Anatole Ertul

Cuto for the Fredrickson-Andersen one spin facilitated model

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Fredrickson and Andersen one spin facilitated model (FA-1f) model is an interacting particle system that belongs to the class of Kinetically Constrained Spin Models. These models are Markov processes that were rst introduced in the 1980's by physicists to model liquid-glass transitions (see [START_REF] Fredrickson | Kinetic Ising model of the glass transition[END_REF] and [START_REF] Jäckle | A hierarchically constrained kinetic Ising model[END_REF]) and that have some analogy with the Glauber dynamics for the Ising model. In a KCSM on a graph G, each vertex can either have a particle or be empty. At rate 1, each site tries to update according to a Bernoulli measure, but eectively does so only if a local constraint is satised. On the graph Z, the constraint can for example be that the site immediately to the right is empty, in which case we get the East model. Another possible constraint is to have either of the adjacent neighboring sites empty, which denes the FA-1f model.

Since the 2000's, the dynamics of KCSM both at and out of equilibrium have been thoroughly studied by the mathematic community. In 2002, Aldous and Diaconis [START_REF] Aldous | The asymmetric one-dimensional constrained Ising model: rigorous results[END_REF] proved that the East model has a positive spectral gap. This result has been generalized to a large class of model in [START_REF] Cancrini | Kinetically constrained spin models[END_REF]. As far as relaxation starting out of equilibrium, studies are made more dicult by the lack of attractiveness of the processes, preventing usual monotonicity and coupling arguments. Partial results can be found for example in [START_REF] Cancrini | Facilitated oriented spin models: some non equilibrium results[END_REF] for the East model and [START_REF] Mountford | Exponential convergence for the Fredrick-sonAndersen one-spin facilitated model[END_REF][START_REF] Blondel | Fredrickson-Andersen one spin facilitated model out of equilibrium[END_REF] for FA-1f at low density. For the East model and the FA-1f model on Z, an interesting topic of study is the motion of the front, i.e. the rightmost empty site of a conguration lled on an innite half-line. First, Blondel [START_REF] Blondel | Front progression in the East model[END_REF] showed that the front has a linear speed in the East model. Based on this work, Ganguly, Lubetzky and Martinelli showed in [START_REF] Ganguly | Cuto for the East process[END_REF] a central limit theorem for the front, which was followed by the proof that the East model exhibits cuto thanks to a simple coupling argument. In [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF], it was shown that the front for the FA-1f process also behaves according to a CLT when the density is below a certain threshold. The aim of this paper is to use this last result to prove a cuto for the FA-1f process. It will require much more work than for East because the main coupling argument [START_REF] Ganguly | Cuto for the East process[END_REF]Section 4.2] no longer holds for FA-1f.

The cuto phenomenon was rst exhibited by Aldous and Diaconis in the context of card shuing [START_REF] Aldous | Shuing cards and stopping times[END_REF]. It consists of a sharp drop of the total variation distance to the equilibrium measure of a Markov process (see [START_REF] Levin | Markov Chains and Mixing Times[END_REF] for an introduction to this phenomenon). Since then, examples and counter-examples of cuto have be shown for a wide variety of processes, but a universal criterion is missing. In 2004, Peres conjectured that a process exhibits cuto if and only if it satises the product condition t rel = o(t mix ), where t rel is the inverse of the spectral gap and t mix is the total variation mixing time of the process. This condition turned out not to be sucient in general but sucient for a large class of processes (see [START_REF] Basu | Characterization of cuto for reversible Markov chains[END_REF]). Recently, several cuto results have been shown for particle systems like the Ising model [START_REF] Lubetzky | Universality of cuto for the Ising model[END_REF], the Asymmetric Simple Exclusion Process [START_REF] Labbé | Cuto phenomenon for the asymmetric simple exclusion process and the biased card shuing[END_REF] or a stratied random walk on the hypercube [START_REF] Ben | Cuto for a stratied random walk on the hypercube[END_REF].

In this paper, we will study in detail the relaxation of the FA-1f process depending on the initial conguration. We mainly rely on a result giving a bound on the mixing time for a conguration with many empty sites. The core of our study will thus be to see how quickly any initial conguration can create enough empty sites. For that, we study the big intervals that are initially lled with particles, and interpret the endpoint of those as fronts going inward at speed v. Thanks to another result referred to as "Zeros Lemma", we will be able to show that the conguration becomes suitable for relaxation after roughly the time it takes for the fronts of the biggest particle cluster to meet. In the end, our main result gives precise bounds on the mixing time of the process for any initial conguration, which is a more complete result than just a cuto statement.

First, we give the denition of the model and our main result in Section 2, then an important coupling with a threshold contact process in Section 3 and results about relaxation to equilibrium in Section 4. In Section 5, we extend the central limit theorem proved in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF] in order to t the actual proof of the main result, which appears in Section 6.

2 Denitions and main result 2.1 FA-1f processes Z -is the set of non-positive integers.

We will encounter three types of FA-1f processes, depending on their state space. Fix q ∈ [0, 1]. The FA-1f process of parameter q on an interval Λ ⊂ Z is given by the generator:

Lf (σ) = x∈Λ r(x, σ)(f (σ x ) -f (σ)),
for any local function f and σ ∈ {0, 1} Z , where σ x is the conguration equal to σ everywhere except at site x. The rate r(x, σ) is given by: r(x, σ) = (1 -σ(x -1)σ(x + 1))(qσ(x) + (1 -q)(1 -σ(x)). σ(x) = 1 is to be interpreted as the presence of a particle at x, and σ(x) = 0 as an empty site. In words, every site makes a ip 0 → 1 at rate 1 -q and 1 → 0 at rate q but only if it satises the kinetic constraint c x (σ) := (1 -σ(x -1)σ(x + 1)) = 1, that is if it has a least one empty neighbor. If Λ has boundaries, we set the conguration to be xed at 0 at the (outer) boundaries. We now x some notations and conventions used in throughout the paper.

FA-1f processes on Z are generally denoted by the letter σ. Their state space is Ω := {0, 1} Z . We dene LO = {σ ∈ Ω | ∃X, ∀x < X, σ(x) = 1} the set of congurations equal to one on an innite left-oriented half line.

FA-1f processes on Z -are generally denoted by the letter η. Their state space is

Ω -:= {η ∈ Ω | η(0) = 0, ∀x > 0, η(x) = 1}. We dene LO -= {σ ∈ Ω -| ∃X, ∀x < X, σ(x) = 1}. Lastly, for a nite interval Λ = [1, L], we dene Ω L = {σ ∈ Ω | σ(0) = σ(L + 1) = 0, ∀x / ∈ [0, L + 1], σ(x) = 1}.
For σ ∈ {0, 1} Z and x ∈ Z, we dene the shifted conguration θ x σ by : ∀y ∈ Z, θ x σ(y) = σ(y + x).

For any σ ∈ LO, we dene the front X(σ) (or sometimes simply written X) as the leftmost zero in σ. When (σ t ) t≥0 is a process, we will write X t instead of X(σ t ) when σ t clear from the context. We also denote by σ the conguration seen from the front, namely θ X(σ) σ. Similarly, we dene the front X(η) and η for any η ∈ LO -.

We also dene µ σ t (resp. μσ t ) the law of the FA-1f process (resp. seen from the front) at time t starting from σ. Whether the process takes place in Z or Z -is implicitly given by the conguration σ. We denote by µ p Λ the Bernoulli product law of parameter p on {0, 1} Λ . The FA-1f process on Λ with parameter q is reversible with respect to µ 1-q Λ . In the following, we shall write µ instead of µ 1-q Λ when q and Λ are clearly xed. Finally we dene, for I = [a, b] ⊂ Λ an interval and > 0 the set:

H(I, ) = H(a, b, ) := {σ ∈ {0, 1} Λ | ∀x ∈ [a, b -+ 1], ∃y ∈ [x, x + -1], σ(y) = 0}. If J = k I k is a union of disjoint intervals, then we dene H(J, ) := k H(I k , ).

Main result

We denote

Ω δ L = {σ ∈ Ω L | σ ∈ H(1, L, δL)}. We also dene, for σ ∈ Ω L , B(σ) := max{h ≥ 0 | ∃x ∈ [0, L -h], σ [x+1,x+h] ≡ 1}
the size of the largest component of occupied sites in σ. The most precise result about the relaxation of the FA-1f process in a nite interval proved in this paper is the following:

Theorem 2.1. There exists q < 1 such that for every q > q, δ ∈ (0, 1) and ε > 0, the following holds. There exist three constants a = a(ε) > 0 and 0 < v = v(q) < v = v(q) such that, if we rst dene the following three times for any σ ∈ Ω L :

t 1 (σ) = B(σ) 2v + (B(σ)) 1/4 ∨ (log L) 9 2v + (log L) 9/4 , t 2 (σ) = B(σ) 2v + 7a vδ √ L, t 3 (σ) = B(σ) 2v - 2a v B(σ),
then:

lim sup L→+∞ sup σ∈Ω δ L ||µ σ t 1 (σ) -µ|| T V = 0, (2.1
)

lim sup L→+∞ sup σ∈(Ω δ L ) c ||µ σ t 2 (σ) -µ|| T V ≤ ε δ 2 . (2.2)
Moreover, for any function

Φ such that Φ(L) -→ L→+∞ +∞, lim inf L→+∞ inf σ∈H(0,L,Φ(L)) c ||µ σ t 3 (σ) -µ|| T V ≥ 1 -ε. (2.3)
Let us give some comments about this statement. First, the two constants v and v have the following interpretations : v is the speed of the infection propagation for a threshold contact process dened in the following paragraph. The constant v correspond to the speed of the front for the FA-1f process starting with an innitly lled half-line. All of those depend on q that we chose here to be greater to a certain threshold necessary for the threshold contact process to survive. As we will see, the value q is roughly equal to 0.76 [START_REF] Brower | Critical exponents for the Reggeon quantum spin model[END_REF]. The rst two equations (2.1) and (2.2) of Theorem 2.1 give an upper bound on the time at which the process is well mixed. The rst one works nicely for initial congurations that already have enough empty sites. Indeed, in this case B(σ) is smaller than δL, making up for the loss in the constant 1/(2v) instead of the expected 1/(2v). The time t 1 has to be bounded from below by (log L) 9 for technical reasons we will see later. This bound is most likely not optimal but still oers a precise upper bound on the mixing time. The second equation handles the congurations that have macroscopic components of occupied sites and has the optimal leading behaviour B(σ) 2v one can expect as shown in the last equation (2.3). The last equation gives a lower bound on the mixing time for conguration with at least an interval of size Φ(L) occupied. Since it is only required that Φ(L) goes to innity, this hypothesis is not really restrictive. It is however necessary for our argument (Theorem 5.1) to work. As a consequence of this theorem, we can conclude that the FA-1f process exhibits a cuto phenomenon.

