David Dereudre

Thibaut Vasseur

Existence of Gibbs point processes with stable infinite range interaction

Keywords: DLR equations, entropy bounds, superstable interaction

Existence of Gibbs point processes with stable infinite range interaction

Introduction

The Gibbs point processes constitute a large class of point processes with interaction between the points. The interaction can be attractive or repulsive, depending on geometrical features, whereas the null interaction is associated to the so-called Poisson point process. The existence of such processes in the infinite volume regime has a long history and is initially related to the existence of thermodynamic behaviours in statistical physics. Now the Gibbs point processes are used in several other applied sciences such as material science, astronomy, epidemiology, plant ecology, seismology, telecommunications, and others. Therefore their existence in the infinite volume regime is also relevant for spatial statistics considerations. In the present paper we give a new proof of the existence of such processes for a large class of infinite range interactions.

The starting point of the theory is an energy function H defined on the space of locally finite configurations in R d . In the following, ω denotes such a point configuration and H(ω) its energy. Then the finite volume Gibbs measure on a bounded set Λ ⊂ R d is simply the probability measure

P Λ = 1 Z Λ e -H π Λ ,
where π Λ is the Poisson point process in Λ with intensity one and Z Λ the normalization constant. The existence of P Λ is guaranteed by the stability condition recalled below. The existence of an infinite volume measure, corresponding to the case "Λ = R d ", is not obvious and can not be achieved by the definition above. In fact, the general strategy is first to obtain a suitable thermodynamic limit for (P Λ ) when Λ tends to R d and then derive a good description of the limiting point by the so-called DLR equations. The first general result in this direction is due to Ruelle in the seventies [START_REF] Ruelle | Superstable interactions in classical statistical mechanics[END_REF][START_REF] Ruelle | Statistical mechanics: Rigorous results[END_REF]. The setting was the pairwise interaction;

H(ω) = x =y∈ω φ(|x -y|),
where the potential φ is assumed regular, which means roughly that φ is summable at infinity (i.e.

+∞ R r d-1 |φ(r)|dr < +∞ for R > 0 large enough) and superstable: there exists two constants A > 0 and B ≥ 0 such that for any bounded set Λ ⊂ R d large enough and for any finite configuration ω in Λ

H(ω) ≥ -B|ω| + A λ d (Λ) |ω| 2 ,
where |ω| denotes the number of points in ω and λ d the Lebesgue measure on R d . Under these two assumptions, Ruelle proved the existence of at least one infinite Gibbs point processes. Similar results has been proved more recently in [START_REF] Kondratiev | Existence of Gibbs state for a non-ideal gas in R d : the case of a pair, long-range interaction[END_REF][START_REF] Kondratiev | Gibbs measures of continuous systems: an analytic approach[END_REF] using functional analysis tools. In any case the superstability assumption is required. As a corollary of our main result we improve these existence results in substituting the superstability assumption by the Stability assumption: there exists a constant B ≥ 0 such that for any finite configuration ω H(ω) ≥ -B|ω|.

Let us note that the difference between stable and superstable potential is in fact weak since any stable potential becomes superstable if a pairwise continuous non-negative and non-null at origin potential is added. However there exists several pairwise stable potential which are not superstable. For instance any continuous, non-negative pairwise potential null at origin is stable without being superstable. Let us note also that several examples of stable (and non-superstable) energy functions have introduced recently in stochastic geometry and spatial statistics [START_REF] Baddeley | Area-interaction point processes[END_REF][START_REF] Dereudre | Existence of Gibbsian point processes with geometry-dependent interactions[END_REF][START_REF] Møller | Power diagrams and interaction processes for unions of discs[END_REF].

A multi-body interaction occurs when the energy function H is decomposed using potentials on pairs, triplets, quadruplets, k-uplets of points (but not only on pairs). Our results do not use and do not depend on such decompositions. Therefore our existence result covers several multi-body interactions, including infinite range cases (examples are given in Section 3). Let us mention that the finite range multi-body interaction have been treated in [START_REF] Belitsky | Uniqueness of Gibbs state for non-ideal gas in R d : the case of multibody interaction[END_REF] using the Dobrushin's criterium.

Our main tool is the compactness of entropy level sets for the local convergence topology. This tool is particularly efficient for proving the tightness of the sequence of finite volume Gibbs point processes. It has been used for the first time in [START_REF] Georgii | Phase transition in continuum Potts models[END_REF] and then a collection of papers followed [START_REF] Dereudre | The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains[END_REF][START_REF] Dereudre | Existence of Gibbsian point processes with geometry-dependent interactions[END_REF][START_REF] Dereudre | Infinite volume continuum random cluster model[END_REF]. Before the present paper, the entropy strategy has been used only in the setting of finite range or random finite range interaction. As far as we know, it is the first time that it is applied in the setting of pure infinite range interaction. Therefore, our main contribution here was to developed a way to control the decay of the interaction adequately with the entropy approach. It is the reason why we introduce the Intensity Regular assumption (see Definition 2.4). It is called Intensity Regular because the decay of the interaction is controlled via the intensity of the process. This choice is directly related to the entropy bounds which gives a uniform control of intensities for finite volume Gibbs processes within the thermodynamic limit. In the setting of pairwise interaction, our definition is similar to the Regular assumption by Ruelle.

As mentioned before, in the setting of pairwise interaction, the entropy strategy improves the existence results in relaxing the superstability assumption. However we lose Ruelle's estimates which ensures existence of moments of any order and some exponential and super-exponential moments. The entropy approach only provides moments of order one. Better estimates have to be obtained by different tools.

In the following Section 2, we introduce the definitions and notations for Gibbs point processes. Our main existence Theorem is also given. Examples of energy functions are presented in Section 3. Finally, Section 4 is devoted to the proof of the theorem.

Notations and results

The real d-dimensional space R d is equipped with the usual Euclidean distance . and its associated Borel σ-algebra. Any set Λ ⊂ R d is assumed measurable.

Finite volume measure

The space of configurations is the set of locally finite subsets of R d

Ω = {ω ⊂ R d : |ω ∩ ∆| < ∞, ∀∆ ⊂ R d , bounded},
where | • | is the cardinal. We denote ω ∩ ∆ by ω ∆ , the union ω ∪ ω of two configurations by ωω , the space of finite configurations by Ω f and the space of configurations in Λ ⊂ R d by Ω Λ .

Our space is equipped with the sigma field F generated by the counting functions N ∆ : ω → |ω ∆ | for all ∆ ⊂ R d bounded. In our setting, a point process is simply a probability measure on (Ω, F). Note that, with this definition, we identify a point process with its distribution. We say that the process has a finite intensity if for all bounded subsets ∆ the expectation E P [|ω ∆ |] is finite. If we denote this expectation µ(∆) then µ is a sigma finite measure on R d called the intensity measure. When µ = i(P )λ d , where λ d is the Lebesgue measure on R d and i(P ) ≥ 0, we simply say that the point process has intensity i(P ). We also introduce

ξ(P ) = sup Λ⊂R d 0<λ d (Λ)<+∞ E P [|ω Λ |] λ d (Λ)
and we say that a probability measure P has a bounded intensity if ξ(P ) < +∞. Obviously, if P has a finite intensity i(P ) then ξ(P ) = i(P ).

A point process P is stationary if for all u ∈ R d , P = P • τ -1 u , where τ u is the translation of vector u. If a stationary point process has a finite intensity then its intensity measure is proportional to the Lebesgue measure and has the form

µ = i(P )λ d .
The most popular point processes are the Poisson point processes. We consider here only the homogeneous (or stationary) case where the intensity has the form µ = zλ d . The process is denoted π z , or simply π if z = 1. Recall briefly that π z is the only point process in R d with intensity µ = i(P )λ d such that any two disjoint regions of space are independent under π z . See the recent book [START_REF] Last | Lectures on the Poisson process[END_REF] on the subject.

Let us now define the interaction between the points. We need to introduce an energy function.

Definition 2.1. An energy function is a measurable function H on the space of finite configurations Ω f with values in R ∪ {+∞} such that:

-H is non degenerate: H(∅) < +∞, -H is hereditary: for all ω ∈ Ω f and x ∈ ω then

H(ω) < +∞ ⇒ H(ω \ {x}) < +∞, -H is stationary: for all ω ∈ Ω f and u ∈ R d H(τ u (ω)) = H(ω).
A crucial assumption is the stability of the energy function.

Definition 2.2. An energy function is said to be stable if there exists a constant

B ≥ 0 such that for all ω ∈ Ω f H(ω) ≥ -B|ω|.
The assumption [Stable] is standard and have been treated deeply in the literature (see for instance Section 3.2 in [START_REF] Ruelle | Statistical mechanics: Rigorous results[END_REF]). Now we can define the Gibbs point processes in finite volume.

Definition 2.3. Let Λ be a bounded subset in R d . The Gibbs point process on Λ for the stable energy function H is the probability measure on Ω Λ defined by

P Λ (dω) = 1 Z Λ e -H(ω) π Λ (dω)
with the normalization constant Z Λ = e -H(ω) π Λ (dω) called the partition function.

We can check with the properties of H that P Λ is well defined (i.e. 0 < Z Λ < +∞). In comparison with the standard formalism of Gibbs measures in statistical physics (see [START_REF] Ruelle | Statistical mechanics: Rigorous results[END_REF] for instance), the activity and inverse temperature parameters are included in the function H here.

Infinite volume measure

Let us turn now to the definition of Gibbs point processes in the infinite volume regime. For the following, we choose the convention ∞ -∞ = 0. We need to introduce the local energy which is given by H ∆ (ω) = H(ω) -H(ω ∆ c ) for a finite configuration ω ∈ Ω f . It represents the contribution of energy coming from ω ∆ in ω (the difference of energies with and without ω ∆ ). We need to extend this definition to an infinite configuration. If (∆ l ) l≥0 is an increasing sequence of subsets in R d with ∆ l ↑ R d , we expect that the following limit exists " lim l→+∞ H l ∆ (ω)" where

H l ∆ (ω) = H(ω ∆ l ) -H(ω ∆ l \∆ ) ∈ R ∪ {+∞}.
In particular, the difference H l+1 ∆ (ω) -H l ∆ (ω) should go to zero as l goes to infinity. Our main assumption is a control of the expectation of this difference for point processes with bounded intensities. Definition 2.4. An energy function is said intensity regular if for all bounded subsets ∆ of R d , we can find an increasing sequence of subsets (∆ l ) l≥0 with ∆ l ↑ R d and ∆ ⊂ ∆ 0 , such that for all configuration ω

H l+1 ∆ (ω) -H l ∆ (ω) ≤ |ω ∆ |G l ∆ (ω ∆ c ),
where G l ∆ is a non negative measurable function on Ω ∆ c such that for any probability measure P with bounded intensity we have

E P G l ∆ (ω ∆ c ) ≤ α l a(ξ(P )),
with a : R + → R + an increasing function and (α l ) l≥0 a positive summable sequence (i.e. l≥0 α l < +∞).

From assumption [intensity regular], we have

+∞ l=0 H l+1 ∆ (ω) -H l ∆ (ω) ≤ |ω ∆ | +∞ l=0 G l ∆ (ω ∆ c ),
and we can control the expectation of the second part of the right-hand side

E P +∞ l=0 G l ∆ (ω ∆ c ) ≤ a(ξ(P )) +∞ l=0 α l < +∞.
Then the local energy is correctly defined for P -almost every configurations ω (with P a probability measure with bounded intensity) by

H ∆ (ω) = H 0 ∆ (ω) + +∞ l=0 H l+1 ∆ (ω) -H l ∆ (ω) .
In addition, if we introduce the function C l ∆ defined on Ω ∆ c by

C l ∆ (ω ∆ c ) = +∞ j=l G j ∆ (ω ∆ c ),
then for all configurations ω ∈ Ω we have the following approximation result

H ∆ (ω) -H l ∆ (ω) ≤ |ω ∆ |C l ∆ (ω ∆ c ), (1)
with

E P [C l ∆ (ω ∆ c )] ≤ a(ξ(P )) ∞ j=l α j . Remark 2.5.
f (ω)P (dω) = f (ω ∆ ω ∆ c ) 1 Z ∆ (ω ∆ c ) e -H ∆ (ω ∆ ω ∆ c ) π ∆ (dω ∆ )P (dω), (2)
with the normalization constant

Z ∆ (ω ∆ c ) = e -H ∆ (ω ∆ ω ∆ c ) π ∆ (dω ∆ ).
The equations (2) for all ∆ and f are called DLR for Dobrushin-Lanford-Ruelle. To be correctly defined we need to check that 0 < Z ∆ (ω ∆ c ) < +∞ for P almost all configuration ω. A lower bound can easily be obtained with

Z ∆ (ω ∆ c ) ≥ e -H ∆ (ω ∆ c ) π ∆ (∅) = e -λ d (∆) > 0.
To have an upper bound, we use assumptions [Stable] and [Intensity Regular]. From inequality (1) and assuming that H(ω ∆ 0 \∆ ) < +∞ we have

H ∆ (ω) ≥ H 0 ∆ (ω) -|ω ∆ |C 0 ∆ (ω ∆ c ) = H(ω ∆ 0 ) -H(ω ∆ 0 \∆ ) -|ω ∆ |C 0 ∆ (ω ∆ c ) ≥ -(B + C 0 ∆ (ω ∆ c ))|ω ∆ | -B|ω ∆ 0 \∆ | -H(ω ∆ 0 \∆ ).
Then we obtain

Z ∆ (ω ∆ c ) ≤ exp B|ω ∆ 0 \∆ | + H(ω ∆ 0 \∆ ) e (C 0 ∆ (ω ∆ c )+B)|ω ∆ | π ∆ (dω ∆ ) = exp B|ω ∆ 0 \∆ | + H(ω ∆ 0 \∆ ) +∞ n=0 λ d (∆)e (C 0 ∆ (ω ∆ c )+B)) n n! e -λ d (∆) = exp B|ω ∆ 0 \∆ | + H(ω ∆ 0 \∆ ) + λ d (∆) e (C 0 ∆ (ω ∆ c )+B)) -1 < +∞.
If H(ω ∆ 0 \∆ ) = +∞ then, by the hereditary property, for all l ≥ 0 we have H(ω ∆ l \∆ ) = +∞ and H l ∆ (ω) = 0 (according to our convention ∞ -∞ = 0 given above), which gives H ∆ (ω) = 0 and Z ∆ (ω ∆ c ) < +∞.

