Percolation Transition For Some Excursion Sets

We consider a random field $(X_n)_{n\in\mathbb{Z}^d}$ and investigate when the set $A_h=\{k\in\mathbb{Z}^d; \vert X_k\vert \ge h\}$ has infinite clusters. The main problem is to decide whether the critical level $$h_c=\sup\{h\in R : P(A_h\text{ has an infinite cluster })>0\}$$ is neither $0$ nor $+\infty$. Thus, we say that a percolation transition occurs. In a first time, we show that weakly dependent Gaussian fields satisfy to a well-known criterion implying the percolation transition. Then, we introduce a concept of percolation along reasonable paths and therefore prove a phenomenon of percolation transition for reasonable paths even for strongly dependent Gaussian fields. This allows to obtain some results of percolation transition for oriented percolation. Finally, we study some Gibbs states associated to a perturbation of a ferromagnetic quadratic interaction. At first, we show that a transition percolation occurs for superstable potentials. Next, we go to the the critical case and show that a transition percolation occurs for directed percolation when $d\ge 4$. We also note that the assumption of ferromagnetism can be relaxed when we deal with Gaussian Gibbs measures, i.e., when there is no perturbation of the quadratic interaction.


Introduction
In the last twenty years, percolation processes have taken a major place in the modelisation of disordered spatial systems, e.g. of inhomogeneous media.Of course, the mathematical study of dependent percolation is not as well advanced as those of Bernoulli percolation.In spite of this, by the appearance of new powerful tools [11] and by its deep relationships with some model of statistical mechanic, dependent percolation became an exciting area of research.We refer the reader to the stimulating book by Georgii, Häggström and Maes [8] for an overview of this large virgin country.
We will concentrate here about the problem of percolation transition for some families of dependent fields.The questions are simple to formulate: • Given a stationary random field (X n ) n∈Z d , for which values of h does the so-called excursion set have an infinite connected component with a positive probability ?• If this happens with positive probability, does it actually happens almost surely ?It is also natural to introduce the critical level h c = sup{h ∈ R; P (the origin belongs to an infinite cluster) > 0}.
We say that there is a percolation transition if h c belongs to the interior of the support of the distribution of a single site variable.In this paper, we will deal with a random field which is obtained as the absolute value of an initial random field.