Theorem 2.2. Let d(t) = sup σ∈Ω Λ ||µ σ t -µ|| T V .
Then for all q > q and ε > 0, there exist α(ε), β(ε) > 0 independent of L, such that for L large enough:

d L 2v -α(ε) √ L ≥ 1 -ε, d L 2v + β(ε) √ L ≤ ε.
3 Coupling with a contact process and consequences

Graphical construction

Although we can easily dene the FA-1f process through its generator, we also provide a more convenient construction called the graphical construction. For each site x ∈ Z, dene a Poisson process T x of parameter 1 and an innite sequence of Bernoulli random variables (β x n ) n≥1 of parameter 1 -q , all of these variables being independent. Given an initial conguration σ 0 , at each increment t of a Poisson process, say the n th increment at site x, we check if the constraint c x (σ) is satised, that is if the site x has at least one empty neighbor. If it does, then we set σ t (x) = β x n . Otherwise, σ t (x) is unchanged.

From this construction, we can dene the standard coupling which simply consists in taking the same set of random variables (T x ) x∈Z and (β x n ) n≥1 for dierent initial congurations. Note that this coupling is not monotone: one can have σ 0 ≤ σ 0 and σ t ≤ σ t even if (σ t ) t≥0 and (σ t ) t≥0 follow the standard coupling (see Figure 3.1). This is an important reason why this dynamics can be dicult to study.

Using the graphical construction, it is standard to show that there is a nite speed of propagation: Proposition 3.1 ([7, Lemma 2.5]). Let (σ t ) t≥0 be a FA-1f process. For t ≥ 0 and two sites x < y, we dene the events: F (x, y, t) = {there is a sequence of successive rings linking x and y in a time interval of length t},

F (x, y, t) = {∃z ∈ [x, y] s.t. F (x, z, t) ∩ F (y, z, t)}.
Then there exists a constant v > 0 such that if |x -y| ≥ vt, then P(F (x, y, t)) ≤ P( F (x, y, t)) ≤ e -|x-y| . This implies a maximum speed for the front: Corollary 3.2. For all σ ∈ LO or η ∈ LO -, and c ≥ v, P(X t -X 0 < -ct) ≤ e -ct .

Contact process

In [START_REF] Durrett | Supercritical contact processes on Z[END_REF], Durett and Grieath introduced the contact process on Z. It is a process on {0, 1} Z that allows sites to ip from 0 (infected site) to 1 at rate 1, and from 1 to 0 at a rate depending on the amount of empty adjacent sites. Based on this process, we introduce the same threshold contact process as in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF], denoted here by (ζ t ) t≥0 and given by the following generator:

∀f local, ∀ζ ∈ Ω, L f (ζ) = x∈Z r (x, ζ)(f (ζ x ) -f (ζ)), where r(x, ζ) = (1 -ζ(x -1)ζ(x + 1))qζ(x) + (1 -q)(1 -ζ(x)
). Note that we took here the convention that 0 is an infected site, which is not the most common one. This process diers from both FA-1f and Durett and Grieath's process in its constraint: a site is free to ip from 0 to 1 at rate p but has to have at least one empty neighbor to ip from 1 to 0 at rate q. This threshold contact process is the one well suited to a coupling with FA-1f as we will explain later. According to the results of Durett and Grieath [START_REF] Durrett | Supercritical contact processes on Z[END_REF], we set λ c the critical parameter of the standard contact process on Z and q = 2λ c 1 + 2λ c > 1/2.

Whenever q ≥ q, then the threshold contact process starting from a single infected site had a positive probability of surviving and will create infected sites in an interval scaling linearly with time [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF]Appendix B].

Let η 0 ∈ LO -and (η t ) t≥0 a FA-1f process starting from η 0 . Let ζ 0 = δ X 0 where X 0 is the front of η 0 and δ X 0 is the conguration (on Z) equal to 0 in site X 0 and equal to 1 everywhere else. Dene (ζ t ) t≥0 the contact process starting from conguration ζ 0 , evolving with respect to the standard coupling with (η t ) t≥0 . Then ∀t ≥ 0, ∀x ∈ Z -, η t (x) ≤ ζ t (x). Indeed, this inequality holds for t = 0. Then, for each ringing of a site x ∈ Z -\{0}, assuming ∀y ∈ Z -, η t -(y) ≤ ζ t -(y), there are three cases:

The kinetic constraint at x is not satised in η t -, in which case it is not satised in ζ t - either. In this case, the value η t (x) does not change and the only possible change for ζ t (x) is 0 → 1, which preserves the order.

The kinetic constraint at x is satised in η t -but not in ζ t -. In this case, a 0 → 1 ip is possible for both congurations, and a 1 → 0 ip is only possible for η t (x), which does not break the inequality.

The constraint is satised for both congurations, and then by construction of the coupling, η t -(x) = ζ t -(x), which again does not change the inequality.

Note also that the behavior of ζ on Z + inuences the behavior on Z -only through the x = 0 site. Since ∀t, η t (0) = 0, we clearly have η t (0) ≤ ζ t (0).

Thanks to this observation, and to the known results on the contact process [7, Appendix B], we can now state a rst important lemma which provides a minimal speed for the front. Lemma 3.3. Let q > q. There exists v > 0 and A, B > 0 such that for every η ∈ LO -, and

t ≥ 0, P(X t -X 0 > -vt) ≤ Ae -Bt
The proof of the result is based on the comparison explained above. Hence, it is identical to the proof of Corollary 4.2 in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF].

Zeros Lemmas

Next, we state a crucial lemma based on the coupling explained above and on results for surviving contact processes. It is the same result as [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF]Corollary 4.3], only extended to the FA-1f process on Z -and [1, L], and it follows from the same coupling argument. Lemma 3.4. Let q > q. There exist c 1 , c 2 > 0 such that for any η 0 ∈ Ω -and x ≤ 0 such that

η 0 (x) = 0, ∀t ≥ 0, P(η t / ∈ H(x -vt, (x + vt) ∧ 0, ) ≤ c 1 t exp(-c 2 (t ∧ )).
For L > 0, if σ 0 ∈ Ω L and x ∈ [0, L + 1] is such that σ 0 (x) = 0, then similarly, ∀t ≥ 0, P(σ t / ∈ H((x -vt) ∨ 0, (x + vt) ∧ (L + 1), ) ≤ c 1 t exp(-c 2 (t ∧ )).

This last lemma can be used with x being the front of a conguration η ∈ LO -(or its boundary). This idea leads to a result we will now refer to as the "Zeros Lemma". We give here two versions of this result. The rst one matches Lemma 4.4 in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF] while the second one is more specic to our needs later on. Lemma 3.5 (Zeros Lemma). Let q > q. Let s, , M, L > 0 and η ∈ LO -.

If L + M ≤ 2vs, there exists c > 0 depending only on q such that:

P η (η s / ∈ H(L, (L + M ) ∧ (-X s ), )) ≤ (L + M ) 2 exp(-c(L ∧ )).
If L + M > 2vs and η0 ∈ H(0, (L + M ) ∧ (-X 0 ), 2vs), then there exists c > 0 such that:

P η (η s / ∈ H(L, (L + M ) ∧ -X s , )) ≤ s 2 L exp(-c(L ∧ )) + M s exp(-c(s ∧ )).
Proof. It is identical to the proof in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF], just taking into account the boundary.

Lemma 3.6 (Zeros Lemma II). Let q > q. Let s, > 0 and η ∈ LO -. Assume η ∈ H(X 0 , y, ) for some y ∈ [X 0 , 0], where X 0 denotes the leftmost zero of η. Then, if 2vs ≥ ,

P η (η s / ∈ H(X s , y, )) ≤ C(1 + |X 0 |) s 2 exp(-c(s ∧ )).
Proof. In the same way as the proof of [7, Lemma 4.4], we use Lemma 3.4 at intermediate times.

Let us x s and dene:

∆ = 2(v -v) ∧ s n = (s -∆)(v -v) 2v∆ ∆ = s -∆ n s i = i∆ for i ∈ [0, n].
For i ∈ [0, n], by Lemma 3.4 and Markov property applied at time s i , there exist c, C > 0 such that:

P η (η s / ∈ H(X s i -v(s -s i ), 0 ∧ (X s i + v(s -s i )), /2)) ≤ C(s -s i )e -c((s-s i )∧( /2)) (3.1)
In words, each position of the fronts on the times s i provides an interval at time s where it is likely to nd enough empty sites. Moreover, we can control the front evolution during the intervals [s i , s i+1 ] with Corollary 3.2 and Lemma 3.3 to nd that:

P η 0 ≤ X s i -X s i+1 ≤ v∆ ≥ 1 -e -c∆ , (3.2)
P η (0 ≤ X s -X sn ≤ v∆) ≥ 1 -e -c∆ . (3.3)
Our choice of ∆ and ∆ ensures that under these events, the intervals at time s previously found overlap, that is to say:

n i=0 [X s i -v(s -s i ), X s i + v(s -s i )] ⊃ [X s + /2, X 0 -vs].
We now take into account the zeros of the initial conguration to make this interval go all the way to the boundary. Let x 0 = X 0 < x 1 < • • • < x n ≤ y the empty sites in η 0 between X 0 and y. By Lemma 3.4, we have for all i,

P η (η s / ∈ H(x i -vs, 0 ∧ (x i + vs), /2)) ≤ Cse -c(s∧( /2)) (3.4) Now since 2vs ≥ , then (x i+1 -vs) -(x i + vs) ≤ 0 if s is large enough.
We can now conclude from equations (3.1), (3.2), (3.3) and (3.4):

P η (η s / ∈ H(0, -X s , )) ≤ Cnse -cs∧( /2) + e -c∆ + ne -c∆ + C|X 0 |se -cs∧( /2) ≤ C(1 + |X 0 |) s 2 e -cs∧ .