Our main result is the following theorem which is proved in Section 4.

Theorem 1. For any energy function H satisfying assumptions [Stable] and [Intensity Regular], there exists at least one stationary Gibbs point process P with finite intensity and locally finite energy (i.e.

P (H(ω ∆ ) < ∞, ∀∆ ⊂ R d , bounded) = 1).
Remark 2.7. (temperedness) Our formalism of Gibbs point processes does not really need the introduction of tempered configurations as it is done usually [START_REF] Ruelle | Superstable interactions in classical statistical mechanics[END_REF][START_REF] Kondratiev | Gibbs measures of continuous systems: an analytic approach[END_REF]. But, the definition of DLR equations requires a space of good configurations. In our case, this space is the space of configurations produced by any point processes with bounded intensity. It is not always obvious to compare with the different definitions of temperedness mentioned above.

Examples

Let Finite range interaction. An energy function H has a finite range if there exists R > 0 such that for all finite configurations ω ∈ Ω f and bounded subset ∆, H ∆ (ω) = H ∆ ω ∆⊕B(0,R) . It is easy to see that a finite range energy function verifies assumption [Intensity Regular], with any increasing sequence of subsets (∆ l ) l≥0 such that ∆ 0 = ∆ ⊕ B(0, R). Therefore in this setting of finite range interaction, only the assumption [Stable] is required to ensure the existence of Gibbs point processes.

Corollary 3.1. For a finite range and stable energy there exists at least one stationary Gibbs point process with finite intensity and locally finite energy.

This result has been proved previously in [START_REF] Dereudre | Existence of Gibbsian point processes with geometry-dependent interactions[END_REF] (See also [START_REF] Dereudre | Introduction to the theory of Gibbs point processes[END_REF] for a simpler and pedagogical proof).

Pairwise interaction An energy function H is pairwise if there exists a symmetric function φ : R d → R ∪ {+∞}, called a potential, such that

H(ω) = {x,y}⊂ω φ(x -y).
We do not assume the finite range property and so the support of φ can be unbounded. However, we still need an assumption on the long range behaviour of φ. The potential φ is regular if there exists a decreasing function ψ : R + → R + such that +∞ A ψ(r)r d-1 dr < +∞ for some A > 0 and |φ(x)| ≤ ψ( x ) for all x ∈ R d . Let us show that this energy function satisfies the assumption [Intensity Regular].

Let ∆ be a bounded subset of R d . We choose z ∈ R d and ρ > 0 such that ∆ ⊂ B(z, ρ) and we introduce ∆ l = B(z, ρ + A + l). We have

H l ∆ (ω) = {x,y}⊂ω∩∆ l {x,y}∩∆ =∅ φ(x -y),
which gives

H l+1 ∆ (ω) -H l ∆ (ω) = x∈∆ y∈∆ l+1 \∆ l φ(x -y). By definition, if x ∈ ∆ and y ∈ ∆ l+1 \ ∆ l then x -y ≥ A + l and |φ(x - y)| ≤ ψ(A + l). Introducing the function G l ∆ (ω ∆ c ) = |ω ∆ l+1 \∆ l |ψ(A + l) we have |H l+1 ∆ (ω) -H l ∆ (ω)| ≤ |ω ∆ |G l ∆ (ω ∆ c
) with

E P G l ∆ (ω ∆ c ) ≤ ξ(P )λ d (∆ l+1 \ ∆ l )ψ(A + l) ≤ ξ(P )c d (ρ + A + l + 1) d -(ρ + A + l) d ψ(A + l) ≤ ξ(P )c d d(ρ + A + l + 1) d-1 ψ(A + l) Since +∞ A
ψ(r)r d-1 dr < +∞ and ψ is decreasing, we have that

l≥0 (ρ + A + l + 1) d-1 ψ(A + l) < +∞.
Hence the energy H satisfies the assumption [Intensity Regular] with a = id and

α l = c d d(ρ + A + l + 1) d-1 ψ(A + l).
Corollary 3.2. For a stable pairwise energy associated to a regular potential, there exists at least one stationary Gibbs point process with finite intensity and locally finite energy.

The stability assumption for a pairwise potential is delicate and has been investigated long time ago. We refer to [START_REF] Ruelle | Statistical mechanics: Rigorous results[END_REF] for several results. Let us mention that a non-negative potential which is equal to zero in the neighbourhood of zero and positive outside this neighbourhood is a simple example of a Stable but not Superstable potential.

Note that the sum of two energy functions satisfying assumption [Intensity Regular], satisfies the assumption [Intensity Regular] as well. The following corollary follows.

Corollary 3.3. For a stable energy function, which is the sum of a finite range energy function and a pairwise energy function associated to a regular potential, there exists at least one stationary Gibbs point process with finite intensity and locally finite energy.

Cloud interaction In this last example, we provide an energy function which is infinite range and not reducible, at any scale, to a pairwise interaction. It is a multibody interaction between a germ-grain interaction (see for instance the Quermass model [START_REF] Dereudre | The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains[END_REF] or the Widom-Rowlinson interaction [START_REF] Widom | New model for the study of liquid-vapor phase transitions[END_REF]) and a pairwise interaction. We call it cloud interaction because each point of the configuration is diluted in a cloud around itself and the pairwise interaction is integrated on this cloud. Precisely for any finite configuration ω

H(ω) = x∈ω L R (ω) φ(x -y)dy where L R (ω) = x∈ω B(x, R
) is the cloud produced by the configuration ω (R > 0 is a fixed parameter) and φ : R d → R ∪ {∞} is a symmetric function. This energy function can be viewed as an approximation of the pairwise interaction introduced above. Indeed H(ω)/R d tends to the pairwise interaction function (times a multiplicative constant) when R goes to zero.

We assume that there exists a decreasing function ψ : R + → R + such that ψ(r)r d-1 dr < +∞ and |φ(x)| ≤ ψ( x ) for all x ∈ R d . This ensures that the energy function is well-defined and satisfies the assumption

[Stable] since |H(ω)| ≤ |ω| |ψ(r)|r d-1 dr.
Note that H is not superstable. Indeed, it would require that for all ω ∈ Ω Λ , H(ω) ≥ -B|ω| + A|ω| 2 /λ d (Λ) with B ≥ 0 and A > 0 which is impossible with the previous upper-bound of H.

Concerning assumption [Intensity Regular], for ∆ a bounded subset of R d , the local energy is given by

H l ∆ (ω) = H(ω ∆ l ) -H(ω ∆ l \∆ ) = x∈ω ∆ l L R (ω ∆ l ) φ( x -y )dy - x∈ω ∆ l \∆ L R (ω ∆ l \∆ ) φ( x -y )dy = x∈ω ∆ l \∆ L R (ω ∆ l )\L R (ω ∆ l \∆ ) φ( x -y )dy + x∈ω ∆ L R (ω ∆ l ) φ( x -y )dy.
We can find z ∈ R d and ρ > 0 such that ∆ ⊂ B(z, ρ) and, for the following, we choose ∆ 0 = B(z, ρ + 2R), which implies that

L R (ω ∆ l ) \ L R (ω ∆ l \∆ ) = L R (ω ∆ 0 ) \ L R (ω ∆ 0 \∆ ) def = L R ∆ (ω).
Using this notation we have

H l ∆ (ω) = x∈ω ∆ l \∆ L R ∆ (ω) φ( x -y )dy + x∈ω ∆ L R (ω ∆ l ) φ( x -y )dy.
The first term corresponds to the interaction of the points outside of ∆ with the cloud created by the points in ∆ and the second corresponds to the interaction of the points in ∆ with the full cloud. We can compute the cost of adding a shell

H l+1 ∆ (ω) -H l ∆ (ω) = x∈ω ∆ l+1 \∆ l L R ∆ (ω) φ( x -y )dy + x∈ω ∆ L R (ω ∆ l+1 )\L R (ω ∆ l )
φ( x -y )dy.

If we choose ∆ l = B(z, ρ + 2R + l), for x ∈ ω ∆ l+1 \∆ l and y ∈ L R ∆ (ω) or for x ∈ ω ∆ and y ∈ L R (ω ∆ l+1 ) \ L R (ω ∆ l ) then we have x -y > A + l. Then we have

|H l+1 ∆ (ω) -H l ∆ (ω)| ≤ |ω ∆ l+1 \∆ l |λ d (L R ∆ (ω)) + |ω ∆ |λ d (L R (ω ∆ l+1 \∆ l )) ψ(A + l).
With the union bound we have

λ d (L R ∆ (ω)) ≤ |ω ∆ |c d R d and λ d (L R (ω ∆ l+1 \∆ l ))) ≤ |ω ∆ l+1 \∆ l |c d R d . If we introduce the function G l ∆ (ω ∆ c ) = 2|ω ∆ l+1 \∆ l |c d R d ψ(A+l) we obtain |H l+1 ∆ (ω)- H l ∆ (ω)| ≤ |ω ∆ |G l ∆ (ω ∆ c ) with E P G l ∆ (ω ∆ c ) ≤ ξ(P )λ d (∆ l+1 \ ∆ l )ψ(A + l) ≤ ξ(P )dc d (ρ + A + l + 1) d-1 ψ(A + l). Since +∞ A ψ(r)r d-1 dr < +∞ and ψ is decreasing it implies l≥0 (ρ + A + l + 1) d-1 ψ(A + l) < +∞.
Hence the energy H satisfies assumption [Intensity Regular].

Corollary 3.4. For the cloud interaction, there exists at least one stationary Gibbs point process with finite intensity and locally finite energy.

4 Proof of the theorem

Construction of an infinite volume measure

The first step of the proof is to build an accumulation point of a sequence of finite volume Gibbs measures. Using entropy bounds and the stability of the energy, we prove the existence of such an accumulation point for the local convergence topology. This strategy and its tools have been used several time in the literature ( [START_REF] Dereudre | The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains[END_REF], [START_REF] Dereudre | Introduction to the theory of Gibbs point processes[END_REF], [START_REF] Dereudre | Existence of Gibbsian point processes with geometry-dependent interactions[END_REF], [START_REF] Dereudre | Infinite volume continuum random cluster model[END_REF]) and we recall here only the main ideas.

For n a positive integer, we denote by Λ n the set ] -n, n] d . We consider the sequence of Gibbs measures in finite volume given by

P n (dω) = P Λn (dω) = 1 Z n e -H(ω) π Λn (dω),
with the normalization constant Z n = e -H(ω) π Λn (dω). Since our tension tool will be defined for stationary measures, we need to modify (P n ) n≥1 . We defined the periodized version P per n by the probability measure

u∈Z d P n • τ -1 2nu
, and the stationnarized version by Given two probabilities measures µ and ν on Ω, we recall that the relative entropy of µ with respect to ν on Λ is defined as

P sta n = 1 λ d (Λ n ) Λn P per n • τ -1 u du.
I Λ (µ|ν) = log f dµ Λ if µ Λ ν Λ and f = dµ Λ dν Λ +∞ otherwise.
Definition 4.2. Let µ be a stationary probability measure with finite intensity on Ω. For z > 0 the specific entropy of µ with respect to π z is defined by

I z (µ|π z ) = lim n→+∞ I Λn (µ |π z ) λ d (Λ n ) = sup Λ⊂R d 0<λ d (Λ)<+∞ I Λ (µ|π z ) λ d (Λ) . (3)
For details we refer to Chapter 15 of [START_REF] Georgii | Gibbs measures and phase transitions[END_REF]. The next result, stated in [START_REF] Georgii | Large deviations and the maximum entropy principle for marked point random fields[END_REF], is our tension tool. In order to apply this proposition in our case, we need to compute the specific entropy of the probability measure P sta n . Using the affine property of the specific entropy, it is well known that

I z (P sta n ) = 1 λ d (Λ n ) I Λn (P n |π z ).
What remains is to compute the relative entropy of the Gibbs measure P n with respect to the Poisson point process π z on Λ n

I Λn (P n |π z ) = log dP n dπ z dP n = log dP n dπ + log dπ dπ z dP n = -log(Z n ) -H(ω) + log e (z-1)λ d (Λn) 1 z |ω| P n (dω) = (z -1)λ d (Λ n ) -log(Z n ) + [-H(ω) -log(z)|ω|] P n (dω).
Under π z , the random variable |ω Λn | follows a Poisson law of parameter zλ d (Λ n ) and so

E π z e |ω Λn | = e -zλ d (Λn) ∞ p=0 (zλ d (Λ n )) p p! e p = exp(zλ d (Λ n )(e -1)).
Then we obtain

i(P sta n )λ d (Λ n ) = E P sta n [|ω Λn |] ≤ (ze + H(∅))λ d (Λ n
), and since ξ( Pn ) ≤ i(P sta n ), we deduce the lemma.