It means that we study
The case of a Gaussian field (X n ) n∈Z d is naturally interesting.It is actually used by physicists to modelize composite media.Excursion sets E h are denoted as one-level cut Gaussian Random Models, whence excursion sets A h corresponds to two-level cut Gaussian Random Model.We refer the reader the reader to Roberts and Teubner [16], Roberts and Knackstedt [15], and the references therein for more information.
The mathematical treatment of the problem was initiated by Molchanov and Stepanov.At the beginning [12] of a cycle of three papers [12,13,14] about dependent percolation, they have formulated a simple criterion to ensure the presence (or absence) of percolation for a low (or high) level h.The study of E h for weakly correlated Gaussian fields was one of their applications.Later, Bricmont, Lebowitz and Maes [1] provided the first example of a percolation transition for a system with infinite susceptibility.The aim of this paper is the mathematical study of A h for some random fields (X n ) n∈Z d .Of course we will deal with Gaussian fields, but our study will not be limited to these fields: we will also study some Gibbs measures associated to a perturbation of a ferromagnetic quadratic interaction.
In section 1, we show how to apply the criterion of Molchanov and Stepanov to a weakly dependent Gaussian fields and obtain the existence of a percolation transition for stationary Gaussian fields with finite susceptibility.
In section 2, we show how some restrictions on the geometry of the percolating cluster allow to replace the Molchanov-Stepanov criterion by a weakened condition.Then, we can ensure the absence of percolation along "reasonable" clusters even if the dependence of the underlying process is strong.
These results are used in section 3 to prove the existence of a percolation transition for dependent oriented percolation on Z d in cases where the Molchanov-Stepanov criterion is not satisfied.An example is also given.
The goal of section 4 is to extend the preceding results to show percolation transition for some Gibbs measures associated to a perturbation of a quadratic interaction.For superstable and ferromagnetic interactions, we show the existence of a percolation transition for (unoriented) site percolation.When the assumption of superstabily is not satisfied, we nevertheless obtain the existence of a percolation transition for directed site percolation when d ≥ 4. Finally, we remark that the assumption of ferromagnetism can also be relaxed when we consider Gaussian Gibbs measures,, i.e. when there is no perturbation of the quadratic interaction.
For some proofs, we will need some results related to the control of the covariance of stationary Gaussian processes with a spectral density which can have some singular points.For readability, these results of Fourier analysis have been reported to a last section.Notations 0.1.Graphs and lattices.
A path from x to y is a sequence of vertices with x as the first element and y as the last one such that each element of the sequence is adjacent in G with the next one.The set of points which can be joint from x is denoted by C G (x).
Let Ω = R E and P be a probability measure on (Ω, B(Ω)).As usually, X k : Ω → R denotes the canonical projection on the k-th component.
Let h be a positive number.For a given digraph (V, E), we will consider the random subgraphs (V, ).We will say that a realization of the field (X k ) k∈E percolates over h (resp.under )) contains at least one infinite cluster.For x ∈ E, we say that a realization of the field (X k ) k∈E percolates over h (resp.under h) from ).We will work here with classical graphs built on x i and y − x 1 = 1}.
We note by µ d the d-dimensional connective constant, that is where c(d, n) is the number of injective paths on L d starting from the origin.
If A if a subset of Z d , we denote by Mod(A) the smallest subgroup of (Z d , +) which contains A. 0.2.Gibbs measures.Let us recall the concept of Gibbs measure.Each ω ∈ Ω = R Z d can be considered as a map from Z d to R. We will denote ω Λ its restriction to Λ.Then, when A and B are two disjoint subsets of Z d and (ω, η) ∈ R A × R B , ωη denotes the concatenation of ω and η, that is the element z ∈ R A∪B such that For finite subset Λ of Z d , we define σ(Λ) to be the σ-field generated by {X i , i ∈ Λ}.
For every finite Λ in Z d , let Φ Λ be a real-valued σ(Λ)-measurable function.The family (Φ Λ ) Λ , when Λ describes the finite subsets of Z d , is called an interaction potential, or simply a potential.For a finite subset Λ of Z d , the quantity is called the Hamiltonian on the volume Λ. Usually, H Λ can be defined only on a subset of R Z d .We suppose that there exists a subset Ω of Ω such that We now define the so called partition function Z Λ : denoting by λ the Lebesgue's measure on the real line, we let By convention, we set exp(−H Λ (η Λ ω Λ c )) = 0 when the Hamiltonian is not defined.
We suppose that for each ω in Ω, we have 0 < Z Λ (ω) < +∞.Then, we can define for each bounded measurable function f and for each ω ∈ Ω, For each ω, we will denote by Π Λ (ω) the measure on Ω which is associated to If a measure µ on Ω is such that µ( Ω) = 1, we say that µ is a Gibbs measure or a Gibbsian field when for each bounded measurable function f and each finite subset Λ of Z d , we have Let J : Z d → R be an even function such that i∈Z d |J(i)| < +∞ and V a continuous function.
Given these parameters, we deal with Gibbsian random fields µ associated to the potential Φ J,V defined on Ω by Then, the corresponding Hamiltonian function is equal to On Ω, H Λ is well defined.It is clear that it could not be possible to take a larger Ω, so this is a canonical choice.
For fixed (J, V ), we denote by G J,V the set of Gibbs measures on R Z d associated to the Hamiltonian given in (3).If G J,V contains more than one point, we say that phase transition occurs.G J,V is a convex set whose extreme points are called pure phases.(For general results on Gibbs measures, see [7].) For z = (z 1 , ..., z d ) ∈ C d and n = (n 1 , ..., n d ) ∈ Z d , we set We introduce Ĵ, the dual function of J, defined on a subset of C d by ( 4) whenever the considered series is absolutely convergent.Since J is summable, it is clear that Ĵ always defines a continuous map on U. 0.3.Miscellaneous.We recall that J ν is the Bessel function of first order with index ν, that is