Relaxation results

We give here several results of relaxation starting out of equilibrium. They build upon the results of [START_REF] Blondel | Fredrickson-Andersen one spin facilitated model out of equilibrium[END_REF]. First, let us recall a proposition about the relaxation of the FA-1f process on a nite interval with zero boundary conditions.

Proposition 4.1 ([7, Corollary 3.3]). Let q > 1/2, L > 0 and f a bounded function with support contained in [1, L] such that µ(f ) = 0. If L ≤ e t α for some α < 1/2, then there exists

c = c (α, q) > 0 such that, if σ ∈ H(0, L + 1, √ t), |E σ [f (σ t )]| ≤ 1 c ||f || ∞ e -c √ t
In words, this proposition states that if the process starts in H(0, L + 1, ), then it is mixed after a time 2 . Note that this holds for q > 1/2 which covers a larger regime than q > q because it does not rely on the coupling with the threshold contact process. Actually, we can combine this proposition with Lemma 3.4 to get a better relaxation speed for the supercritical regime and have a mixing after a time ∼ 2v instead of 2 : Proposition 4.2. Let q > q, 0 < β < 1/2, L > 0, > 0 and f a bounded function with support contained in [1, L]

such that µ(f ) = 0. If L ≤ e
α for some α < β, then there exist c, C > 0 depending only on q and α such that, if σ ∈ H(0, L + 1, ) and

t = 2v + β , |E σ [f (σ t )]| ≤ C||f || ∞ Le -c β/2 . Proof. Set t 1 = 2v . Let 0 = x 1 < • • • < x p = L + 1 a sequence of sites in [0, L + 1] such that: ∀1 ≤ i ≤ p, σ(x i ) = 0, ∀1 ≤ i ≤ p -1, x i+1 -x i ≤ , p ≤ 2L .
Thanks to Lemma 3.4, we have

P σ t 1 / ∈ H(0, L + 1, β/2 ) ≤ P p i=1 {σ t 1 / ∈ H((x i -vt 1 ) ∨ 0, (x i + vt 1 ) ∧ (L + 1), β/2 )/2} ≤ p i=1 P σ t 1 / ∈ H(x i -vt 1 , x i + vt 1 , β/2 /2) ≤ Cpt 1 e -c(t 1 ∧( β/2 /2)) ≤ C Le -c β/2 .
Now we can use Proposition 4.1 (with σ t 1 ∈ H(0, L + 1, √ β ), and L < e β with β < 1/2) and Markov property:

|E σ [f (σ t )]| ≤ E σ [f (σ t )1 σt 1 ∈H(0,L+1, β/2 ) ] + ||f || ∞ P σ t 1 / ∈ H(0, L, β/2 ) ≤ E σ [1 σt 1 ∈H(0,L+1, β/2 ) |E σt 1 [f (σ t-t 1 )]|] + C ||f || ∞ Le -c β/2 ≤ 1 c ||f || ∞ e -c β/2 + C ||f || ∞ Le -c β/2 .
A notable result comes from the particular case = L + 1 and β = 1/4. Indeed, any conguration σ ∈ Ω L belongs to H(0, L + 1, L + 1).

Proposition 4.3. There exist

c, C > 0, such that for any L > 0, if t = L+1 2v + (L + 1) 1/4 , sup σ∈LO [0,L] ||µ σ t -µ|| T V ≤ CLe -cL 1/8 .
Remark 4.4. This results proves that the mixing time is always bounded by L 2v + o(L). This gives a rst scale for t mix but the constant 1 2v will turn out to be too large in general, as we will see in the nal section. Now, we give another result that deals with the relaxation for the FA-1f process on Z far from the front: Proposition 4.5 ([7, Theorem 5.1]). Let q > q, α < 1/2 and δ > 0. There exists c > 0 such that for all t ≥ 0, for any M ≤ e δt α , any f with support in [0, M ], such that µ(f

) = 0 and ||f || ∞ ≤ 1, for all σ ∈ LO such that σ ∈ H(vt, M + (4v -v)t, √ t), then |E [f (θ 3vt σt )] | ≤ e -c √ t .
The same kind of result holds for a process on Z -: Proposition 4.6. Let q > q, α < 1/2 and δ > 0. There exists c > 0 such that for any f with support

[0, M ], µ(f ) = 0 and ||f || ∞ ≤ 1, for any t ≥ 0, M ≤ e δt α , for all η ∈ LO -such that η ∈ H(vt, M + (4v -v)t, √ t) and X 0 ≤ -M -(3v -v)t, then |E [f (θ 3vt ηt )] | ≤ e -c √ t .
Proof. The proof is almost identical to that of the previous proposition. However, in this case we have to make sure that the interval [3vt, 3vt + M ] seen from the front does not go out the domain Z -. Recall that with probability 1 -O(e -Bt ) we have X t -X 0 ≤ -vt. Then as long as -X 0 + vt ≥ 3vt + M , we have X t + 3vt + M ≤ 0 with probability 1 -O(e -Bt ).

5 Central limit theorem for FA-1f on Z -

We now aim to prove a central limit theorem for the front of the FA-1f process on Z -. Our result diers from the one proved in [7, Theorem 2.2] in two ways. First, it studies an FA-1f process on the half-line as opposed to Z. This change forces us to take into account the boundary condition in the origin, tough we will see that the front gets far from the origin so this barely makes a dierence in the proof. The second change is a uniformity result with respect to the initial conguration. This requires to study carefully the original proof and tweak a few key arguments in subsections 6.2 and 6.3. For a rst read, one can take Theorem 5.1 for granted and skip to the next section in order to get to the main result.

Theorem 5.1. Let q > q. There exist v > 0 and s ≥ 0 such that for all η ∈ LO -,

X t -vt √ t L -→ t→∞ N (0, s 2 ).
The constants v and s are the same as in [7, Theorem 2.2]. Moreover, this convergence is uniform in η in the following sense: for all real numbers a < b,

sup η∈LO - P η a ≤ X t -vt √ t ≤ b -P(a ≤ N ≤ b) -→ t→∞ 0, (5.1)
with N ∼ N (0, s 2 ) Remark 5.2. In this theorem, the variance s 2 of the normal law can be zero. For this more favorable case, the law N (0, 0) is simply δ 0 , so that we have sup

η∈LO - P η -a ≤ X t -vt √ t ≤ a -1 -→ t→∞ 0 for all a > 0.
To prove this theorem, we will rst study the convergence of the law behind the front.

Theorem 5.3. Let q > q. The process seen from the front has a unique invariant measure ν. This measure is the same as in [7, Theorem 2.1]. There exist d * , c > 0 such that for all η 0 ∈ LO -, for t large enough,

||μ η 0 t -ν|| [0,d * t] ≤ exp -ce (log t) 1/4 ,
where μη 0 t is the distribution of the conguration seen from the front at time t, starting from η 0 and || • -• || Λ is the total variation distance on Λ.

Convergence behind the front

In this subsection, we prove Theorem 5.3 with a similar coupling as in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF]. Let us x η 0 ∈ LO -. We pick σ 0 ∈ LO an arbitrary conguration such that X(η 0 ) = X(σ 0 ). We will now prove that there exist d * > 0 and c > 0 such that

||μ η 0 t -μσ 0 t || [0,d * t] ≤ exp -ce (log t) 1/4 . (5.2)
Throughout the proof, c, C denote generic constants independent of σ 0 , η 0 . These constants may vary from line to line. Before diving into the complete coupling, we shall dene a coupling that comes into play during the proof: the Λ-maximal coupling. If Λ ⊂ Z is nite, and µ, µ are two probability measures on Ω -, we dene the Λ-maximal coupling between µ and µ as follow:

1. we sample (σ, σ ) Λ×Λ according to the maximal coupling, i.e. the one that achieves the total variation distance of the marginals of µ and µ on Ω Λ , 2. we sample σ Z -\Λ and σ Z -\Λ independently according to their conditional distributions µ(.|σ Λ ) and µ (.|σ Λ ).

