Olivier Couronn
email: couronne@clipper.ens.fr

POISSON APPROXIMATION FOR LARGE CLUSTERS IN THE SUPERCRITICAL FK MODEL

Keywords: 1991 Mathematics Subject Classification. 60K35, 82B20 FK model, ratio weak mixing

Using the Chen-Stein method, we show that the spatial distribution of large finite clusters in the supercritical FK model approximates a Poisson process when the ratio weak mixing property holds.

Introduction.

We consider here the behaviour of large finite clusters in the supercritical FK model. In dimension two and more, their typical structure is described by the Wulff shape [START_REF] Alexander | Stability of the Wulff minimum and fluctuations in shape for large finite clusters in two-dimensional percolation[END_REF][START_REF] Alexander | Cube-root boundary fluctuations for droplets in random cluster models[END_REF][START_REF] Alexander | The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation[END_REF][START_REF] Bodineau | The Wulff construction in three and more dimensions[END_REF][START_REF] Cerf | Large deviations for three-dimensional supercritical percolation[END_REF][START_REF] Cerf | On the Wulff crystal in the Ising model[END_REF][START_REF] Cerf | Phase coexistence in Ising, Potts and percolation models[END_REF]. An interesting issue is the spatial distribution of these large finite clusters. Because of their rarity, a Poisson process naturally comes to mind. Indeed, we prove that the point process of the mass centers of large finite clusters sharply approximates a Poisson process. Furthermore, considering large finite clusters in a large box such that their mean number is not too large, we observe Wulff droplets distributed according to this Poisson process.

Redig and Hostad have recently studied the law of large finite clusters in a given box [START_REF] Van Der Hofstad | Maximal clusters in non-critical percolation and related models[END_REF]. Their aim was different, in that they obtained accurate estimates on the law of the maximal cluster in the box, but the intermediate steps are similar. In the supercritical regime they considered only Bernoulli percolation and not FK percolation.

As in [START_REF] Abadi | Exponential distribution for the occurrence of rare patterns in Gibbsian random fields[END_REF][START_REF] Chazottes | Occurrence, repetition and matching of patterns in the lowtemperature Ising model[END_REF][START_REF] Ferrari | Poisson approximation for large-contours in low-temperature Ising models[END_REF][START_REF] Van Der Hofstad | Maximal clusters in non-critical percolation and related models[END_REF], our main result is based on a second moment inequality. We have to control the interaction between two clusters. For this, we suppose that ratio weak mixing holds [START_REF] Alexander | On weak mixing in lattice models[END_REF]. This property allows us to apply the Chen-Stein method in order to get the approximation by a Poisson process.

The ratio weak mixing holds in dimension two as soon as dual connectivities are exponentially decreasing [START_REF] Alexander | On weak mixing in lattice models[END_REF]. For dimensions at least three, we prove that ratio weak mixing holds for p close enough to 1. Hence our main results are valid in all dimensions for p large enough.

The following section is devoted to the statement of our results. In section 3, we define the FK model. We recall the weak and the ratio weak mixing properties and we state a perturbative mixing result in section 4. Section 5 contains the definition of our point process and the description of the Chen-Stein method. The core of the article is section 6, where we study a second moment inequality. In section 7, we deal with the probability of having a large finite cluster with its center at the origin. In section 8, we treat the case of distant clusters and we finish the proof of Theorem 1. The proof of Theorem 3 is done in section 9, and the proof of the perturbative mixing result is done in section 10.

2. Statement of the results. We consider the FK measure Φ on the d-dimensional lattice Z d and in the supercritical regime. The point p c stands for p g in dimension two, and for p slab c in dimensions three and more. For q ≥ 1 we let U(q) be the set such that there exists a unique FK measure on Z d of parameters p and q if p is not in U(q). By [START_REF] Grimmett | The random cluster model[END_REF] this set is at most countable.

Let Λ be a large box in Z d . We fix n an integer and we consider the finite clusters of cardinality larger than n. We call them n-large clusters. Let C be a finite cluster. The mass center of C is

M C = 1 |C| x∈C x ,
where ⌊x⌋ denotes the site of Z d whose coordinates are the integer part of those of x. We define a process X on Λ by

X(x) = 1 if x is the mass center of a n-large cluster C 0 otherwise.
Let λ be the expected number of sites x in Λ such that X(x) = 1. We denote by L(X) the law of a process X. For Y a process on Λ, we let ||L(X) -L(Y )|| T V be the total variation distance between the laws of the processes X and Y [START_REF] Arratia | Two moments suffice for Poisson approximations: The Chen-Stein method[END_REF].