Weakly Dependent Gaussian Fields
A natural approach to generate some dependent random fields is to use Gaussian fields.In their pioneering paper [12], Molchanov and Stepanov consider Gaussian variables with a bounded spectral density as an illustration of their criterion.They proved that for large h, {k ∈ Z d ; X k ≥ h} does not percolate.By a symmetry argument, their result also implies the existence of a percolation transition.
In the present paper, we will consider the problem of percolation for {k ∈ Z d ; X k ≥ h}.Of course, by a symmetry argument, the two problems are equivalently difficult for large h, but completely different for small (but positive) h.
At first, let us recall the Molchanov-Stepanov criterion.In this proposition, two vertices i and j are said to be adjacent if i − j 1 = 1 and to be * -adjacent if i − j ∞ = 1.

Proposition 1 (Molchanov and Stepanov
).There exists two finite constants c dis d and c agr d only depending from the dimension such that for each {0, 1}-valued random field (X k ) k∈Z d , we have the following results: • If there exists C > 0 such that for each connected set A, we have has almost surely only finite clusters.
• If there exists C > 0 such that for each * -connected set A, we have has almost surely at least one infinite cluster.
We well need some lemmas.Note that some of them (that is Lemma 2 and Lemma 3) were (at least implicitly) used by Molchanov and Stepanov in their proof of the absence of an infinite cluster in {k; X k ≥ h} for large h.We recall that we consider here {k; X k ≥ h}, not {k; X k ≥ h}.
1.1.Some lemmas.We begin with an elementary but useful lemma: Lemma 1.Let X be a R n -valued Gaussian vector with covariance matrix C.
where the λ i 's are the eigenvalues of C.
where ρ(C) is the spectral radius of C.
Lemma 2. Let X be a R n -valued Gaussian vector with covariance matrix C and where h(x) = 1 2 (x − ln x − 1).h is increasing and positive on ]1, +∞[, with +∞ as limit at +∞. Proof.For each α > 0, we have We choose α = 1 ρ (C) − 1 a 2 and get Note that the proof of Lemma 2 follows the standard of the theory of large deviations.h naturally appears as the function associated to a χ 2 distribution in Chernof's theorem.Now, we can claim the lemma which contains the half part of our first result about percolation transition.Lemma 3. Let (X n ) n∈Z be a centered stationary Gaussian field with bounded spectral density g Then, for each x 2 > g ∞ , we have Proof.Let T = (R/2πZ) d and M g be the Toeplitz operator: 2 (T) → 2 (T) defined by M g (f ) = gf .If A ⊂ Z d and P A is the orthogonal projection from 2 (T) into L = Lin{exp(i .|k); k ∈ A}, then the matrix of covariance of the vector X = (X k ) k∈A is also the matrix of the restriction of P A M g to L. Therefore , it just remains to apply lemma 2 We will now turn to the reverse side of the percolation transition.
Lemma 4. Let X be a n-dimensional Gaussian vector with positive definite covariance matrix C. Let us denote by g(C) the spectral gap i.e. the smallest eigenvalue of C.Then, for each a 2 < g(C), we have . h is positive and decreasing on (0, 1), with an infinite limit at 0. Proof.For each α > 0, we have Then, we choose α = 1 a 2 − 1 g(C) and get Lemma 5. Let (X n ) n∈Z be a centered stationary Gaussian field with variance σ 2 > 0 and with finite susceptibility, i.e. such that Then, there exists f : (0, σ 2 ) → (0, +∞) such that for each finite set A ⊂ Z d , we have f is positive and decreasing on (0, 1), with an infinite limit at 0.
Theorem 1.Let (X n ) n∈Z be a centered stationary Gaussian field with finite susceptibility, i.e. such that Then, let us define • For each a > h + there is almost surely not percolation over level a, whence there is almost surely percolation over level a for a < h − .• For each a < h − there is almost surely not percolation under level a, whence there is almost surely percolation under level a for a > h − .
Proof.Since (X n ) n∈Z d has a finite susceptibility, it has a spectral density, and therefore has a spectral measure without atoms.Then, by a result of Maruyama and Fomin (see for example [19], lecture 13), it follows that the law of (X n ) n∈Z d is ergodic under the group of translations of Z d .Since the existence of an infinite cluster is a translation-invariant event, it follows that the existence of a percolating cluster is a deterministic event.By monotonicity, {a ≥ 0; P (|C X a + (0)| = +∞) = 0} and {a ≥ 0; P (|C X a − (0)| = +∞) = 0} are intervals.It follows that when a < a + (resp.when a > a − ), we have . In both cases, the probability of percolating is positive, and then equal to one.As in the case of independent percolation, the almost sure absence of percolation from the origin imply the almost sure absence of percolation from everywhere using the stationarity of (X n ) n∈Z d and the denumerability of Z d .
By lemma 5, if a is so small enough that f (a) > c dis d , then the Criterion of Molchavov and Stepanov says that there is no percolation under a.Again by lemma 5, if a is so small enough that f (a) > c agr d , then the Criterion of Molchavov and Stepanov says that there is percolation over a.Since X has a finite susceptibility, it has a spectral density.Then, by lemma 3, if a is so large enough that h( a 2 g ∞ ) > c dis d , then the Criterion of Molchavov and Stepanov says that there is no percolation over a.Similarly, if a is so large enough that h( a 2 g ∞ ) > c agr d , then the Criterion of Molchavov and Stepanov says that there is percolation under a.

