Olivia Peytral-Rieu

David Dubuc

Katia Grenier

Microwave Microfabricated Sensor Dedicated to the Dielectric Characterization of Biological Microtissues

Microwave dielectric spectroscopy is a promising technique that allows intracellular, non-invasive, label-free and cost-effective biological studies. To complement the sensing possibilities of this technique at this frequency range, which have been previously demonstrated at the single cell, cellular suspensions and organs levels, a microwave-based sensor is designed, fabricated and evaluated to dielectrically characterize 3D biological microtissues. Such a biological model is indeed of high interest for biological investigations, as it constitutes an intermediate entity between 2D cell cultures and organs, exhibiting a biological response close to the one obtained with tissue physiology. The proposed sensor here the development of one of the first microfabricated device for spheroids studies. A discussion on the way of analyzing the data is proposed thanks the measurements of different fixed spheroids and polystyrene beads.

Introduction

Electrical approaches, especially dielectric spectroscopy, have been proven to be useful for a broad range of research purposes including biological researches [START_REF] Stoy | Dielectric properties of mammalian tissues from 0.1 to 100 MHz; a summary of recent data[END_REF], [START_REF] Mehrotra | EM-wave biosensors: A review of RF, microwave, mmwave and optical sensing[END_REF] as it they are non-invasive and non-destructive. These techniques use the interaction between electromagnetic fields and materials for biological investigations, while being labelfree and cost-effective [START_REF] Thielecke | Evaluation of impedance spectroscopy for the characterization of small biological samples in tissue-based test systems[END_REF], [START_REF] Groeber | Impedance Spectroscopy for the Non-Destructive Evaluation of in Vitro Epidermal Models[END_REF]. Among these techniques, the microwave dielectric spectroscopy, attracts increasing attention, as its frequency range enables intracellular investigations. This ability is indeed due to the fact that electromagnetic waves may by-pass the capacitive cellular membrane, giving access to the interior of cells. Previous investigations on microwave dielectric spectroscopy demonstrated the possible detection of various molecules such as amino acids, carbohydrates, proteins, or nucleic acids in aqueous solutions, as well as cells in solutions [5]- [START_REF] Poiroux | Label-free detection of mitochondrial activity with Microwave Dielectric Spectroscopy Research Article To cite this version[END_REF], at the individual cell level and with organs [8]- [START_REF] Lazebnik | Physics in Medicine & Biology Tissue-mimicking phantom materials for narrowband and ultrawideband microwave applications Tissue-mimicking phantom materials for narrowband and ultrawideband microwave applications[END_REF]. It may also provide different dielectric properties depending on chemically or electrically based stimuli applied to cells or yeasts. It therefore makes this analyzing technique as a promising method for drug testing notably [START_REF] Osterberg | Microwave Sensing of Yeast Cell Species and Viability[END_REF], [START_REF] Artis | Microwaving biological cells: Intracellular analysis with microwave dielectric spectroscopy[END_REF]. So far, another important biological model is still missing from the analyzing arsenal of microwave dielectric spectroscopy. 3D cell aggregates (also called spheroids or micro-tissues) constitutes however a well-established model, which is located in-between the in vivo and the 2D in vitro cell culture models. In addition to mimicking the particular defense of genuine tumors through biomechanisms, they present a convenient array of advantages for accurate studies. They present i) a quick maturation time, ii) reduced production costs, iii) in vivo-like configurations in terms of complexity, size and organization [START_REF] Nath | Three-dimensional culture systems in cancer research: Focus on tumor spheroid model[END_REF]. For example, when microtissues exhibit diameters greater than 200 µm, they present an onion configuration with different layers. A hypoxic internal core is formed (composed of dead or dying cells) due to the limited oxygen diffusion allowed by the upper cell layers, whereas the external layer present healthy cells. This may notably lead to the triggering of the differentiation of certain cells among the different spheroid cell layers, making the external layer cells invasive for instance [START_REF] Veelken | Single cell-based automated quantification of therapy responses of invasive cancer spheroids in organotypic 3D culture[END_REF]. Different microwave-based sensors have already been defined for the analysis of biological objects ranging from individual cells to entire organs of patients [START_REF] O'rourke | Dielectric properties of human normal, malignant and cirrhotic liver tissue: In vivo and ex vivo measurements from 0.5 to 20 GHz using a precision open-ended coaxial probe[END_REF], [START_REF] Tamra | Impact of a chemical stimulus on two different cell lines through microwave dielectric spectroscopy at the single cell level[END_REF]- [START_REF] Lazebnik | Highly Accurate Debye Models for Normal and[END_REF]. However, there is a gap in the literature for the study of mid-size microtissues with microwave dielectric spectroscopy. Here, a microfabricated device dedicated to the microwave characterization of such microtissues is introduced. It enables to fill the gap previously mentioned by characterizing the dielectric properties of spheroids with the extraction of capacitive and conductive contrasts for first polystyrene beads and then fixed spheroids. The microfabrication of this device is presented and maintained simple due to an appropriate design, which allies both microwave and fluidic considerations and constraints.

