Variational characterisation of Gibbs measures with Delaunay triangle interaction

This paper deals with stationary Gibbsian point processes on the plane with an interaction that depends on the tiles of the Delaunay triangulation of points via a bounded triangle potential. It is shown that the class of these Gibbs processes includes all minimisers of the associated free energy density and is therefore nonempty. Conversely, each such Gibbs process minimises the free energy density, provided the potential satisfies a weak long-range assumption.


Introduction
It is well-known that stationary renewal processes with a reasonable spacing distribution can be characterised as Gibbs processes for an interaction between nearest-neighbour pairs of points [16,Section 6]. Here we consider an analogue in two dimensions, viz. Gibbsian point processes on 2 with an interaction depending on nearest-neighbour triples of points, where the nearest-neighbour triples are defined in terms of the Delaunay triangulation. Recall that the Delaunay triangulation is dual to the Voronoi tessellation, in the sense that two points are connected by a Delaunay edge if and only if their Voronoi cells have a common edge. Since the Voronoi cell of a point consists of the part of space that is closer to this point than to any other point, this means that the Delaunay graph defines a natural nearest-neighbour structure between the points. (Of course, the analogy with renewal processes does not reach too far because the independence of spacings under the Palm distribution, which is characteristic of one-dimensional renewal processes, is lost in two dimensions due to the geometric constraints.) There is a principal difference between the Delaunay interactions considered here and the pair interactions that are common in Statistical Physics. Namely, suppose a point configuration ω is augmented by a new particle at x. In the case of pair interactions, x is subject to some additional interaction with the particles in ω, but the interaction between the particles of ω is not affected by x. In the Delaunay case, however, the particle at x not only gives rise to some new tiles of the Delaunay triangulation, but also destroys some other tiles that were present in the triangulation of ω. This so-called non-hereditary nature of the Delaunay triangulation blurs the usual distinction between attractive and repulsive interactions and makes it difficult to use a local characterisation of Gibbs measures in terms of their Campbell measures and Papangelou intensities. Such a local approach to the existence of Gibbs measures for Delaunay interactions was used in the previous work [2; 3; 5; 7] and made it necessary to impose geometric constraints on the interaction by removing triangles with small angles or large circumcircles.
In this paper we address the existence problem from a global point of view, which is based on stationarity and thermodynamic quantities such as pressure and free energy density. Specifically, we show that all minimisers of the free energy density are Gibbsian, which implies the existence of Delaunay-Gibbs measures because the entropy density has compact level sets. The converse part of this variational principle is harder and requires the comparison of different boundary conditions in the thermodynamic limit. It is here that the non-hereditary nature of the interaction shows up again, but it can be controlled with the help of stationarity and an additional condition which is much weaker than the geometric constraints mentioned above. In contrast to [7], however, we need to assume throughout that the interaction potential is bounded, and therefore do not cover hard-core interactions that forbid particular shapes of the tiles. We note, however, that some ideas of the present paper can be used to establish the existence of Delaunay-Gibbs measures in a more general setting that includes also the hard-core case, see [8]. The extension of the variational principle to such interactions is left to future work. As a final comment, let us emphasise that we make repeated use of Euler's polyhedral formula and the resulting linear complexity of the Delaunay triangulations, and are therefore limited to two dimensions, as was already the case in the previous papers mentioned above.

Configurations and Delaunay triangulations
A subset ω of 2 is called locally finite if card(ω ∩ ∆) < ∞ for all bounded ∆ ⊂ 2 ; each such ω is called a configuration. We write Ω for the set of all configurations ω. The configuration space Ω is equipped with the σ-algebra that is generated by the counting variables N ∆ : ω → card(ω ∩ ∆), with ∆ an arbitrary bounded Borel subset of 2 .
For each Borel set Λ ⊂ 2 we write Ω Λ = {ω ∈ Ω : ω ⊂ Λ} for the set of configurations in Λ, pr Λ : ω → ω Λ := ω ∩ Λ for the projection from Ω to Ω Λ , ′ Λ = |Ω Λ for the trace σ-algebra of on Ω Λ , and Λ = pr −1 Λ ′ Λ for the σ-algebra of events in Ω that happen in Λ only. For each configuration ω ∈ Ω we consider the Delaunay triangulation D(ω) associated to ω. By definition, is uniquely defined and determines a triangulation of the convex hull of ω whenever ω is in general circular position, in that no four points of ω lie on a circle that contains no further points of ω inside [17]. If this is not the case, one can apply some determistic rule to make the Delaunay triangulation unique. Indeed, let be the set of all triangles (or tiles) in 2 where '≺' stands for the lexicographic order in 2 . The triangles in can be compared by the lexicographic order of ( 2 ) 3 , and this in turn induces a lexicographic order on finite collections of triangles. Now, if n ≥ 4 points of ω lie on a circle with no points inside then the associated Delaunay cell is a convex polygon having these n points as vertices.
To define a unique triangulation of this polygon one can then simply take the smallest among all possible triangulations. Conflicts with other possible polygons cannot arise because the tessellations inside and outside a fixed convex polygon of n ≥ 4 points on a circle do not depend on each other.
Let us note that the prescription ω → D(ω) is a mapping from Ω to the set Ω( ) of all locally finite subsets of . If Ω( ) is equipped with the σ-algebra ( ) that is defined in analogy to , one can easily check that this mapping is measurable.
Next we assign to each tile τ ∈ a centre and a radius. Specifically, for every τ ∈ we write c(τ) for the centre and ̺(τ) for the radius of the circumscribed disc B(τ). The centres allow us to consider D(ω) as a germ-grain system, i.e., as a marked point configuration of germs in 2 and marks in the space 0 = {τ ∈ : c(τ) = 0} of centred tiles, by considering the mapping from Ω to the point configurations on 2 × 0 . Here we write τ − c(τ) := { y − c(τ) : y ∈ τ} for the shifted tile.
A crucial fact we need in the following is the linear complexity of Delaunay triangulations, which is expressed in the following lemma. This result follows directly from Euler's polyhedral formula, and is the main reason why we need to confine ourselves to two spatial dimensions; see