Reasonable percolating sets
2.1.The concept of reasonable sets.We will define some families of subset of where Of course, every subset of The following remark is fundamental: if A is the support of a path in the oriented graph a piece of a bad configuration a piece of a good configuration When the decay of the correlation of the Gaussian process is too slow, one is not able to prove the absence of percolation.Nevertheless, we will see that we can sometimes prove the absence of an infinite reasonable path.
For each y ∈ Ã and k ∈ Z + , we will define By the definition of Ã, b n (x) = 0 for n < r.Then, We will denote by r ε the smallest r which can enjoy this property and Ãε the relative set.
We can now prove the existence of g.We take ε ∈ (0, Let x 2 ∈ (0, σ 2 − ε): by lemma 4, we have Then, we can define g by g : (0, σ 2 ) → (0, +∞) Similarly, using the fact that ρ(C) ≤ φ(0) + ε, we can define f by We will say that a realization of the field (X k ) k∈E exhibits a s, K-reasonable has at least one infinite connected set which belongs to M s,K .We denote by R h s,K this event.For x ∈ E, we say that a realization of the field (X k ) k∈E exhibits a s, K-reasonable percolation over h (resp.under h) from x if (V, E X h + ) (resp.(V, E X h − )) has at least one infinite connected set which belongs to M s,K and contains x.We also denote by R h s,K (x) this event.
Theorem 3. Let (X n ) n∈Z d be a centered Gaussian process, σ 2 ∈ (0, +∞), s ∈ [1, +∞) and φ : Z + → R + such that the following assumptions hold: Then, let us define for K ∈ (0, +∞) • For each a > h + (s, K) there is almost surely no s, K-reasonable percolation over level a. • For each a < h − (s, K) there is almost surely no s, K-reasonable percolation under level a.
If there is s, K reasonable percolation over a from x, there exists a self avoiding avoiding walk starting from x and whose support S is such that Let ε > 0. There exists k ε such that for each n ∈ Z + and each x ∈ Z d , the number of self-avoiding walks starting from x is less k ε (µ d + ε) n .Therefore, by lemma 2, we have a) , where f has been defined in lemma 2 If a is so large enough that f (a) > ln(µ d + ε), then ∀x ∈ Z d P (R a s,K (x)) = 0 and then P (R a s,K ) = 0. Since ε is arbitrary, it follows that P (R a s,K (x)) = 0 holds for each x as soon as f (a Similarly, where g has been defined in lemma 2.