4.2 Some remarks on the local energy Lemma 4.5. If P is a point process with bounded intensity, then for P almost all ω the following equivalence holds for all l ≥ 0

H ∆ (ω) = +∞ ⇐⇒ H l ∆ (ω) = +∞. Proof.
Let ω be a configuration. With our convention ∞ -∞ = 0 and the intensity regular assumption, for l ≥ 1 having H l ∆ (ω) = +∞ implies that H l-1 ∆ (ω) = +∞ and H l+1 ∆ (ω) = +∞. Then by induction we deduce that for all l ≤ 1 we have H l ∆ (ω) = +∞ if and only if H 0 ∆ (ω) = +∞. The local energy is defined by

H ∆ (ω) = H 0 ∆ (ω) + +∞ l=0 H l+1 ∆ (ω) -H l ∆ (ω) ,
and since for a point process P with bounded intensity the second part of the right hand side is finite we deduce that for P -almost all ω, H ∆ (ω) = +∞ if and only if H 0 ∆ (ω). Lemma 4.5 ensures that the event {H ∆ (ω) = +∞} is local. In particular, the hard-core part of the energy has to be finite range.

For a finite configuration ω ∈ Ω f , the sum in the definition of the local energy H ∆ (ω) is a finite sum, and then Lemma 4.5 also holds. Hence we can write

H ∆ (ω) = H l ∆ (ω) = H(ω ∆ l ) -H(ω ∆ c l ) = H(ω) -H(ω ∆ c
) for some l ≥ 0 (depending on ω). Moreover, thank to hereditary of the energy and our convention ∞ -∞ = 0, we have for all ω ∈ Ω f

H(ω) = H ∆ (ω) + H(ω ∆ c ).
This equality is useful for proving the DLR equations for finite volume Gibbs measures.

The DLR equation

We prove in this section that the accumulation point P satisfies the DLR equations stated in Definition 2.6. By a standard monotone class argument we can replace the class of bounded measurable functions by the class of bounded local functions. Let f be a bounded local function, ∆ be a bounded measurable subset of R d , we have to show

f dP = f ∆ dP with f ∆ (ω) = f (ω ∆ ω ∆ c ) 1 Z ∆ (ω ∆ c ) e -H ∆ (ω ∆ ω ∆ c ) π ∆ (dω ∆ ).
We fix > 0.

Step 1. There exists K > 0 such that for all k ≥ K , we have

P (|ω ∆ | > k) ≤ ε, this implies that for all k ≥ K , f ∆ (ω)P (dω) -f ∆ (ω)1 |ω ∆ |≤k P (dω) ≤ f ∞ ε. (4)
The introduction of this indicator function will be usefull in step 4.

Step 2 We approximate f ∆ by f l ∆,k which corresponds to the approximation of the local energy H ∆ by H l ∆ and a restriction to configurations having less than k points in ∆, which means

f l ∆,k (ω) = 1 Z l ∆,k (ω ∆ c ) f (ω ∆ ω ∆ c )e -H l ∆ (ω ∆ ω ∆ c ) 1 |ω ∆ |≤k π ∆ (dω ∆ ), with the normalization constant Z l ∆,k (ω ∆ c ) = e -H l ∆ (ω ∆ ω ∆ c ) 1 |ω ∆ |≤k π ∆ (dω ∆ )
. We prove that we can find K ≥ K and l (depending on K) such that :

f ∆ (ω)1 |ω ∆ |≤K P (dω) -f l ∆,K (ω)1 |ω ∆ |≤K P (dω) ≤ 6 f ∞ ε. (5)
We introduce an event A ε such that P (A ε ) ≤ ε. Its precise definition is given later. We have

f ∆ (ω)1 |ω ∆ |≤k P (dω) -f l ∆,k (ω)1 |ω ∆ |≤K P (dω) ≤ A c ε |f ∆ (ω) -f l ∆,k (ω)|1 |ω ∆ |≤k P (dω) + 2 f ∞ ε. (6)
We must estimate the approximation error

f ∆ (ω) -f l ∆,k (ω) = 1 Z ∆ (ω ∆ c ) f (ω ∆ ω ∆ c ) e -H ∆ (ω ∆ ω ∆ c ) -e -H l ∆ (ω ∆ ω ∆ c ) 1 |ω ∆ |≤k π ∆ (dω ∆ ) + 1 Z ∆ (ω ∆ c ) - 1 Z l ∆,k (ω ∆ c ) f (ω ∆ ω ∆ c )e -H l ∆ (ω ∆ ω ∆ c ) 1 |ω ∆ |≤k π ∆ (dω ∆ ).
Since the difference between the normalization constants is

Z l ∆,k (ω ∆ c ) -Z ∆ (ω ∆ c ) = e -H l ∆ (ω ∆ ω ∆ c ) -1 |ω ∆ |≤k e -H ∆ (ω ∆ ω ∆ c ) π ∆ (dω ∆ ),
we obtain the upper-bound

|f ∆ (ω) -f l ∆,k (ω)| ≤ 2 f ∞ Z ∆ (ω ∆ c ) e -H ∆ (ω ∆ ω ∆ c ) -e -H l ∆ (ω ∆ ω ∆ c ) 1 |ω ∆ |≤k π ∆ (dω ∆ ) = 2 f ∞ Z ∆ (ω ∆ c ) e -H ∆ (ω ∆ ω ∆ c ) -e -H l ∆ (ω ∆ ω ∆ c ) 1 |ω ∆ |≤k π ∆ (dω ∆ ) + 2 f ∞ Z ∆ (ω ∆ c ) e -H ∆ (ω ∆ ω ∆ c ) 1 |ω ∆ |>k π ∆ (dω ∆ ). (7)
Using Lemma 4.5, the inequality |e b -e a | ≤ |b -a|e |b-a|+a and the approximation (1) we obtain the upper-bound

e -H l ∆ (ω ∆ ω ∆ c ) -e -H ∆ (ω ∆ ω ∆ c ) 1 |ω ∆ |≤K ≤ KC l ∆ (ω ∆ c )e KC l ∆ (ω ∆ c )-H ∆ (ω ∆ ω ∆ c ) . (8)
By combining ( 7) and ( 8) we have

|f ∆ (ω) -f l ∆,k (ω)| ≤ 2 f ∞ KC l ∆ (ω ∆ c )e KC l ∆ (ω ∆ c ) + 2 f ∞ Z ∆ (ω ∆ c ) e -H ∆ (ω ∆ ω ∆ c ) 1 |ω ∆ |>k π ∆ (dω ∆ ). ( 9
)
Now we need to control the two terms of the right part of [START_REF] Georgii | Large deviations and the maximum entropy principle for marked point random fields[END_REF]. For the second one, by the dominated convergence theorem, we can find K ≥ K such that

1 Z ∆ (ω ∆ c ) e -H ∆ (ω ∆ ω ∆ c ) 1 |ω ∆ |>K π ∆ (dω ∆ )P (dω) ≤ ε.
Once K is chosen, we can control the first term. If W : R + → R + is the inverse function of x → xe x (W is the so-called Lambert function), using Markov's inequality we have

P KC l ∆ (ω ∆ c )e KC l ∆ (ω ∆ c ) > ε = P (KC l ∆ (ω ∆ c ) > W (ε)) ≤ KE P [C l ∆ (ω ∆ c )] W (ε)
.

As E P [C l ∆ (ω ∆ c )] goes to zero when l goes to infinity, we can choose l (depending on K) such that P KC l ∆ (ω ∆ c )e KC l ∆ (ω ∆ c ) > ε ≤ ε.

Introducing the event A ε = {KC l ∆ (ω ∆ c )e KC l ∆ (ω ∆ c ) > ε} in [START_REF] Dereudre | Infinite volume continuum random cluster model[END_REF] and with our choice of K and l, we have finally the approximation [START_REF] Dereudre | Existence of Gibbsian point processes with geometry-dependent interactions[END_REF].

According to Lemma 4.4, the point processes ( Pn ) n≥1 have uniformly bounded intensities and, as mentionned in Remark 2.5, the expectations E Pn [C l ∆ (ω ∆ c )] is bounded from above uniformly in n ≥ 1. Hence l could be chosen such that for all n ≥ 1

Pn

KC l ∆ (ω ∆ c )e KC l ∆ (ω ∆ c ) > ε ≤ ε, (10)
it will be useful later, in Step 4.

Step 3. For n large enough (depending on K and l) we have

f l ∆,K (ω)1 |ω ∆ |≤K P (dω) -f l ∆,K (ω)1 |ω ∆ |≤K Pn (dω) ≤ f ∞ ε. (11)
It is simply a consequence of the local convergence of the sequence ( Pn ) n≥1 to P .

Step 4. For all n ≥ 1 we show the approximation

f l ∆,K (ω)1 |ω ∆ |≤K Pn (dω) -f ∆,K (ω)1 |ω ∆ |≤K Pn (dω) ≤ 4 f ∞ ε, (12)
where

f ∆,K (ω) = 1 Z ∆,K (ω ∆ c ) f (ω ∆ ω ∆ c )e -H ∆ (ω ∆ ω ∆ c ) 1 |ω ∆ |≤K π ∆ (dω ∆ ),
with the normalization constant Z ∆,K (ω

∆ c ) = e -H ∆ (ω ∆ ω ∆ c ) 1 |ω ∆ |≤K π ∆ (dω ∆ ).
Similarly to the upper-bounds ( 7) and ( 8) we obtain

|f l ∆,K (ω) -f ∆,K (ω)| ≤ 2 f ∞ Z ∆,K (ω ∆ c ) |e -H ∆ (ω ∆ ω ∆ c ) -e -H l ∆ (ω ∆ ω ∆ c ) |1 |ω ∆ |≤K π ∆ (dω ∆ ) ≤ 2 f ∞ e KC l ∆ (ω ∆ c ) KC l ∆ (ω ∆ c ).
From our previous choice of K and l in estimate [START_REF] Kondratiev | Existence of Gibbs state for a non-ideal gas in R d : the case of a pair, long-range interaction[END_REF], and using a restriction to a specific event as in [START_REF] Dereudre | Infinite volume continuum random cluster model[END_REF], we obtain the approximation [START_REF] Last | Lectures on the Poisson process[END_REF].

Using the local convergence of (P n ) n≥1 to P again, we have, for n large enough

f (ω)1 |ω ∆ |≤K Pn (dω) -f (ω)1 |ω ∆ |≤K P (dω) ≤ f ∞ ε.
With our choice of K we have P (|ω ∆ | > K) ≤ ε and we obtain [START_REF] Ruelle | Superstable interactions in classical statistical mechanics[END_REF].

Conclusion

Gathering approximations (4), ( 5), ( 11), ( 12), ( 13) and ( 14), we have finally

f ∆ dP -f dP ≤ 16 f ∞ ε.
The inequality is true for every > 0, this ends the proof of Theorem 1.

  us give examples of energy functions satisfying assumptions [Stable] and [Intensity Regular] of Theorem 1. Since the assumption [Stable] is studied deeply in the literature, we focus mainly on interesting examples satisfying assumption [Intensity Regular]. In the following, B(z, ρ) is the ball of center z and radius ρ, ⊕ is the Minkowski sum of two sets and c d = λ d (B(0, 1)).

Definition 4 . 1 .

 41 A function f is said local if there exists a bounded set ∆ such that f is F ∆ measurable (i.e. for all configurations ω in Ω, we have f (ω) = f (ω ∆ )). A sequence of measures (µ n ) converges to µ for the local convergence topology if for all bounded local functions f f dµ n → n→+∞ f dµ.

Proposition 4 . 3 .

 43 For any z > 0 and c > 0, the set of probability measures {µ stationary with finite intensity, I z (µ) ≤ c} is compact and sequentially compact for the local convergence topology.

  Assuming that a is an increasing function in the Defintion 2.4 allows to control the approximation (1) for a sequence (P n ) n≥1 of probability measures. Indeed, if (P n ) has a uniformly bounded intensity (i.e. ξ(P n ) ≤ ξ for all n ≥ 1), then the expectation E Pn [C l ∆ (ω ∆ c )] is uniformly bounded from above. We are now able to give the definition of infinite volume Gibbs point processes.Definition 2.6. A probability measure P on Ω with bounded intensity is a Gibbs point process for an energy function H, satisfying assumptions [Stable] and [Intensity Regular], if for all bounded subsets ∆ and all bounded measurable functions f we have

University of Lille, david.dereudre@univ-lille.fr

University of Lille, thibaut.vasseur@univ-lille.fr

Acknowledgement

This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01) and by the CNRS GdR 3477 GeoSto.

The normalization constant can easily be bounded from below Z n = e -H dπ ≥ e -H(∅) e -λ d (Λn) .

Then, using the stability of H, we have

If we choose z > 0 such that B -log(z) ≤ 0, we have I z (P sta n ) ≤ z + H(∅) for all n ≥ 1. According to Proposition 4.3 we can exhibit a sub-sequence of (P sta n ) n≥1 which converges to a stationary measure P with finite intensity. To simplify the notations, we can suppose that we have changed the indexation of the sequence (Λ n ) n≥1 such that (P sta n ) n≥1 converges locally to P . We can prove that P is also an accumulation point of the sequence

See Lemma 3.5 [START_REF] Dereudre | The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains[END_REF] for details. We need to verify that P has locally finite energy. If ∆ is bounded subset of R d , the event {H(ω ∆ ) < +∞} is a locall event and

because by hereditary we have Pn (H(ω ∆ ) = +∞) = 0. We deduce, using again hereditary, that

Let us finish this section by giving the crucial property of uniform control of intensities for the sequence ( Pn ) Lemma 4.4. We can find ξ ≥ i(P ) such that for all integers n ≥ 1, ξ( Pn ) ≤ ξ.

Proof. We use the entropic inequality gdµ ≤ I(µ|ν) + log e g dν to obtain

Using the expression of the specific entropy as a supremum (3), we have

Step 5. We use the DLR equations for finite volume Gibbs processes to prove that

and if we denote

we have the approximation

Let us detail the term

For u ∈ Λ * n , using the fact that τ u (ω)

Then, by a similar calculation, we find

But we can write the measure in finite volume as

, and integration with respect of the measure π τ -1 u (∆) will give the normalization constant (thanks to the indicator function introduce in Step 1). After simplification we have for the translated

This DLR type equation is then verified for the mixture P * n f ∆,K (ω)1 |ω ∆ |≤K P * n (dω) = f (ω)1 |ω ∆ |≤K P * n (dω).