Theorem 1. Let q ≥ 1 and p > p c with p / ∈ U(q). Let Φ be the FK measure on Z d of parameters p and q. We suppose that Φ is ratio weak mixing. There exists a constant c > 0 such that: for any box Λ, letting X be defined as above, and letting Y be a Bernoulli process on Λ with the same one-dimensional marginals as X, we have for n large enough

||L(X) -L(Y )|| T V ≤ λ exp(-cn (d-1)/d ).
As a corollary, the number of large clusters in Λ is approximated by a Poisson variable.

Corollary 2. Let Φ be as in Theorem 1. Let N be the number of large finite clusters whose mass centers are in the box Λ. Let Z be a Poisson variable of mean λ, and let c > 0 be the same constant as in Theorem 1. Then for any A ⊂ Z + and for n large enough,

|P (N ∈ A) -P (Z ∈ A)| ≤ λ exp -cn (d-1)/d .
We provide next a control of the shape of the large finite clusters. Here we consider a sequence of boxes (Λ n ) n . If the size of Λ n is not too large, that is of order less than exp(ρn (d-1)/d ) for a certain constant ρ, then the energy created by 2

the n-large clusters of Λ n dominates a term of entropy. In this case we can assert that the shape of these n-large clusters are close to the Wulff shape. More precisely, let W be the Wulff crystal, let θ be the density of the infinite cluster, and let L d (•) be the Lebesgue measure on R d . Let

W = 1 θL d (W) 1/d W
be the renormalized Wulff crystal. For l > 0, let V ∞ (C, l) be the neighbourhood of C of width l for the metric | • | ∞ . For two sets A and B, the notation A △ B stands for the symmetric difference between A and B.

Theorem 3. Let Φ be as in Theorem 1. Let f : N → N be such that f (n)/n → 0 and f (n)/ ln n → ∞ as n goes to infinity. Let (Λ n ) n be a sequence of boxes in Z d , and let λ n be the expected number of mass centers of n-large clusters in Λ n . For all δ > 0, there exists c > 0 such that if lim sup 1/n (d-1)/d ln λ n ≤ c.

lim sup n→∞ 1 n (d-1)/d ln Φ L d x∈Λn X(x)=1 (x + W ) △ n -1 C n-large MC ∈Λn V ∞ (C, f (n)) ≥ δ {x : X(x) = 1} < 0.
For clarity, we omit the subscript n on X. Remark : Consider a sequence (Λ n ) n such that |Λ n | ≃ exp(ρn (d-1)/d ) and let w 1 > 0 be such that [START_REF] Cerf | The Wulff crystal in Ising and Percolation models, Saint-Flour lecture notes[END_REF]:

P (n ≤ |C(0)| < ∞) ≈ exp(-w 1 n (d-1)/d ).
On the one hand we need ρ ≥ w 1 in order to have some n-large clusters in Λ n . On the other hand the condition on λ n in theorem 3 may be rewritten as ρ ≤ c + w 1 .

The ratio weak mixing property is a key hypothesis in our results. The following proposition allow us to apply the three preceding results for p large enough in all dimensions.

Proposition 4. Let d ≥ 3 and q ≥ 1. There exists p 0 < 1 such that Φ satisfies the ratio weak mixing property for p > p 0 .

FK model.

We consider the lattice Z d with d ≥ 2. We turn it into a graph by adding bonds between all pairs x, y of nearest neighbours. We write E for the set of bonds and we let Ω be the set {0, 1} E . A bond configuration ω is an element of Ω. A bond e is open in ω if ω(e) = 1, and closed otherwise.

A path is a sequence (x 0 , . . . , x n ) of distinct sites such that x i , x i+1 is a bond for each i, To define the FK measure, we first consider finite volume FK measures. Let Λ be a box included in Z d . We write E(Λ) for the set of bonds x, y with x, y ∈ Λ.

0 ≤ i ≤ n -1. A subset ∆ of Z d is connected if for every x,
Let Ω Λ = {0, 1} E(Λ) be the space of bonds configuration in Λ. Let F Λ be its σ-field, that is the set of subsets of Ω Λ . For ω in Ω Λ , we define cl(ω) as the number of clusters of the configuration ω.

