Continuum percolation for Gibbsian point processes with attractive interactions

We study the problem of continuum percolation in infinite volume Gibbs measures for particles with an attractive pair potential, with a focus on low temperatures (large $\beta$). The main results are bounds on percolation thresholds $\rho_\pm(\beta)$ in terms of the density rather than the chemical potential or activity. In addition, we prove a variational formula for a large deviations rate function for cluster size distributions. This formula establishes a link with the Gibbs variational principle and a form of equivalence of ensembles, and allows us to combine knowledge on finite volume, canonical Gibbs measures with infinite volume, grand-canonical Gibbs measures.


Introduction
The present article is concerned with percolation properties for Gibbsian point processes. We are interested in infinite volume Gibbs measures (in the sense of the Dobrushin-Lanford-Ruelle conditions) for particles in R d interacting via an attractive, finite range pair potential. The dimension is two or higher, d ≥ 2. Around each particle x of a random configuration ω, draw a ball of radius R, for some fixed R > 0. Percolation occurs if the region in R d covered by the union of such balls, ∪ x∈ω B(x, R), has an unbounded connected component, with positive probability. In the notation of Meester and Roy [MR96], our problem is a Boolean percolation model (X, ρ) driven by a Gibbsian point process X and with deterministic radius ρ = R.
This problem has been studied before [M75,Z08,PY09,Ar12], for both repulsive pair potentials and potentials with an attractive part. Mürmann [M75] investigated finite-range potentials and gave a sufficient condition for the absence of percolation; a different proof, with an extension to tempered boundary conditions for attractive potentials, was given by Zessin [Z08]. His proof builds on integration by parts for Gibbsian measures. Pechersky and Yambartsev [PY09] proved a criterion for absence of percolation with the help of a coupled branching process; their result does not require that the potential has compact support. In addition, Pechersky and Yambartsev gave a sufficient condition for the presence of percolation, valid in dimension 2, for attractive potentials with possibly unbounded support. An analogous result, for hard spheres in dimension 2 was shown by Aristoff [Ar12].
The cited works all formulate criteria in terms of the activity z -i.e., the intensity parameter of some a priori Poisson point process -rather than the density ρ, which for interacting particles is a non-trivial function of z. As a consequence, for attractive potentials, the cited results cannot distinguish between two very different physical pictures. First, percolation might be a high-density effect, as expected for hard spheres; second, it might be an energetic effect -at low temperature, it may happen that the interaction favors the formation of large connected components, which because of entropy may coexist with large almost empty regions of space containing a few small components. In such a situation the density threshold for percolation may be very small, as suggested by recent results [JKM11] on large deviations for cluster size distributions in the canonical ensemble.
The aim of the present article is, therefore, to give bounds on percolation and non-percolation thresholds in terms of the density rather than the activity. Our main result is Theorem 3.6 below, valid for finite-range, attractive potentials. Quickly summarized, Theorem 3.6 states that there are curves ρ ± (β) such that if P β,ρ is a shift-invariant Gibbs measure at density ρ and inverse temperature β, the following holds.
The key technical tool for the proof of Proposition 3.7 is a variational formula for a large deviations rate for cluster size distributions in the canonical ensemble (Theorem 3.2); the large deviations principle was investigated in [JKM11]. The variational formula establishes a relation with the Gibbs variational principle and allows us to apply a form of equivalence of ensemble [G95]. This in turn enables us to combine the knowledge for grand-canonical, infinite volume Gibbs measures in [M75,PY09] with results on the canonical ensemble [JKM11].
We conclude the introduction with a word of caution on the physical interpretation of percolation. We should stress that a percolation transition need not be a phase transition -for the ideal gas at activity z (Poisson point process with intensity z), the pressure is an analytic function no matter the value of z, even though there is a percolation transition at high enough z [MR96]. Nevertheless, for attractive pair potentials and at low temperature, the percolation transition might coincide with a phase transition. In fact, for nearest neighbor attractive lattice gases (or Ising model), and temperatures below some threshold T + , the percolation transition and the phase transition coincide (see the review by Georgii, Häggström and Maes [GHM01]). In dimension two, T + equals the Curie temperature T C , but in higher dimensions T + < T C [ABL87], illustrating again that percolation and phase transition in general do not coincide. For the reader's convenience, we summarize some relevant results in Appendix A.