Stationary point processes and their tile distribution
Let P Θ be the set of all probability measures P on (Ω, ) that satisfy the following two properties: (I) P has a finite intensity z(P) = |∆| −1 N ∆ d P < ∞. Here, ∆ ⊂ 2 is any bounded Borel set of non-zero Lebesgue measure |∆|. (∆ can be arbitrarily chosen due to stationarity.) Each P ∈ P Θ is called a stationary point process on 2 with finite intensity. For Λ ⊂ 2 , we write P Λ := P • pr −1 Λ for the projection image of P on (Ω Λ , ′ Λ ), which can of course be identified with the restriction P| Λ of P to the events in Λ.
Every P ∈ P Θ defines a germ-grain modelP, namely the distribution of P under the mapping D defined in (2.2). That is,P is a stationary marked point process on 2 with mark space 0 . LetP 0 be the associated Palm measure on 0 × Ω and µ P =P 0 ( · × Ω) the associated mark distribution, or centred tile distribution, on 0 . By definition, for all nonnegative measurable functions f on 2 × 0 . For each P ∈ P Θ , µ P has total mass µ P = 2z(P), as follows from Euler's polyhedral formula; see, for example, [17, Eq. Let us say a measure P ∈ P Θ is tempered if We write P tp Θ for the set of all tempered P ∈ P Θ . Of course, (2.4) is equivalent to the condition So, P is tempered if and only if the last expression is finite. A sufficient condition for temperedness will be given in Proposition 4.9.
The most prominent members of P tp Θ are the Poisson point processes, which will take the role of reference processes for the models we consider. Recall that the Poisson point process Π z with intensity z > 0 is characterised by the following two properties: (P1) For every bounded Borel set ∆, the counting variable N ∆ is Poisson distributed with parameter z|∆|.
(P2) Conditional on N ∆ = n, the n points in ∆ are independent with uniform distribution on ∆, for every bounded Borel set ∆ and each integer n.
The temperedness of Π z follows from Proposition 4.3.1 of [17], or Proposition 4.9 below.
Another type of measures in P tp Θ are the stationary empirical fields that are defined as follows. Let Λ ⊂ 2 be an open square of side length L, and for ω ∈ Ω Λ let ω Λ,per = {x + Li : x ∈ ω, i ∈ 2 } be its periodic continuation. The associated stationary empirical field is then given by It is clear that R Λ,ω is stationary. In addition, it is tempered because 2̺(τ) ≤ diam Λ for each triangle τ ∈ D(ω Λ,per ). Finally, one easily finds that see the proof of the similar result in [14, Remark 2.3 (3)]. In the following we will often consider the probability kernel R Λ : ω → R Λ,ω = R Λ,ω Λ , where ω may be taken from either Ω or Ω Λ depending on the context.

The topology of local convergence
In contrast to the traditional weak topology on the set P Θ of stationary point processes, we exploit here a finer topology, which is such that the intensity is a continuous function, but nonetheless the entropy density has compact level sets.
Let denote the class of all measurable functions f : Ω → which are local and tame, in that there exists some bounded Borel set The topology T of local convergence on P Θ is then defined as the weak* topology induced by , i.e., as the smallest topology for which the mappings P → f d P with f ∈ are continuous. By the definition of the intensity, it is then clear that the mapping P → z(P) is continuous.
A further basic continuity property is the fact that the centred tile distribution µ P depends continuously on P. Let 0 be the class of all bounded measurable functions on the space 0 of centred tiles, and T 0 the associated weak* topology on the set ( 0 ) of all finite measures on 0 . (This is sometimes called the τ-topology.) Proposition 2.2. Relative to the topologies T and T 0 introduced above, the mapping P → µ P from P Θ to ( 0 ) is continuous.
This result will be proved in Section 4.1. It takes advantage of the linear complexity of finite Delaunay triangulations, and therefore relies on the planarity of our model.