Throughout this section, µ denotes the Bernoulli product measure on Z -: µ = µ 1-q Z -. For t > 0, let us dene the following quantities :

ε = v 2(v + v) , t 0 = (1 -ε)t, ∆ 1 = exp (log t) 1/4 , ∆ 2 = (log t) 3/4 , ∆ = ∆ 1 + ∆ 2 .
The time interval [0, t] is divided in several sub-intervals. First, during a time t 0 = (1 -ε)t (which is the longest phase), we aim to use the Zeros Lemma, which provides a good amount of empty sites behind the front. We then divide the remaining time εt into N steps of length ∆, with N = εt ∆ and a remainder step of length εt -N ∆. Each of these steps of length ∆ consists in two steps of respective lengths ∆ 1 and ∆ 2 . We dene t n = t 0 + n∆ and

s n = t n + ∆ 1 , n ≤ N . See Figures 5.1 and 5.2 2 σ ∈ LO, η ∈ LO - (1 -ε)t t 0 = Z 0 t 0 ∆ T ∆ ∆ T S S ∆ t N t Figure 5.1: [7]
Coupling of the evolutions from distinct initial congurations. The labels T correspond to trials (Steps 1-2, detailed in Figure 5.2), where the coupling attempts to match the two congurations. After the rst success (label S), the standard coupling maintains the matching up to time t. We dene the interval

t n = t 0 + n∆ s n+1 = t n + ∆ 1 t n+1 = s n+1 + ∆ 2 ∆ 1 = e (log t) 1/4 t ∆ 2 = (log t) 3/4 Z n H( √ ∆ 1 ) v∆ 1 vt n indep coupling maximal coupling Λ n Z n H( √ ∆ 2 /2) √ ∆ 2 /2 v∆ 1 v∆ 1 ≥ vs n+1 -(v + v)∆ 1 standard coupling maximal coupling standard coupling x ≤ √ ∆2 2 ≥ I n+1 = vt n+1 -(v + v)∆(n +
Λ n = [3v∆ 1 , vs n -(v + v)∆ 1 ] and the distance d n = vt n -(v + v)∆n.
Lastly, let us introduce the following notation. For (A n ) a sequence of measurable subsets of LO × LO -, we write :

P An (•) := sup (σ,η)∈An P (•| (σ tn , ηtn ) = (σ, η)) .
Depending on the context, t n can be replaced by s n in the denition above.

We are now ready to explain the coupling between σ t and η t that will lead to Theorem 5.2.

At time t 0 , we sample σ t 0 and η t 0 independently according to the laws µ η 0 t 0 and µ σ 0 t 0 .

Suppose the coupling at time t n constructed. If the congurations σtn and ηtn coincide on the interval [1, d n ], then we let the evolve according to the standard coupling until time t.

If not, we proceed in two steps. With high probability, both congurations will have zeros behind the front. More precisely, let Z n be the event:

Z n = {σ tn , ηtn ∈ H(v∆ 1 , vt n , ∆ 1 ) and (X(σ tn ) -X(σ 0 )), (X(η tn ) -X(η 0 )) ≤ -vt n }.
From both the Zeros Lemma, Lemma 3.3 and their analogs in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF], we have for t large enough,

P(Z n ) ≥ 1 -2(vt n ) 2 e -c √ ∆ 1 -Ce -c tn ≥ 1 -Ct n 2 e -c √ ∆ 1 .
The last term of the rst line comes from the fact that we ask for the inequalities X(σ tn ) ≤ X(σ 0 ) -vt n and X(η tn ) ≤ X(η 0 ) -vt n . Note that none of the constants c, C depend on η 0 or σ 0 .

In the following, it will be useful to also introduce the event

Z n = {σ tn , ηtn ∈ H(0, 3v∆ 1 , 2v∆ 1 )}.
We can already note that Z n ⊂ Z n for t large enough.

At time s n , we sample σsn and ηsn according to the Λ n -maximal coupling between the laws μσt n ∆ 1 and μηt n ∆ 1 . Let Q n = {σ sn = ηsn on Λ n }. First, note that by denition of the maximal coupling and by the Markov property, we have, for (σ, η) ∈ LO × LO -:

P(Q c n |σ tn = σ, η tn = η) = ||μ σ ∆ 1 -μη ∆ 1 || Λn ≤ ||μ σ ∆ 1 -µ|| Λn + ||μ η ∆ 1 -µ|| Λn .
Since the interval Λ n is "far from the front", then these two distances are small uniformly in σtn , ηtn as long as the event Z n is satised. Let us justify this claim by applying Proposition 4.6.

Let η ∈ LO -be an initial conguration such that η ∈ H(v∆ 1 , vt n , √ ∆ 1 ). Assume also that X(η) ≤ -vt n . Let us check the hypotheses of Proposition 4.6. Take

M = vs n -(v + v)∆ 1 -3v∆ 1 . Then M +(4v-v)∆ 1 = vt n -v∆ 1 ≤ vt n , which guarantees that we do have η ∈ H(v∆ 1 , M + (4v -v)∆ 1 , √ ∆ 1 )
. Next, since X(η) ≤ -vt n , we nd:

-M -(3v -v)∆ 1 = -vt n + (v + v)∆ 1 ≥ -vt n ≥ X(η).
Finally, we do have M ≤ e ∆ α 1 for α = 1/4 for example. We can now apply Proposition 4.6 that gives ||μ η ∆ 1 -µ|| Λn ≤ Ce -c √ ∆ 1 with C, c independent of η. We can handle the term ||μ σ ∆ 1 -µ|| Λn the same way in order to nd in the end:

P Zn (Q c n ) ≤ Ce -c √ ∆ 1 .
Next, if the event Q n occurs, we dene x as the distance between the front and the leftmost zero of σsn (or ηsn ) located in Λ n . If there is none, dene x as the distance between the front and the right boundary of Λ n . Let β be a Bernoulli random variable (independent of everything else constructed so far) such that P(β = 1) = e -2∆ 2 . The event {β = 1} has the same probability as the event that two independent Poisson clocks do not ring during a time ∆ 2 . We now construct the congurations at time t n+1 .

If β = 1, we consider that, seen from the front at time s n , the clocks associated with the sites 0 and x do not ring during ∆ 2 . First, we sample the conguration at the left of 0 according to the standard coupling, with empty boundary condition. Let ξ n+1 be the common increment of the front during this time, namely ξ n+1 = X t n+1 -X sn .

Next, we x σt n+1 (-ξ n+1 ) = ηt n+1 (-ξ n+1 ) := 0 and σt n+1 (x -ξ n+1 ) = ηt n+1 (xξ n+1 ) := σsn (x) and sample the congurations σt n+1 and ηt n+1 according to the maximal coupling on [1 -ξ n+1 , x -1 -ξ n+1 ], with boundary conditions 0 on 0 and σsn (x) on x. To the right of this interval, we sample the congurations according to the standard coupling with the appropriate boundary conditions.

If β = 0, then we let evolve (σ sn , ηsn ) for a time ∆ 2 via the standard coupling conditioned to have at least one ring either at x or at 0.

First, let us notice that x is likely to be not too far right inside Λ n . Indeed, under the event Z n the law of the conguration in Λ n is close to the product measure at time s n .

Let

B n := {x ≤ 3v∆ 1 + √ ∆ 2
2 }. Then, by Proposition 4.5:

P Zn (B c n ∩ Q n ) ≤ P Zn (σ sn ) [3v∆ 1 ,3v∆ 1 + √ ∆ 2 2 ] ≡ 1 (5.3) ≤ p √ ∆ 2 2 + O(e -c √ ∆ 1 ). (5.4)
Now, denote by Z n the event σsn , ηsn ∈ H

√ ∆ 2 2 , 3v∆ 1 , √ ∆ 2 2
, and recall that Z n ⊂ Z n . Using the second case of Zeros Lemma 3.5 and its equivalent [7, Lemma 4.4] (with L = √ ∆ 2 /2, L + M = 3v∆ 1 , = L), we nd that:

P Zn ((Z n ) c ) ≤ 2∆ 2 1 √ ∆ 2 exp(-c ∆ 2 ) + ∆ 1 3v∆ 1 - √ ∆ 2 2 exp(-c∆ 2 ) (5.5) ≤ O(e -c √ ∆ 2 ). (5.6)
With the event Z n , we are now able the use Proposition 4.1 to couple the conguration at time t n on the interval (5.7)

J n := [1 -ξ n+1 , x -ξ n+1 -1]. First, note that Z n ∩ B n ⊂ {σ sn , ηsn ∈ H 0, x, √ ∆ 2 }. Let L = 3v∆ 1 + √ ∆ 2 2 ≤ e ∆ 1/
Again, this bound does not depend on y or the initial congurations. This in turn shows that:

P Z n ,Bn (σ t n+1 = ηt n+1 on J n ) = O(e -c √ ∆ 2 ).
(5.8)

Under the event Q n , the congurations seen from the front at time s n coincide on the interval Λ n . On the right part of this interval, namely [x, vs n -(v + v)∆ 1 ], we let the congurations evolve according to the standard coupling. The matching of the congurations will then persist on the sub-interval

[x -ξ n+1 , vs n -(v + v)∆ 1 -ξ n+1 -v∆ 2 ] w.h.p. thanks to Proposition 3.1. Note that vs n -(v + v)∆ 1 -v∆ 2 = vt n+1 -v∆ ≥ d n+1 . Denote by K n the interval [x -ξ n+1 , d n+1 ].
We found that:

P Qn σt n+1 = ηt n+1 on K n | β = 1 ≤ P(F (0, v∆ 2 , ∆ 2 ) + P(ξ n+1 ≥ 0) = O(e -c∆ 2 ). (5.9)
Finally, notice that if the clock attached to the site 0 (i.e the front at time s n ) does not ring, and we let the congurations at the left of this site evolve according to the standard coupling, we naturally get:

P Qn σt n+1 = ηt n+1 on [0, -ξ n+1 ] | β = 1 = 0.
(5.10)

From Equations (5.8), (5.9) and (5.10), we can conclude that:

P Qn,Z n ,Bn σt n+1 = ηt n+1 on [1, d n+1 ] | β = 1 = O(e -c √ ∆ 2 ). (5.11)
We are now ready to estimate the probability that the coupling "succeeds". Let M n = {σ tn = ηtn on [1, d n ]}. At time t n+1 , depending on the coupling at time t n , we can write:

P(M c n+1 ) ≤ P(M c n+1 ∩ M c n ∩ Z n ) + P(Z c n ) + P(M c n+1 ∩ M n ) ≤ P(M c n+1 |M c n ∩ Z n )P(M c n ) + P(Z c n ) + P(M c n+1 ∩ M n ). (5.12)
Let us focus on the term P(M c n+1 |M c n ∩ Z n ). We condition on the events that can happen at time s n :

P(M c n+1 |M c n ∩ Z n ) ≤ P Zn,M c n (M c n+1 |Z n ∩ Q n ∩ B n ) + P Zn ((Z n ) c ) + P Zn (Q c n ) + P Zn (B c n ∩ Q n ).
Out of all these terms, only the rst one is (a priori) not vanishing when t goes to innity according to equations (5.3), (5.5), (5.1). Thanks to equation (5.11), for t large enough:

P Zn,M c n (M c n+1 |Z n ∩ Q n ∩ B n ) ≤ 1 -P(β = 1) 1 -P M c n ∩Zn M c n+1 |{β = 1} ∩ Z n ∩ Q n ∩ B n (5.13) ≤ 1 - 1 2 e -2∆ 2 .
(5.14)

Back to the initial inequality (5.12), we can plug our estimate for the rst term, and easily bound the remaining two terms to nd :

P(M c n+1 ) ≤ 1 - 1 2 e -2∆ 2 P(M c n ) + Ce -c∆ (5.15)
It only remains to solve this recursive equation with the values of N, ∆ 1 , ∆ 2 we chose. We nd:

P(M c N ) ≤ 1 - 1 2 e -2∆ 2 N + C2 -N +1 e -c∆-2(N -1)∆ 2 (5.16)
≤ O(e -c∆ 1 ).