Materials & Methods

The studied object: fixed spheroids

Multicellular three-dimensional (3D) tumor spheroids are recognized as superior models for the preclinical evaluation of anti-cancer therapies due to their better and more advanced representation of in vivo tumors. The organization of cells within the 3D spheroidal structure means that they are in direct contact with each other and the secreted extracellular matrix (ECM), allowing them to utilize alternative cell growth and survival signaling mechanisms not readily observed in 2D monolayers [START_REF] Costa | 3D tumor spheroids: an overview on the tools and techniques used for their analysis[END_REF], [START_REF] Nath | Three-dimensional culture systems in cancer research: Focus on tumor spheroid model[END_REF]. This study is employing microtissues constituted of HEK293T cells. They are fixed using paraformaldehyde and stored in Phosphate Buffer Solution (Gibco Phosphate Buffered Saline 1X in our case, called PBS in the rest of this article).

Presentation of the microwave sensor and its microfabrication process

As shown with the schematic of Figure 1, the microwave biosensor is composed of a coplanar waveguide with a capacitive gap located in the center of the structure.

Coplanar are widened at the RF input and output to allow coplanar probe positioning with a pitch of 150µm. The size of the coplanar waveguide has been defined to exhibit a 50 Ω characteristic impedance. It also includes a capacitive gap of 10 µm width at the center of the structure. The biological entity under test is localized just above this sensing gap, within a dedicated place. The sensor also includes an open fluidic microchannel with two reservoirs placed apart. Due to the open fluidic configuration, the process only includes two mains steps, one to realize the metallic coplanar waveguide, and a second one for the fluidic part respectively. Figure 2 shows the fabrication flow diagram of the biosensor. The biosensor is fabricated on a lossless dielectric substrate (Corning Eagle XG series). The glass substrate is cleaned using a piranha bath during five minutes followed by an oxygen plasma treatment for fifteen minutes. A mask is photolithographically applied to a 2.5-µm-thick AZ NLOF-photoresists to define the biosensor electrodes structures. A Titanium and Gold (Ti/Au) layer is then deposited by evaporation. With an acetone bath applied during one hour, the NLOF photoresist is removed to obtained only the gold electrodes through a lift off process. The final step consists in the deposition by gravitation of a 500 µm thick SU8-3050 layer, which is then photopatterned and cured. This layer enables the creation of (i) the fluidic channel, where the spheroid is placed and maintained in PBS, and (ii) two fluidic reservoirs placed apart, which prevents evaporation in the fluidic channel. For testing, the device under test is connected to a Vector Network Analyzer using two coplanar probes and two coaxial cables. Before measurements, a preliminary SOLT calibration step is done. Microwave measurements are then performed from 500 MHz to 20 GHz. A protocol has then been established. First, the device is measured empty of any liquid or object, allowing to obtain the S parameters of the device itself. The liquid host medium (i.e. PBS in our case) is then loaded in the device and measured. This measurement serves afterwards as a reference one. Finally, the object under study is loaded and measured. Dielectric data are then extracted with an electrical model, which is presented in the Figure 3. Y1 represents the admittance of the fluidic walls, Y2 the admittance of the sensing zone above the electrodes. Parameters of interest here are the capacitance C and the conductance G. Extracted from the above-mentioned S parameters, CPBS and GPBS, as well as CSO and CSO are introduced and correspond to the electrical parameters when the device is loaded with the reference fluid or with the studied object (SO) in liquid respectively. Further details of these calculations may be found in [START_REF] Peytral-Rieu | Microwave Sensor Dedicated to the Determination of the Dielectric Properties of 3D Biological Models from 500MHz to 20GHz[END_REF]. Based on these electrical elements, capacitive and conductive contrasts are calculated between the system loaded with the studied object and the reference fluid, using equations ( 1) and ( 2) respectively. Such contrasts are used to better highlight small dielectric variations.