The remainder of the article is organized as follows. In Sections 2 and 3 we formulate the setting and our results. Section 4 defines the topology of local convergence and summarizes continuity properties of important functions such as the relative entropy rate. Sections 5 to 7 are devoted to the proofs.

Setting
2.1. Pair potential. The pair potential is a function v : [0, ∞) → R ∪ {∞} that serves to define the total energy of an N -particle configuration By a slight abuse of notation, we shall drop the subscript and write U (x) instead of U N (x).
Assumption 1. The pair potential satisfies the following basic assumptions: • Either v is everywhere finite or there is a r hc > 0 such that v(r) = ∞ for r < r hc and v(r) < ∞ for r > r hc . (We impose no condition on v(r hc ).) • v has compact support: • v has an attractive tail: for suitable r 0 < r 1 and all r ∈ (r 0 , r 1 ), v(r) < 0.
If r hc > 0, we say that v has a hard core. If v has compact support, we shall also say that v has finite range. For k ∈ Z d , let C(k) be the unit cube [k 1 + 1) × · · · × [k d , k d + 1) and Every superstable interaction is stable, since Eq. (1) implies in particular that U (x 1 , . . . , x N ) ≥ −bN . If v satisfies Assumption 1 and is non-integrably divergent at the origin, then v is superstable; see [R70, R69] for a proof and other sufficient conditions. Assumption 3. The potential is integrable in {v < ∞}: |x|>r hc |v |x| |dx < ∞.
Assumption 4. There is an r min > 0 such that for every N ∈ N, U has a minimizer (x 1 , . . . , x N ) ∈ (R d ) N with interparticle distance bounded from below by r min : In addition, v is Hölder-continuous in [r min , ∞).
A sufficient condition for the lower bound on interparticle distances is that v(r)/r d → ∞ as r → 0, as can be shown along [T06,Lemma 2.2].
Assumption 5. There is a C > 0 such that for every N ∈ N, U has a minimizer (x 1 , . . . , x N ) ∈ (R d ) N with diameter bounded by CN 1/d : Under Assumption 1, this condition is trivially fulfilled in dimension 1. In dimension 2, sufficient conditions are given, for example, in [T06], where much more is proven on the ground states. To the best of our knowledge, there is no result in dimension 3 or higher; in fact, providing upper bounds on interparticle distances for Lennard-Jones type interactions seems to be a non-trivial problem in non-linear optimization, see the article by Blanc [B04] and the references therein.
Let us briefly comment on our conditions on the pair potential. Assumption 1 simply defines the class of pair potentials we are interested in. Superstability as in Assumption 2 is a standard condition that ensures the existence of infinite volume Gibbs measures, see the next subsection. The integrability assumption 3 will allow us to use a bound on Mayer expansions going back to Brydes and Federbush [BF78]. Assumptions 4 and 5 are needed for precise statements about Gibbs measures at low temperature β −1 and densities above some threshold of the form exp(−βν * ).

Infinite volume Gibbs measures.
Let Ω be the set of locally finite point configurations, with B(0, r) the open ball of radius r centered at the origin. We equip Ω with the σ-algebra F generated by the counting variables N B (ω) := |ω ∩ B|, B ⊂ R d Borel-measurable, and denote the probability measures on (Ω, F ) with the letter P. The following subsets of P will be relevant for us: shift-invariant measures P θ , tempered measures, and infinite-volume Gibbs measures G(β, µ); we proceed with their definition. For The collection of shift-invariant measures is denoted P θ . We say that P ∈ P is tempered if for P -almost all ω, ∃t(ω) > 0 ∀ℓ ∈ N : where C(k) is the unit cube [k 1 , k 1 + 1) × · · · × [k d , k d + 1). Fix β > 0 and µ ∈ R. Let Λ ⊂ R d be a bounded Borel set and ζ ∈ Ω a configuration satisfying the temperedness condition (2). For n ∈ N, define the measure Q n on Λ n as the measure with Lebesgue density The image of Q n under (x 1 , . . . , x n ) → {x 1 , . . . , x n } is a measure on Ω which we denote again by Q n . Let Q 0 be the probability measure on Ω which gives probability 1 to the event that ω = ∅. Define P β,µ,Λ|ζ by The normalization Ξ Λ|ζ (β, µ) is defined by the requirement that P β,µ,Λ|ζ is a probability measure on Ω.