(5.17)

Let us now conclude the proof. At time t N , with probability P(M N ) the congurations σt N and ηt N match on the interval [1, d N ]. From time t N to time t, we let the congurations evolve according to the standard coupling. In the same way as we did before, we can estimate the interval on which the congurations are equal at time t given the event M N .

P(σ t = ηt on [1, d * t]) = P(σ t = ηt on [1, d * t] | M N )P(M N ) (5.18) ≥ P({X t ≤ X t N } ∩ F (d N , d * t, t -t N ) c )P(M N ). ( 5
d N -d * t ≥ vt 4 -vt ≥ v∆ ≥ v(t -t N ), (5.20
)

which implies P(F (d N , d * t, t -t N )) = O(e -ct ). The event {X t ≤ X t N } has probability 1 - O(e -c∆
) so we nd in the end the announced estimate:

P(σ t = ηt on [1, d * t]) = O(e -c∆
).

(5.21)

The existence of an invariant measure ν comes from the compacity of the set of probability measures on Ω. The uniqueness and the convergence estimate of Theorem 5.3 come from Equation (5.21).

A Central Limit Theorem

From Theorem 5.2, we will now get a central limit theorem as in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF]. Here, we state a general central limit theorem that we will apply to the increments of th front in the next paragraph. It resembles [7, Theorem A.1] but adds a uniformity result.

Theorem 5.4. Let (σ t ) be a Markov process (in continuous time) on a probability space (Ω, F, P) and (F t ) the adapted ltration. Let (X i ) i≥1 be real random variables satisfying the following hypotheses:

1. (a) sup

σ∈Ω sup n∈N E σ [X 2 n ] < ∞; (b) for every i ≥ 1, X i is measurable w.r.t. F i ; (c) for every k, n ≥ 1, f : R n → R measurable such that sup σ E σ [f (X 1 , ..., X n )] < ∞, for all initial σ, we have the Markov property E σ [f (X k , ..., X k+n-1 ) | F k-1 ] = E σ k-1 [f (X 1 , ..., X n-1 )];
(5.22)

2. There exists a decreasing function Φ, constants C, c * ≥ 1 and v ∈ R and a measure ν such that

(a) lim n→∞ e (log n) 2 Φ(n) = 0; (b) for every i ≥ 1, E ν [X i ] = v; (c) for every k, f : R → R s.t. e -|x| f (x) ∈ L 1 (R), sup σ |E σ [f (X k )] -E ν [f (X 1 )]| ≤ CΦ(k); (5.23) (d) for every k, n and f such that f : R → R s.t. e -|x| f (x) ∈ L 1 (R), sup σ |Cov σ [f (X k ), f (X n )] -Cov ν [f (X 1 ), f (X n-k+1 )]| ≤ CΦ(k); (5.24) sup σ |Cov σ [f (X k ), f (X n )]| ≤ C sup σ E σ [f (X 1 ) 2 ] 1/2 Φ(n -k);
(5.25) (e) for every k, n such that k ≥ c * n and any bounded function

F : R n → R, sup σ |E σ [F (X k , ..., X k+n-1 )] -E ν [F (X 1 , ..., X n )]| ≤ C||F || ∞ Φ(k).
(5.26) Then there exists s ≥ 0 such that

n i=1 X i -vn √ n L -→ N (0, s 2 ).
Moreover, this convergence holds uniformly in the initial conguration in the following sense:

∀a < b, sup σ |P a ≤ n i=1 X i -vn √ n ≤ b -P (a ≤ N ≤ b) | -→ n→∞ 0,
where N ∼ N (0, s 2 ).

Note that the hypothesis 2.(d) diers from the one in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF]. It is the correct hypothesis and should be corrected in the original article. It was however veried in the application therein, see [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF]Lemma 7.3].

Proof. We rst focus on bounded random variables. Let (X i ) i≥1 , satisfying the hypotheses.

Let us dene like in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF]

Y i := X i -E ν [X 1 ], n = n 1/3 , S n = n i=1 Y i , S j,n = n i=1 1 |k-j|≤ln Y i and nally α n = n i=1 E σ [Y j S j,n ]
. and assume each Y i is bounded. We follow the same line of arguments, replacing the "Bolthausen Lemma" [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF]Lemma A.3] with an adapted version: Lemma 5.5. Let (ν σ n ) n≥0 be a family of probability measures on R such that:

1. sup σ sup n |x| 2 dν σ n (x) < ∞ 2. ∀R > 0, lim n→∞ sup σ sup |λ|≤R (iλ -x)e iλx dν σ n (x) = 0
Then for all σ ∈ Ω, (ν σ n ) n≥0 converges to the standard normal law. Furthermore, for all continuous bounded function f :

lim n→∞ sup σ |ν σ n (f ) -µ(f )| = 0,
where µ = N (0, 1).

Moreover, for all real numbers a < b,

lim n→∞ sup σ ν σ n (1 [a,b] ) -µ(1 [a,b] ) = 0
The proof of this result can be found in Appendix A. Looking at the proof of Theorem in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF], the distributions of ( S n √ α n ) n≥0 satisfy the stronger hypotheses (in particular item 2) of the above Lemma. Consequently, we have the conclusion of Theorem 5.4 for bounded variables.

We now give details on how to extend the bounded case to the general case. We no longer assume the (Y i ) i≥0 to be bounded. Let a ∈ R be a real number and ε > 0. Let N ≥ 0 be an integer that will be xed later, and dene the truncation operator T N (x) := max{min(x, N ), -N }, and the remainder R N (x) := x -T N (x).

Then we can estimate:

P σ n i=1 Y i √ n ≥ a = P σ n i=1 (T N (Y i ) -E ν [T N (Y i )]) + (R N (Y i ) -E ν [R N (Y i )]) √ n ≥ a ≤ P σ n i=1 (T N (Y i ) -E ν [T N (Y i )]) √ n ≥ a -ε + 2P σ |R N (Y i ) -E ν [R N (Y i )]| √ n ≥ ε .
(5.27)

We rst handle the last term:

P σ |R N (Y i ) -E ν [R N (Y i )]| √ n ≥ ε ≤ 1 ε 2 n Var σ n i=1 R N (Y i ) ≤ 1 ε 2 n   n i=1 Var σ (R N (Y i )) + i =j Cov σ (R N (Y i ), R N (Y j ))   .
For any i, note that we can easily bound

E σ [R N (Y i ) 2 ]: E σ (R N (Y i ) 2 ) = E σ [(Y i -N ) 2 1 Y 1 ≥N ] + E σ [(Y i + N ) 2 1 Y 1 ≤-N ] ≤ 4 √ 2(E σ [Y 4 i ] + N 4 ) 1/2 P σ (|Y i | ≥ N ) 1/2 ≤ 4 √ 2(C + N 4 ) 1/2 P σ (|Y i | ≥ N ) 1/2 ≤ C (C + N 4 ) 1/2 e -N/4 ,
where C does not depend on σ or i, and the last bound comes from the exponential Chebychev inequality and hypothesis 2.(c).

From this we can conclude that

1 ε 2 n n i=1 Var σ (R N (Y i )) ≤ Cε -2 e -cN , with c, C > 0 independent of n, N, σ.
From assumption 2.(d) (equation (5.25)), we get that:

n i =j Cov σ (R N (Y i ), R N (Y j )) ≤ n i=1 ∞ j=1 sup σ E σ [R N (Y j ) 2 ]Φ(|j -i|) ≤ Ce -cN n i=1 ∞ j=1 Φ(|j -i|) ≤ C ne -cN ,
where we used the bound on E σ [R N (Y j ) 2 ] previously found for the second inequality, and the fact that j Φ(j) < ∞ for the last one.

Putting the last two inequalities into (5.27), we get:

P σ n i=1 Y i √ n ≥ a ≤ P σ n i=1 (T N (Y i ) -E ν [T N (Y i )]) √ n ≥ a -ε + Cε -2 e -cN .
Let N be such that Cε -2 e -cN ≤ ε. We now use our theorem on the bounded variables T N (Y i ) to nd:

lim sup n→∞ sup σ P σ n i=1 Y i √ n ≥ a -P(N ≥ a -ε) ≤ ε,
which in turn shows that lim sup

n→∞ sup σ P σ n i=1 Y i √ n ≥ a -P(N ≥ a) ≤ 0.
It now remains to prove the reverse inequality.

P σ n i=1 Y i √ n ≥ a ≥ P σ n i=1 T N (Y i ) -E ν [T N (Y i )] √ n ≥ a + ε and n i=1 R N (Y i ) -E ν [R N (Y i )] √ n ≤ ε ≥ P σ n i=1 T N (Y i ) -E ν [T N (Y i )] √ n ≥ a + ε -P σ n i=1 R N (Y i ) -E ν [R N (Y i )] √ n ≥ ε
From here, we bound the last term as we have done previously, and nd this time:

lim sup n→∞ sup σ P σ n i=1 Y i √ n ≥ a -P(N ≥ a) ≥ 0,
which concludes the proof.