The entropy density
Recall that the relative entropy (or Kullback-Leibler information) of two probability measures P, Q on any measurable space is defined by It is well-known that I(P; Q) ≥ 0 with equality precisely for P = Q. For a point process P ∈ P Θ and a bounded Borel set Λ in 2 , we write I Λ (P, Π z ) = I(P Λ ; Π z Λ ) for the relative entropy of the restrictions P Λ and Π z Λ . By the independence properties of Π z , these quantities are superadditive in Λ, which implies that the limit exists and is equal to the supremum of this sequence. For our purposes, it is sufficient to take this limit along a fixed sequence of squares; for example, one can take squares with vertex coordinates in + 1/2. The claim (2.8) then follows from the well-known analogous result for lattice models [10, Chapter 15] by dividing 2 into unit squares. I z is called the (negative) entropy density with reference measure Π z .
We set I = I 1 . Each I z differs from I only by a constant and a multiple of the particle density. In fact, an easy computation shows that I(P) = I z (P) + 1 − z + z(P) log z for all z > 0 and P ∈ P Θ . (2.9) A crucial fact we need later is the following result obtained in Proposition 2.6 of [14].

Lemma 2.3.
In the topology T , each I z is lower semicontinuous with compact level sets {I z ≤ c}, c > 0.

Triangle interactions
This paper is concerned with point processes with a particle interaction which is induced by the associated Delaunay triangulation. We stick here to the simplest kind of interaction, which depends only on the triangles that occur in each configuration. Specifically, let ϕ : 0 → be an arbitrary measurable function. It can be extended to a unique shift-invariant measurable function ϕ on via ϕ(τ) := ϕ(τ − c(τ)), τ ∈ . Such a ϕ will be called a triangle potential. We will assume throughout that ϕ is bounded, in that |ϕ| ≤ c ϕ (2.10) for some constant c ϕ < ∞. In Theorem 3.4 we will need the following additional condition to prove the temperedness of Gibbs measures. Let us say that a triangle potential ϕ is eventually increasing if there exist a constant r ϕ < ∞ and a measurable nondecreasing function ψ : This condition is clearly satisfied when ϕ is constant for all triangles τ with sufficiently large radius ̺(τ).
with β > 0. Of course, many variants are possible; e.g., one can replace the barycentre by the centre of the inscribed circle. By way of contrast, to penalise regular configurations one can replace the ϕ i 's by their negative.
The triangle potentials ϕ i above are not eventually increasing. But each triangle potential ϕ can be modified to exhibit this property by setting with r > 0 and K a suitable constant. When K is large, one has the additional effect of favouring small circumcircles.
Remark 2.5. The type of interaction introduced above is the simplest possible that is adapted to the Delaunay structure. In particular, we avoid here any explicit interaction ψ along the Delaunay edges. This has two reasons: First, we might add a term of the form 1 2 e⊂τ: card e=2 ϕ edge (e) to the triangle interaction ϕ. Such a term would take account for an edge interaction ϕ edge whenever D(ω) is a triangulation of the full plane. Secondly, we often need to control the interaction over large distances; the condition of ϕ being eventually increasing is tailored for this purpose. It is then essential to define the range in terms of triangles rather than edges. Namely, if a configuration ω is augmented by a particle at a large distance from ω, the circumcircles of all destroyed triangles must be large, but their edges can be arbitrarily short. So, a large-circumcircle assumption on the triangle potential allows to control this effect, but a long-edge asumption on an edge potential would be useless.

Results
Let ϕ be a fixed triangle potential. We assume throughout that ϕ is bounded, see (2.10), but do not require in general that ϕ is eventually increasing. To introduce the Hamiltonians and Gibbs distributions for ϕ we first need to introduce a class of 'reasonable' configurations.

Definition. Let us say a configuration
We write Ω * for the set of all admissible configurations.
In Corollary 4.2 we will show that P(Ω * ) = 1 for all P ∈ P Θ with P({ }) = 0. Suppose now that ω ∈ Ω * and Λ ⊂ 2 is a bounded Borel set. The Hamiltonian for ϕ in Λ with boundary condition ω is then defined by where ζ ∈ Ω is arbitrary. Since ω ∈ Ω * , the sum is finite, so that the Hamiltonian is well-defined. Note also that H Λ,ω ( ) = 0 in general. Moreover, Lemma 2.1 shows that whereΛ(ω) is as above and c ϕ from (2.10). This in turn implies that for each z > 0 the associated partition function is finite. We can therefore define the Gibbs distribution with activity z > 0 by The measure G Λ,z,ω depends measurably on ω and thus defines a probability kernel from (Ω * , Λ c ) to (Ω, ).
Definition. Let P be a probability measure on (Ω, ) which is of first moment, in that N Λ d P < ∞ for all bounded Borel sets Λ in 2 . P is called a Gibbs point process for the Delaunay triangle potential ϕ and the activity z > 0, or a Delaunay-Gibbs measure for short, if P(Ω * ) = 1 and P = P(dω) G Λ,z,ω for all bounded Borel sets Λ ⊂ 2 . We write G Θ (z, ϕ) for the set of all stationary Gibbs measures for ϕ and z, and G tp Θ (z, ϕ) for the set of all tempered stationary Gibbs measures; recall Eq. (2.4).
The above definition corresponds to the classical concept of a Gibbs measure, which is based on the location of points. We note that an alternative concept of Gibbs measure that considers the location of Delaunay triangles has been proposed and used by Zessin [21] and Dereudre [7].
Intuitively, the interaction of a configuration in Λ with its boundary condition ω reaches not farther than the setΛ(ω) above, which guarantees some kind of quasilocality. So one can expect that a limit of suitable Gibbs distributions G Λ,z,ω as Λ ↑ 2 should be Gibbsian, and the existence problem reduces to the question of whether such limits exist. Our approach here is to take the necessary compactness property from Lemma 2.3, the compactness of level sets of the entropy density. In fact, we even go one step further and show that the stationary Gibbs measures are the minimisers of the free energy density. Since such minimisers exist by the compactness of level sets, this solves in particular the existence problem. The free energy density is defined as follows; recall the definition of the centred tile distribution µ P before (2.3).
Definition. The energy density of a stationary point process P ∈ P Θ is defined by where ∆ is an arbitrary bounded Borel set of positive Lebesgue measure. The free energy density of P relative to Π z is given by I z (P) + Φ(P).
The definition of Φ is justified by Proposition 3.6 below. Here are some crucial facts on the free energy density, which will be proved in Subsection 4.1.