Remark. The Palm measure is the continuum analogue of a simple lattice object: For a shift-invariant, tempered measure P , we define the expected energy per unit volume, or energy density, as U (P ) takes values in R ∪ {∞}. For superstable potentials as in Eq. (1), the energy is bounded from below: U (P ) ≥ −b 2 /4a. The entropy per unit volume, or entropy density is defined as a relative entropy rate. Let Q ∈ P θ be the Poisson point process with intensity 1 and P ∈ P θ . For Λ = [−L, L] d ⊂ R d , let P Λ be the image of P under the projection map ω → ω ∩ Λ. Define Q Λ in a similar way. The relative entropy is if P Λ has Radon-Nikodym derivative dP Λ /dQ Λ = f , and I(P Λ ; Q Λ ) := ∞ if P Λ s not absolutely continuous with respect to Q Λ . The limit exists for all P ∈ P θ , see [GZ93]. Continuity properties of U (P ) and S(P ) with respect to a suitable topology on P are recalled in Section 4 below.
Remark. The additive constant 1 is included in Eq. 6 for aesthetic reasons; if we did not include it, we would need an additive constant in Eq. (9).
Example. Let P be a Poisson point process with intensity parameter z. Then 2.4. Cluster densities. Let r 1 > 0 be the range of the potential as in Assumption 1. Fix R ≥ r 1 . For ω ∈ Ω, let G ω be the graph with vertex set ω and edge set For P ∈ P θ and k ∈ N, the expected number of k-clusters per unit volume or k-cluster density is The last identity holds for every Borel set C, compare Eq. (5). Note that for every P ∈ P θ , ∞ k=1 kρ k (P ) ≤ ρ(P ). For later purpose we also define finite volume empirical densities as Thus ρ k,Λ (ω) is the number of k-clusters in ω ∩ Λ, divided by the volume |Λ|. We have, for all ω ∈ Ω,

2.5.
Large deviations for cluster size distributions. Finally we recall a large deviation principle shown in [JKM11]. Fix R > r 1 . For Λ = [0, L] d and N ∈ N, let be the canonical partition function and P β,N,Λ the probability measure on Λ N with . , x N } is a measure on (Ω, F ), for which by a slight abuse of notation we use the same letter P β,Λ,N . Equip R N + with the product topology and the associated Borel σ-algebra, and let We are interested in the behavior of ρ Λ in the thermodynamic limit for ρ > 0. First we recall the definition of the free energy f (β, ρ) and the closepacking density ρ cp . Consider the limit along (8). It is well-known [R69] that the limit exists and is finite when the density ρ is strictly smaller than the close-packing density ρ cp > 0, and is infinite when ρ > ρ cp . When the potential has no hard core (r hc = 0), we have ρ cp = ∞ and the limit f (β, ρ) is finite for all ρ > 0.
The following holds [JKM11]: for all β > 0 and ρ ∈ (0, ρ cp ), in the limit (8), the random variable ρ Λ (ω) satisfies a large deviation principle with speed β|Λ|. The rate function is of the form Note that the rate function f (β, ρ, ·) (but not the free energy f (β, ρ)!) depends on the connectivity radius R ≥ r 1 ; to lighten notation, however, we leave the Rdependence implicit.

Results
Here we formulate our main results. Section 3.1 provides a variational characterization of percolation. Sections 3.2 and 3.3 formulate bounds on percolation thresholds in terms of the chemical potential µ = β −1 log z and the density ρ. Section 3.1 on the one hand and Sections 3.2 and 3.3 on the other hand are logically independent, except for Proposition 3.7.
Throughout the remainder of the article we shall assume without further mention that the pair potential satisfies Assumptions 1 and 2, and the connectivity radius R is larger or equal to the potential range r 1 .

Variational characterization of percolation.
Our first result expresses the rate function f (β, ρ, (ρ k )) of [JKM11] in terms of a variational problem.
with the convention min ∅ = ∞.