Microwave measurements protocol and electrical modeling

∆𝐶 = 𝐶 𝑆𝑂 -𝐶 𝑃𝐵𝑆

Results & Discussions

To validate the sensor and verify the accuracy of the technique, measurements of two different beads are done first. Figure 4 presents the corresponding results in terms of capacitance and conductance versus frequency (Figure 4 a and b), as well as the capacitive and conductive contrasts (Figure 4 c andd). Due to the low permittivity of polystyrene beads close to 2.56 compare to the PBS liquid, which exhibits a permittivity close to the one of water, a large variation between the reference liquid and the beads is obtained. The extracted capacitive contrasts are close to 60 fF below 5 GHz, whereas the conductive contrasts at 20 GHz reach a value of 10 mS, in relative value. Three different fixed spheroids are then characterized. Results are also indicated in Figure 4. As expected, the differences between the spheroids and the PBS are much smaller than for the polystyrene beads.

each spheroid presents a specific dielectric response due to variations in terms of size and shape. It is indeed difficult to produce microtissues perfectly round and exhibiting the same diameter. The discrimination between the host medium and the spheroids is even weaker as they have been fixed in a formaldehyde solution. This fixation creates indeed the permeabilization of the cells membranes, leading to the equilibrium between the intracellular medium and the extracellular one [START_REF] Mortensen | The effect of fixation conditions on the ultrastructural appearance of gastrin cell granules in the rat gastric pyloric antrum[END_REF], [START_REF] Virtanen | the Effect of Fixation on and Morphology in Scanning Electron Microscopy[END_REF]. The consecutive contrast is consequently lower that those, which would have been reached with unfixed microtissues. The choice of such fixed spheroids is due to the fact that we were looking to establish a proof of concept, namely the first validation of the sensor to characterize 3D biological spheroids. It was consequently easier to prepare fixed spheroids, which may be stored for weeks and used at convenience for experiments. The extracted electrical contrasts are then close to the reference level (PBS corresponds to the zero level), with values ranging from 8 to 18 fF at 5 GHz for the capacitive contrast, and between 0.7 to 2.6 mS for the conductive contrast at 20 GHz.

Conclusions

A microfabricated device enabling the microwave characterization of microtissues is presented. It exhibits a simple fabrication process. The successful microwave measurements of both polystyrene beads and then spheroids with a diameter close to 300 µm demonstrate the possible dielectric analysis of such biological models.

Figure 1 .

 1 Figure 1. Scheme of the miniature sensor dedicated to the microwave characterization of a spheroid. (a) Global scheme; (b) zoom vision of the black square in the middle of the device.

Figure 2 :

 2 Figure 2: Microfabrication flow diagram of the biosensor with, in pink, the representation of the NLOF photoresist, in yellow, the conductive layer in gold, in brown, its adhesion titanium layer, and in green, a thick and patterned SU-8 polymer layer.