Proposition 3.1.
Relative to the topology T on P Θ , Φ is continuous, and each I z + Φ is lower semicontinuous with compact level sets. In particular, the set M Θ (z, ϕ) of all minimisers of I z + Φ is a non-empty convex compact set, and in fact a face of the simplex P Θ .
Next we observe that the elements of M Θ (z, ϕ) are nontrivial, in that the empty configuration ∈ Ω has zero probability; this result will also be proved in Subsection 4.1.
Our main result is the following variational characterisation of Gibbs measures.
The proof will be given in Subsections 4.2 and 4.5. Theorem 3.3 raises the problem of whether all stationary Gibbs measures are tempered. It is natural to expect that G Θ (z, ϕ) = G tp Θ (z, ϕ), but we did not succeed to prove this in general. In fact, we even do not know whether G tp Θ (z, ϕ) is always non-empty. But we can offer the following sufficient condition, which will be proved in Subsection 4.6.

Theorem 3.4. Suppose ϕ is eventually increasing and let z > 0. Then every stationary Gibbs measure is tempered, so that
Combining Theorems 3.3 and 3.4 we arrive at the following result.
The proof of Theorem 3.3 is based on an analysis of the mean energy and the pressure in the infinite volume limit when Λ ↑ 2 . For simplicity, we take this limit through a fixed reference sequence, namely the sequence of open centred squares. We shall often write n when we refer to Λ n . That is, we set ω n = ω Λ n , P n = P Λ n , R n,ω = R Λ n ,ω , H n,ω = H Λ n ,ω , and so on. We also write v n = |Λ n | = (2n + 1) 2 for the Lebesgue measure of Λ n . Our first result justifies the above definition of Φ(P). Besides the Hamiltonian (3.1) with configurational boundary condition ω, we will also consider the Hamiltonian with periodic boundary condition, namely Moreover, if P is tempered then Finally we turn to the pressure. Let  and lim inf where G n,z,per = Z −1 n,z,per e −H n,per Π z n is the Gibbs distribution in Λ n with periodic boundary condition, I z,ϕ = I z + Φ + p(z, ϕ) is the excess free energy density, and the closure cl and the interior int are taken in the topology T . Since ϕ is bounded so that Φ is continuous, this is a direct consequence of Theorem 3.1 of [14]. Remark 3.9. Marked particles. Our results can be extended to the case of point particles with marks, that is, with internal degrees of freedom. Let E be any separable metric space, which is equipped with its Borel σ-algebra and a reference measure ν, and Ω the set of all pairsω = (ω, σ ω ) with ω ∈ Ω and σ ω ∈ E ω . In place of the reference Poisson point process Π z , one takes the Poisson point process Π z on Ω with intensity measure zλ ⊗ ν, where λ is Lebesgue measure on 2 . Forω ∈ Ω let Of course, the centre, radius and circumscribed disc of a marked triangleτ are still defined in terms of the underlying τ. In the germ-grain representation, 0 is replaced by the set 0 of all centredτ. The tile distribution µP of a stationary point processP on Ω is a finite measure on 0 and is defined by placing bars in (2.3). A triangle potential is a bounded function ϕ on 0 . Such a ϕ is eventually increasing if ϕ(τ) = ψ(̺(τ)) for some nondecreasing ψ when ̺(τ) is large enough. It is then easily seen that all our arguments carry over to this setting without change.
Remark 3.10. Particles with hard core. There is some interest in the case when the particles are required to have at least some distance r 0 > 0. This is expressed by adding to the Hamiltonian (3.1) a hard-core pair interaction term H hc Λ,ω (ζ) which is equal to ∞ if |x − y| ≤ r 0 for a pair {x, y} ⊂ ζ Λ ∪ ω Λ c with {x, y} ∩ Λ = , and zero otherwise. Equivalently, one can replace the configuration space Ω by the space Ω hc = ω ∈ Ω : |x − y| > r 0 for any two distinct x, y ∈ ω of all hard-core configurations. The free energy functional on P Θ then takes the form F hc z := for P ∈ P Θ with Palm measure P 0 ; here we use the convention ∞ 0 = 0. We claim that our results can also be adapted to this setting. In particular, the minimisers of F hc z are Gibbsian for z and the combined triangle and hard-core pair interaction, and the tempered Gibbs measures for this interaction minimise F hc z . We will comment on the necessary modifications in Remarks 4.4 and 4.12.
Combining the extensions in the last two remarks we can include the following example of phase transition. where β > 0 is the inverse temperature and r 1 > 0 is an arbitrary interaction radius. If one adds a hard-core pair interaction with range r 0 < r 1 / 2 as in Remark 3.10, this model is similar to the model considered in [4]. (Instead of a triangle potential, these authors consider an edge potential along the Delaunay edges that do not belong to a tile τ of radius ̺(τ) > r 1 .) Using a random cluster representation of the triangle interaction as in [15] and replacing edge percolation by tile percolation one finds that the methods of [4] can be adapted to the present model. Consequently, if z and β are sufficiently large, then the simplex M Θ (z, ϕ) = G Θ (z, ϕ) has at least q distinct extreme points.

