 such that for N large enough, for any discrete interval {k 1 , . . . , k 2 } ⊂ B K0 (recall (4.8)), we have on the event

Recall that ω(0) denotes the zero-range configuration mapped from the initial exclusion configuration η(0).

On E c N , we have y k2+1 -y k1 N (u k2+1 -u k1 ) + 2 log 2 N . Moreover, using (C.3) and Assumption (H2), there exists

Putting those bounds together, we obtain that on

For N large enough, the right hand side above is less than 1, therefore condition (i) of Definition 4.1 holds. C.3. Supercritical phase. We now prove that condition (ii) of Definition 4.1 holds. With k * = K0v * v (the microscopic site corresponding to the macroscopic critical point v * ), note that on the event E c N , we have

Therefore, by Assumption (H2), there exists c 2 = c 2 (ρ ini ) such that |u k * -u * | c 2 log 2 N/N . To prove that condition (ii) holds, we need to consider sites in

The case where x 0 sits in the bulk of A K0 , i.e. when there is a macroscopic region around x 0 /N in which the density in ω(0) is larger that 1 + ε for some ε > 0, follows from the same arguments as in [START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF]Lemma 4.10]. This is the easiest case and we do not detail it here. The hardest case is when x 0 is close to the interface, so the density around is close to 1 and particles are not as much in excess. To avoid burdensome notations, we will only detail the proof that site x 0 := k * -K0 satisfies (ii), which would adapt straightforwardly to other sites in A K0 . We therefore prove that there exists

Then lim

Furthermore,

and, developing the function v -1 at v * , we obtain

Recall that we must have v 1 2 , the four equations and bounds above together yield on E c N , since

for N large enough.

Assuming we are on G N , we define ω by keeping from ω(0) only the n := 10 K0 α K0 particles closest to site x 0 in Λ + . This configuration obviously satisfies both x∈Λ + ω x = |Λ + |α K0 and ω ω(0), so that we only need to check

i.e. that those particles are on average closer to x 0 than they are to the other extremity of Λ + . Denote k > x 0 the zero-range site where the n-th particle to the right of x 0 is found (i.e. in the exclusion configuration, there are k empty sites between the empty site y x0 and the n-th particle to the right of y x0 ). so that only remain the terms in O( 3 K0 /K 0 ), which rewrite

Since the constant in parenthesis is negative, and since 3 K0 /K 0 N , this proves the result.

As already pointed out, we will not detail the general cases x ∈ A K0 , we simply sketch out why the problem is the same. Consider a macroscopic point v ∈ [v * , v] and consider the zero-range configuration in a mesoscopic box of size 10 K0 to the right of site Kv/v. If v ∈ (v * , v) then the zero-range density α v = 1/(1 -ρ ini (u(v))) in a mesoscopic box Λ v,K0 of size 10 K0 is strictly larger than 1. In particular, since on E c N , α K0 α vN = 1 + o(1), the majority of the n = 10 K0 α K0 particles closest to x to its right are closer to x than to the other extremity of Λ v,K0 . The only problematic cases are therefore close to the boundaries 0, v * . We treated the most extreme of those cases, in which the site x considered is in the subcritical phase, and at a distance K0 of the supercritical phase, the other cases can be treated analogously.

Appendix D. Existence of macroscopic interfaces: proof of Proposition 2.5

In this section, we lay out the proof for the existence of macroscopic interfaces for the weak solution (in the sense of Definition 2.1) of (2.7). The proof we present here is adapted from Meirmanov [START_REF] Meȋrmanov | The Stefan Problem[END_REF] to our periodic setting. It contains no significant mathematical novelty w.r.t. [START_REF] Meȋrmanov | The Stefan Problem[END_REF]; we include it here for the sake of completeness. The main difficulty of the proof is that the interface speeds diverge as t → 0 + . To solve this issue, we approximate the initial profile ρ ini by

. We first claim that, thanks to the discontinuity of the density at the interfaces, the Stefan problem with initial condition ρ ini,n admits a classical solution.

Then there exists a classical solution (ρ, u -, u + ) to the Stefan problem (2.7) with initial data ρ ini , i.e. ρ : R + × T → [0, 1], u ± : R + → T such that (1) u -(resp. u + ) is non-decreasing (resp. non-increasing), with u -(0) = 0 and u + (0) = u * ;

(2) there exists τ ∈ R + ∪ {∞} such that u -(t) = u + (t) iff t τ , and u ± are constant on [τ, ∞); the time τ is called the merging time;

.

We defer the proof of Lemma D.1 to see how this result can lead us to Proposition 2.5. Let us denote (ρ n , u n -, u n + ) the 6 classical solutions provided by Lemma D.1 when ρ ini = ρ ini,n . We will show that the interfaces u n ± converge, and that the limits satisfy the properties required in Proposition 2.5. To that end, we exploit a monotonicity property of the interfaces defined by Lemma D.1: Lemma D.2. Let ρ ini,> , ρ ini,< two initial profiles satisfying the assumptions of Lemma D.1, such that ρ ini,< ρ ini,> .