Oriented site percolation for Gaussian fields
We will consider here the problem of percolation on the oriented lattice − → L d .
Theorem 4. Let (X n ) n∈Z d be a centered stationary Gaussian field with bounded spectral density g.Then, the covariance function is We suppose moreover that Then, let us define h − = sup{a ≥ 0; P ({directed percolation happens under a}) = 0}. Then, • For each a < h − there is almost surely no directed percolation under level a, whence there is almost surely directed percolation under level a for a > h − .
Proof.The fact that 0 < h − (1, d) has already be proved in Theorem 3. Since every It remains to prove that h − (s, K) < +∞.
We will now consider the restriction of X to a two-dimensional quarter plane: let us denote (Y k,l ) k,l∈Z+ = (X k,l,0,...,0 ) k,l∈Z+ .We also put Y k,l = +∞ when k < 0 or l < 0. Of course, percolation in the quarter plane will imply percolation in the whole space for the initial process.
Let Z 2 * = Z 2 + (1/2, 1/2).For a finite subset A of Z 2 , let us recall a notion of Peierls contours associated to A. Let a, b be two neighbors in Z 2 and i and j be two points a, b ∈ Z 2 * such that the quadrangle aibj is a square.We say that the segment joining a and b is drawn if |A ∩ {a, b}| = 1.Drawn segments form a finite family of closed, non self-intersecting, piecewise linear curves, that are called Peierls contours.If i and j are two neighbors in Z 2 separated by a contour γ, say If A is a finite L d -connected set connected, then, there exists a unique Peierls contour γ A such that A remains in the bounded connected component of R 2 \γ A .
For each contour γ, let us also define We can see that if γ is just a simple closed curve with length l(γ), we have Proof.On each vertex of the dual lattice Z 2 * which is a piece of the curve γ, let us draw an arrow in such a way that γ is described with the inside of γ on the left, and the outside of γ on the right -thus, the arrows indicate how to draw the curve anti-clockwise.Since γ is a simple closed curve, there is as many ↑ and ← as ↓ and →.Then, there is exactly l(γ)/2 ↑ and →.Each point at the right of a ↑ or over a → belongs to F γ .Therefore, since every point is surrounded by at most one ↑ and one →, it follows that |F γ | ≥ l(γ)/4.

Let us consider the random set
It follows from a straightforward computation that that h − ≤ 3, 57 g 1/2 ∞ .

An example.
Let A be a symmetric positive definite matrix with spectral gap g(A) ≥ 1 π and consider the ellipsoid Let (X n ) n∈Z d be a stationary Gaussian process with the indicatrix of E as spectral density.
We claim that the assumptions of theorem 4 are fulfilled by the process (X n ) n∈Z d and that, moreover we have ) The graph of c when Proof.Since X is a stationary process, we only computes The result follows of lemmas 9 and 10 of section 5 with d 0 = 0.