Energy and free energy
We begin with the proof of Proposition 2.2, which states that the centred tile distribution µ P depends continuously on P. The continuity of the energy density Φ and the lower semicontinuity of the free energy density I z + Φ then follow immediately.
Proof of Proposition 2.2: Let (P α ) be a net in P Θ that converges to some P ∈ P Θ . We need to show that g dµ P α → g dµ P for all g ∈ 0 . We can assume without loss of generality that 0 ≤ g ≤ 1.
We first consider the case that g has bounded support, in that g ≤ 1{̺ ≤ r} for some r > 0. Let ∆ ⊂ 2 be any bounded set of Lebesgue measure |∆| = 1. Also, let In view of (2.3), we then have g dµ Q = f dQ for all Q ∈ P Θ , and in particular for Q = P α and Q = P. By the bounded support property of g, f depends only on the configuration in the r-neigbourhood ∆ r := { y ∈ 2 : | y − x| ≤ r for some x ∈ ∆} of ∆. That is, f is measurable with respect to ∆ r . Moreover, f ≤ 2N ∆ r by Lemma 2.1, so that f ∈ . In the present case, the result thus follows from the definition of the topology T .
If g fails to be of bounded support, we can proceed as follows. Let ǫ > 0 be given and r > 0 be so large that µ P (̺ > r) < ǫ.
Since µ P α = 2z(P α ) → 2z(P) = µ P and µ P α (̺ ≤ r) → µ P (̺ ≤ r) by the argument above, we have µ P α (̺ > r) → µ P (̺ > r). We can therefore assume without loss of generality that µ P α (̺ > r) < ǫ for all α. Using again the first part of this proof, we can thus write Since ǫ was chosen arbitrarily, the result follows. ◊ We now turn to the properties of the free energy density.
Proof of Proposition 3.1: As ϕ belongs to 0 , the continuity of Φ follows immediately from Proposition 2.2. By Lemma 2.3, we can also conclude that I z + Φ is lower semicontiuous. Moreover, hypothesis (2.10) implies that the level set {I z + Φ ≤ c} is contained in {I z ≤ c + 2c ϕ z(·)}, which by (2.9) coincides with the compact set {I z ′ ≤ c + z ′ − 1} for z ′ = z exp(2c ϕ ).
Let P = δ ∈ P Θ be the Dirac measure at the empty configuration. Then µ P ≡ 0 and thus Φ(P) = 0. On the other hand, I z (P) = z. This means that I z + Φ is not identically equal to +∞ on P Θ and thus, by the compactness of its level sets, attains its infimum. To see that the minimisers form a face of P Θ , it is sufficient to note that I z + Φ is measure affine; cf. Theorem (15.20) of [10]. ◊ Next we show that the minimisers of the free energy are nondegenerate.

Proof of Proposition 3.2:
The second statement follows from the first because Θ (z, ϕ) is a face of P Θ . For, suppose there exists some P ∈ Θ (z, ϕ) with P({ }) > 0. Then δ appears in the ergodic decomposition of P, which would only be possible if δ ∈ Θ (z, ϕ).
To prove the first statement we note that Φ(Π u ) ≤ c ϕ µ Π u = 2c ϕ u for all u > 0. Therefore, if z > 0 is given and u is small enough then so that δ is no minimiser of the free energy. ◊ Finally we show that the energy density Φ is the infinite volume limit of the mean energy per volume.