Figure 3 :

 3 Figure 3: Electrical model of the device.

( 1 )

 1 ∆𝐺 = 𝐺 𝑆𝑂 -𝐺 𝑃𝐵𝑆[START_REF] Mehrotra | EM-wave biosensors: A review of RF, microwave, mmwave and optical sensing[END_REF]

Figure 4 :

 4 Figure 4: Measurements performed with different polystyrene beads and spheroids. Extracted (a) capacitance and (b) conductance obtained for the different objects. (c) Complex electrical representation of the capacitance and conductance, at 5 GHz and 20 GHz respectively. (d) Capacitive and (e) conductive contrasts. (f) Complex electrical representation of the capacitive and the conductive contrasts at 5 GHz and 20 GHz respectively. The tested spheroids and polystyrene beads present diameters between 200µm and 360µm.

Acknowledgements

This work was partly supported by LAAS-CNRS micro and nanotechnologies platform member of the French RENATECH network.


David Dereudre
email: david.dereudre@univ-valenciennes.fr

Remy Drouilhet
email: remy.drouilhet@upmf-grenoble.fr

Hans-Otto Georgii
email: georgii@math.lmu.de

Existence of Gibbsian point processes with geometry-dependent interactions

Keywords: Gibbs measure, hypergraph, Delaunay mosaic, Voronoi tessellation, entropy. MSC.Primary 60K35; Secondary: 60D05, 60G55, 82B21

We establish the existence of stationary Gibbsian point processes for interactions that act on hyperedges between the points. For example, such interactions can depend on Delaunay edges or triangles, cliques of Voronoi cells or clusters of k-nearest neighbors. The classical case of pair interactions is also included. The basic tools are an entropy bound and stationarity.

Introduction

Recent developments in statistical physics, stochastic geometry and spatial statistics involve Gibbs point processes with interactions depending on the local geometry of configurations in R d . A prominent class of such interactions is based on the nearest-neighbor graph coming from the Delaunay triangulation. In biology, such systems are used to model interacting cells in tissues or foams [START_REF] Farhadifar | The influence of cell mechanics, cell-cell interactions, and proliferation on epithelial packing[END_REF], [START_REF] Lautensack | 3D image analysis of open foams using random tessellations[END_REF]. In spatial statistics and stochastic geometry, structured point patterns, point processes lying along fibers or regular Delaunay tessellations have been studied via geometric Gibbs modifications; see [START_REF] Bertin | Existence of "nearest-neighbour" spatial Gibbs models[END_REF], [START_REF] Dereudre | Gibbs Delaunay tessellations with geometric hard core conditions[END_REF] and [START_REF] Ripley | Modelling spatial patterns (with discussion)[END_REF]. A probabilistic motivation comes from stationary renewal processes: Since these can be characterized as Gibbs processes for interactions between nearest-neighbor pairs of points [START_REF] Holley | Nearest neighbour birth and death processes on the real line[END_REF]Section 6], Gibbs processes on R d with Delaunay tile interaction can be viewed as a multi-dimensional counterpart [START_REF] Dereudre | Variational characterisation of Gibbs measures with Delaunay triangle interaction[END_REF].

In this paper we consider general geometry-dependent interactions that are defined on a hypergraph structure E. For every point configuration ω in R d , E(ω) denotes a set of hyperedges on ω, and the formal Hamiltonian of ω is given by

H(ω) = η∈E(ω) ϕ(η, ω)
with a potential ϕ(η, ω) which only depends on η and the points of ω in some neighborhood of η. This locality of ϕ will be called the finite horizon property, see (2.1) below. This general setting includes all the above-mentioned cases and also the classical many-body interactions of finite range that are familiar from statistical mechanics [START_REF] Ruelle | Statistical Mechanics. Rigorous results[END_REF].