The next elementary proposition establishes a relation between cluster densities and percolation, valid for general shift-invariant point processes. Proposition 3.3. Let P ∈ P θ . The following statements are equivalent: 1 Georgii states his result under the additional assumption that the potential is non-integrably divergent at the origin. A close inspection of the proof shows, however, that we can dispense with this condition because our potentials have finite range; see the comment in [G95] before Lemma 7.3.
A quantitative relation is given in Eq. (20) below. We shall refer to both (2) and (3) as P (there is an infinite cluster) > 0, or more briefly as percolation. An immediate consequence of Theorem 3.2 and Prop. 3.3 is the following characterization of percolation in shift-invariant Gibbs measures. Set The Gibbs variational principle implies that G θ (β, ρ) consists of the minimizers of U (P )−β −1 S(P ) under the constraint ρ(P ) = ρ; compare with the proof of Theorem 3.2 below.

⇔(4).
A similar characterization holds for non-percolation. We note that Corollary 3.4 establishes a relation between percolation for infinite volume Gibbs measures and cluster size distributions in finite volume canonical ensembles as examined in [JKM11].
3.2. Percolation thresholds: grand-canonical ensemble. Set E 1 := 0 and Let r 0 < r 1 as in Assumption 1 and suppose that v is continuous in (r 0 , r 1 ). Set Figure 2 in [PY09]. Note e ∞ ≤ −dm. Consider the conditions Set In addition, for every µ < e ∞ and sufficiently large β, there is a unique (β, µ)-Gibbs measure P ; it is shift-invariant, has no infinite cluster (Palmost surely), and satisfies (2) Suppose that v is continuous in (r 0 , r 1 ). Then We conjecture that for every fixed R ≥ r 1 , under suitable conditions on v, we have µ − (β; R) = µ + (β; R) for sufficiently large β and See Appendix A for the corresponding lattice gas result.
If in addition v satisfies Assumptions 4 and 5, then ρ m is larger than the preferred ground state density ρ 0 , and in particular, bounded away from zero; this follows from Theorem 3.2 in [J12]. Moreover, we expect that for every R ≥ r 1 , as β → ∞, and for ρ − (β; R) < ρ < ρ + (β; R) and very large β, there should be non-ergodic Gibbs measures with percolation probability strictly between 0 and 1. This is what happens for lattice gases (see Appendix A). For continuum systems, we have no proof of this conjecture; we have, however, a result pointing in the right direction: Proposition 3.7. Suppose that v satisfies Assumptions 4 and 5. Let R ≥ r 1 . There are β 0 , ρ 0 , C > 0 such that for all β ≥ β 0 , all ρ ≤ ρ 0 and all P ∈ G θ (β, ρ), the following holds: if ρ = exp(−βν) > exp(−βν * ), then ∀K ∈ N : Thus at densities above exp(−βν * ), the fraction of particles in finite-size clusters is small.

Topology on P and continuity properties
In this section we specify the topology on P that we use, recall some continuity properties of the functionals to be studied, and explain why the variational problems considered in this article have minimizers. We follow [GZ93,G94].
Let M(Ω) be the set of finite measures on (Ω, F ). The topology τ L of local convergence on M(Ω) is defined as follows. Let L be the class of measurable functions f : Ω → R that are local and tame, i.e., f ∈ L if and only if there is a Borel subset B ⊂ R d and a constant c > 0 such that f is a function of ω B alone and for all ω ∈ Ω, |f (ω)| ≤ c(1 + N B (ω)). Then τ L is the smallest topology with respect to which all maps of the form P → Ω f (ω)P (dω), f ∈ L, are continuous.
The following holds [GZ93,G94]: • P and P θ are closed subsets of M(Ω). We endow them with the traces of the topology τ L . • The Palm measure P θ → M(Ω), P → P • is continuous.
Proof. Let g k (ω) := 1(|C ω (0)| = k). The function g k is local and bounded, thus in particular, tame. Therefore, by definition of τ L , P → Ω g k dP is continuous. Since P → ρ k (P ) is the composition of the latter map with the continous map P → P • , it follows that P → ρ k (P ) is continuous. Now we can easily check that the variational problem in Theorem 3.1 admits a minimizer.