Proof of Proposition 3.6:
We begin with the case of periodic boundary conditions. For every P ∈ P Θ , we have v −1 n H n,per d P = Φ(R n ) d P = Φ(PR n ). It is easy to see that PR n → P, cf. Remark 2.4 of [14]. Since Φ is continuous, it follows that Φ(PR n ) → Φ(P).
Next we consider the case of configurational boundary conditions and suppose that P is tempered. Applying (2.3) we obtain for each n is the ̺(τ)-neigbourhood of Λ n . Now, for each τ we have In view of (2.4) and (2.10), we can apply the dominated convergence theorem to conclude that as desired. ◊

The variational principle: first part
In this section we shall prove that each minimiser of the free energy is a Delaunay-Gibbs measure. We start with an auxiliary result on the 'range of influence' of the boundary condition on the events within a bounded set. Let ∆ ⊂ 2 be a bounded Borel set and ω ∈ Ω. Writing B(x, r) for the open disc in 2 with center x and radius r, we let be the set of possible radii of circumcircles hitting ∆ in the Delaunay triangulation of a configuration of the form ζ ∆ ∪ ω ∆ c with arbitrary ζ, and r ∆ (ω) = sup ∆ (ω). Let ∆ 2r = x∈∆ B(x, 2r) be the open 2r-neigbourhood of ∆. We then observe the following.
Proof: (a) Let˜ ∆ (ω) be defined as ∆ (ω), except that x is required to belong to 2 . Theñ and the last set certainly belongs to ∆ 2r \∆ .
has measure zero for all r > 0 and P ∈ P Θ . Now, if {r ∆ = ∞} occurs then ∆ is hit by a sequence of discs B n with N B n = 0 and diam(B n ) → ∞. Select points z n ∈ B n ∩ ∆. Passing to a subsequence one can assume that the sequence (z n ) converges to some z. Let ψ n be the direction from z to the centre of B n and ψ an accumulation point of the sequence (ψ n ). Finally, let C be a cone with apex z, direction ψ and opening angle less than π. It is then clear N C = 0. On the other hand, Poincaré's recurrence theorem implies that N C = ∞ almost surely on {N B(0,r) ≥ 1}. This contradiction gives the desired result. ◊ As an immediate consequence we obtain that each nondegenerate stationary point process is concentrated on the set Ω * of admissible configurations.  We are now ready to show that the minimisers of I z + Φ are Gibbsian.
One also needs to modify the proof of Proposition 3.2, in that the Poisson processes Π u should be replaced by the Gibbs measure P u with activity u and pure hard-core interaction. P u is defined as the limit of the Gibbs distributions G hc n,u,per for the periodic hard-core Hamiltonians H hc n,per . By Proposition 7.4 of [12], P u exists and satisfies Together with (2.9) we find that which is strictly less than z when u is small enough. Since Φ hc (P u ) = Φ hc (δ ) = 0, it follows that the minimisers of I z + Φ + Φ hc are non-degenerate. No further changes are required for the proof of the first part of the variational principle.

Boundary estimates
We now work towards a proof of the reverse part of the variational principle. In this section, we control the boundary effects that determine the difference of H n,per and H n,ω . The resulting estimates will be crucial for the proof of Proposition 4.11. For every ω ∈ Ω and every Borel set ∆ let be the set of all triangles τ ∈ D(ω) for which B(τ) crosses the boundary of ∆. We start with a lemma that controls the influence on S ∆ when two configurations are pasted together.
Proof: Let ∆, ζ and ω be given. If ζ is empty, there exists no x ∈ τ ∆ ⊂ ζ ∆ , so that the statement is trivially true. So let ζ ∈ Ω * and suppose there exists some τ ∈ S ∆ (ζ ∆ ∪ ω ∆ c ) with τ ∆ = . Let x ∈ τ ∆ . Since ζ ∂ ∆ = , x does in fact belong to the interior of ∆. This implies that B(τ ′ ) ∩ ∆ = for each τ ′ ∈ D(ζ) containing x. Therefore we only need to show that B(τ ′ ) \ ∆ = for at least one such τ ′ . Suppose the contrary. Then B(τ ′ ) ⊂ ∆ whenever x ∈ τ ′ ∈ D(ζ). This means that the Delaunay triangles containing x are completely determined by ζ ∆ . This gives the contradiction and the proof is complete. ◊ The following proposition is the fundamental boundary estimate. It bounds the difference of Hamiltonians with periodic and configurational boundary conditions in terms of S n := cardS Λ n .
Proof: Let n, ζ, ω be fixed and In view of (3.1) and (3.5) we have H n,ω (ζ) = τ∈A ϕ(τ) and H n,per (ζ) = τ∈B ϕ(τ). Since ϕ is bounded by c ϕ , we only need to estimate the cardinalities of A \ B and B \ A. We note that A \ B ⊂ S Λ n (ζ Λ n ∪ ω Λ c n ) and B \ A ⊂ S Λ n (ζ n,per ). So we can apply Lemma 4.5 to both ∆ = Λ n and ∆ = Λ c n to obtain that the set of points belonging to a triangle in A \ B is contained in the set of points belonging to a triangle of S Λ n (ζ) ∪ S Λ n (ω). Hence, card τ∈A\B τ ≤ 3(S n (ζ) + S n (ω)). By Lemma 2.1, it follows that card(A \ B) ≤ 6(S n (ω) + S n (ζ)).
To estimate the cardinality of B \ A we may assume that ζ n = . The periodic continuation ζ n,per then contains a lattice, and this implies that every triangle of D(ζ n,per ) has a circumscribed disc of diameter at most 2v n . Hence, each τ ∈ B \ A is contained in Λ 5n+2 , the union of 5 2 translates of Λ n (up to their boundaries). Applying Lemma 4.5 to each of these translates we conclude that the number of points that belong to a triangle of B \ A is bounded by 3 · 5 2 S n (ζ). Using Lemma 2.1 again we find that card(B \ A) ≤ 150(S n (ω) + S n (ζ)), and the result follows with γ = 156. ◊ The following immediate corollary will be needed in the proof of Theorem 3.3.
The next proposition exhibits the fundamental role of the temperedness condition (2.4) combined with stationarity for controlling the boundary effects. Then S n ≤ i∈I n \I n−1 where I n = Λ n ∩ ( 2 + ( 1 2 , 1 2 )). Note that card I n = v n . As in (4.2) we have The last term is finite by the temperedness of P. So, each Z i is P-integrable. Since Z i = Z 0 • ϑ i , the two-dimensional ergodic theorem implies that v −1 n i∈I n Z i converges to a finite limitZ, both P-almost surely and in L 1 (P). This implies that v −1 n i∈I n \I n−1 Z i tends to zero P-almost surely and in L 1 (P). The result thus follows from (4.3). ◊