For p ∈ [0, 1] and q ≥ 1, the FK measure in Λ with parameters p, q and free boundary condition is the probability measure on Ω Λ defined by

∀ ω ∈ Ω Λ Φ f,p,q Λ (ω) = 1 Z f,p,q Λ e∈E(Λ) p ω(e) (1 -p) 1-ω(e) q cl(ω) ,
where Z f,p,q Λ is the appropriate normalization factor. We also define FK measures for arbitrary boundary conditions. For this, let ∂Λ be the boundary of Λ,

∂Λ = {x ∈ Λ such that ∃ y / ∈ Λ, x, y is a bond}.
For a partition π of ∂Λ, a π-cluster is a cluster of Λ when we add open bonds between the pairs of sites that are in the same class of π. Let cl π (ω) be the number of π-clusters in ω. To define Φ π,p,q Λ we replace cl(ω) by cl π (ω) and Z f,p,q Λ by Z π,p,q Λ in the above formula.

There exists a countable subset U(q) in [0, 1] such that the following holds. As Λ grows and invades the whole lattice Z d , the finite volume measures converge weakly toward the same infinite measure Φ p,q ∞ for all p / ∈ U(q) [START_REF] Grimmett | The random cluster model[END_REF]. We will always suppose that this occurs, that is p / ∈ U(q). We shall drop the superscript and the subscript on Φ p,q ∞ , and simply write Φ. It is known that the FK measure Φ is translation-invariant.

The measure Φ verify the finite energy property: for each p in (0, 1), there exists δ > 0 such that for every finite-dimensional cylinders ω 1 and ω 2 that differ by only one bond, [START_REF] Abadi | Exponential distribution for the occurrence of rare patterns in Gibbsian random fields[END_REF] Φ(ω 1 )/Φ(ω 2 ) ≥ δ.

The random cluster model has a phase transition. There exists p c ∈ (0, 1) such that there is no infinite cluster Φ-almost surely if p < p c , and an infinite cluster Φ-almost surely if p > p c . Other critical points have been introduced in order to work with 'fine' properties. In dimension two, we define p g as the critical point for the exponential decay of dual connectivities, see [START_REF] Couronné | Surface order large deviations for 2D FK-percolation and Potts models[END_REF][START_REF] Grimmett | The random cluster model[END_REF]. In three and more dimensions, let p slab c be the limit of the critical points for the percolation in slabs [START_REF] Pisztora | Surface order large deviations for Ising, Potts and percolation models[END_REF]. For brevity, p c will stand for p g in dimension two, and for p slab c in dimensions three and more. It is believed that p c = p c in all dimensions and for all q ≥ 1, but in most cases we know only that p c ≥ p c .

We now state Theorem 17 of [START_REF] Cerf | The Wulff crystal in Ising and Percolation models, Saint-Flour lecture notes[END_REF], applied to FK measures. If q ≥ 1, p > p c and p / ∈ U(q), there exists

w 1 > 0 such that (2) lim 1 n (d-1)/d ln Φ n ≤ |C(0)| < ∞ = -w 1 ,
where C(0) is the cluster of the origin.

Mixing properties.

Let x and y be two points in Z d and let (x i ) d i=1 and (y i ) d i=1 be their coordinates. Write |x -y| 1 = d i=1 |x iy i |. Definition 5. Following [START_REF] Alexander | Mixing properties and exponential decay for lattice systems in finite volumes[END_REF], we say that Φ satisfies the weak mixing property if for some c, µ > 0, for all sets Λ, ∆ ⊂ Z d ,

(3) sup Φ(E | F ) -Φ(E) : E ∈ F Λ , F ∈ F ∆ , Φ(F ) > 0 ≤ c x∈Λ,y∈∆ e -µ|x-y| 1 .
Definition 6. Following [START_REF] Alexander | Mixing properties and exponential decay for lattice systems in finite volumes[END_REF], we say that Φ satisfies the ratio weak mixing property for some

c 1 , µ 1 > 0, for all sets Λ, ∆ ⊂ Z d , (4) sup Φ(E ∩ F ) Φ(E)Φ(F ) -1 : E ∈ F Λ , F ∈ F ∆ , Φ(E)Φ(F ) > 0 ≤ c 1 x∈Λ,y∈∆ e -µ 1 |x-y| 1 ,
Roughly speaking, the influence of what happens in ∆ on the state of the bonds in Λ decreases exponentially with the distance between Λ and ∆.