Since we have the approximation

we obtain finally (13).

Step 6. We show the last approximation


David Dereudre

Pierre Houdebert

Sharp phase transition for the continuum Widom-Rowlinson model

Keywords: Gibbs point process, DLR equation, Boolean model, continuum percolation, random cluster model, Fortuin-Kasteleyn representation

The Area-interaction model (or the continuum one-color Widom-Rowlinson model) is a Gibbs point process with the formal Hamiltonian H(ω) = Volume(∪ x∈ω B 1 (x), where ω is a locally finite configuration of points and B 1 (x) denotes the unit closed ball centred at x. Two other parameters, the activity z > 0 related to the intensity of the process and the inverse temperature β ≥ 0 related to the strength of the interaction allow to tune the model for different features. In the present paper we investigate two kinds of phase transition for this model. First we show that it exhibits a sharp phase transition of percolation for the graph which connect points with distance smaller than r > 0. It means that for any β > 0, there exists 0 < z a c (β, r) < +∞ such that an exponential decay of connectivity at distance n occurs in the subcritical phase (i.e. z < z a c (β, r)) and a logarithmic lower bound of the connection at infinity holds in the supercritical case (i.e. z > z a c (β, r)). Secondly we study a standard liquid-gas phase transition related to the uniqueness/non-uniqueness of Gibbs states of the model depending on the parameters z, β. An old result claims that a non-uniqueness regime occurs for z = β large enough. We improve it by showing that the non-uniqueness holds if and only if z = β highlighting that the phase transition is sharp at this critical value. We show also that this critical value z = β corresponds to the percolation threshold z a c (β, r) for β large enough, providing a straight connection between these two notions of phase transition.

Introduction

The bicolor Widom-Rowlinson model is a model introduced in the late 60' by Widom and Rowlinson [START_REF] Widom | New model for the study of liquid-vapor phase transitions[END_REF] exactly for giving a new representation of areainteraction measures (or one-color Widom-Rowlinson models). It can be defined simply as the reunion of two Poisson Boolean models (with deterministic radii equal to 0.5) conditioned on a hard-core non overlapping condition between the two Boolean models. A formal definition using standard DLR formalism is given in Definition 2.2.

Its interest comes first from its applicability in the description of a binary gas, but also from the fact that it was the first continuum model for which a phase transition was rigorously proved, see [START_REF] Ruelle | Existence of a phase transition in a continuous classical system[END_REF].

Percolation theory is linked in statistical mechanics to the phase transition. For instance in [START_REF] Chayes | The analysis of the widomrowlinson model by stochastic geometric methods[END_REF] the authors gave a new proof of the phase transition of the Widom-Rowlinson model at large activity. Their proof relies on a percolationtype argument applied to the so-called Continuum Random Cluster Model. This argument was generalised to the case of unbounded radii in [START_REF] Houdebert | Percolation results for the Continuum Random Cluster Model[END_REF]. Furthermore percolation is also used to proved uniqueness of Gibbs measure. The technique, called disagreement percolation, relies on the construction of a good coupling of Gibbs instances with different boundary conditions. In a recent paper [START_REF] Hofer-Temmel | Disagreement percolation for marked Gibbs point processes[END_REF] this technique was used to prove a general result of uniqueness for a general class of interactions, non necessarly with finite range.

In this paper we are interested in two very linked topics: the uniqueness/non uniqueness of the area-interaction measures and some percolation properties of area-interaction measures, namely the percolation sharpness.

Percolation and uniqueness are naturely related in statistical mechanics. Indeed often the non uniqueness seems to come from a dependance "at infinity", which is possible only if percolation occurs. More rigoursly the absence of percolation is used, with disagreement percolation arguments, to proves uniqueness of Gibbs measure. This technique, introduced by van den Berg and Maes [START_REF] Van Den Berg | Disagreement percolation in the study of Markov fields[END_REF], relies on the construction of a good disagreement coupling comparing the influence of two boundary conditions. See [START_REF] Hofer-Temmel | Disagreement percolation for marked Gibbs point processes[END_REF] for a recent result on disagreement percolation.

Preliminaries

Space

Let us consider the state space S := R d with d ≥ 1 being the dimension. Let Ω be the set of locally finite configurations ω on S. This means that #(ω ∩ Λ) < ∞ for every bounded Borel set Λ of R d , with #ω being the cardinal of the configuration ω. We write ω Λ as a shorthand for ω ∩ Λ. The configuration space is embedded with the usual σ-algebra F generated by the counting variables. To a configuration ω ∈ Ω we associate the germ-grain structure

B r (ω) := x∈ω B r (x),
where B r (x) is the closed ball centred at x with radius r > 0.

Poisson point processes

Let π z be the distribution on Ω of the homogeneous Poisson point process with intensity z > 0. Recall that it means

• for every bounded Borel set Λ, the distribution of the number of points in Λ under π is a Poisson distribution of mean zL d (Λ), where L d stands for the usual d-dimensional Lebesgue measure;

• given the number of points in a bounded Λ, the points are independent and uniformly distributed in Λ.

We refer to [START_REF] Daley | An introduction to the theory of point processes[END_REF] for details on Poisson point processes.

For Λ ⊆ R d bounded, we denote by π z Λ the restriction of π z on Λ. For simplicity the special case of the Poisson point process of unit intensity (i.e. z = 1) is denoted by π, and its restriction by π Λ .

Area-interaction measures

The area-interaction measures (or the one-color Widom-Rowlinson models) are defined through the stardard Gibbs Dobrushin-Lanford-Ruelle formalism precribing the conditional probabilities. For a bounded Λ ⊆ R d , we define the Λ-Hamiltonian

H Λ (ω) := L d ( B 1 (ω Λ ) \ B 1 (ω Λ c ) ).
(2.1)

The area specification on a bounded Λ ⊆ R d with boundary condition ω Λ c is defined by

P z,β Λ,ω Λ c (dω Λ ) := z #ω Λ e -βH Λ (ω Λ ∪ω Λ c ) Z area (z, β, Λ, ω Λ c ) π Λ (dω Λ ) (2.2)
with the standard partition function

Z area (z, β, Λ, ω Λ c ) := Ω z #ω Λ e -βH Λ (ω Λ ∪ω Λ c ) π Λ (dω Λ ) (2.3)
which is always non-degenerate.

Definition 2.1. A probability measure P on Ω is an area-interaction measure of activity z and inverse temperature β, written P ∈ G area z,β , if for every bounded Borel set Λ ⊆ R d and every bounded measurable function f ,

Ω f dP = Ω Ω f (ω Λ ∪ ω Λ c )P z,β
Λ,ω Λ c (dω Λ )P (dω).

(2.4)

The equations (2.4), for all bounded Λ, are called DLR equations, after Dobrushin, Lanford and Ruelle. Those equations prescribe the conditional probabilities of a Gibbs measure.

Remark 2.1. There is several possible form for the Hamiltonian H Λ , all of which defining the same specification. The nice property about our definition of H Λ is the additivity, in the sense that H Λ (ω) can be seen as the sum of the contribution of each points, with respect to the already considered ones.

Stochastic domination

Let us discuss stochastic domination, which is going to be a key element of several proofs of the paper. Recall that an event E ∈ F is said increasing if for ω ∈ E and ω ⊇ ω , we have ω ∈ E. Finally if P and P are two probability measures on Ω, the measure P is said to stochastically dominate the measure P , written P P , if P (E) ≤ P (E) for every increasing event E ∈ F. The following proposition is a direct application of the classical Georgii and Küneth stochastic domination result [30, Theorem 1.1] and gives standard stochastic dominations. Proposition 2.1. For every bounded Λ ⊆ R d ,

• for every boundary condition ω Λ c and every z, β we have

π ze -βv d Λ P z,β Λ,ω Λ c (dω Λ ) π z Λ , (2.5)
where v d is the volume of the unit ball in dimension d.

This implies in particular that every P ∈ G area (z, β) satisfies

π ze -βv d P π z . (2.6)
• For every boundary conditions ω 1 Λ c ⊆ ω 2 Λ c , every z 1 ≤ z 2 and every β 1 ≥ β 2 we have

P z 1 ,β 1 Λ,ω 1 Λ c (dω Λ ) P z 2 ,β 2 Λ,ω 2 Λ c (dω Λ ).
(2.7)

Free and wired measures

Two particular area-interaction measures are constructed as follows. Consider the increasing sequence Λ n :=] -n, n] d and consider the free and wired areainteraction measures on the bounded box Λ n , denoted by P z,β n,f ree and P z,β n,wired and defined as

P z,β n,f ree dω Λn := z #ω Λn e -βL d ( B 1( ω Λn ) ) Z area (z, β, n, f ree) π Λn (dω Λn ); (2.8) P z,β n,wired dω Λn := z #ω Λn e -βL d ( B 1( ω Λn )∩Λn-1 ) Z area (z, β, n, wired) π Λn (dω Λn ); (2.9)
where Z area (z, β, n, f ree) and Z area (z, β, n, wired) are the normalising constants. The measure P z,β n,f ree is simply P z,β Λn,∅ , whereas P z,β n,wired is the limiting case where the boundary condition would be filled with points on the boundary of Λ n .

From [30, Theorem 1.1] we get the following proposition.

Proposition 2.2. For every n and every z, β we have

• P z 1 ,β 1 f ree ∈ G area z 1 ,β 1 and P z 1 ,β 1 wired ∈ G area z 1 ,β 1 ; • P z 1 ,β 1 f ree P z 2 ,β 2 f ree and P z 1 ,β 1 wired P z 2 ,β 2 wired ; • P z 1 ,β 1 f ree P P z 1 ,β 1 wired for all P ∈ G area z 1 ,β 1 .
As a consequence of the first item of Proposition 2.3, we know that the set of area-interaction measures G area (z, β) is never empty. From the last item of Proposition 2.3, the question of uniqueness of the area-interaction measure translates to the question of the equality of measures P z 1 ,β 1 f ree = P z 1 ,β 1 wired . The next Proposition is stating that this equality happens for a lot of parameters (z, β). Proposition 2.4. For all β > 0, the set {z > 0, P z,β f ree = P z,β wired } is at most countable.

The proof of this proposition is related to standard differentiability/convexity arguments of the pressure function. See for instance Theorem 3.34 in [START_REF] Friedli | Statistical mechanics of lattice systems[END_REF] for a proof in the case of Ising model. An direct adaptation for the area-interaction measure is achievable and omitted here for brevity.

Bicolor Widom-Rowlinson representation of area-interaction measures

The bicolor Widom-Rowlinson model is simply defined as the reunion of two Poisson Boolean models (with deterministic radii equal to 0.5) conditioned on a hard-core non overlapping condition between the two Boolean models. A formal definition using standard DLR formalism is given below.

Definition 2.2. Let ω := (ω 1 , ω 2 ) denotes a couple of configurations. Let A := {(ω 1 , ω 2 ) ∈ Ω 2 , B 1/2 (ω 1 ) ∩ B 1/2 (ω 2 ) = ∅} be the event of authorised (couple of ) configurations. Let π z 1 ,z 2 := π z 1 ⊗ π z 2 .
Then a probability measure P on Ω 2 is a Widom-Rowlinson measure with parameters z 1 , z 2 , written P ∈ G wr z 1 ,z 2 , if for every bounded Λ ⊆ R d and every bounded measurable function f ,

Ω 2 f dP = Ω 2 Ω 2 f (ω Λ ∪ ω Λ c ) 1 A (ω Λ ∪ ω Λ c ) Z(Λ, z 1 , z 2 , ω Λ c ) π z 1 ,z 2 Λ (dω Λ )P (dω), (2.10)
with Z(Λ, z 1 , z 2 , ω Λ c ) being the standard partition function associated to the Widom-Rowlinson interaction.

The following identifications between the bicolor and the one-color Widom-Rowlinson models are standard; the proof is omitted and we refer to [START_REF] Chayes | The analysis of the widomrowlinson model by stochastic geometric methods[END_REF][START_REF] Georgii | The random geometry of equilibrium phases[END_REF][START_REF] Widom | New model for the study of liquid-vapor phase transitions[END_REF] for details. Proposition 2.5.

• Let P ∈ G wr z 1 ,z 2 . Then the first marginal of P is an area-interaction measure of activity z = z 1 and inverse temperature β = z 2 . The analog is true for the second marginal.

• Let P ∈ G area z,β . Then the measure P := π β R d \B 1 (ω 1 ) (dω 2 )P (dω 1 ) is a Widom-Rowlinson measure: P ∈ G wr (z 1 = z, z 2 = β).

• The sets G area z,β and G wr (z 1 = z, z 2 = β) are in bijection. By symmetry of G wr (z 1 , z 2 ) with respect to z 1 , z 2 , the sets G area z,β and G area β,z are in bijection as well; this property is called the duality property.

Percolation

The theory of percolation studies the connectivity in random structures. Formally the percolation is defined as follows.

Definition 2.3. Let r > 0;

• two sets Λ 1 , Λ 2 ⊆ R d are said to be r-connected in ω, written Λ 1 ←→ Br(ω) Λ 2 (or Λ 1 ←→ r Λ 2 when there is no possible confusion) if B r (ω) has a connected component overlapping both Λ 1 and Λ 2 ;
• a configuration ω is said to r-percolate if the germ-grain structure B r (ω) has at least one unbounded connected component;

• a probability measure P on Ω is said to r-percolate (respectively do not percolate) if P ({ω r-percolates}) = 1 (respectively P ({ω r-percolates}) = 0).