Temperedness and block average approximation
Our first result in this subsection is a sufficient condition for temperedness in terms of vacuum probabilities. For P ∈ P Θ let V k (P) = ess sup P( be the essential supremum of the conditional probability that Λ k contains no particle given the configuration outside.
is tempered.
Proof: For each k ≥ 1 we consider the shifted squares Λ k (i) = Λ k + (2k + 1)i, i ∈ 2 , as well as the event A k = N Λ k (i) = 0 for all i ∈ 2 with i ∞ ≤ 1 .
Since P = δ , it is clear that P(A k ) → 1 as k → ∞. Thus we can write with the convention A 0 = . Now, if ω ∈ A k then each circumscribed disc containing 0 of a triangle τ ∈ D(ω) has a diameter not larger than 2 2v k , so that each such τ in fact belongs to D(ω Λ 7k+3 ).
Lemma 2.1 thus shows that the number of such τ is at most 2N 7k+3 (ω). The last sum is therefore not larger than In view of the stationarity of P, the last integral is bounded by V k−1 (P) v 7k+3 z(P) = 7 2 v k V k−1 (P) z(P). So we arrive at the estimate card τ ∈ D(ω) : 0 ∈ B(τ) P(dω) ≤ 2 · 7 2 3 2 z(P) k≥1 v k V k−1 (P) .
Together with (2.5) and assumption (4.5), this implies the temperedness of P because v k ∼ v k−1 as k → ∞. ◊ The second result concerns the approximation of stationary measures in terms of tempered ergodic measures. This approximation uses the block average construction first introduced by Parthasarathy for proving that the ergodic measures are dense in P Θ ; cf. [10,Theorem (14.12)], for example. Proposition 4.10. Let z > 0 and Q ∈ P Θ be such that I z (Q) + Φ(Q) < ∞. Then for each ǫ > 0 there exists some tempered Θ-ergodicQ ∈ P Θ such that I z (Q) + Φ(Q) < I z (Q) + Φ(Q) + ǫ Proof: Let Q ∈ P Θ be given. We can assume that Q = δ because otherwise we can chooseQ = Q. For n ≥ 1 let Q iid n denote the probability measure on Ω relative to which the particle configurations in the blocks Λ n + (2n + 1)i, i ∈ 2 , are independent with identical distribution Q n = Q • pr −1 Λ n . (In particular, this means that the boundaries of these blocks contain no particles.) Consider the spatial average It is clear that Q iid-av n ∈ P Θ . It is also well-known that Q iid-av n is Θ-ergodic; cf. [10,Theorem (14.12)], for example. By an analogue of [10,Proposition (16.34)] or [14,Lemma 5.5], its entropy density satisfies Next, the same argument as in [14,Lemma 5.7] shows that Q iid-av n → Q in the topology T . By Proposition 2.2, Φ is continuous. We thus conclude that Φ(Q iid-av n ) → Φ(Q), whence Φ(Q iid-av n ) < Φ(P)+ǫ for large n. It remains to prove that each Q iid-av n is tempered. Let n be fixed and k ≥ ℓ(2n+1) for some ℓ ≥ 1. We claim that V k (Q iid-av n ) ≤ q v ℓ−1 with q = Q(N n = 0). Indeed, for each x ∈ Λ n we have Λ k + x ⊃ Λ n+(ℓ−1)(2n+1) , and the latter set consists of v ℓ−1 = (2ℓ − 1) 2 distinct blocks as above. Letting g be any nonnegative Λ c k -measurable function and using the independence of block configurations, we thus conclude that which proves the claim. Now, we have q < 1 because Q = δ . It follows that for some constant C n < ∞. Together with Proposition 4.9, this gives the temperedness of Q iid-av n . ◊