In dimension two, the measure Φ is ratio weak mixing as soon as p > p c [START_REF] Alexander | Mixing properties and exponential decay for lattice systems in finite volumes[END_REF], but such a result is not available in dimension larger than three. We provide a perturbative mixing result, which is valid for all dimensions larger than three, and which is similar to the weak mixing property. Lemma 7. Let d ≥ 3 and q ≥ 1. There exists p 1 < 1 and c > 0 such that: for all p > p 1 , all connected sets Γ, ∆ with Γ ⊂ ∆, every boundary conditions η, ξ on ∆, every event E supported on Γ,

|Φ η,p,q ∆ (E) -Φ ξ,p,q ∆ (E)| ≤ 2|∂∆| exp -c inf |x -y| 1 , x ∈ Γ, y ∈ ∂∆ .
We are not aware of a particular reference for this result, and we give a sketch of the proof in Section 10.

The Chen-Stein method.

From the percolation process, we want to extract a point process describing the occurrence of large finite clusters. For a point x in R d , let ⌊x⌋ denotes the site of Z d whose coordinates are the integer parts of those of x. Assume that C is a finite subset of Z d . Then the mass center of C is

M C = 1 |C| x∈C x .
Let n ∈ N. A n-large cluster is a finite cluster of cardinality larger than n. Let Λ be a box in Z d . We define a process X on Λ by

X(x) = 1 if
x is the mass center of a n-large cluster C 0 otherwise.

In order to apply the Chen-Stein method, we define for x, y in Z d , p x = Φ(X(x) = 1),

p xy = Φ ∃ C, C ′ two clusters such that: C ∩ C ′ = ∅, n ≤ |C|, |C ′ | < ∞, M C = x and M C ′ = y ,
and we let B x = B(x, n 2 ) be the box centered at x of side length n 2 . Let λ be the expected number of sites x in Λ such that X(x) = 1. We have λ = x∈Λ p x and, because of the translation-invariance of Φ, for each site x in Λ

(5)

λ = |Λ| • p x .
We introduce three coefficients b 1 , b 2 , b 3 by:

b 1 = x∈Λ y∈Bx p x p y , b 2 = x∈Λ y∈Bx\x p xy , b 3 = x∈Λ E E X(x) -p x |σ(X(y), y / ∈ B x .
Let Z 1 and Z 2 be two Bernoulli processes on Λ. The total variation distance between the laws of the processes Z 1 and Z 2 [START_REF] Arratia | Two moments suffice for Poisson approximations: The Chen-Stein method[END_REF] is

||L(Z 1 ) -L(Z 2 )|| T V = 2 sup P (Z 1 ∈ A) -P (Z 2 ∈ A) , A subset of {0, 1} Λ .
Let Y be a Bernoulli process on Λ such that the Y (x)'s are iid and

P (Y (x) = 1) = p x .
The Chen-Stein method provides a control of the total variation distance between X and Y in terms of the b i 's. Indeed we apply Theorem 3 of [START_REF] Arratia | Two moments suffice for Poisson approximations: The Chen-Stein method[END_REF] to obtain that ( 6)

||L(X) -L(Y )|| T V ≤ 2(2b 1 + 2b 2 + b 3 ) + 4 x∈Λ p 2 x .
To prove Theorem 1, we shall provide an upper bound on each term b i . The ratio weak mixing property is essential to our proof of the bound of b 2 . Nevertheless, we believe that one can prove the following inequality, without any mixing assumption:

(7) Φ n ≤ C(x) < ∞, n ≤ C(y) < ∞, C(x) ∩ C(y) = ∅ ≤ Φ(2n ≤ C(0) < ∞).
Let us give now an upper bound on p x . By [START_REF] Grimmett | Percolation. Second Edition[END_REF], there exists a constant c > 0 such that:

Φ(n ≤ |C(0)| < ∞) ≤ exp -cn (d-1)/d . 6
But

p x ≤ k≥n Φ ∃ C, |C| = k, M C = x ≤ k≥n y∈B(x,2k) Φ |C(y)| = k ≤ k≥n (2k) d exp -ck (d-1)/d .
Hence there exists a constant c > 0 such that for n large enough (8) p x ≤ exp(-cn (d-1)/d ).