In the next proposition we state the standard percolation phase transition of the Poisson Boolean model. See for instance [START_REF] Duminil-Copin | Subcritical phase of ddimensional Poisson-Boolean percolation and its vacant set[END_REF] for a modern proof. Proposition 2.6. For every r > 0, there exists 0 < z p c (r) < ∞, called rpercolation threshold of the Poisson Boolean model, such that • for every z < z p c (r), the measure π z does not r-percolate, and we have the existence of c := c(r, z) > 0 such that

π z 0 ←→ r ∂Λ n ≤ exp(-cn), (2.11)
where ∂Λ n is the boundary of the set

Λ n =] -n, n] d ;
• For every z > z p c (r), the measure π z r-percolates, and we have the existence of c := c (r) > 0 such that for z in a neighbourhood of z p c (r)

π z 0 ←→ r ∞ ≥ c (z -z p c (r)).
(2.12)

Concerning area-interaction measures such a behaviour is not proven, and is one of the questions investigated in this paper. But as a consequence of the Propositions 2.1, 2.3, 2.4 and 2.6 we have the existence of a non degenerate percolation threshold, common to all area-interaction measures. This is stated in the following Proposition.

Proposition 2.7. For all β > 0 and r > 0, there exists 0 < z a c (β, r) < ∞ such that

• for all z < z a c (β, r), any area-interaction measure P ∈ G area z,β almost never r-percolates, i.e P ({ω r-percolates}) = 0;

• for all z > z a c (β, r), any area-interaction measures P ∈ G area z,β almost surely r-percolates, i.e P ({ω r-percolates}) = 1.

Proof. The fact that both the free and wired measures have the same threshold is a consequence of Proposition 2. [START_REF] Baddeley | Variational estimators for the parameters of Gibbs point process models[END_REF]. Then, from the stochastic dominations (2.13), we have the following bound on the percolation threshold of the area-interaction measures: for all r > 0 and β ≥ 0,

z p c (r) ≤ z a c (β, r) ≤ z p c (r) exp(βv d ).

Results

Let us now present our results related to the two kinds of phase transition for the area-interaction measures. The proofs are given in the following sections.

Sharp phase transition of Percolation

The first result proves a sharp phase transition of percolation for the areainteraction measures in the spirit of Proposition 2.6 for the Boolean model. That means exponential decay of connectivity at distance n in the subcritical phase and a local linear lower bound on the supercritical case of the connection at infinity.

Theorem 1. Let β ≥ 0.

1. For all z < z a c (β, r), there exists α 1 = α 1 (z, β, d, r) > 0 such that for all P ∈ G area z,β and all n,

P 0 ←→ r ∂Λ n ≤ exp(-α 1 n). (3.1)
2. There exists α 2 = α 2 (β, d, r) such that for all z > z a c (β, r) small enough and all P ∈ G area z,β ,

P 0 ←→ r ∞ ≥ α 2 (z -z a c ). (3.2)
The proof of this theorem relies on the theory of randomised algorithms developed by Duminil-Copin, Raoufi and Tassion in a series of papers [START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and Potts models via decision trees[END_REF][START_REF] Duminil-Copin | Exponential decay of connection probabilities for subcritical Voronoi percolation in R d[END_REF][START_REF] Duminil-Copin | Subcritical phase of ddimensional Poisson-Boolean percolation and its vacant set[END_REF]. The main ingredient, and our main contribution with respect to watch was already done, is the proof of an OSSS-type inequality which gives a control of the variance of a function f by a bound depending on the influence of each point of the process. The proof of this inequality relies on a procedure, sampling an area-interaction configuration using a dominating Poisson configuration. This procedure is in some sense monotonic with respect to the domination Poisson configuration.

As usual the threshold z a c (β, r) is mainly intractable excepted when r ≥ 1/2 and β is large (see proposition below). We give also some qualitative properties of the function β → z a c (β, r). Proposition 3.1. For any r > 0, the function β → z a c (β, r) is a bi-Lipschitz increasing one-to-one map from R + to [z p c (r), +∞). Moreover for every r ≥ 1/2, there exists 0 < βr < ∞ such that for β > βr , then z a c (β, r) = β. The proofs of Theorem 1 and Proposition 3.1 are given in the following Section 4.

Sharp liquid-gas phase transition

The other question of interest is the Sharp liquid-gas phase transition for which there are several definitions based either on the regularity of the partition functions or the uniqueness/non uniqueness of Gibbs measures. Here we say that a sharp liquid-gas phase transition occurs at temperature 1/β if there exists only one value z such that the Gibbs measures are not unique. This phenomenon is conjectured for several models but there does no exist complete rigorous proof in the continuum. Here we improve existing results for the area-interaction measures.

Already known results

Several results are already known on this subject. First, it is well-known that the set of gibbs measures is generally reduced to a singleton when the parameters z or/and β are small enough (see for instance [START_REF] Ruelle | Statistical mechanics: Rigorous results[END_REF]). As a consequence of a recent disagreement percolation result [START_REF] Hofer-Temmel | Disagreement percolation for marked Gibbs point processes[END_REF], explicit bounds related to 1-percolation threshold of the Poisson Boolean model are given. Proposition 3.2. Recall that z p c (r) is the percolation threshold of the Poisson Boolean model of constant radii r. Then for every z < z p c (1) and every β ≥ 0, there is an unique area-interaction measure. Moreover, by duality, for every β < z p c (1) and every z ≥ 0, the uniqueness occurs as well.

In addition, a Fortuin-Kasteleyn representation and the percolation properties of the Continuum Random Cluster Model developed allow to prove a non uniqueness result of the symmetric bicolor Widom-Rowlinson model [START_REF] Chayes | The analysis of the widomrowlinson model by stochastic geometric methods[END_REF]. This result translates directly, thanks to Proposition 2.5, to a non uniqueness result of the area-interaction measure. Proposition 3.3. There exists z < ∞ such that for all z > z

• the measure P z,z wired does 1/2-percolate;

• the measure P z,z f ree does not 1/2-percolate;

hence we have P z,z f ree = P z,z wired , and therefore #G area (z, z) > 1. So in the symmetric case z = β, a standard phase transition is already known, where uniqueness is obtain at low activity z and non-uniqueness at large activity. However it is not proved that there exists a threshold between both regimes. As far as we know, this conjecture is still open today.

In the non-symmetric case z = β, very few is known expect from Proposition 3.2. In particular, the sharp phase transition around the symmetric case z = β was unknown. Our results in the next section solve partially this question.

[Je pense que la figure est inutile ici!!! La deuxième est super! ]

z β z p c (1)
•

Uniqueness z small Prop 3.2 z = β • z
Non uniqueness symmetric case Prop 3.3 ? ?

Figure 1: What is already proved

New results about uniqueness

It is conjectured that the non-uniqueness holds only for z = β larger than a certain threshold β * > 0. We do not solve this conjecture here but we show in corollary 3.1 that for β large enough the non-uniqueness holds only for z = β.

Actually we succeed to prove uniqueness in a larger domain drawn in Figure 3.2.2.

Our main Theorem, given below, ensures the uniqueness as soon as the area-measures do not 1-percolate.

Theorem 2. For all β ≥ 0 and z < z a c (β, 1), we have P z,β f ree = P z,β wired , and therefore there is uniqueness of the area-interacton measure. By duality the result holds also for all z ≥ 0 and β < z a c (β, 1).

The proof of this theorem relies on a generalisation of the disagreement percolation thechnique, relying on the construction of a coupling, called disagreement coupling comparing the influence of the boundary condition to a domination Poisson point process. Using the monotonicity of the area interaction, see Proposition 2.1, a better dominating measure is an area-interaction measure but with higher activity parameter. The dominating measure is not a Poisson point process and therefore the construction of the disagreement coupling is more elaborate, even though it still relies on the original and brillant idea of van den Berg and Meas [START_REF] Van Den Berg | Disagreement percolation in the study of Markov fields[END_REF]. The proof of Theorem 2 is done in Section

[?]. z β z p c (1)
•

z p c (1) • z = β • z Non uniqueness symmetric case Prop 3.3 β → z a c (β, 1)
uniqueness Th 2

Proofs related to Percolation results

In this section we give the proofs of Theorem 1 and Proposition 3.1 involving the sharp phase transition of percolation.

Proof of Theorem 1

First let us note that it is enough to prove Theorem 1 for the wired areainteraction measure P z,β wired . Indeed, recall that from Proposition 2.3 we have the sandwich domination: for all P ∈ G area z,β , P z,β f ree P P z,β wired . Therefore the equation (3.1), i.e. the exponential decay of connectivity when z < z a c (β, r), translates directly from P z,β wired to all P ∈ G area z,β . For the equation (3.2), consider z > z > z a c (β, r) such that P z ,β f ree = P z ,β wired . From Proposition 2.4 the parameter z can be considered as close to z as we need. But once again from Proposition 2.3 we have

P 0 ←→ r ∞ ≥ P z,β f ree 0 ←→ r ∞ ≥ P z ,β f ree 0 ←→ r ∞ = P z ,β wired 0 ←→ r ∞ ≥ α 2 (z -z a c ) -→ z →z α 2 (z -z a c ),
and Equation (3.2) is proved. Throught the remainder of this section the parameters β > 0 and r > 0 are fixed and might be omitted from notations and we consider only the wired case. Let µ z n := P z,β 3n+2,wired . We are considering, for r > 0 fixed, the connection probability

θ n (z) = µ z n (0 ←→ r ∂Λ n ), (4.1)
where ∂Λ n is the boundary of Λ n =] -n, n] d .

Remark 4.1. The term 3n + 2 was chosen for several reasons. First the term "+2" is there to ensure that the wired measure µ z n is well defined, even for n = 0. Second the factor 3 is there to ensure a good inclusion of boxes in (4.40). Lemma 4.1. For each z, the sequence (θ n (z)) converges and we have

θ(z) := lim n→∞ θ n (z) ≤ P z,β wired 0 ←→ r ∞ . (4.2)
Proof. The event {0 ←→ r ∂Λ n } depends only on the points inside Λ n+r , which is included in Λ 3n+2 as soon as n ≥ r/2. Therefore, using Proposition 2.2 we have for such n:

θ n (z) = P z,β 3n+2,wired (0 ←→ r ∂Λ n ) ≥ P z,β 3(n+1)+2,wired (0 ←→ r ∂Λ n ) ≥ P z,β 3(n+1)+2,wired (0 ←→ r ∂Λ n+1 ) = θ n+1 (z).
Hence the sequence is decreasing for n large enough and the convergence follows.

For the inequality, just notice that for any k

θ(z) = lim n→∞ θ n (z) ≤ lim n→∞ P z,β n,wired (0 ←→ r ∂Λ k ) = P z,β wired (0 ←→ r ∂Λ k ).
Letting k go to infinity yelds the result.

Recall that z a c = z a c (β, d, r) ∈]0, ∞[ is the percolation threshold of the wired (and free as well) area-interaction measure, defined in Proposition 2.7.

Theorem 3.

1. For all z < z a c , there exists

α 1 = α 1 (z, β, d, r) > 0 such that, for all n, θ n (z) ≤ exp(-α 1 n). ( 4

.3)

2. There exists α 2 = α 2 (β, d, r) such that for all z > z a c small enough,

θ(z) ≥ α 2 (z -z a c ). (4.4)
Before proving this theorem, let us see quickly how it leads to the proof of Theorem 1. The equation (4.4) together with Lemma 4.1 implies the equation (3.2), while (3.1) is a consequence of (4.3) and Proposition 2.2.

The proof of Theorem 3 relies on the theory of randomised algorithms popularized in [START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and Potts models via decision trees[END_REF][START_REF] Duminil-Copin | Exponential decay of connection probabilities for subcritical Voronoi percolation in R d[END_REF][START_REF] Duminil-Copin | Subcritical phase of ddimensional Poisson-Boolean percolation and its vacant set[END_REF]. First we are going to prove in Section 4.2 a generalized version of the OSSS inequality, first introduced in [START_REF] O'donnell | Every decision tree has an influential variable[END_REF], satisfied by P z,β Λ,ω Λ c for every cube Λ and every boundary condition ωΛ c . This inequality, valid in particular for the wired measure P z,β 3n+2,wired , is then applied in Section ?? in order to get the result.

OSSS inequality

Introduction of the formalism

In this section Λ is a fixed cube of side length c > 0. Let > 0 such that c/ is a positiv integer. We are dividing the box Λ into small cubes of size . Let t := (c/ ) d be the total number of such cubes. Therefore a configuration ω in Λ can be written as the collection (ω e ) e∈E where E := Λ ∩ (Z + 1/2) d and ω e := ω B e , where B e = e ⊕ ] -1/2, 1/2] d . For an enumeration (e 1 , . . . , e t ) of the cubes, we wrote e [i] = (e 1 , . . . , e i ) and ω e [i] = (ω e 1 , . . . , ω e i ).

Consider a boolean function f : Ω → {0, 1} and consider a decision tree T determining the value f (ω). A decision tree queries the configuration ω one cube after the other. Hence to a random configuration is associated a random ordering of the cubes e = (e 1 , . . . , e t ). It starts from a deterministic cube e 1 and looks at the configuration ω e 1 . Then it chooses a second cube e 2 depending on e 1 and ω e 1 and carries out. At step i > 1 the cubes e [i-1] have been visited and the configuration ω e [i-1] is known. The next cube e i to be explored is then expressed as a deterministic function of what have been already explored, i.e.

e i = φ i (e [i-1] , ω e [i-1] ). (4.5)
We then define the (random) time τ that the algorithme takes to determine the function f , meaning

τ (ω) = min{i ≥ 1, ∀ω , ω e [i] = ω e [i] ⇒ f (ω) = f (ω )}. (4.6)
Theorem 4 (OSSS inequality). If the function f is increasing, then

Var P z,β Λ, ωΛ c (f ) ≤ 2 e∈E δ(e, T )Cov P z,β Λ, ωΛ c (f, #ω e ) + O( d ), (4.7)
where δ(e, T ) := P z,β n (∃i ≤ τ, e i = e) is called the revealment of e and is the probability that the cube e is needed to determine the value of the function f .