Proof of Theorem 5.1

The aim of the following four lemmas is to justify the various hypotheses of Theorem 5.4 on the increments ξ n := X n -X n-1 , where X n denotes the front at time n of an FA-1f process on Z - started at an arbitrary conguration η ∈ LO -. They can be proved in the exact same way as in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF], with a minor change explained in the proof of Lemma 5.7. In the following, ν denotes the measure dened in Theorem 5.3.

Lemma 5.6. For f :

Z → R such that e -|x| f (x) 2 ∈ L 1 , we have sup η∈LO - E η [f (ξ 1 ) 2 ] = c(f ) < ∞.
(5.28) Lemma 5.7. There exists γ > 0 such that for f :

Z → R with e -|x| f (x) 2 ∈ L 1 (R), sup η∈LO - |E η [f (ξ n )] -E ν [f (ξ 1 )]| ≤ C(f )e -γe (log n) 1/4 . (5.29)
Proof. Here, we dene as in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF], the conguration Φ t (η) to be equal to η on [X(η), X(η) + d * t] and 1 elsewhere (except at the origin). Let X, (resp. X) the front of the conguration starting form η (resp. Φ t (η)), both being coupling via the standard coupling. In our case, it is possible that the interval [X(η), X(η)+d * t] contains the origin, which can mess up the original argument.

To prevent this, let us note the following fact:

if t ≥ 0, η ∈ LO -, the event X = X is a subset of A t := F (0, d * t, 1) ∪ {X t ≥ -vt}.
For η ∈ LO -, f an appropriate function, and n ≥ 0, we get:

E η [f (ξ 1 )] -E Φ n-1 (η) [f (ξ 1 )] = E[(f (X) -f ( X)1 X = X ] ≤ E (f (X) -f ( X) 2 P(A n-1 ). Now note that P(A n-1 ) ≤ e -d * (n-1) 2 + Ae -B(n-1) = O(e -cn
). From there, we can conclude like in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF].

Lemma 5.8. There exists γ > 0 such that, for f :

N → R, e -|x| f (x) 2 ∈ L 1 (R) and j < n two positive integers, 1. sup η∈LO - |Cov η [f (ξ j ), f (ξ n )]| ≤ C(f )e -γe (log(n-j)) 1/4
, and the same holds for the covariance under ν;

2. for j ≥ v d * (n -j), sup η∈LO - |Cov η [f (ξ j ), f (ξ n )] -Cov ν [f (ξ 1 ), f (ξ n-j+1 )]| ≤ C(f )e -γe (log j) 1/4 .
Lemma 5.9. For any k, n ∈ N such that d * (k -1) ≥ vn and any bounded function

F : R n → R sup η∈LO - |E η [F (ξ k , ..., ξ k+n-1 )] -E ν [F (ξ 1 , ..., ξ n )]| = O ||F || ∞ e -γe (log k) 1/4 .
From here, we can apply Theorem 5.4 and conclude exactly like in [START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF].

6 Cut-o for FA-1f

We are now ready to prove the main result. Let Λ = [1, L], we x q > q the parameter of the FA-1f process. Every constant introduced in this section implicitly depends on q.

We split the proof into two cases. First, we tackle initial congurations with macroscopic sub-intervals of occupied sites. In this section, we shall call such an interval a particle cluster. To tackle these, we need to study closely the fronts of each particle cluster that are going inward. We will see that after the time it takes for them to meet, the conguration created enough empty sites to relax to equilibrium quickly. Next, we study the initial congurations with no such particle cluster, that is congurations in H(0, L + 1, ) for a certain threshold (L). We handle the latter category with softer arguments using the coupling of FA-1f and the contact process.

Congurations with macroscopic particle clusters

Dene, for any L > 0 and 0 < δ < 1,

Ω δ = {σ ∈ Ω Λ | σ ∈ H(0, L + 1, δL)}. For σ ∈ Ω Λ , recall that B(σ) := max{h ≥ 0 | ∃x ∈ [0, L -h], σ [x+1,x+h] ≡ 1}
is the size of the largest particle cluster in σ. The following result holds. Proposition 6.1. Let δ > 0, σ 0 ∈ (Ω δ ) c and ε > 0. There exists a = a(ε) > 0 such that,

if t = B(σ 0 ) 2v + 7av vδ √ L, then: ||µ σ 0 t -µ|| T V ≤ v 2 v 2 δ 2 ε + F (L),
with F (L) independent of σ 0 and lim

L→∞ F (L) = 0.
The proof of this proposition is divided into four lemmas. To make things easier, let us rst introduce the sketch of the proof in words. In the case we are studying now, the initial conguration has at least one macroscopic particle cluster of size ≥ δL. If we watch the evolution of the process in such an interval during a short period of time, we see that the two endpoints of this particle cluster behave just like fronts of FA-1f processes in half-lines (one left-oriented, the other one right-oriented). As a consequence, thanks to the Central Limit Theorem 5.1 we can nd an explicit time after which the fronts are much closer to each other, at a certain threshold distance d of order √ L. At this point, the space between the original positions of the fronts and the current one should contain a lot of zeros thanks to the Zeros Lemma 3.5. We repeat this argument for every initial particle cluster, and thanks to appropriate choices of constants and coupling arguments handling the space in between the clusters, we end up with a conguration in H(Λ, d) with high probability. From there, it will only remain to apply the relaxation result from Section 4 to conclude.

Before diving into the proof, we need to set a general framework. The following lemmas aim to explain in detail how a conguration with some particle clusters can turn the smallest of them into an interval with a lot of zeros. Let us x an arbitrary set of disjoint intervals Λ 1 , . . . , Λ r ⊂ Λ. We dene their endpoints:

Λ k = [a k , b k ]. Fix ε > 0, and let d = 6a
√ L, with a a constant depending on ε that will be determined later. We now dene a class of congurations that we consider in our proof. We say that a conguration σ ∈ Ω Λ is in the class C if for all 1 ≤ k ≤ r, there exist

X k < Y k ∈ Λ k such that 1. σ(X k ) = σ(Y k ) = 0, 2. σ (X k ,Y k ) ≡ 1, 3. σ ∈ H(a k , X k , d) ∩ H(Y k , b k , d).
Note that X k and Y k are uniquely determined if σ ∈ C \ H(Λ k , d). If σ ∈ H(Λ k , d), there can be multiple choices of X k and Y k , but this case will be excluded in the following. Let us introduce a few functions on C (see Figure 6.1). Fix σ ∈ C.

κ = {k | σ / ∈ H(Λ k , d)} is the set of indices such that Y k -X k > d, p(σ) = #κ is the cardinality of κ, (σ) = min k∈κ (Y k -X k ), t(σ) = (σ) 2v -2a v (σ). We set t(σ) = 0 if p(σ) = 0.
Having set that, we dene the random variable F (σ) = σ t(σ) . That is, F (σ) is the result of the FA-1f process with initial conguration σ ∈ C after a time t(σ). The aim of the random function F is to turn the smallest particle cluster [X k , Y k ] of size ≥ d into an interval in H(., d).

Λ 1 Note that this construction depends on the family of intervals Λ 1 , ..., Λ r that is for now arbitrary.

Λ 2 Λ 3 σ ∈ C X 1 Y 1 X 3 Y 3 ≤ d
Let σ ∈ C and (σ t ) t≥0 be a FA-1f process starting from σ. Dene X k t (resp. Y k t ) as the position of the rightmost (resp. leftmost) zero in [0,

M k ] (resp. [M k , L + 1]) in σ t , with M k := 1 2 (X k + Y k ). We have X k 0 = X k and Y k 0 = Y k .
As long as X k t and Y k t do not meet, we think of X k t as the "front" of a conguration on the (right oriented) half-line. Let R t = {∀k ≤ p(σ), ∀s < t, X k s < M k -1 and Y k s > M k + 1} the event that none of the "fronts" reaches the midpoint of the interval. In particular, under R t , the front have not met. The following Lemma localizes these fronts at time t(σ) using Theorem 5.1. Lemma 6.2. There exists a > 0 depending only on ε such that, with the notations previously introduced, and d = 6a √ L, the following holds. For all σ ∈ C such that p(σ) ≥ 1,

P ∀k ∈ κ, X k t(σ) ∈ [X k + vt(σ) -a (σ), X k + vt(σ) + a (σ)], Y k t(σ) ∈ [Y k + vt(σ) -a (σ), Y k + vt(σ) + a (σ)], R t(σ) ≥ 1 -p(σ)ε + F (L) (6.1)
with F (L) independent of σ such that lim L→∞ F (L) = 0.

In particular, for k 0 such that Y k 0 -X k 0 = (σ), the event above implies that:

0 ≤ Y k 0 t(σ) -X k 0 t(σ) ≤ d
From now on, we call

I k = [X k + vt(σ) -a (σ), X k + vt(σ) + a (σ)]
and

J k = [Y k + vt(σ) -a (σ), Y k + vt(σ) + a (σ)]. M k 0 σ σ t(σ) 6a (σ) ∼ vt(σ) X k 0 X k 0 t(σ) Y k 0 Y k 0 t(σ)
. Dene R k t = {∀s ≤ t, X k s < M k -1 and Y k s > M k + 1} such that R t = 1≤k≤p(σ) R k t .
The key remark for what follows is that if the event R k t occurs, the fronts X k s , Y k s behave like independent fronts of FA-1f processes on Z -for all s ≤ t. To justify that, let us dene auxiliary processes (σ k t ) t≥0 and (σ k t ) t≥0 on Z -as follows.