Proof. If U (P ) − β −1 S(P ) = ∞ for every P ∈ A, there is nothing to show. If U (P ) − β −1 S(P ) < ∞ for some P ∈ A, let (P n ) be a minimizing sequence. The sequence (U (P n ) − β −1 S(P n )) n∈N is bounded from above and, because U (P ) is bounded from below, S(P n ) is bounded from below too. Since the superlevel sets {S ≥ −c} are τ L -sequentially compact, there is a subsequence P nj converging to some P ∈ P θ . The continuity of the maps ρ(·) and ρ k (·) ensures that P ∈ A, and the lower semi-continuity of U and −S shows that P is a minimizer.

Proof of Theorem 3.1
The main idea for the proof of Theorem 3.1 is to apply a large deviations principle for the stationary empirical field proven in [G94,GZ93] and the contraction principle [DZ98,Section 4.2.1]. Two complications stand in our way. First, the large deviations principle in [G94,GZ93] was shown in the grand-canonical rather than the canonical ensemble. Second, the cluster size densities can only be expressed as functions of the stationary empirical field if we modify their definition and, loosely speaking, define them with periodic boundary conditions; this yields a modified variable ρ per Λ . In order to circumvent these difficulties, we proceed as follows: • We show first that the large deviations principle in the canonical ensemble for ρ Λ implies a large deviations principle in the grand-canonical ensemble (Lemma 5.1). • We apply the contraction principle and show that ρ per Λ satisfies a large deviations principle with convex rate function (Lemma 5.2). • Next we show that (truncations of) ρ Λ and ρ per Λ , in the grand-canonical ensemble, are exponentially equivalent [DZ98, Section 4.2.2]; this follows from Ruelle's superstability bounds [R70]. As a consequence, the grandcanonical rate functions for ρ Λ and ρ per Λ must be equal (Lemma 5.3). • Taking Legendre transforms, we deduce the desired identity for the canonical rate function (Lemma 5.4). For the purpose of this section it is most convenient to work with measures that are not normalized and to suppress the β-dependence in the notation. Let . , x N ))dx 1 · · · dx N be a measure on Λ N with total mass Z Λ (β, N ). The image of Q can N,Λ under the map Λ N → Ω, (x 1 , . . . , x N ) → {x 1 , . . . , x N } is a measure on (Ω, F ), for which we use the same letter Q can N,Λ . Furthermore define We use the same letter for the measure on Ω and the measure on disjoint unionṡ ∪ N ≥0 Λ N . Λ 0 is a dummy space corresponding to ω = {∅}: the event that there is no point at all has Q µ,Λ -measure 1. Remember that R N + is equipped with the product topology and the corresponding Borel σ-algebra.
Next we define the stationary empirical field and the modified cluster size distribution ρ per Λ . For ω ∈ Ω and Λ = [0, L] d , let ω per Λ := ∪ k∈Z d θ Lk (ω ∩ Λ) be the periodic continuation of ω ∩ Λ. The translation invariant empirical field is Remark. The stationary empirical field associates with every configuration ω a probability measure supported on configurations with the same relative coordinates as ω ∩ Λ, but randomized center of mass. For example, in dimension d = 1, if ω consists of two particles 0, ǫ, then R Λ,ω describes configurations {x, x + ǫ} with x having uniform Lebesgue density 1/L.

Proof of Prop. 3.3.
(2) ⇒ (3) By definition of our probability space Ω, for every configuration ω and every bounded set A the number of particles in A is finite. Therefore, if ω has a cluster with infinitely many particles, this cluster has infinite diameter.
(3) ⇒ (2) If in a configuration ω there is a cluster with infinite diameter, then this cluster must contain infinitely many particles -otherwise it would have diameter bounded by R times the cardinality of the cluster.
(1) ⇔ (2) Let The function is the limit of local, measurable functions and therefore measurable. Eq. (5) gives is the expected number of particles in [0, 1] d belonging to an infinite cluster. If ρ(P ) − ∞ k=1 kρ k (P ) > 0, it follows right away that with positive probability, there is an infinite cluster. If ρ(P ) − ∞ k=1 kρ k (P ) = 0, using shift-invariance, we see that for every unit cube C(k), k ∈ Z d (see p. 3) the probability that the cube intercepts an infinite cluster is zero; it follows that the probability that there is an infinite cluster vanishes.