Proof of Theorem 4

The original proof of the OSSS inequality, see [START_REF] O'donnell | Every decision tree has an influential variable[END_REF] or [START_REF] Duminil-Copin | Lectures on the Ising and Potts models on the hypercubic lattice[END_REF] for more probabilistic version, uses the product structure of the space considered (the Bernoulli percolation model). But µ z n is not a product measure and we need a more elaborate method. For this we will generalised the idea from [22, Lemma 2.1] which sampled a finite family of dependant random variables, one after the other in a random order, using independant uniform variables.

In the continuum setting of point processes this simple idea is much harder to implement. We will use ideas from the theory of stochastic domination. Indeed the stochastic domination P z,β Λ,ω Λ c ≺ π z Λ from Proposition 2.1 implies, using Strassen's Theorem, that a configuration ω ∼ P z,β Λ,ω Λ c can be obtained from a dominating Poisson configuration ω D ∼ π z Λ by a random thinning of the dominating configuration, deciding for each x ∈ ω D if it belongs to the thinned configuration ω. We will use an explicite form of the thinning probability proved in [START_REF] Hofer-Temmel | Disagreement percolation for marked Gibbs point processes[END_REF].

To formalize the thinning decision we are adding to each point of the configuration ω D an independent uniform mark between 0 and 1. The mark configuration is denoted by ω D,U and its law is simply a marked Poisson Point process.

Even if ω D,U is a marked onfiguration on Λ, the sampling procedure we will construct below needs as many dominating configurations ω D,U as there is cubes e i in Λ, meaning t. Let us wrote ω D,U,⊗ = (ω D,U,1 , . . . , ω D,U,t ). The law of ω D,U,⊗ is a product of marked Poisson point processes, which satisfies a classical FKG inequality for the order defined now. Definition 4.1. For two marked configuration ω D,U , ω D,U , we say that ω D,U is smaller than ω D,U , writing ω D,U ≤ ω D,U , if for each (x, u) ∈ ω D,U there is a (x, u) ∈ ω D,U with u ≤ u.

We write ω D,U,⊗ ≤ ω D,U,⊗ if for all i we have ω D,U,i ≤ ω D,U,i Proposition 4.1. Let e = (e 1 , . . . , e t ) be a random sequence such that for all i, ω D,U,i is independent of e [i] (in particular the construction (4.5) follows this property). Then there exists an increasing function F e such that

ω = F e (ω D,U,⊗ ) ∼ P z,β Λ,ω Λ c .
Proof. The configuration ω is sampled in each cube e i one by one. At each step i the configuration ω e [i-1] is already sampled and we are sampling ω e i . The sampling procedure is taken from [41, Proposition 4.1] which gives explicitly the thinning probability for sampling a Gibbs point process dominated by a Poisson point process.

We are going to sample a configuration (which we wrote ω ) on

∆ i := Λ \ (B e 1 ∪ • • • ∪ B e i-1 )
according to the specification P z,β

∆ i ,ω Λ c ∪ωe [i-1]
and then only keep the points inside B e i by setting ω e i = ω e i .

To sample ω we consider the dominating configuration ω D,U,i

∆ i
restricted to the region where we sample the configuration. Consider on ∆ i a lexicographic order, which orders the points (x, u) ∈ ω D,U,i ∆ i . The marks play no role in the ordering of the configuration. The configuration ω is then constructed inductively the following way:

• at the beginning of the induction we set ω = ∅.

• Then we consider each point (x, u) ∈ ω D,U,i ∆ i one after the other with respect to the lexicographic order, and if

p i x, ω ∪ ω e [i-1] ∪ ωΛ c ≤ u (4.8)
we add x to the configuration ω , i.e. ω = ω ∪ {x}. The function p i , whose expression comes from [41, Proposition 4.1], is defined as

p i (x, ω) = e -βH {x} (x∪ω) × Z area (z, β, ∆ i ∩]x, ∞[, ω ∪ x) Z area (z, β, ∆ i ∩]x, ∞[, ω) , (4.9)
where the interval ]x, ∞[ is defined with respect to the lexicographic ordering on ∆ i , and where ω is a configuration such that ω

∩ ∆ i ∩]x, ∞[= ∅.
And finally at the end we set ω e i = ω e i .

From the DLR equations (2.4), the assumption on e and the Proposition 4.1 from [START_REF] Hofer-Temmel | Disagreement percolation for marked Gibbs point processes[END_REF], the sampled configuration has the law of P z,β

∆ i ,ω Λ c ∪ωe [i-1]
.

In order to prove that F e is increasing, It remains to prove that p i (x, ω) is increasing in ω. Let us write ∆ x i := ∆ i ∩]x, ∞[. The function p i can be rewritten as

p i (x, ω) = z #γ e -βH ∆ x i (γ∪ω) e -βH {x} (x∪γ∪ω) π ∆ x i (dγ) Z area (z, β, ∆ x i , ω) (4.10)
= e -βH {x} (x∪γ∪ω) P z,β

∆ x i ,ω (dγ). (4.11)
Now using the fact that the integrated function is increasing in ω with the stochastic domination from Proposition2.1, we have that the function p e i is increasing in ω.

Remark 4.2. The Proposition 4.1 is the main improvement from the theory of randomized algorithm from Duminil-Copin, Raoufi and Tassion [START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and Potts models via decision trees[END_REF][START_REF] Duminil-Copin | Exponential decay of connection probabilities for subcritical Voronoi percolation in R d[END_REF][START_REF] Duminil-Copin | Subcritical phase of ddimensional Poisson-Boolean percolation and its vacant set[END_REF].

Considering the assumption on the e, the Proposition 4.1 applies in particular when e is independante of ω D,U,⊗ or when e is constructed from ω D,U,⊗ as in (4.5).

The proof of Proposition 4.1 relies only on the fact that the function H {x} (x∪ ω), often called local energy, is

• uniformly bounded from below;

• decreasing with respect to ω. So Proposition 4.1, and more generally Theorem 3 would trivially generalised to every Gibbs measure whose interaction satisfies those two properties. While the first property is a standard property satisfied by most interaction consider in the literrature, the second property, related to the monotony of the Gibbs specification, is less common. To the best of our knowledge the area-interaction is the only interaction considered in the literrature which satisfies this property. Now consider two independent configurations ω D,U,⊗ and ω D,U,⊗ . The random ordering of cubes e = (e 1 , . . . , e t ) consider starting now is constructed from ω D,U,⊗ with (4.5). We write ω = F e (ω D,U,⊗ ) and ω = F e ( ω D,U,⊗ ). Thanks to Proposition 4.1, those are two realisations of P z,β Λ,ω Λ c , which are independent even though they are constructed from the same e. Now wrote for i ≤ τ = τ (ω) γ i = F e ( ω D,U,1 , . . . , ω D,U,i , ω D,U,i+1 , . . . , ω D,U,τ , ω D,U,τ +1 , . . . , ω D,U,t ).

Then we have

Var P z,β Λ, ωΛ c [f ] ≤ E P z,β Λ, ωΛ c [ |f -E P z,β Λ, ωΛ c [f ]| ] = E[ |f (γ 0 ) -E[f (γ τ )]| ], (4.12)
where in the right hand side the expectation are with respect to the two independent marked Poisson realisations ω D,U,⊗ and ω D,U,⊗ from which the γ i are constructed. Then

Var P z,β Λ, ωΛ c [f ] ≤ E[ |E f (γ 0 )|ω D,U -E f (γ τ )|ω D,U | ] (4.13) ≤ E[ |f (γ 0 ) -f (γ τ )| ] (4.14) ≤ i=1...t E[ |f (γ i ) -f (γ i-1 )| 1 i≤τ ] (4.15) = i=1...t e∈E E[E[|f (γ i ) -f (γ i-1 )| | ω D,U,1...i-1 ] 1 i≤τ 1 e i =e ], (4.16)
where ω D,U,1...i-1 is a short-hand for (ω D,U,1 , . . . , ω D,U,i-1 ).

Lemma 4.2. On the events {i ≤ τ } and {e i = e} we have

E[|f (γ i ) -f (γ i-1 )| | ω D,U,1...i-1 ] ≤ 2Cov P z,β Λ, ωΛ c (f, #ω e ) + O( 2d
|f (γ i ) -f (γ i-1 )| ≤ (f (γ i ) -f (γ i-1 )) 1 #γ i e ≥1 -1 #γ i-1 e ≥1 + 1 #(γ i e ∪γ i-1 e )>1 , (4.21)
where we used the monotonicity of Proposition 4.1 and the fact that if #γ i-1 e = #γ i e = 0, then #γ i-1 = #γ i . But using the stochastic domination we have the following easy bound

E 1 #(γ i e ∪γ i-1 e )>1 | ω D,U,1...i-1 ≤ π 2z (#ω e > 1) = O( 2d ). (4.22)
Using Proposition 4.1 we have

E f (γ i-1 )1 #γ i-1 e ≥1 | ω D,U,1...i-1 = E f (γ i )1 #γ i e ≥1 | ω D,U,1...i-1 (4.23) = E P z,β Λ, ωΛ c [f (ω)1 ωe≥1 ] (4.24)
Now since the functions f (γ i-1 ) and 1 γ i e ≥1 are increasing, we are using the second part of Proposition 4.1 and the FKG inequality to obtain

E f (γ i-1 )1 #γ i e ≥1 | ω D,U,1...i-1 ≥ E f (γ i-1 ) | ω D,U,1...i-1 E 1 #γ i e ≥1 | ω D,U,1...i-1 (4.25) = E P z,β n [f (ω)] E P z,β n [1 ωe≥1 ] . (4.26)
The same is true for

E f (γ i )1 #γ i-1 e ≥1 | ω D,U e [i-1] ≥ E P z,β n [f (ω)] E P z,β n [1 ωe≥1 ] , (4.27)
and therefore

Var P z,β Λ, ωΛ c (f ) ≤ 2 e∈E δ(e, T )Cov P z,β Λ, ωΛ c (f, 1 ωe≥1 ) + O( d ) (4.28) ≤ 2 e∈E δ(e, T )Cov P z,β Λ, ωΛ c (f, #ω e ) + O( d ), (4.29)
where the last inequality comming from the use of the FKG inequality applied to f and #ω e -1 ωe≥1 . The result is proved

Proof of Theorem 3

We are now going to prove Theorem 3 by applying Theorem 4. We need the following classic Lemma in the theory of randomised algorithms (see Lemma 3 in [START_REF] Duminil-Copin | Exponential decay of connection probabilities for subcritical Voronoi percolation in R d[END_REF] for instance).

[Il me semble que l'on peut importer directement ce lemme] Lemma 4.3. Consider a converging sequence of increasing differentiable function

g n : [z min , z max [→ [0, 1] satisfying for all n ≥ 1 g n (z) ≥ α 3 n Σ n (z) g n (z), (4.30)
where c > 0 is a positive constant and

Σ n (z) = n-1 i=0 g i (z).
Then there exists z ∈ [z min , z max ] such that:

• For every z < z, there exists c 1 := c 1 (z) > 0 such that for all n,

g n (z) ≤ exp(-c 1 n). (4.31)
• For every z > z,

g(z) := lim n→∞ g n (z) ≥ α 3 (z -z) . (4.32)
Therefore in order to prove Theorem 3 it is sufficient to prove that the functions g n = θ n satisfies the assumptions of Lemma 4.3. By the construction the functions z → θ n (z) are increasing. The following lemma proves the differentiability of the functions θ n . Lemma 4.4. For all functions f and all z > 0,

d dz P z,β n,wired (f ) = 1 z Cov P z,β n,wired (f, #ω). (4.33)
The proof of this result is done in the annex Section 6. Therefore the only remaining task in order to prove Theorem 3 is to prove that the functions θ n satisfy (4.30). This is done using the OSSS inequality (4.7) to the wired measure P z,β n,wired for the Boolean function f (ω) = 1 0←→ r ∂Λn (ω) to the well chosen algorithms from [START_REF] Duminil-Copin | Exponential decay of connection probabilities for subcritical Voronoi percolation in R d[END_REF][START_REF] Duminil-Copin | Subcritical phase of ddimensional Poisson-Boolean percolation and its vacant set[END_REF]. where the sum goes over all ẽ such that the cube B ẽ contains a point a distance exactly r from B e . The cardinality of this summation set is bounded by a constant α1 depending on the radius r and the dimension d.

This proposition uses the now standard algorithms used in [START_REF] Duminil-Copin | Lectures on the Ising and Potts models on the hypercubic lattice[END_REF][START_REF] Duminil-Copin | Exponential decay of connection probabilities for subcritical Voronoi percolation in R d[END_REF][START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and Potts models via decision trees[END_REF]. The proof of Proposition 4.2 is done in Section 6. The particular bound (4.34) is inspired by [START_REF] Duminil-Copin | Subcritical phase of ddimensional Poisson-Boolean percolation and its vacant set[END_REF], where the use of the neighbouring cubes prevents for augmenting the size of cube from B ẽ to B ẽ ⊕ B(0, r).