∀x ≤ 0, σk

0 (x) = σ(-x) if -x ≤ X k , 1 if -x > X k .
(σ k t ) t≥0 is the FA-1f process on Z -starting from σk 0 constructed with the standard coupling with respect to (σ t (-.)) t≥0 . We denote by Xk t its front at time t.

∀x ≤ 0, σk 0 (x) = σ(x + L + 1) if x + L + 1 ≥ Y k , 1 if x + L + 1 < Y k .
(σ k t ) t≥0 is the FA-1f process on Z -starting from σk 0 constructed with the standard coupling with respect to (θ L+1 σ t ) t≥0 . We denote by Y k t its front at time t. Thanks to Theorem 5.1, we have, for N ∼ N (0, s 2 ),

Let

P σk 0 Xk t ∈ -I k = P σk 0 Xk t -Xk 0 -vt(σ) (σ)/2v ∈ [-2va, 2va] -→ L→∞ P (N ∈ [-2va, 2va]) (6.3) since (σ) ≥ d → ∞ as L → ∞.
This convergence is furthermore uniform in σ. We choose a such that P (N ∈ [-2va, 2va]) ≥ √ 1 -ε. Note that this choice only depends on ε. This proves that

P σk 0 Xk t ∈ -I k = √ 1 -ε + F 1 (L) (6.4)
with F 1 (L) independent of σ and lim L→∞ F 1 (L) = 0. We handle the term in Y k t the same way and nd the same kind of equation. Note here that if s 2 = 0, our estimate is even better as we would get P σk

0 Xk t ∈ -I k = 1 -F 1 (L).
Next, we show that if the fronts end up in the expected intervals at time t(σ), then Rk t(σ) is very likely. We denote M k = -M k and we divide the time interval [0, t(σ)] into intermediate

times 0 < s 1 < ... < s n = t(σ) with n = 2vt(σ) a (σ) -2 such that ∆ := s i+1 -s i ≤ a 2v (σ)- 1 v . Notice that: {∃s < t(σ), Xk s ≤ M k + 1} ⊂ {∃i, Xk s i ≤ M k + a 2 (σ)} ∪ {∃s < t(σ), Xk s ≤ M k + 1 and ∀i, Xk s i > M k + a 2 (σ)}.
First, with Proposition 3.3 and with our choice of ∆, we have that the second event is unlikely because of nite speed of propagation:

P σ ∃s < t(σ), Xk s ≤ M k + 1 and ∀i, Xk s i > M k + a 2 (σ) ≤ n-1 i=0 P σ ∃s ∈ [s i , s i+1 ], | Xk s -Xk s i+1 | ≥ a 2 (σ) -1 ≤ n-1 i=0 P σ ∃s ∈ [s i , s i+1 ], | Xk s -Xk s i+1 | ≥ v(s i+1 -s i ) ≤ Ce -c √ (σ) ≤ Ce -c √ d for some C, c > 0.
Then, by the Markov property at time s i , the probability that Xk

s i ≤ M k + a 2 (σ) while Xk t(σ)
is to the right of that is bounded by the probability that the front is going backward for a time at least ∆. This gives:

P σ Xk t(σ) ∈ -I k and ∃i, Xk s i ≤ M k + a 2 (σ) ≤ n-1 i=0 P σ ( Xk t(σ) > Xk s i ) ≤ nAe -B∆ ≤ Ae -B √ d .
Combining the previous inequalities shows that:

P σ Xk t(σ) ∈ -I k and ( Rk t ) c ≤ Ce -c √ d , (6.5)
with C, c > 0 independent of σ, L, ε.

Combining Equations (6.2), (6.4) and (6.5), since d = 6a √ L, we get

P σ X k t ∈ I k , Y k t ∈ J k and R k t ≥ 1 -ε + F (L),
with F (L) going to 0 as L → ∞, uniformly in σ. By a union bound over all boxes B k (σ), we get the expected result since the number of boxes is bounded by δ -1 .

We have now localised the fronts in F (σ), so we can use Lemma 3.5 to show that behind those fronts, with high probability, there are a lot of zeros (see Figure 6.3). Lemma 6.3. Let σ ∈ C such that p(σ) ≥ 1. With the previous notations, for all k ∈ κ,

P σ X k t(σ) ∈ I k , Y k t(σ) ∈ J k , R t(σ) and σ t(σ) / ∈ H(X k , X k t(σ) , d) ∩ H(Y k t(σ) , Y k , d) ≤ CL 9/4 exp(-c √ L)
for some C, c > 0 independent of every other parameter.

Proof. Given the event R t(σ) , we can use the previous coupling argument to show that:

P σ X k t(σ) ∈ I k , Y k t(σ) ∈ J k , R t(σ) and σ t(σ) / ∈ H(X k , X k t(σ) , d) ∩ H(Y k t(σ) , Y k , d) = P σk 0 Xk t ∈ -I k , σk t(σ) / ∈ H( Xk t , Xk 0 , d), Rt(σ) P σk 0 Y k t ∈ θ -L (J k ), σk t(σ) / ∈ H( Y k t , Y k 0 , d), Řt(σ) . (6.6) σ σ t(σ) X 1 Y 1 X 2 Y 2 X 3 Y 3 X 1 t(σ) Y 1 t(σ) X 3 t(σ) Y 3 t(σ) ≤ 6a √ L Figure 6.3:
The expected behaviour of the fronts during time t(σ). Dashed zones are in H(., d).

We see that at time t(σ), the conguration lost at least one particle cluster.

By Lemma 3.6 (with y = Xk 0 ), we have that:

P σk 0 σk t(σ) / ∈ H( Xk t , Xk 0 , d) ≤ CL 3/2 exp(-cL).
The same inequality holds for σk and therefore we get the claimed result. Now, we show that if an interval already contains enough zeros, it most likely will also at time t(σ). This will ensure that once a cluster has disappeared and has left a lot of empty sites, the region will keep as many empty sites during any other iteration of F . It is actually a very general result that only relies on Lemma 3.4. Lemma 6.4. Let σ ∈ Ω Λ , x, y ∈ Λ, d > 0 such that σ ∈ H(x, y, d). Then:

P σ (F (σ) / ∈ H(x, y, d)) ≤ CL exp(-c √ L), (6.7)
with c, C independent of d, x, y, σ.

Proof. We simply use the comparison with a supercritical contact process (Lemma 3.4) for every initial zero in σ (a,b) with = d/2.

So far, we have proved that with high probability after a time t(σ), every front moves through a distance ∼ (σ) 2v (Lemma 6.2), while creating zeros behind them (Lemma 6.3). This results in one box k 0 initially "blocked" (Y k 0 -X k 0 > d) to become lled with zeros. Meanwhile, every box Λ k that satises σ ∈ H(Λ k , d) stays in H(Λ k , d) after a time t(σ) (Lemma 6.4). As illustrated in Figure 6.4, all of this combines into a nice result about F (σ).

Lemma 6.5. For any σ ∈ C such that p(σ) ≥ 1,

P (F (σ) ∈ C and p(F (σ)) < p(σ)) ≥ 1 -p(σ)ε -F 2 (L)
with F 2 (L) independent of σ (but depending on r) and lim L→∞ F 2 (L) = 0.

Before getting to the proof of Proposition 6.1, let us make a nal remark. On the event appearing in Lemma 6.2, we can bound the size of every particle cluster in F (σ):

∀k ∈ κ, Y k t(σ) -X k t(σ) ≤ Y k -X k -(σ) + 2a √ L (6.8)
Note that it is possible that in one iteration of F , two or more particle clusters go from a size > d to a size ≤ d.

Let us now prove Proposition 6.1.

σ σ t(σ) Λ 1 Λ 2 Λ 3 X 1 Y 1 X 3 Y 3 X 3 t(σ) Y 3 t(σ) ≤ d
Figure 6.4: The expected behaviour of the whole conguration during time t(σ). Here, the interval Λ 2 kept its zeros and the particle cluster in Λ 1 shrunk to a size ≤ d.

Proof of Proposition 6.1. Fix δ > 0, σ 0 ∈ (Ω δ ) c and ε > 0. We also dene a as in Lemma 6.2.

Throughout the proof, F (L) will denote a function that may depend on δ but not on σ such that lim L→∞ F (L) = 0. Let Λ 1 , ...Λ r be the particle clusters of σ 0 of size ≥ v v B(σ 0 ) (with their neighboring sites that are empty in σ). More precisely, the intervals

Λ k = [a k , b k ] are such that b k -a k ≥ v v B(σ 0 ), σ 0 (a k ) = σ 0 (b k ) = 0 and σ 0 (a k ,b k ) ≡ 1.
Dening the Λ k denes the class C and we have of course σ 0 ∈ C and p(σ) = r. Note already that even though r depends now on σ, it can simply be bounded by v vδ , which only depends on δ. We aim to prove that after a time

t 1 = B(σ 0 ) 2v + 3ar √ L, the conguration is in H(Λ, d) with d = 6a √ L.
To do that, we rst focus on the particle clusters. We dene now the iterations of F starting from σ 0 . For all 1 ≤ k ≤ r, σ (i) := F (σ (i-1) ) with σ (0) := σ 0 . If we ever have σ (i) / ∈ C, then set for example F (σ (i+1) ) = σ (i) and t(σ (i) ) = 0. We will not encounter this case later on. By Lemma 6.5, we have by induction that P p(σ (r) ) = 0 ≥ 1 -r 2 ε + F (L).