Let (ρ > , u > -, u > + ), (ρ < , u < -, u < + ) be the associated classical solutions with merging times τ > , τ < respectively.

Then

A consequence of Lemma D.2 is that ρ n , u n -, u n + are monotone in n. Since they are also bounded, they have limits which we call ρ, u -, u + respectively. The monotone convergence Theorem straightforwardly yields that ρ, thus defined, is the weak solution of 2.1 with initial profile ρ ini . Letting τ := inf{t 0 : u + (t) = u -(t)}, the properties of Proposition 2.5 are simple consequences of the above construction. Following [START_REF] Meȋrmanov | The Stefan Problem[END_REF]Theorem 2,p. 151], one can actually show that (ρ, u -, u + ) is also a classical solution with initial profile ρ ini , however since we do not require it here, we will not expand further.

We now conclude by giving the proofs of Lemmas D.1 and D.2.

Proof of Lemma D.1. This is very close to [28, Lemma 3, p.151] and seems to be a standard result for free boundary problems. We sketch here a proof for completeness, mainly taken from [2] and adapted to our periodic setting. Part of the statement is that the derivatives in (5) and ( 6) are well defined. The main idea is to construct the interfaces as solutions to a fixed point problem.

Fix T > 0 and let M = sup{|( ρ ini ) (u)|, u ∈ T \ {0, u * }}. Let U be the space of functions u -, u + : [0, T ] → T which satisfy the following conditions:

• u ± are Lipschitz-continuous, with Lipschitz constant bounded by M , • u -(resp. u + ) is non-decreasing (resp. non-increasing), 6 It is easy to check that classical solutions are also weak solutions in the sense of Definition 2.1, and the uniqueness of ρ n is therefore guaranteed by Proposition 2.3.

• u -(0) = 0, u + (0) = u * , and u -u + .

Any such function is differentiable almost everywhere in [0, T ]. With a slight abuse of notations, we denote by u ± ∞ M the maximal Lipschitz constant of u ± . Note that U is a convex compact subset of the Banach set C([0, T ]) × C([0, T ]) endowed with the norm

For any (u -, u + ) ∈ U, let τ := inf{t 0 : u -(t) = u + (t)} and ρ : [0, T ] × T → [0, 1] be defined as follows: first, ρ 0 (u) = ρ ini (u) for any u ∈ T, and

(D.1)

Note that up to time τ , assuming u ± are fixed, ρ is the solution to a Dirichlet problem (D.1) with moving boundaries. It is then standard to show (see [2,Lemma 4.1] for instance) that ρ is well defined, and that its spatial derivatives are continuous up to the boundaries {(t, u ± (t)), t ∈ (0, T )}. Consider the transformation T : U → U defined as follows: first, let

and define

τ * := inf t 0 :

Then, let

Then a fixed point for this transformation also yields the desired classical solution to our Stefan problem. By Schauder's fixed point Theorem, it is therefore enough to show that T is continuous w.r.t. |||•|||. In turn, by regularity of ρ ini , and since ρ ini is bounded away from 1 2 in (u -, u + ), it is enough to show that the application

To that aim, let t T , fix (u -, u + ) ∈ U, and define

is the length of the diffusive phase, so that in particular g 0

)) = 0 and ∂ 2 u g s (u) = 0, a second integration by parts shows that the second term in the right hand side vanishes. Consequently, recalling that ∂ u H(ρ s (u)) = 4∂ u ρ s (u) and ρ s (u -(s)) = 1 2 , we have The first identity follows from the fact that ρ s (u -(s)) = ρ s ( u -(s)) = 1 2 , whereas the second follows from the maximum principle applied to ∂ u ρ in the moving boundary domain {(t, u), t T, u ∈ [u + (t), u -(t)]}. We can now apply the maximum principle to ρ -ρ in the domain Λ t := (s, u) : s ∈ (0, t), u ∈ (u + (s), u -(s)) ∩ (s, u) : s ∈ (0, t), u ∈ ( u + (s), u -(s) , to obtain that for all (s, u) ∈ Λ t , ρ s (u) -ρ s (u)

M max u -(s) -u -(s) , u + (s) -u + (s) .

Denote by g, g the functions given by (D.2) resp. for (u -, u + ), ( u -, u + ). In particular, since |ρ s (u)| 1, |g s (u)| 1 and g s (u) is uniformly continuous in (u -, u + ), we obtain as wanted that Letting ε → 0, we obtain that ρ T , g 0 for any non-negative g ∈ C 2 (T). We now choose non-negative functions g k ∈ C 2 (T) converging in L 2 (T) to 1 {ρ T <0} as k → ∞, to obtain that ρ T 0 a.e., which concludes the proof.