Oriented and unoriented percolation for Gibbs measures
We introduce the following assumptions: (1) (H 1 ) V is even.
The main idea of this section is to compare non-Gaussian Gibbs measures to Gaussian Gibbs measures for which the results of the preceding sections apply.We will use a lemma which in the spirit of a lemma due to van Beijeren and Sylvester [21] related to stochastic domination for finite Gibbs measures with the same ferromagnetic interaction and different reference measures.
Let us first recall the concept of domination for finite measures on a partially ordered set E .We say that a measure µ dominates a measure ν, if ≤ f dµ µ(E) holds as soon as f in an increasing function.We also write ν ≺ µ.
4.1.First results.Lemma 6.Let J = (J(i, j)) i,j∈Λ be a symmetric positive definite matrix satisfying to Let also be ν 1 and ν 2 two even measures which have a bounded density with respect to Lebesgue's measure Then, we can define for each bounded function f : R Λ → R: where ν is the measure on (0, +∞) defined by dν(x) = exp(− c 2 x 2 )dν(x).Then, it follows that for all even bounded functions F i : R → R, nonnegative, and monotone increasing on [0, +∞), we have Since the reader can found in [21] the proof of an analogous lemma in a more general context, we will omit these one.
The existence of Gibbs measures for superstable Hamiltonians is well known since the works of Doss and Royer [5], and Cassandro et al. [2].
However, we will see that in our case, lemma 6 and the assumptions of ferromagnetism will allow some comparisons which give as the existence of a Gibbs measure as the results of percolation.
Since we will have to compare some measures associated to the same two-body interaction J but with different self-interactions, we will denote by Π J,V Λ (ω) the Gibbs measure on Λ associated to the Hamiltonian defined in (3) and with boundary condition ω.
It is easy easy to see that the matrix J Λ = (J(i, j)) i,j∈Λ is positive definite.
Let us also prove that ν1 ≺ ν2 .It is equivalent to proof (see for example [21]) that It is known (see for example [4], chapter 13 or [9,10]) that the sequence (µ 0 n ) n≥1 converges to the stationary centered Gaussian measure with spectral density 1 Ĵ .It follows that this sequence is tight.Let K be a compact subset of R Z d such that Then, it follows from the general theory of Gibbs measure -see for example Georgii [7] -that every limit point of this sequence is an extremal Gibbs measure for the Hamiltonian H J,V Λ with Lebesgue's as reference measure.
Lemma 7. We suppose here that (H 1 ), (H 3 ), (H 5 ), (H 6 ) are fulfilled.Let µ V be an extremal Gibbs measure which is obtained as a limit point of the sequence considered in theorem 5.Then, for each finite set Λ, we have Then, for each x 2 ≥ 1 γ , we have Proof.By lemma 6, we have where µ 0 is the stationary centered Gaussian measure with spectral density 1 Ĵ .Now, the result follow from lemma 3.
The goal of the next lemma is to compare (when a is small) the random field 1 1 {|X k |≤a} with a product of Bernoulli measures.It is a classical method in the study of dependent percolation.
Lemma 8. We suppose here that (H 1 ), (H 2 ), (H 3 ), (H 4 ) are fulfilled.Let µ be a Gibbs measure for the considered Hamiltonian and (X n ) n∈Z d be a random field with P X = µ.

2
) cosh ηx dλ(x) -the same arguments than in theorem 5 apply.Then By an usual coupling technique, (13) implies that See, for example, Russo [18] (lemma 1) for a proof of a more general result or Liggett, Schonmann and Stacey [11] for deeper results.
• For each a > h + there is almost surely not percolation over level a, whence there is almost surely percolation over level a for a < h − .• For each a < h − there is almost surely not percolation under level a, whence there is almost surely percolation under level a for a > h − .
Proof.Since the existence of percolation is a tail event, the 0 − 1 behavior follows from the fact that the tail σ-field is trivial under extremal Gibbs measures.If p c denotes the critical probability for independent Bernoulli site percolation, it follows from lemma 8 that h − ≤ p c χ < +∞ and h + ≥ 1−p c χ > 0. The facts that h − < +∞ and h + > 0 follows from lemma 7 together with the criterion of Molchanov and Stepanov.

4.3.
Transition of oriented percolation in the critical case.We will consider here the problem of transition percolation on the oriented lattice − → L d .The first theorem is related to Gaussian and non-Gaussian Gibbs measures associated to ferromagnetic Hamiltonians Theorem 7. We suppose here that (H 1 ), (H 3 ), (H 4 ), (H 5 ) are fulfilled.We suppose moreover that one of the following assumptions holds: Then, there exists at least one extremal Gibbs measure which is obtained as a limit point of the sequence considered in theorem 5. Let P = µ V be such a measure and define h + = inf{a ≥ 0; P ( directed percolation happens over a) = 0}.
• For each a > h + there is almost surely not percolation over level a, whence there is almost surely percolation over level a for a < h + .
Proof.Since the existence of percolation is a tail event, the 0 − 1 behavior follows from the fact that the tail σ-field is trivial under extremal Gibbs measures.If p c denotes the critical probability for independent Bernoulli directed site percolation, it follows from lemma 8 that h + ≥ 1−p c χ > 0.
If Ĵ does not vanish on U, it is clear that (H 2 ) holds.Moreover, it follows that J is an invertible element in the Banach Algebra [6].It follows that the Fourier coefficients of 1 Ĵ form a sequence which belongs to A: Let us study the second case.We define f (θ 1 , . . ., θ d ) = Ĵ(e iθ 1 , . . ., e iθ d ).
For each k ≤ 3, we have with D k 0 R = 0 for k ∈ {0, 1, 2}.By the triangular inequality, f (θ) = 0 if and only if the complex numbers −J(n)e i n,θ belong to R + .Then, θ is orthogonal to Mod({n, J(n) = 0}) as soon as f (θ) = 0.It follows that f only vanishes on (2πZ) d .By the same arguments, Q is positive definite.Then, we can apply Theorem 9 with N = 3.