There is a principal difference between the geometric interactions considered here and the classical many-body interactions. Namely, suppose a particle configuration ω is augmented by a new particle at x. In the case of a many-body interaction, this is only influenced by an additional interaction term between x and the particles of ω, and the interaction between the particles of ω is not affected by x. In other words, the classical many-body interactions are additive. In our setting, the new particle at x typically alters the hyperedges around x completely: some hyperedges are created and some others are destroyed. This means that both E(ω ∪ {x}) \ E(ω) and E(ω) \ E(ω ∪ {x}) are nonempty, so that H(ω ∪ {x}) and H(ω) each contain terms that are not present in the other. This phenomenon blurs the usual distinction between attractive and repulsive interactions. Moreover, if the potential ϕ(η, ω) is allowed to take the value ∞ (hard-exclusion case), we arrive at the so-called non-hereditary situation that a configuration ω is excluded although ω ∪ {x} is possible. This last case makes it difficult to use an infinitesimal characterization of Gibbs measures in terms of their Campbell measures and Papangelou intensities. Nevertheless, such an infinitesimal approach was first used to prove the existence of Gibbs measures for Delaunay interactions by requiring geometric constraints on the interaction [START_REF] Bertin | Existence of "nearest-neighbour" spatial Gibbs models[END_REF][START_REF] Bertin | Existence of Delaunay pairwise Gibbs point process with superstable component[END_REF][START_REF] Bertin | R-local Delaunay inhibition model[END_REF][START_REF] Dereudre | Gibbs Delaunay tessellations with geometric hard core conditions[END_REF]. A quite different global approach, first used in [START_REF] Georgii | Phase transition in continuum Potts models[END_REF] and based on stationarity and thermodynamic quantities such as pressure and free energy density, recently allowed to prove the existence of Gibbsian Delaunay tessellations for general bounded interactions without any geometric restrictions [START_REF] Dereudre | Variational characterisation of Gibbs measures with Delaunay triangle interaction[END_REF], and a similar approach could also be applied to quermass-interaction processes [START_REF] Dereudre | The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains[END_REF].

In this paper we address the existence problem for the general formalism of hypergraph interactions introduced here. Our approach is global as in [START_REF] Dereudre | The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains[END_REF][START_REF] Dereudre | Variational characterisation of Gibbs measures with Delaunay triangle interaction[END_REF] and leads to a significant improvement of the existing results. In particular, we establish the existence of Gibbsian Delaunay tessellations for non-bounded and hard-exclusion potentials. In the classical context of stable many-body interaction of finite range, our results permit to relax the superstability assumption. Basic ingredients of the proof are an entropy bound to exploit the compactness of the level sets of the entropy density, and a somewhat delicate control of the range of the interaction, which takes advantage of stationarity.

The general setting of Gibbs measures for hypergraph interactions is introduced in Section 2. Section 3 contains our assumptions and the existence theorems. Section 4 offers a series of examples that includes many-body interactions of finite range as well as interactions on Delaunay tiles and Voronoi cells and between k-nearest neighbors. The proofs of the main results follow in Section 5, and an appendix is devoted to measurability questions.

Preliminaries

Point configurations and hyperedge interactions

Consider the Euclidean space R d of arbitrary dimension d ≥ 1. Subregions of R d will typically be denoted by Λ or ∆ and will always be assumed to be Borel with positive Lebesgue measure |Λ| resp. |∆|. We write ∆ ⋐ R d if ∆ is bounded. A configuration is a subset ω of R d which is locally finite, in that ω ∩∆ has finite cardinality N ∆ (ω) = #(ω ∩∆) for all ∆ ⋐ R d . The space Ω of all configurations is equipped with the σ-algebra F that is generated by the counting variables N ∆ with ∆ ⋐ R d . It will often be convenient to write ω ∆ in place of ω ∩ ∆. As usual, we take as reference measure on (Ω, F) the Poisson point process Π z of an arbitrary intensity z > 0. Recall that Π z is the unique probability measure on (Ω, F) such that the following holds for all ∆ ⋐ R d : (i) N ∆ is Poisson distributed with parameter z|∆|, and (ii) conditional on N ∆ = n, the n points in ∆ are independent with uniform distribution on ∆, for each integer n ≥ 1.