Percolation properties of Gibbs measures
In this section we prove Theorems 3.5 and 3.6 and Proposition 3.7. The proofs are a combination of results from [M75,Z08,PY09,JKM11], what is known from cluster expansions [R69, R70], and equivalence of ensembles as in [G95].
7.1. Grand-canonical ensemble. For the proof of percolation, it is convenient to discretize space. Fix ℓ > 0. For k ∈ Z d , let C(k) := [k 1 ℓ, (k 1 +1)ℓ)×· · ·×[k N ℓ, (k N + 1)ℓ). Let C be the collection of cubes C(k), k ∈ Z d . We say that two cubes are nearest neighbors if their centers have Euclidean distance ℓ. A collection R ⊂ C of cubes is connected if any two cubes in R can be joined by a path (C 1 , . . . , C n ) of cubes in R such that for every j, the cubes C j and C j+1 are nearest neighbors. The following lemma is a variant of well-known contour arguments [R69,Section 5.3].
Lemma 7.1. Let P be a probability measure on {0, 1} C . There is a constant α d > 0 such that the following holds: if for some α > α d , some A > 0, and all connected subsets R ⊂ C, P (∀C ∈ R : ω C = 0) ≤ A exp(−α|R|), then P -almost surely, there is an infinite connected set W ⊂ C such that ω C = 1 for all C ∈ W.
We refer to cubes C with ω C = 0 as empty, and a cube with ω C = 1 as occupied.
Proof of Lemma 7.1. Let F be the collection of the (d − 1)-dimensional closed faces of the cubes in C. For example, [0, ℓ] d−1 × {0} ∈ F . A set Γ ⊂ F is a contour if R d \ ∪ F ∈Γ F splits into exactly two connected components, one finite (inside) and one infinite (outside). Let Int Γ ⊂ C be the collection of cubes that are inside Γ, and ∂ int Γ ⊂ Int Γ the cubes in the interior of Γ that touch the contour (sharing a face, an edge or a corner), i.e., Note that ∂ int Γ is connected and |∂ int Γ| ≥ c|Γ|, for a suitable Γ-independent constant c.
The Borel-Cantelli lemma shows that P -almost surely, C 0 is enclosed in only finitely many contours with empty boundary ∂ int Γ. Thus we can pick a cell adjacent from outside to the union of all such contours. This cell cannot be empty, and it cannot be surrounded by another contour with empty boundary. It follows that it must be a member of an infinite connected set of occupied cells.
We partition R d into the cubes C(k) of side-length ℓ, as described before Lemma 7.1. Let R ⊂ C be a finite connected collection of cubes. Set n := |R|. Consider the events Ω 0 (R) that there is no particle in Λ R := ∪ C∈R C and Ω 1 (R) that every cube C ∈ R contains exactly one particle, and this particle has distance < ǫ to the center of the cube.
Proof of (1) in Theorem 3.5. For β > 0 and k ∈ N, define the cluster partition functions and let R cl (β) be the radius of convergence of ∞ k=1 z k Z cl k (β), compare [M75,Prop. 3.3]). Let P ∈ G(β, µ) be a Gibbs measure that can be obtained as a limit of finite volume, grand canonical Gibbs measures with empty boundary conditions. Suppose that z < R cl (β). Then P (there is an infinite cluster) = 0 [M75, Theorem 3.1]. Mürmann's proof moreover yields, for every cube Λ ⊂ R d and all k ∈ N, the bound . Let ξ > 0 be as a in the proof of Lemma 5.3. If P is shift-invariant, Eq. (5) and Ruelle's superstability bounds [R70] show that as |Λ| → 0. It follows that ρ k (P ) ≤ z k Z cl k (β).