6. Second moment inequality. In this section we bound the term p xy with the help of the ratio weak mixing property. First we introduce a local version of p xy . We define p xy by

p xy = Φ ∃ C, C ′ two clusters such that n ≤ |C| < n 2 , n ≤ |C ′ | < n 2 , M C = x, and M C ′ = y .
The distance between two sets Γ and ∆ ⊂ Z d is

d(Γ, ∆) = inf{|x -y| 1 , x in Γ, y in ∆},
and it is the length of the shortest path connecting Γ to ∆. We divide the term p xy into two parts. Let µ 1 be the constant appearing in the definition of the ratio weak mixing property and let K > 5/µ 1 . We define p c xy by First we focus on p d xy . We have

p c xy = Φ ∃ C, C ′ two clusters such that d(C, C ′ ) ≤ K ln n, n ≤ |C| < n 2 , n ≤ |C ′ | < n 2 , M C = x,
p d xy ≤ C,C ′ distant Φ(C and C ′ are clusters),
where the sum is over the couples (C, C ′ ) of connected subsets of Z d such that

n ≤ |C| < n 2 , n ≤ |C ′ | < n 2 , M C = x, M C ′ = y, and d(C, C ′ ) > K ln n.
Let c 1 , µ 1 be the constants appearing in the definition of the ratio weak mixing property. Let (C, C ′ ) be a couple appearing in the sum above. We have u∈C,v∈C ′ e -µ1|u-v| ≤ n 4 exp(-µ 1 K ln n), so for n large enough c 1 u∈C,v∈C ′ e -µ1|u-v| ≤ 1.

So for n large enough Φ(C and

C ′ are clusters) ≤ 2Φ(C is a cluster) • Φ(C ′ is a cluster),
by the ratio weak mixing property (4). Hence there exists c > 0 such that for n large enough ( 9)

p d xy ≤ u∈B(x,2n 2 ),v∈B(y,2n 2 ) 2Φ(n ≤ |C(u)| < ∞) • Φ(n ≤ |C(v)| < ∞) ≤ exp(-cn (d-1)/d ).
Now we consider p c xy . We have

p c xy ≤ C,C ′ close Φ(C and C ′ are clusters),
where the sum is over the couples (C, C ′ ) of subsets of Z d such that

n ≤ |C| < n 2 ,n ≤ |C ′ | < n 2 , M C = x, M C ′ = y, and d(C, C ′ ) ≤ K ln n.
For n large enough, the event {C and C ′ are clusters} is F B(x,3n 2 ) -measurable. So we only consider bond configurations in B(x, 3n 2 ). We give a deterministic total order on the pairs (u, v) of Z d in such a way that if

|u 1 -v 1 | 1 < |u 2 -v 2 | 1 , then (u 1 , v 1 ) < (u 2 , v 2 )
. Let (C, C ′ ) be a pair of sets appearing in the above sum. Take a configuration ω in B(x, 3n 2 ) such that C and C ′ are clusters in ω. We change the configuration ω as follows.

To start with, we take the pair (u, v) such that u ∈ C, v ∈ C ′ and (u, v) is the first such pair for the order above. For 0 ≤ i ≤ d, we define t i the point whose di first coordinates are equal to those of u, and the others are equal to those of v. Hence t 0 = u, t d = v, and t i and t i+1 differ by only one coordinate. We consider the shortest path (u 0 , . . . , u k ) connecting u to v through the t

i 's. It is composed of the segments [t i , t i+1 ] for 0 ≤ i ≤ d -1.
We open all the bonds u i , u i+1 for i = 0 . . . k -1. In the same time, we close all the bonds incident to u i for i = 1 . . . k -1 distinct from the previous bonds u j , u j+1 . Let ω be the new configuration in B(x, 3n 2 ). We denote by C the set

C ∪ C ′ ∪ {u i } k-1 i=1 . By construction, C is a cluster in ω. We have 2n ≤ C < 4n + K ln n.
The number of bonds we have changed is bounded by 2dK ln n. By the finite energy property (1): Φ( ω) ≥ n 2dK ln δ Φ(ω), for a certain constant δ in (0, 1). Now we control the number of antecedents by our transformation. Take a configuration ω of B(x, 3n 2 ). To get an antecedent of ω, we have to (a) choose two sites u, v in B(x, 3n 2 ), with |u -v| 1 ≤ K ln n (b) take the path connecting u to v along the coordinate axis (c) choose the state of the bonds that have an endpoint on this path. In step (a), we have less than (3n 2 ) d (2K ln n) d choices. In step (b) we have just one choice. In step (c) the number of choices is bounded by 2 2dK ln n . Hence for n large enough the number of antecedents of ω is bounded by n 4dK .