Cette phrase n'est pas clair, on va voir si tu comprends un truc Using Theorem 4 and summing over s between 0 and n -1 we get

θ n (z)(1 -θ n (z)) ≤ 1 n e∈E n-1 s=0 ẽ µ z n (B ẽ ←→ r ∂Λ s )ds Cov µn (f, #ω e ) + O( d ), (4.35)
and letting goes to 0 we obtain

θ n (z)(1 -θ n (z)) ≤ 1 n e∈E ẽ n-1 s=0 µ z n (ẽ ←→ r ∂Λ s )ds Cov µn (f, #ω e ). (4.36)
But we have is the only step of the proof of Theorem 3 where we used the fact that µ z n is a wired area-interaction measure. The purpose of summing up to n/2 is to ensure that Λ 3s+2 (ẽ) ⊆ Λ 3n+2 and therefore µ z n µ z s . This is why we considered Λ 3n+2 in the definition on µ z n .

n-1 s=0 µ z n (ẽ ←→ r ∂Λ s )ds ≤ n-1 s=0 µ z n (ẽ ←→ r ∂Λ s-||ẽ|| (ẽ))ds (4.37) ≤ 2 n-1 s=0 µ z n (ẽ ←→ r ∂Λ s (ẽ))ds (4.
Hence from (4.36), (4.40) and Lemma 4.4 we obtain

θ n (z)(1 -θ n (z)) ≤ 4 Σ n (z) n e∈E Cov µn (f, #ω e ) ≤ 4 Σ n (z) n Cov µn (f, #ω) (4.41) ≤ 4 Σ n (z) n zθ n (z). (4.42)
Now consider 0 < z min < z a c < z max . Then for z min ≤ z ≤ z max we have that 1 -θ n (z) ≥ c > 0, where c can be chosen uniformly in n. Therefore

θ n (z) ≥ c 4z max n Σ n (z) θ n (z)
and equation (4.30) is fulfilled. From Lemma 4.3 we get the existence of a threshold z but from the conclusion of Lemma 4.3 this threshold has to be the percolation threshold z a c . The proof of Theorem 3 is finished.

Proof of proposition 3.1

We are in this section proving proposition 3.1, meaning the regularity of the function β → z a c (β, r) and that for all r ≥ and β large enough (depending on r) we have z a c (β, r) = β.

Regularity of the threshold

The regularity of the function β → z a c (β, r) is a consequence of uniform properties of the Papangelou intensity of the area-interaction measure which is defined for every x ∈ R d and ω ∈ Ω by

γ z,β (x, ω) = ze -βH {x} (ω∪x) = ze -βL d (B 1 (x)\B 1 (ω)) .
Let z, β be in compact set and u be in [0, 1] then by simple geometric considerations there exists a positive constant c > 0 such that

γ z+u,β+cu (x, ω ) ≥ γ z,β (x, ω),
for any ω ⊂ ω . Then we deduce by Theorem 1.1 in [START_REF] Georgii | Stochastic comparison of point random fields[END_REF] that the areainteraction measure with parameter z + u, β + cu stochastically dominates the area-interaction measure with parameter z, β. By duality that proves also that the area-interaction measure with parameter z, β stochastically dominates the area-interaction measure with parameter z +uc, β +u. Involving the percolation threshold z a c (β, r), that implies that for β in a compact set and for h > 0 small enough c < z a c (β + h, r) -z a c (β + h, r) < 1/c. It is sufficient to claim that the function β → z a c (β + h, r) is locally bi-Lipschitz increasing one-to-one map. The global property is a consequence of the explicit value of z a c (β + h, r) for β large enough.

Explicit value for the threshold

To prove this result we are going to use the well-known Fortuin Kasteleyn representation of Widom-Rowlinson measures (and therefore area-interaction measures as well) by Continuum Random Cluster Measures, defined as follows.

Definition 4.2. A stationary measure P crc on Ω is a Continuum Random Cluster measure of activity z, if for all bounded Λ ⊆ R d and all bounded measurable function f ,

f dP crc = f (ω Λ ∪ ω Λ c ) 2 N Λ cc (ω Λ ∪ω Λ c ) Z crc (z, Λ, ω Λ c ) π z Λ (dω Λ )P crc (dω), (4.43)
where

N Λ cc (ω) = lim ∆→R d N cc (ω ∆ ) -N cc (ω ∆\Λ ) with N cc (ω ∆ )
counting the number of connected components of B 1/2 (ω ∆ ), and Z crc (z, Λ, ω Λ c ) being the standard non-degenarate partition function.

The existence of Random Cluster measures for every activity z was proved in [START_REF] Dereudre | Infinite volume continuum random cluster model[END_REF]. In [START_REF] Houdebert | Percolation results for the Continuum Random Cluster Model[END_REF] the author proves that for z > z every Continuum Random Cluster measures 1/2-percolates, which as a consequence gives the non uniqueness of area-interaction measures in the symmetric case for z = β > z. The other consequence is that z a c (β, 1/2) = β for β > z. The result is therefore already proved for r = 1/2. Remark 4.4. In the setting of the present article, the percolation result of [START_REF] Houdebert | Percolation results for the Continuum Random Cluster Model[END_REF] is trivial since there is a sandwich domination inequality between Continuum Random Cluster measures and two Poisson point processes. The hard work of [START_REF] Houdebert | Percolation results for the Continuum Random Cluster Model[END_REF] was to consider random unbounded radii.

From now on let z > z and let P be a Continuum Random Cluster measure of activity z, which is therefore 1/2-percolating by the choice of z. Consider the measure P thin obtained from P crc by removing all points x ∈ ω belonging to the (unique) infinite component of B 1/2 (ω). Then as a consequence of the Fortuin-Kasteleyn representation, stated for instance in [START_REF] Houdebert | Percolation results for the Continuum Random Cluster Model[END_REF], we have the following domination:

P z,z f ree P thin . (4.44)
By construction P thin does not 1/2-percolates, since the 1/2 infinite connected component was removed. We are proving in the following lemma that by considering z large enough, this removed infinite connected component prevents r-percolation in P thin .

Lemma 4.5. There exists z ≥ z such that for all z > z, P thin does not rpercolate.

This lemma implies that z a c (β, 1) ≥ β for β > z, which together with the fact that z a c (β, 1) ≤ z a c (β, 1/2) ≤ β, proves Theorem ??.

Proof of Lemma 5.1. We are divinding the space R d into unit squares C y := y⊕] -1/2, 1/2] d , with y ∈ Z d . We are going to prove that a lot of cubes are filled by balls from B 1/2 (ω), where ω ∼ P crc . By applying a well-known result from Liggett,Schonmann and Stacey [START_REF] Liggett | Domination by product measures[END_REF], the infinite connected component of B 1/2 (ω) will be (for large activities) very thick. This will prevent P thin to 1-percolate. By stationarity of P crc we are only considering the conditional probability

p z (ω Λ c 0 ) = P crc (C 0 ⊆ B 1/2 (ω)|ω Λ c 0 ), (4.45)
where Λ 0 :=] -2, 2] d . For C 0 to be 1/2-covered, it is sufficent that ω has enough nicely placed points. Therefore we are once again diving C 0 into smaller cubes Ci , 1 ≤ i ≤ k, of side smaller than 1 2 √

d . If all those small cubes contains a point then C 0 will be 1/2-covered. Using the union bound we have

1 -p z (ω Λ c 0 ) ≤ k i=1 P crc (#ω Ci = 0|ω Λ c 0 ). (4.46)
But

P crc (#ω Ci = 0|ω Λ c 0 ) = 1 #ω Ci =0 2 N Ci cc (ω Ci ∪ω Λ 0 \ Ci ∪ω Λ c 0 ) Z crc z, Ci , ω Λ 0 \ Ci ∪ ω Λ c 0 ) π z Ci (dω Ci )P crc (dω Λ 0 |ω Λ c 0 ) (4.47) = e -zL d ( Ci ) Z crc z, Ci , ω Λ 0 \ Ci ∪ ω Λ c 0 ) P crc (dω Λ 0 |ω Λ c 0 ), (4.48)
where we used (5.9) and the fact that N Ci cc (∅) = 0. But we also have the following bound on N Ci cc (ω):

#ω Ci (1 -c d ) ≤ N Ci cc (ω) ≤ #ω Ci , (4.49)
where c d is the kissing number in dimension d, which is always larger than 2. This implies

P crc (#ω Ci = 0|ω Λ c 0 ) ≤ e -zL d ( Ci ) e zL d ( Ci )(1-2 1-c d ) = e -2 1-c d zL d ( Ci ) , (4.50)
and therefore

p z (ω Λ c 0 ) ≥ 1 -ke -2 1-c d zL d ( Ci ) . (4.51)
The bound (5.17) is uniform in ω Λ c 0 and goes to 1 as z goes to infinity. Therefore by applying the result of Liggett Schonmann and Stacey [START_REF] Liggett | Domination by product measures[END_REF], we have for z large enough that the set of completely covered (by balls of radii 1/2) boxes C y does percolate, with respect to site percolation on the standard square latice Z d . Let C inf inite be the set of sites y ∈ Z d belonging to the infinite connected component. Now consider the set C f inite = Z d \ C inf inite of sites y ∈ Z d not belonging to the infinite cluster of this site percolation. By considering z large enough we have that C f inite only contains bounded connected components, with respect to the site percolation. Now consider the configuration ω f inite obtained from ω by removing the points x in the infinite connected component of B 1/2 (ω). By construction we have that B 1/2 (ω f inite ) is entirely contained in the cubes C y for y ∈ C f inite . Now consider what happens if we double the radii from 1/2 to 1. The ω f inite cannot 1-percolates, because otherwise the the infinite connected component would need to cross cubes from C inf inite , which is not possible by construction.

This implies that z large enough the measure P thin does not 1-percolate.

5 Preuve des résultats d'unicités

Proof of Theorem 2

The proof relies on a disagreement coupling method. In this section we are considering z < z a c (β, r), and we want to prove P z,β f ree = P z,β wired . To do so we are considering z < z < z a c (β, r) such that P z ,β f ree = P z ,β wired . Such a z exists thanks to Proposition 2.4.

Definition 5.1. A disagreement coupling family

P dcf Λ,ω 1 Λ c ,ω 2 Λ c ,ω D Λ c
index by a bounded Λ ⊆ R d and three configurations satisfying

ω 1 Λ c ⊆ ω 2 Λ c ⊆ ω D Λ c
, is a coupling of three marginals, with canonical variables ξ 1 , ξ 2 and ξ 3 , satisfying

∀1 ≤ i ≤ 2 : P dcf Λ,ω 1 Λ c ,ω 2 Λ c ,ω D Λ c (ξ i = dω ) = P z,β Λ,ω i Λ c
(dω ) (5.1a)

P dcf Λ,ω 1 Λ c ,ω 2 Λ c ,ω D Λ c (ξ 3 = dω ) = P z ,β Λ,ω D Λ c
(dω ) (5.1b)

P dcf Λ,ω 1 Λ c ,ω 2 Λ c ,ω D Λ c (ξ 1 ∪ ξ 2 ⊆ ξ 3 ) = 1 (5.1c) P dcf Λ,ω 1 Λ c ,ω 2 Λ c ,ω D Λ c ∀x ∈ ξ 1 ξ 2 |B 1 (x) ←→ B 1 (ξ 3 ) B 1 (ω D Λ c ) = 1 (5.1d) Proposition 5.1.
If there exists a disagreement coupling family, then P z,β f ree = P z,β wired .

Proof. Let E be an event, that without loss of generality, only depends on the configurations inside a given bounded Λ. Then for Λ ⊆ Λ n we have

|P z,β f ree (E) -P z,β wired (E)| (5.2) ≤ |P z,β Λn,ω 1 Λ c (E) -P z,β Λn,ω 2 Λ c
(E)|P z,β f ree (dω 1 )P z,β wired (dω 2 ).

≤ ω 1 ⊆ω 2 ⊆ω D P dcf Λn,ω 1 Λ c n ,ω 2 Λ c n ,ω D Λ c n Λ ←→ B 1 (ξ 3 ) Λ n-2 φ 1 ω 2 (ω 1 )φ 2 ω D (ω 2 )P z ,β wired (dω D ), (5.6)
where the last inequality comes from the existence of the disagreement coupling family and the property (5.1d). Therefore

|P z ,β f ree (E) -P z ,β wired (E)| ≤ P z ,β Λn,ω D Λ c Λ ←→ B 1 (ξ) Λ n-2 P z ,β wired (dω D ) (5.7) = P z ,β wired Λ ←→ B 1 (ξ) Λ n-2 -→ n→∞ 0, (5.8)
where the convergence is a consequence of z < z a c (β, r).

It remains to prove the existence of the disagreement coupling family .

Theorem 5. There exists a disagreement coupling family P dcf

Λ,ω 1 Λ c ,ω 2 Λ c ,ω D Λ c
.

The construction of the disagreement coupling family is a generalisation of the one made in [START_REF] Hofer-Temmel | Disagreement percolation for marked Gibbs point processes[END_REF], where in the case the dominating measure is a Poisson point process. The coupling is sampled starting from the balls close to the boundary of Λ, and going inductively inside Λ.

Proof. The coupling is constructed inductively. Recall that Λ ⊆ R d is bounded and that ω 1

Λ c ⊆ ω 2 Λ c ⊆ ω D Λ c . Define the disagreement zone Γ = {x ∈ Λ, ||x; ω D Λ c || ≤ 2}
as the region where a point x of the point process would be directly 1-connected to the dominante boundary condition ω D Λ c (i.e. the ball B 1 (x) would overlap B 1 (y) for at least one y ∈ ω D Λ c ). The induction will be made with respect to the disagreement zone Γ in the following way.

• If Γ = ∅, let us first sample ξ 3 ∼ P z ,β Λ,ω D Λ c
. We are then sampling

ξ 2 ∼ P z,β Λ,ω 2 Λ c
as a thinning of ξ 3 . Then sample

ξ 1 ∼ P z,β Λ,ω 1 Λ c as a thinning of ξ 2 .
This procedure is possible, since the conditions z < z and

ω 1 Λ c ⊆ ω 2 Λ c ⊆ ω D
Λ c implies, thanks to Proposition 2.1, the following domination:

P z,β Λ,ω 1 Λ c P z,β Λ,ω 2 Λ c P z ,β Λ,ω D Λ c
.