It follows that c
It follows from theorem 2 that there exists a function f : (c 0 , +∞) → (0, +∞) having an infinite limit at +∞ and such that By lemma 6, it follows that Now, we proceed as in Theorem 3 to prove that the is no 1, d reasonable percolation over a for large (but finite) a.Then, there is also no oriented percolation over a for large (but finite) a. Precisely, h + ≤ inf{x; f (x) > ln µ d }.
Example: Let m ≥ 0 and define J by By a direct computation, we see that Ĵ ≥ m on U.Then, the assumptions of Theorem 7 are fulfilled as soon as m > 0 or d ≥ 4 When V = 0, these models are called harmonic model.For m > 0 and V = 0, it is the so-called harmonic model with mass, whence for m = 0, it is the so-called massless harmonic model.
It is interesting to compare these results with those that were obtained by Bricmont, Lebowitz and Maes in [1].They have shown the existence of a (unoriented) percolation transition for {k ∈ Z d ; X k ≥ h} for the harmonic model with mass in each dimension and for the massless harmonic model when d = 3.
Remark: in the ferromagnetic case, c n is proportional to the Green function associated to the random walk associated to the measure µ defined by µ(0) = 0 and µ(n) = − J(n) J(0) for n = 0.If we know that this random walk is aperiodic and that µ * k (n) = o( n 2−d ) holds for each k ≥ 1, then it follows from a result by Spitzer [20] that for d ≥ 3, we have c n ∼ n, Q −1 n 1−d/2 for a suitable definite matrix Q.Then, +∞ k=1 sup{|c n |; n ≥ k} < +∞ holds for d ≥ 4 and not for d = 3.Note that the estimate of Spitzer is more precise that ours, but it requires an assumption of aperiodicity that sometimes fails, for example for one-range interations.
The last theorem shows that for Gaussian Gibbsian fields, the assumption of ferromagnetism can be relaxed.Theorem 8. We suppose here that (H 1 ), (H 2 ), (H 3 ), (H 4 ) are fulfilled.We suppose moreover that Let P = µ 0 be the extremal Gibbs measure which is obtained as a limit point of the sequence considered in theorem 5.Then, let us define h + = inf{a ≥ 0; P ( directed percolation happens over a) = 0}.
• For each a > h + there is almost surely not percolation over level a, whence there is almost surely percolation over level a for a < h + .
Moreover each the following assumptions imply that (H 2 ) and (H 7 ) hold: is a definite positive quadratic form Proof. Since the skeleton of this proof is essentially the same as those of the previous theorem, we will omit it.