Finally,

C,C ′ close Φ(C and C ′ are clusters) ≤ n 4dK • n 2dK ln δ C Φ( C is a cluster),
where the sum is over connected subsets C of Z d such that 2n ≤ | C| < 5n and C is contained in B(x, 3n 2 ). This sum is bounded by

|B(x, 3n 2 )| • Φ(2n ≤ |C(0)| < 5n).
Thus by ( 2), there exists c 2 > w 1 such that for n large enough,

p c xy ≤ exp(-c 2 n (d-1)/d ). (10)
To conclude, remark that

p xy -p xy ≤ Φ ∃ C a cluster such that n 2 ≤ |C| < ∞, M C = x .
By [START_REF] Bodineau | The Wulff construction in three and more dimensions[END_REF], there exists c such that for n large enough the difference between p xy and p xy is bounded by exp(-cn 2(d-1)/d ). So by [START_REF] Cerf | Large deviations for three-dimensional supercritical percolation[END_REF] there exists c > 0 such that p xy ≤ p c xy + exp(-cn). Since in [START_REF] Cerf | On the Wulff crystal in the Ising model[END_REF] the constant c 2 is strictly larger than w 1 , there exists c 3 > w 1 such that for n large enough [START_REF] Cerf | Phase coexistence in Ising, Potts and percolation models[END_REF] p xy ≤ exp(-c 3 n (d-1)/d ).

7. A control of p x . We compare p x and Φ(n ≤ |C(0)| < ∞).

Lemma 8. If q ≥ 1, p > p c , and p / ∈ U(q), then

lim 1 n (d-1)/d ln p x = -w 1 .
We note that in [START_REF] Van Der Hofstad | Maximal clusters in non-critical percolation and related models[END_REF], the authors take the left endpoints of the clusters instead of the mass center and get the same limit.

Proof of Lemma 8. We begin with a lower bound for p x . We recall that for all x in Z d , p x = Φ(X(0) = 1). Let α > 1. Because of (2), we have

lim 1 n (d-1)/d ln Φ(n ≤ |C(0)| < ∞) = lim 1 n (d-1)/d ln Φ(n ≤ |C(0)| < n α ). Then Φ(n ≤ |C(0)| < n α ) ≤ x∈B(0,n α ) Φ(n ≤ |C(0)| < n α , M C = x) ≤ |B(0, n α )|Φ(X(0) = 1).
We give next an upper bound:

Φ(X(0) = 1) = Φ(∃C a cluster, M C = 0, n ≤ |C| < n α ) + Φ(∃C a cluster, M C = 0, n α ≤ |C| < ∞) ≤ x∈B(0,n α ) Φ(n ≤ |C(x)| < ∞) + k≥n α Φ ∃C a cluster, |C| = k, C ∩ B(0, 2k) = ∅ ≤ |B(0, n α )|Φ(n ≤ |C(x)| < ∞) + k≥n α |B(0, 2k)|Φ(|C(0)| = k).
Finally, we use the limit (2) to get

lim 1 n (d-1)/d ln p x = lim 1 n (d-1)/d ln Φ(n ≤ |C(0)| < ∞) = -w 1 .

Proof of Theorem 1.

We recall that Λ is a box and λ is the expected number of the mass centers in Λ of n-large clusters. We write F Bx Λ for the σ-field F Λ\Bx . First, we bound the term

E E X(x) -p x |F Bx Λ .
Let X(x) be equal to 1 if x is the mass center of a cluster C, with C such that n ≤ |C| < n 2 /4, and equal to 0 otherwise. Let p x = Φ( X(x)). We have

(12) E E X(x) -p x |F Bx Λ ≤ E E X(x) -X(x)|F Bx Λ +E E X(x) -p x |F Bx Λ + E E p x -p x |F Bx Λ .
Since the quantity X(x) -X(x) is always positive,

E E X(x) -X(x)|F Bx Λ = E E X(x) -X(x)|F Bx Λ = p x -p x .
We have also

E E p x -p x |F Bx Λ = p x -p x . But p x -p x = Φ(∃ C a cluster, n 2 /4 ≤ |C| < ∞, M C = x),
so by [START_REF] Bodineau | The Wulff construction in three and more dimensions[END_REF] there exists c > 0 such that p xp x ≤ exp(-cn 2 ). The variable X(x) is F B(x,n 2 /4) -measurable. The distance between B(x, n 2 /4) and the complementary region of B x is of order n 2 . If Φ is weak mixing, or by lemma 7 if p is close enough to 1, there exists a constant c > 0 such that for n large enough