Next, let Ω f ⊂ Ω denote the set of all finite configurations ω (which means that #(ω) < ∞), and

F f = F| Ω f the trace σ-algebra of F on Ω f . The product space Ω f × Ω carries the product σ-algebra F f ⊗F. For each Λ ⊂ R d we write Ω Λ = {ω ∈ Ω : ω ⊂ Λ} for the set of all configurations in Λ, pr Λ : ω → ω Λ = ω ∩ Λ for the projection from Ω to Ω Λ , F ′ Λ = F| Ω Λ for the trace σ-algebra of F on Ω Λ , and
F Λ = pr -1 Λ F ′ Λ ⊂ F
for the σ-algebra of all events that happen in Λ only. The reference measure on (Ω Λ ,

F ′ Λ ) is Π z Λ := Π z •pr -1 Λ . Finally, let Θ = (ϑ x ) x∈R d be the shift group, where ϑ x : Ω → Ω is the translation by the vector -x ∈ R d . By definition, N ∆ (ϑ x ω) = N ∆+x (ω) for all ∆ ⋐ R d .
The interaction of points to be considered in this paper will depend on the geometry of their location. This geometry will be described in terms of a hypergraph, and the interaction potential will be defined on the hyperedges. Definition.

• A hypergraph structure is a measurable subset E of Ω f × Ω such that η ⊂ ω for all (η, ω) ∈ E. If (η, ω) ∈ E,
we say that η is a hyperedge of ω, and we write η ∈ E(ω).

• A hyperedge potential is a measurable function ϕ from a hypergraph structure E to R ∪ {∞}.

• A hyperedge potential ϕ (or, more explicitly, the pair (E, ϕ))

is called shift-invariant if (ϑ x η, ϑ x ω) ∈ E and ϕ(ϑ x η, ϑ x ω) = ϕ(η, ω) for all (η, ω) ∈ E and x ∈ R d .
• Let us say that ϕ (or the pair (E, ϕ)) satisfies the finite horizon property if for each (η, ω) ∈ E there exists some ∆ ⋐ R d such that (2.1) (η, ω) ∈ E and ϕ(η, ω) = ϕ(η, ω) when ω = ω on ∆.

We will assume throughout this paper that the hyperedge potential ϕ under consideration is shift-invariant and exhibits the finite horizon property. Moreover, for notational convenience we set ϕ = 0 on E c . Since E is measurable, this does not affect the measurability of ϕ.

The domain E of ϕ can be considered as a rule that turns each configuration ω into a hypergraph (ω, E(ω)). Both E and ϕ are not affected by translations. Moreover, the presence of a hyperedge η ∈ E(ω) and the value of ϕ(η, ω) can be determined by looking at ω in a (sufficiently large but) bounded neighborhood ∆ of η, called the horizon of η in ω, which in general depends on both η and ω. Note that in general there is no minimal such horizon. To obtain a standard choice of ∆ one can take the closed ball B η,ω = B(g η , r η,ω ) with center at the gravicenter g η of η and radius r η,ω chosen smallest possible. Finally, we note that the concept of hypergraph structure is similar to that of a cluster property as introduced in [START_REF] Zessin | The Gibbs cluster process[END_REF]. Here are two examples the reader might keep in mind. Further examples will follow in Section 4.

Example 2.1. Many-body interactions of bounded range. Let r > 0 and LC r = (η, ω) : η ⊂ ω, diam(η) ≤ r, ω ∈ Ω be the locally complete graph. Thus, for each ω ∈ Ω, LC r (ω) consists of all hyperedges between points of distance at most r. If we assume that ϕ(η, ω) only depends on η, we are in the classical situation of many-body interactions of range r. The finite horizon property for (η, ω) ∈ LC r then holds for arbitrary ∆ ⊃ η. If LC r is restricted to hyperedges of cardinality two, we arrive at the familiar pair interactions of statistical mechanics.