(22) In order to go from shift-invariant limits of finite volume Gibbs measures to general Gibbs measures, we use the theory of Mayer expansions. It is well-known that for every β > 0, some strictly positive R May (β) > 0, and activities z = exp(βµ) < R May (β), there is a unique Gibbs measure P β,µ ∈ G(β, µ); furthermore, the pressure and the correlation functions admit absolutely convergent expansions in powers of z (with temperature-dependent coefficients) [R70,Theorem 5.7]. The measure P β,µ is shift-invariant. Since every finite volume Gibbs measure converges to an infinite volume Gibbs measure, this limit must be P β,µ , and we can apply the previous considerations to P β,µ . Thus we see that when z < min(R cl (β), R May (β)), there is a unique Gibbs measure. It is shift-invariant, has no infinite cluster (P -almost surely), and satisfies Eq. (22).
Proof of (1) in Theorem 3.6. Remember R May (β) and R cl (β) from the proof of (1) in Theorem 3.5. For exp(βµ) < R May (β), the pressure is differentiable in µ and ρ(β, µ) is well-defined. Fix ǫ > 0 and let β ǫ > 0 large enough so that for β ≥ β ǫ , exp(β[e ∞ − ǫ]) < min(R May (β), R cl (β)). For β ≥ β ǫ and ρ < ρ(β, e ∞ − ǫ), we know that ρ = ρ(β, µ) for a unique µ; for this µ, G θ (β, ρ) = G(β, µ) = {P } with a unique Gibbs measure P . From the proof of Theorem 3.5, we see that P assigns probability zero to the event that there is an infinite cluster. As a consequence, ρ − (β; R) ≥ ρ(β, e ∞ − ǫ). Since Formally, we have which determines the external magnetic field h of the Ising model in terms of the chemical potential µ of the lattice gas as h = (µ + dJ)/2. The parameter J > 0 measures both the strength of attraction between particles and the strength of the ferromagnetic coupling between spins. The relevant notion of percolation is dependent site percolation. Dependent refers to the underlying measure on {0, 1} Z d , which is a Gibbs measure rather than a product of Bernoulli measures, and site percolation refers to the notion of connectivity -two occupied lattice sites x, y ∈ Z d are connected if there is a path (x 1 , . . . , x n ) of nearest neighbor (|x j+1 − x j | = 1) joining x 1 = x and x n = y, such that each site of the path is occupied, n xj = 1. In the Ising picture, we are interested in percolation of +clusters.
Our first remark is that at low temperature, the phase transition and the percolation transition coincide; this observation, as alluded to in the introduction, is the driving motivation for the present article's investigation. Fix a temperature T > 0 and vary the external field h (or the chemical potential µ). At temperatures above the Curie temperature T C > 0, there is no phase transition (the pressure stays analytic and the Gibbs measure is unique); at T < T C , there is a first-order phase transition at h = 0. On the other hand, let P + T,h be the Gibbs measure with + boundary conditions. We know that for all T > 0, there is a threshold h(T ) ∈ R such that the P + T,h -probability of having an infinite cluster is 0 at external fields h < h(T ), and 1 at fields h > h(T ). Furthermore, there is a temperature T + > 0 such that h(T ) = 0 for T < T + and h(T ) < 0 for T > T + [ABL87], and it is known that T + < T C in high dimensions. Thus at temperatures above the Curie temperature, there is a percolation transition but no phase transition; at temperatures between T + and T C , as h is increased, the percolation transition happens before the phase transition; and below T + , the percolation transition coincides with the phase transition.
The next observation is that the value at which the transition takes place is consistent with our grand-canonical Theorem 3.5: indeed, the zero external field h = 0 corresponds, in the lattice gas picture, to a chemical potential µ = −dJ, which can be interpreted as a ground state energy per particle -if in Z d every lattice site is occupied, the energy per particle is e ∞ = −dJ.
Finally, this agreement of thresholds extends to the canonical ensemble. At T < T C and external field h = 0, there are two shift-invariant Gibbs measures P ± T,h , with magnetizations m ± (T ). Low temperature contour expansions show that as T → 0, m ± (T ) = ±1+O(exp(−J/T )). Transforming as ρ ± (T ) = (m ± (T )+1)/2, we obtain that the corresponding curves for the lattice gas satisfy ρ − (T ) = O(exp(−J/T )) and ρ + (T ) = 1 + O(exp(−J/T )), which compares nicely with Theorem 3.6: for T < T + , the magnetizations m ± (T ) of the Ising model are also percolation thresholds (this is not true for T + < T < T C ).