E E X(x) -p x |F Bx Λ ≤ exp(-cn 2 ).
Putting together the estimates of the three terms on the right-hand side of ( 12), we conclude that there exists c > 0 such that for n large enough ( 13)

E E X(x) -p x |F Bx Λ ≤ exp(-cn 2 ).
Now observe that |Λ| = λp -1

x . Using inequality [START_REF] Cerf | Phase coexistence in Ising, Potts and percolation models[END_REF] and the limit of Lemma 8, there exists c > 0 such that

b 2 ≤ λp -1 x exp -c 3 n (d-1)/d ≤ λ exp -cn (d-1)/d .
Because of ( 13), there exists c > 0, c ′ > 0 such that

b 3 ≤ λp -1 x exp(-cn 2 ) ≤ λ exp(-c ′ n 2 ).
The term b 1 is controlled by Lemma 8. We apply finally the Chen-Stein inequality [START_REF] Alexander | The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation[END_REF] to obtain Theorem 1.

Proof of Theorem 3.

The Wulff crystal is the typical shape of a large finite cluster in the supercritical regime. The crystal is built on a surface tension τ . The surface tension is a function from S d-1 , the (d -1)-dimensional unit sphere of R d , to R + . It controls the exponential decay of the probability for having a large separating surface in a certain direction, with all bonds closed. We refer the reader to [START_REF] Cerf | Large deviations for three-dimensional supercritical percolation[END_REF][START_REF] Cerf | The Wulff crystal in Ising and Percolation models, Saint-Flour lecture notes[END_REF] for an extended survey of this function.

In the regime p > p c and p / ∈ U(q), the surface tension is positive, continuous, and satisfies the weak simplex inequality. We denote by W the Wulff shape associated to τ , W = {x ∈ R d , x.u ≤ τ (u) for all u in S d-1 }.

The Wulff shape is a main ingredient in the proof of [START_REF] Alexander | On weak mixing in lattice models[END_REF]. Let θ = Φ(0 ↔ ∞) be the density of the infinite cluster. Let f : N → N, such that f (n)/n → 0 and f (n)/ ln n → ∞ as n goes to infinity. Let x and y be two points of R d , and let (x i ) d i=1 and (y i ) d i=1 be their coordinates. We write |x -y| ∞ = max 1≤i≤d |x iy i |. We define a neighbourhood of a cluster C by

V ∞ (C, f (n)) = {x ∈ R d , ∃ y ∈ C, |x -y| ∞ ≤ f (n)}.
Let (Λ n ) n be a sequence of boxes in Z d , and let λ n be the expected number of mass centers of n-large clusters in Λ n . In Theorem 3, we consider the event ( 14)

L d x∈Λn X(x)=1 (x + θL d (W) -1/d W △ n -1 C n-large MC ∈Λn V ∞ (C, f (n)) ≥ δ {x : X(x) = 1} . It is included in the event there exists C a n-large cluster such that M C ∈ Λ n , L d M C + θL d (W) -1/d W △ n -1 V ∞ (C, f (n)) ≥ δ .
Taking the logarithm of its probability and dividing by n (d-1)/d , we may show that for n large it is equivalent to the logarithm divided by n (d-1)/d of the following quantity:

λ n Φ L d M C(0) + θL d (W) -1/d W △ n -1 V ∞ (C(0), f (n)) ≥ δ n ≤ |C(0)| < ∞ .
By [START_REF] Cerf | Large deviations for three-dimensional supercritical percolation[END_REF][START_REF] Cerf | The Wulff crystal in Ising and Percolation models, Saint-Flour lecture notes[END_REF], there exists c > 0 such that if lim sup 1/n (d-1)/d ln λ n ≤ c, then the inequality in Theorem 3 holds.

A perturbative mixing result.

First we prove lemma 7, following the proof of the uniqueness of the FK measure for p close enough to 1 in [START_REF] Grimmett | The stochastic random-cluster process and the uniqueness of random-cluster measures[END_REF]. The difference is that we consider not just one but two independent FK measures. The idea of using two independent copies of a measure comes from [START_REF] Georgii | The random geometry of equilibrium phases, Phase Transit[END_REF]. Then the proof of proposition 4 follows.

Proof of lemma 7.