From ξ 1 , ξ 2 and ξ 3 we are only keeping the points inside Γ. The induction then goes on with Λ ← Λ ∩ Γ c with the new boundary conditions

ω 1 Λ c ∪Γ = ω 1 Λ c ∪ ξ 1 Γ , ω 2 Λ c ∪Γ = ω 2 Λ c ∪ ξ 2 Γ and ω D Λ c ∪Γ = ω D Λ c ∪ ξ 3 Γ .
• If Γ = ∅. This is the terminal step of the induction. In this case we have

P z,β Λ,ω 1 Λ c = P z,β Λ,ω 2 Λ c
. Therefore we simply sample ξ 3 ∼ P z ,β

Λ,ω D Λ c
, and then

sample ξ 2 = ξ 1 ∼ P z,β Λ,ω 2 Λ c
as a thinning of ξ 3 .

It is easy to see that the induction terminates almost surely. Indeed if at one step the sampled configuration ξ 3 is empty (which happens with positive bounded from below probability) then at the following step we will have Γ = ∅. Therefore the number of steps is dominated by a geometric random variable, which is almost surely finite.

Finally the construction ensures that all properties of (5.1) are fullfilled.

aucune idée du titre

So to conclude the proof of Theorem ??, it remains to prove the following lemma.

Proposition 5.2. There exists 0 < β < ∞ such that for β > β, we have z a c (β, 1) = β.

Proof. To prove this result we are going to use the well-known Fortuin Kasteleyn representation of Widom-Rowlinson measures (and therefore area-interaction measures as well) by Continuum Random Cluster Measures, defined as follows.

Definition 5.2. A stationary measure P crc on Ω is a Continuum Random Cluster measure of activity z, if for all bounded Λ ⊆ R d and all bounded measurable function f ,

f dP crc = f (ω Λ ∪ ω Λ c ) 2 N Λ cc (ω Λ ∪ω Λ c ) Z crc (z, Λ, ω Λ c ) π z Λ (dω Λ )P crc (dω), (5.9)
where N Λ cc (ω) = lim ∆→R d N cc (ω ∆ ) -N cc (ω ∆\Λ ) with N cc (ω ∆ ) counting the number of connected components of B 1/2 (ω ∆ ), and Z crc (z, Λ, ω Λ c ) being the standard non-degenarate partition function.

The existence of Random Cluster measures for every activity z was proved in [START_REF] Dereudre | Infinite volume continuum random cluster model[END_REF]. In [START_REF] Houdebert | Percolation results for the Continuum Random Cluster Model[END_REF] the author proves that for z > z every Continuum Random Cluster measures 1/2-percolates, which as a consequence gives the non uniqueness of area-interaction measure in the symmetric case for z = β > z.

Remark 5.1. In the setting of the present article, the percolation result of [START_REF] Houdebert | Percolation results for the Continuum Random Cluster Model[END_REF] is trivial since there is a sandwich domination inequality between Continuum Random Cluster measures and two Poisson point processes. The hard work of [START_REF] Houdebert | Percolation results for the Continuum Random Cluster Model[END_REF] was to consider random unbounded radii.

From now on let z > z and let P be a Continuum Random Cluster measure of activity z, which is therefore 1/2-percolating by the choice of z. Consider the measure P thin obtained from P crc by removing all points x ∈ ω belonging to the (unique) infinite component of B 1/2 (ω). Then as a consequence of the Fortuin-Kasteleyn representation, stated for instance in [START_REF] Houdebert | Percolation results for the Continuum Random Cluster Model[END_REF], we have the following domination: P z,z f ree P thin .

(5.10)

By construction P thin does not 1/2-percolates, and from the following lemma it will not 1-percolate for z large enough.

Lemma 5.1. There exists z ≥ z such that for all z > z, P thin does not 1percolate.

Before proving the lemma let us see how it leads to the wanted result. With (5.10) it implies that P z,z f ree does not 1-percolate and therefore z a c (β, 1) ≥ β for β > z. But since β > z we also know that P z,z wired is 1/2-percolating, implying that z a c (β, 1) ≤ z a c (β, 1/2) ≤ β, and therefore z a c (β, 1) = β. Remark 5.2. The following proof extends naturally to every radius r ≥ 1/2. This means that all percolation threshold z a c (β, r) are blalvla je ne sais pas comment le dire.

Proof of Lemma 5.1. We are divinding the space R d into unit squares C y := y⊕] -1/2, 1/2] d , with y ∈ Z d . We are going to prove that a lot of cubes are filled by balls from B 1/2 (ω), where ω ∼ P crc . By applying a well-known result from Liggett,Schonmann and Stacey [START_REF] Liggett | Domination by product measures[END_REF], the infinite connected component of B 1/2 (ω) will be (for large activities) very thick. This will prevent P thin to 1-percolate.

By stationarity of P crc we are only considering the conditional probability

p z (ω Λ c 0 ) = P crc (C 0 ⊆ B 1/2 (ω)|ω Λ c 0 ), (5.11)
where Λ 0 :=] -2, 2] d . For C 0 to be 1/2-covered, it is sufficent that ω has enough nicely placed points. Therefore we are once again diving C 0 into smaller cubes Ci , 1 ≤ i ≤ k, of side smaller than 1 2 √

d . If all those small cubes contains a point then C 0 will be 1/2-covered. Using the union bound we have

1 -p z (ω Λ c 0 ) ≤ k i=1 P crc (#ω Ci = 0|ω Λ c 0 ). (5.12) But P crc (#ω Ci = 0|ω Λ c 0 ) = 1 #ω Ci =0 2 N Ci cc (ω Ci ∪ω Λ 0 \ Ci ∪ω Λ c 0 ) Z crc z, Ci , ω Λ 0 \ Ci ∪ ω Λ c 0 ) π z Ci (dω Ci )P crc (dω Λ 0 |ω Λ c 0 ) (5.13) = e -zL d ( Ci ) Z crc z, Ci , ω Λ 0 \ Ci ∪ ω Λ c 0 ) P crc (dω Λ 0 |ω Λ c 0 ), (5.14)
where we used (5.9) and the fact that N Ci cc (∅) = 0. But we also have the following bound on N Ci cc (ω):

#ω Ci (1 -c d ) ≤ N Ci cc (ω) ≤ #ω Ci , (5.15)
where c d is the kissing number in dimension d, which is always larger than 2. This implies

P crc (#ω Ci = 0|ω Λ c 0 ) ≤ e -zL d ( Ci ) e zL d ( Ci )(1-2 1-c d ) = e -2 1-c d zL d ( Ci ) , (5.16)
and therefore p z (ω Λ c 0 ) ≥ 1 -ke -2 1-c d zL d ( Ci ) .

(5.17)

The bound (5.17) is uniform in ω Λ c 0 and goes to 1 as z goes to infinity. Therefore by applying the result of Liggett Schonmann and Stacey [START_REF] Liggett | Domination by product measures[END_REF], we have for z large enough that the set of completely covered (by balls of radii 1/2) boxes C y does percolate, with respect to site percolation on the standard square latice Z d . Let C inf inite be the set of sites y ∈ Z d belonging to the infinite connected component. Now consider the set C f inite = Z d \ C inf inite of sites y ∈ Z d not belonging to the infinite cluster of this site percolation. By considering z large enough we have that C f inite only contains bounded connected components, with respect to the site percolation. Now consider the configuration ω f inite obtained from ω by removing the points x in the infinite connected component of B 1/2 (ω). By construction we have that B 1/2 (ω f inite ) is entirely contained in the cubes C y for y ∈ C f inite . Now consider what happens if we double the radii from 1/2 to 1. The ω f inite cannot 1-percolates, because otherwise the the infinite connected component would need to cross cubes from C inf inite , which is not possible by construction.

This implies that z large enough the measure P thin does not 1-percolate. • Let X 0 = ∅ ⊆ (Z + 1/2) d ∩ Λ n+1 , representing the already explored cubes (which is empty at the beginning of the algorithm);

• let Z 0 = ∂Λ s the region known to be covered by r-balls (from the already explored cubes);

• let Z0 = ∅ ⊆ R d the region known not to be covered by r-balls.

At the step i + 1 of the algorithm we take a cube e ∈ (Z + 1/2) d ∩ Λ n+1 \ X i such that:

1. ||B e ; Z i || ≤ r: this cube can be connected to the already explored connected components of ∂Λ s (recall that ||A; B|| stands for the distance between the two bounded subset A, B of R d );

2. (Z i ∪ Zi ) ∩ B e = B e : the covering of the cube is not entirely known.

If no such e exists, then the algorithm stops as the connected components of ∂Λ s have been entirely explored. If there is several e satisfying the condition, we choose one using a deterministic (but not important) rule.

We then look at the cubes around B e (starting from the closest) until the state of all points of B e (covered by a r-ball or not) is known. Those are the cubes revealed by the algorithm.

Then we set • X i+1 = X i ∪ {e};

• Z i+1 = Z i ∪ {new points known to be covered by a r-ball};

• Zi+1 = Zi ∪ {new points known NOT to be covered by a r-ball}.

The bound (4.34) from Proposition 4.2 follows directly from the definition of the algorithms. i . From (4.30) we obtain

T n (z) ≥ α 3 z log n n i=1 g i (z) Σ i (z) ≥ α 3 log Σ n+1 (z) -log Σ 1 (z) z log n , (6.4)
where the last inequality uses the standard comparison

g i Σ i ≥ Σ i+1 Σ i dt t = log Σ i+1 -log Σ i . (6.5)
For z ∈]z, z[ and integrating (6.4) between z and z yields

and letting z go to z, we obtain the expected result.

Proof of Lemma 4.4

We have P z,β n,wired (f ) = f (ω)

z #ω h(ω)
Z area (z,β,n,wired) π Λn (dω), where h(ω) := e -βL d (B(ω Λn ,1)∩Λ n-1 ) .

Using a standard derivative theorem we obtain

Figure 2 :

 2 Figure 2: J'ai beaucoup galéré pour cette figure, et je la trouve un peu suggestive mais c'est peut-être mon esprit mal tourné...[LOL non ca va!!!! Peut-on mettre un point d'interogation dans la zone du milieu?] Uniqueness/non-uniqueness regimes for the area-interaction measures with parameters z, β.

Proposition 4 . 2 .

 42 For 0 ≤ s ≤ n, there exists an algorithm T s such that we have δ(e, T ) = 0 if e ∈ Λ n+1 and otherwiseδ(e, T s ) ≤ ẽ µ z n B ẽ ←→ r ∂Λ s ,(4.34)

rRemark 4 . 3 .

 43 s stands for the floor function of the absolute value of s, and where Λ s (e) = e ⊕ Λ s . Now using the stochastic domination between wired measures µ ∂Λ s )ds ≤ 4Σ n (z). (4.40) Equation (4.40)

(5. 3 )1 ω 2

 32 Now by the stochastic domination P z,β f ree P z,β wired P z ,β wired , we have from Strassen's theorem the existence of thinning probabilities φ 1 ω 2 , φ 2 ω D such that |P z ,β f ree (E) -P z ,β wired ((ω 1 )φ 2 ω D (ω 2 )P z ,β wired (dω D ) (5.5)

2 Definition 6 . 1 (

 261 Let us define the algorithm T s which explores the r-connected components of ∂Λ s . Definition of the algorithm).

6. 2 ••

 2 Proof ofLemma 4.3 Let z := inf{z ≥ z min , lim sup n→∞ Let z < z. Then (4.30) transforms intog n (z) ≥ α 3 z n Σ n (z)g n (z). (6.1)Fix δ > 0 and consider z = z -δ and z = z -2δ. We are going to prove the exponential decay at z . First by definition of z there exists N and α such that Σ n (z) ≤ n 1-α for n ≥ N . Integrating the inequality g n ≥ α 3 z n αg n between z and z yields that for n ≥ N ,g n (z ) ≤ exp(-δ α 3 z n α). (6.2)This implies that for all n, Σ n (z ) is bounded by a quantity Σ < ∞. By the monotonicity of the functions g n , integrating the inequality g n ≥ α 3 z n Σ g n between z and z gives the wanted result: for all n g n (z ) ≤ exp(-Let z > z. Define T n (z) := 1 log n n i=1 g i (z)

T

  n (z) -T n (z ) ≥ α 3 log z z log Σ n+1 (z ) log n . (6.6)Letting n go to infinity, we obtaing(z) -g(z ) ≥ α 3 log z z lim sup n→∞ log Σ n (z ) log n ≥ α 3 log z z ,

  n,wired (f (.)#(.)) -d dz Z area (z, β, n, wired) Z area (z, β, n, wired) P z,β n,wired (f (#(.)) -d dz Z area (z, β, n, wired) Z area (z, β, n, wired) , (f (.)#(.)) -P z,β n,wired (#(.))P z,β n,wired (f ) , (6.10) proving the result.

  From Proposition 2.2 and using carathéodory's extension theorem we get the existence of P z,β f ree and P z,β wired . Those probability measures are, thanks to [11, Theorem 11.1.VII], weak limits of the sequences P z,β For every z 1 ≤ z 2 and β 1 ≥ β 2 ,

	They are also stationary.	n,f ree n	and P z,β n,wired n	.
	Proposition 2.3.
	• P z,β n,f ree	P z,β n+1,f ree ;
	• P z,β n+1,wired |Λn	P z,β n,wired ,

where P z,β n+1,wired |Λn stands for the measure P z,β n+1,wired restricted to Λ n .

Acknowledgement: This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01), the GDR 3477 Geosto, the ANR project PPPP (ANR-16-CE40-0016) and the ANR project MALIN (ANR-16-CE93-0003).