The variational principle: second part
In this section we will complete the proof of the variational principle. The essential ingredient is the following counterpart of Proposition 3.7 involving configurational instead of periodic boundary conditions. We only state the lower bound we need.
for every ergodic Q ∈ P tp Θ . We can assume without loss that the right-hand side is finite. Now, since I z (Q) is finite, Q is locally absolutely continuous with repect to Π z . We fix some ǫ > 0, let f n = dQ n /dΠ z n , and consider for every ω ∈ Ω * the set Then for sufficiently large n we have It is therefore sufficient to show that for P-almost every ω, Q(A n,ω ) → 1 as n → ∞. By the ergodic theorem, Φ(R n ) converges to Φ(Q) in Q-probability; cf. Remark 2.4 of [14]. By McMillan's theorem [9; 18], Q(v −1 n log f n < I z (Q) + ǫ) → 1 when n tends to infinity. Moreover, Propositions 4.6 and 4.8 imply that, for P-almost all ω, |H n,ω − H n,per |/v n converges to 0 in L 1 (Q). This gives the result. ◊ We can now show that every tempered stationary Gibbs measure minimises the free energy density.
Therefore I z (P) ≤ −Φ(P) + min[I z + Φ], as required. ◊ Remark 4.12. In the hard-core setting of Remark 3.10, a slight refinement of Proposition 4.10 is needed. Namely, under the additional assumption that Φ hc (Q) = 0 one needs to achieve that also Φ hc (Q) = 0. To this end we fix an integer k > r 0 /2 and define Q iid n in such a way that the particle configurations in the blocks Λ n + (2n + 1)i, i ∈ 2 , are independent with identical distribution Q n−k , rather than Q n . This means that the blocks are separated by corridors of width 2k > r 0 that contain no particles. It follows that Φ hc (Q iid-av n ) = 0, and it is still true that lim sup n→∞ I z (Q iid-av n ) ≤ I z (Q); cf. [12, Lemma 5.1]. We thus obtain the refined Proposition 4.10 as before.
A similar refinement is required in the proof of Proposition 4.11. One can assume that Φ hc (P) = 0 and Φ hc (Q) = 0, and in the definition of A n,ω one should introduce an empty corridor at the inner boundary of Λ n to ensure that H hc n,ω = H hc n,per = 0 on A n,ω , see [12,Proposition 5.4] for details. In the proof of the second part of Theorem 3.3, one then only needs to note that Φ hc (P) = 0 when P is a Gibbs measure P for the combined triangle and hard-core pair interaction. The proof of Theorem 3.4 carries over to the hard-core case without any changes.

Temperedness of Gibbs measures
Here we establish Theorem 3.4. By Proposition 4.9 it is sufficient to show the following. Proposition 4.13. Let ϕ be bounded and eventually increasing, z > 0, and P be any stationary Gibbs measure for ϕ and z. Then there exists a constant C > 0 such that for all k ≥ 1.
To prove this we need an auxiliary result which states that the radii of all circumcircles in the Delaunay tessellation must decrease when a point is added to the configuration. Specifically, let ω ∈ Ω * and x ∈ 2 \ ω be such that ω ∪ {x} is in general circular position and x is not collinear with two points of ω. We consider the sets of ∆ x (ω) is covered by the discs B(τ) with τ ∈ C + x (ω), which by definition are free of particles. Consequently, ∆ o x (ω) contains no particle of ω, so that the vertices of each τ ∈ C x (ω) belong to the boundary ∂ ∆ x (ω). Next, Lemma 2.1 shows that cardC + x (ω) = cardC x (ω) + 2 , (4.9) and for every Λ ∋ x we have Here is the monotonicity result announced above.
The case k = 0 is more complicated because the tiles τ ∈ C (0) x (ω) are not naturally associated to a tile of C + x (ω). To circumvent this difficulty we define an injectionĨ from C (0) x (ω) \ {τ x } to C (2) x (ω) such that ̺(Ĩ(τ)) ≤ ̺(τ). Each triangle τ ∈ C (0) x (ω) different from τ x can then be mapped to the triangle I(τ) = e 2 (Ĩ(τ)) ∪ {x}, provided the latter belongs to C +∧ x (ω); otherwise I(τ) remains undefined. This completes the construction of I. (Note that τ x does not necessarily belong to C (0) x (ω). However, if it does we have no useful definition ofĨ(τ x ).) To constructĨ we turn C x (ω) into the vertex set of a graph G x (ω) by saying that two tiles are adjacent if they share an edge. The set C (2) x (ω) then coincides with the set of all leaves of G x (ω), and C (0) x (ω) is the set of all triple points (= points of degree 3) of G x (ω). Consider a fixed τ * ∈ C (0) x (ω) \ {τ x }. Since τ * ⊂ ∂ ∆ x (ω), the set ∆ x (ω) \ 〈τ * 〉 splits into three connected components.