Let ∆ be a connected subset of Z d . There is a partial order in Ω ∆ given by ω ω ′ if and only if ω(e) ≤ ω ′ (e) for every bond e. A function f

: Ω ∆ → R is called increasing if f (ω) ≤ f (ω ′ ) whenever ω ω ′ .
An event is an element of Ω ∆ . An event is called increasing if its characteristic function is increasing. For a pair of probability measures µ and ν on (Ω ∆ , F ∆ ), we say that µ (stochastically) dominates ν if for any F ∆ -measurable increasing function f the expectations satisfy µ(f ) ≥ ν(f ) and we denote it by µ ν. Let P p be the Bernoulli bond-percolation measure on Z d of parameter p. The FK measures on ∆ dominate stochastically a certain Bernoulli measure restricted on E(∆):

(15) Φ η,p,q ∆ P p/[p+q(1-p)] E(∆) .
For (ω 1 , ω 2 ) ∈ Ω 2 , we call a site x white if ω 1 (e)ω 2 (e) = 1 for all bond e incident with x, and black otherwise. We define a new graph structure on Z d . Take two sites x and y and label x i , y i their coordinates. If max i=1...d |x iy i | = 1, then x, y is a ⋆-bond and y is a ⋆-neighbour of x. A ⋆-path is a sequence (x 0 , ..., x n ) of distinct sites such that x i , x i+1 is a ⋆-bond for 0 ≤ i ≤ n -1.

For any set V of sites, the black cluster B(V ) is the union of V together with the set of all x 0 for which there exists a ⋆-path x 0 , . . . , x n such that x n ∈ V and x 0 , . . . , x n-1 are all black. Let Γ, ∆ be two connected sets with Γ ⊂ ∆. The 'interior boundary' D(B(∂∆)) of B(∂∆) is the set of sites x satisfying:

(a) x / ∈ B(∂∆) (b) there is a ⋆-neighbour of x in B(∂∆) (c) there exists a path from x to Γ that does not use a site in B(∂∆). Let I be the set of sites x 0 for which there exists a path x 0 , . . . , x n with x n ∈ Γ, x i / ∈ B(∂∆) for all i, see figure 1. Pick η, ξ two boundary conditions of ∆. For brevity let P = Φ η,p,q ∆ × Φ ξ,p,q ∆ . We shall write X, Y for the two projections from Ω ∆ × Ω ∆ to Ω ∆ . Then for any E ∈ F Γ , we have by the claims above P(X ∈ E, K Γ,∆ ) = P(Y ∈ E, K Γ,∆ ) = P(Φ w,p,q I (E)1 KΓ,∆ ).

Hence

|Φ η,p,q ∆ (E) -Φ ξ,p,q ∆ (E)| ≤ 1 -P(K Γ,∆ ) .

Because of inequality [START_REF] Ferrari | Poisson approximation for large-contours in low-temperature Ising models[END_REF] and by the stochastic domination result in [START_REF] Liggett | Domination by product measures[END_REF], the process of black sites is stochastically dominated by a Bernoulli site-percolation process whose parameter is independent of Γ, ∆, η, ξ and decreases to 0 as p goes to 1. There exists p 1 < 1 such that this Bernoulli process is subcritical for the ⋆-graph structure of Z d and for p ≥ p 1 . Hence there exists c > 0 such that for p > p 1 , for all Γ, ∆, η, ξ,

P(K Γ,∆ ) ≥ 1 -|∂∆| exp -c d(Γ, ∂∆) .
Proof of proposition 4. The domination inequality [START_REF] Ferrari | Poisson approximation for large-contours in low-temperature Ising models[END_REF] implies that for p large enough, the mesures Φ η,p,q ∆ have exponentially bounded controlling regions in the terminology of [START_REF] Alexander | On weak mixing in lattice models[END_REF]. Thus by theorem 3.3 of [START_REF] Alexander | On weak mixing in lattice models[END_REF], lemma 7 implies the ratio weak mixing property for the mesures Φ η,p,q ∆ .

  y in ∆, there exists a path included in ∆ connecting x and y. If all bonds of a path are open in ω, we say that the path is open in ω. A cluster is a connected component in Z d when we keep only open bonds. It is usually denoted by C. Let x be a site. We write C(x) for the cluster containing x.

  and M C ′ = y .We define also p d xy byp xy = Φ ∃ C, C ′ two clusters such that d(C, C ′ ) > K ln n, n ≤ |C| < n 2 , n ≤ |C ′ | < n 2 , M C = x,and M C ′ = y . The superscripts c and d stand for close and distant. So p xy = p c xy + p d xy and we study separately these two terms.

figure 1 :

 1 figure 1: The set I inside ∆