Note that p(σ (r) ) = 0 simply means that σ (p) ∈ r k=1 H(Λ k , d). Let T = r-1 i=0 t(σ (i) ), that is the time at which we reach the conguration σ (r) . Our goal now is to bound T from above. To do that, we use Equation (6.8) and by induction we get

P T ≤ B(σ 0 ) 2v + 2ar √ L ≥ 1 -r 2 ε + F (L).
Now using the exact same argument as in Lemma 6.4, as long as T ≤ B(σ 0 ) 2v + 2ar √ L and σ T ∈ r k=1 H(Λ k , d), then with high probability these zeros will remain at time t 1 = B(σ 0 ) 2v + 3ar √ L. This leads to the result we were looking for, that at time t 1 , with high probability, every initial particle cluster contains a lot of zeros:

P σ t 1 ∈ r k=1 H(Λ k , d) ≥ 1 -r 2 ε + F (L).
(6.9)

We can now handle the rest of Λ. Initially, we have zeros in Λ \ r k=1 Λ k that are spaced at most v v B(σ 0 ) apart. This threshold was precisely chosen so that we can again use the supercritical contact process to ensure that at time t 0 = B(σ 0 ) 2v , we have:

P σ 0 σ t 0 / ∈ H(Λ \ r k=1 Λ k , d) ≤ Crt 0 e -c √ L .
Again, since t 0 ≤ t 1 , this property persists at time t 1 . Putting this together with equation ( 6 . The fact that all the generic constants c, C used in the proof do not depend on σ 0 and that the CLT used is uniform in the initial conguration guarantees that the bound above is also uniform in σ 0 .

Conclusion

We can now prove Theorem 2.1.

Proof of Theorem 2.1. First, we proved Equation In this case, t 1 = (log L) 9 2v + (log L) 9/4 so we get our result.

If B(σ) > (log L) 9 , then by using Proposition 4.2 with = B(σ) and β = 1/4, we have: Proof. We prove the result in two parts. First, we show that the convergence of the characteristic functions of the ν σ n happens uniformly in σ. Then, we show how it proves the result. In the following, we will denote by Φ σ n and Φ the characteristic functions of ν σ n and µ respectively:

|E σ [f (σ √ L 2v +L 1/4 )]| ≤ C||f || ∞ Le -cB(σ) 1/8 ≤ C||f || ∞ Le -c(log L) 9/
Φ σ n (λ) = e iλx dν σ n (x), Φ(λ) = e -λ 2 /2 . Condition 2. implies: ∀ε > 0, ∀R > 0, ∃n 0 ≥ 0, ∀n ≥ n 0 , ∀σ, ∀|λ| ≤ R, (iλ -x)e iλx dν σ n (x) ≤ ε. The convergence of the characteristic functions is uniform in σ. Note that due to our strong hypothesis, the convergence is local-uniform rather than just pointwise.

We now show that it implies the theorem. For that, we follow a standard proof of Lévy's continuity theorem. Hence, for n large enough, we get |ν σ n (1 (-∞,a] ) -µ(1 (-∞,a] )| ≤ C ε, which concludes the proof.

Figure 3 . 1 :

 31 Figure 3.1: An example of non monotonicity via the standard coupling. The middle site could update in σ -but not in σ + .

Figure 5 . 2 : [ 7 ]

 527 Figure 5.2: [7] Coupling of the evolutions from distinct initial congurations: Steps 1 and 2.

4 2 .P

 2 Then by Proposition 4.1, we nd that for any y ∈ [3v∆ 1 , 3v∆ 1 + σt n+1 = ηt n+1 on [1 -ξ n+1 , y -ξ n+1 -1] | x = y and σsn ηsn ∈ H(0, y, ∆ 2 ) = O(e -c √ ∆ 2 ).

Figure 6 . 1 :

 61 Figure 6.1: A conguration in the set C. Dashed zones represent zones in H(., d). Here, we have p(σ) = 2 and (σ) = Y 1 -X 1 .

Figure 6 . 2 :

 62 Figure 6.2: The expected behaviour of the smallest cluster of σ during time t(σ).

Proof

( 2 . 2 )

 22 with Proposition 6.1 in the previous subsection. Next, let δ > 0, σ ∈ Ω δ L andt 1 (σ) = B(σ) 2v + (B(σ)) 1/4 ∨ (log L) 9 2v + (log L) 9/4 .If B(σ) ≤ (log L)9 , then we use the fact than σ ∈ H(Λ, (log L)9 ) to use Proposition 4.2 with = (log L)9 and β = 1/4. This way, we get that:|E σ [f (σ (log L) 9 2v +(log L) 9/4 )]| ≤ C||f || ∞ Le -c(log L) 9/8 ,which goes to zero as L → ∞ uniformly in σ.

8 ,

 8 which again leads to our result.At last, letΦ : N → N such that Φ(L) -→ L→+∞ +∞. Let σ ∈ H(0, L + 1, Φ(L)) c. Let Λ 1 be the largest particle cluster in σ:

Let ε > 0 ,

 0 R > 0 and x such a n 0 . Then ∀n, ∀|λ| ≤ R, ∀σ, |iλΦ σ n (λ) + i(Φ σ n ) (λ)| ≤ ε, i.e. |λΦ σ n (λ) + (Φ σ n ) (λ)| ≤ ε.By Grönwall's inequality, we get:|Φ σ n (λ) -Φ(λ)| ≤ εe -λ 2 /2 λ 0 e s 2 /2 ds.The function D + (λ) := e -λ 2 /2 λ 0 e s 2 /2 ds being bounded by 1, we nd||Φ σ n -Φ|| [-R,R] ≤ ε, i.e. sup σ ||Φ σ n -Φ|| [-R,R] -→ n→∞ 0.

2 )

 2 Let f ∈ C c (R), and m ∈ R. Let g m (xet f m = f * g m . We will show that supσ |ν σ n (f m ) -µ(f m )| -→ n→∞ 0. f m ∈ L 1 so there exists ϕ m ∈ L 1 such that f m = ϕ m . Then ν σ n (f m ) = R ϕ m (x)dν σ n (x) = R ϕ m (λ)Φ σ n (λ)dλ. Fix ε > 0. First, note that ϕ m (λ)Φ σ n (λ) ≤ ϕ m (λ) and, since ϕ m ∈ L 1 , there exists R > 0 such that ∀σ, R\[-R,R] ϕ m (λ)Φ σ n (λ) ≤ ε. Then: R ϕ m (λ)Φ σ n (λ)dλ -R ϕ m (λ)Φ(λ)dλ = R\[-R,R] ϕ m (λ)(Φ σ n (λ) -Φ(λ))dλ + [-R,R] ϕ m (λ)(Φ σ n (λ) -Φ(λ))dλ ≤ 2ε + C.||Φ σ n -Φ|| [-R,R] .and since g is continuous and bounded, sup σ |ν σ n (g) -µ(g)| -→ n→∞ 0.

  .19) It only remains to chose a d * such that the event F (d N , d * t, t -t N ) has low probability. Let d * = v 4 . Then using t -t N ≤ ∆ and replacing ε and d * by their values, we nd:

  Rk t = {∀s ≤ t, Xk s > -M k + 1}, and Řk t = {∀s ≤ t, Ys > M k -L}.Then, for all intervals B, B ⊂ Λ, we have: We apply this with the intervals I k and J k . The two factors in the right hand side are similar so we are study rst P σk

	P σ X k t ∈ B, Y k t ∈ B and R k t = P σk 0	Xk t ∈ -B and Rk t P σk 0	Y k t ∈ θ -L-1 (B ) and Řk t .
				(6.2)
	0	Xk t ∈ -I k and Rk t . Note that
	P σk

0 Xk t ∈ -I k and Rk t = P σk 0 Xk t ∈ -I k -P σk 0 Xk t ∈ -I k and ( Rk t ) c

  (σ t 1 ∈ H(Λ, d)) ≥ 1 -r 2 ε. (6.10)From there, it only remains to apply Proposition 4.2 with = d and β = 1/4 to nd that for any local function f , at time t := t 1 + (σ t )]| ≤ r 2 ε.

										.9)
	nally gives:
		lim inf
					6a √ 2v	L	+ (6a	√	L) 1/4 ,
			lim sup
	Finally, note that r ≤	vL vB(σ 0 )	≤	v vδ	so we have t ≤	B(σ 0 ) 2v	+	7a √ vδ	L

L→∞ P L→∞ |E[f

Institut Camille Jordan, Université Lyon 1, France. ertul@math.univ-lyon1.fr

Credit to Aurelia Deshayes for the illustrations[START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF] Figures 4 & 5]

Acknowledgments. The author would like to thank Oriane Blondel for her many useful advice and Fabio Toninelli for his careful reading. This project has been supported by the ANR grant LSD (ANR-15-CE40-0020).

. By Lemma 6.2, we see that with t = B(σ) 2v -2a v B(σ), σ) , (6.11) with F (L) going to 0 when L → +∞ uniformly in σ (it is coming from the Central Limit Theorem). This event implies that

which goes to zero when L → ∞.

Putting together Equations (6.11) and (6.12) concludes the proof of Equation (2.3) and thus the proof of Theorem 2.1.

Now we can conclude the proof of the cuto.

Proof of Theorem 2.2. The proof of the lower bound is straightforward by taking initial conguration 1 L and applying Equation (2.3).

Appendix A : An extension of Bolthausen's Lemma

In this appendix we generalize the lemma used in [8, Lemma 2], by handling a family of measures converging to the standard normal law. We show how a "uniform" convergence in the parameter in the hypotheses translates in the distribution convergence. Let (Ω, F, P) be a probability space. Lemma 6.6. Let (ν σ n ) n≥0 be a family of probability measures on R indexed on σ ∈ Ω such that:

Then for all σ ∈ Ω, (ν σ n ) n≥0 converges to the standard normal law. Furthermore, for all continuous bounded function f :

where µ = N (0, 1).

Moreover, for all real numbers a < b,

The RHS tends to zero uniformly in σ. Now, note that:

The convergence ν σ n (f ) -→ µ(f ) does happen uniformly, but for the set of functions:

We will now extend this result to the set C c (R) and then C b (R).

which concludes the proof of (6.13).

It remains to extend to functions of the type 1 [a,b] to get (6.14). It suces to study the case of 1 (-∞,a] . Let ε > 0. We dene two smooth functions f and g such that:

We can estimate:

) -ν σ n (f )|. Since f is continuous and bounded, sup