Some Fourier asymptotics
In this section, we will prove a theorem to estimate the asymptotic behavior of the Fourier coefficients of some quite smooth functions which only have a singularity in a single point.
Then, Using some methods of integration such as those which are described in Rudin [17], it is not so hard to proof that if g • . is integrable, we have where G d , which only depends from the dimension, can be computed with the choice λ = 0 and g(r) = exp(− r 2 2 ).Now, if we take g(r) = r −d0 1 1 [0,1] (x), the desired formula follows from a last change of variable.Now, we have to estimate C( x ) for large x.Lemma 10.Let is semi-convergent an can be computed.More generally, if ν and µ are complex numbers such that .
This integral is sometimes called Weber's integral, who computed its values when ν is an integer.For a proof or an historic, see [22], §13.24, page 391.
Here, we have ν Let us consider the case γ ≥ 1 2 .Then, ( 14) diverges, but I α (x) can be estimated using some asymptotics for Bessel's functions.Bessel's functions are related to Hankel's functions H 1  ν and H 2 ν by the formula We have on C\R − the following asymptotics (see for example [3], chap.XV.). Similarly The norm of the integral appearing in the last part of the preceding identity can be bounded as previously, and we finally get Theorem 9. Let d ≥ 2 and f : R d → R + with (2πZ) d as period and such that the following holds: We also define α the following asymptotics holds: where I d,d 0 has been defined in lemma 10.It can be precised as follows: 2 and N ≥ d + 1, we have the equivalent, when n goes to +∞: Proof.By the separation theorem, there exists an infinitely smooth function h which is identically 0 on The function 1 − h f can be periodized into a C N -smooth function, because it coincides with the periodic function 1 f on the boundary of [−π, π] d .Its Fourier coefficients , which represent the second part of the sum in (15) are O( n −N ).
Let us now study the first term of the sum.Since the support of h resides inside A −1 B(0, 1), this integral can be written as Then h 1 is C ∞ -smooth, is identically 1 on B(0, 1 2 ), whence f 1 only vanishes at 0 and satisfies to (16) f 1 (x) = x d0 + R 1 (x), where R 1 follows the same assumptions as R. We put The control of the first part is done by using lemmas 9 and 10.It remains to control B(0,1) e i A −1 n,x dx We will use Green's formula, which will be used here as a multidimensional integration by parts.
If we suppose that V is a volume with an infinitely smooth orientable boundary, and that u is C 2 -smooth and φ C 1 -smooth from V into R, we have N (x) denotes the unitary vector which is normal to ∂V and is oriented to the outside of V and σ is the surface measure on ∂V .
Taking u(x) = − 1 n 2 e i n,x , we get grad u = − in n 2 e i n,x and ∆u = e i n,x , hence (17) V φe i n,x dx = − i n ∂V φ v, N (x) e i n,x dσ(x) + i n V D 1 x φ(v)e i n,x dx, with v = n n .Iterating this process, we get for a C N -smooth φ: x φ(v ⊗k ) v, N (x) e i n,x dσ(x) We will take here φ = s, V = V r n = {x ∈ R d r n ≤ x ≤ 1}, where r n is a sequence which is to determinate and will have a null limit.
To control the partial derivatives, we will need the following lemma: Lemma 11.The exist a constant K such that For readability, we will prove it later and admit it for a short time.Then, Let us now expand u x (h): we have In R[[x]] holds We have also We can also deduce that for each n ≥ 1 Then, by the inequality of Cauchy-Schwarz, we have It follows that we can find K u in order to have for each k ≤ N The homogeneous component whose degree related to h is n writes u x (h n ≥ a + k + b is a necessary condition for the sum not to be zero.If it holds , the inequalities ( 20) and (21) show that we can find K such that for |x| < 1 2 , we can write: (22) |Φ n a,b,k (h)| ≤ K x (a+k)d 0 +bN −n h n .Composing these two developments, we get Φ n a,b+1,k (h) x (k+1)d0 ( x d0 + R 1 (x)) a+b+1 Applying the bounds found in (22) and the fact that R 1 (x) is o( x d 0 ), we can see that each term of the sum if bounded by which is manifestly what we wanted to prove.

2 (
t)t γ dt, with α = d − d 0 and γ = d 2 − d 0 .Proof.By invariance under the group of isometries, it is easy to see that C(n) only depends on n 2 .

1 nr N −2d 0 n n N − 1 k=0(
)e i n,x dx | ≤ K S d r d−n r n ) −k + K B d n N r −2d0 nwhere S d is the area of the d-dimensional unit sphere and B d the volume of the d-dimensional unit ball.Also, integrating the inequality(11) with k = 0, we getB(0,rn) |s(x) dx| ≤ K d + N − 2d 0 r d+N −2d0 nMaking the product, we getF A,B (h 1 , h 2 ) =

0
v x (h ⊗j p ) j p !where the sum runs over the positive integers satisfying to