Example 2.2. Delaunay potentials. The set Del of Delaunay hyperedges consists of all pairs (η, ω) with η ⊂ ω for which there exists an open ball B(η, ω) with ∂B(η, ω) ∩ ω = η that contains no points of ω. For k = 1, . . . , d+1 we write Del k = {(η, ω) ∈ Del : #η = k} for the set of all Delaunay simplices with k vertices. Clearly, Del and Del k are hypergraph structures. It is possible that the convex hull of a set η ∈ Del(ω) is not a simplex, namely when η consists of four or more points on a sphere with no point inside. However, this is an exceptional case, which occurs only with probability zero for our Poisson reference measure Π z . Note that B(η, ω) is only uniquely determined when #η = d+1 and η is affinely independent.

The simplest class of Delaunay hyperedge potentials consists of pair interactions of the form ϕ(η, ω) = φ(|x -y|) for η = {x, y} ∈ Del 2 (ω). Such a ϕ satisfies the finite horizon property (2.1) with ∆ = B(η, ω) for any ball B(η, ω) as above.

An example of a potential ϕ(η, ω) on Del 2 which does not only depend on η but also on ω is ϕ(η, ω) = φ(Vor ω (x), Vor ω (y)). Here we write Vor ω (x) for the Voronoi cell associated to a point x ∈ ω, viz. the set

(2.2) Vor ω (x) := y ∈ R d : |x -y| ≤ |x -y| ∀ x ∈ ω
of all points of R d which are closer to x than to all other points of ω. It is well-known that the Voronoi cells form a tessellation [START_REF] Schneider | Stochastic and integral geometry[END_REF]. Also, any two points of a configuration are connected by a Delaunay edge if and only if their Voronoi cells have a non-trivial intersection. That is,

(2.3) {x, y} ∈ Del 2 (ω) ⇐⇒ #(Vor ω (x) ∩ Vor ω (y)) > 1.
This reveals that the Delaunay graph is a nearest-neighbor graph. The potential above satisfies the finite horizon property (2.1) with ∆ equal to the closure of the set ξ∈Del(ω), ξ∩η =∅ B(ξ, ω), provided the cells Vor ω (x) and Vor ω (y) are bounded. The proviso is necessary because unbounded Voronoi cells, which can occur at the "boundary" of ω, are not protected by the points in ∆. So we must exclude from Del 2 (ω) all edges {x, y} for which Vor ω (x) ∪ Vor ω (y) is unbounded.

Gibbs measures for hyperedge potentials

Our objective here is to introduce the concept of a Gibbsian point process for an activity z > 0 and a hyperedge potential ϕ defined on a hypergraph structure E. First we will introduce the Hamiltonian for a bounded region Λ ⋐ R d with configurational boundary condition ω ∈ Ω. This requires to consider the set of hyperedges η in a configuration ω for which either η itself or ϕ(η, ω) depends on the points of ω in Λ. Specifically, we set

(2.4) E Λ (ω) = η ∈ E(ω) : ϕ(η, ζ ∪ ω Λ c ) = ϕ(η, ω) for some ζ ∈ Ω Λ .
Recall the convention that ϕ = 0 on

E c . So, if η ∈ E(ω) \ E(ζ ∪ ω Λ c ) for some ζ ∈ Ω Λ then either η / ∈ E Λ (ω)
, or η is irrelevant for ω, in that ϕ(η, ω) = 0. The Hamiltonian in Λ with boundary condition ω is then given by the formula

(2.5) H Λ,ω (ζ) := η∈E Λ (ζ∪ω Λ c ) ϕ(η, ζ ∪ ω Λ c ) for ζ ∈ Ω Λ ,
provided this sum is well-defined. As usual, we also consider the associated partition function

Z z Λ,ω := e -H Λ,ω (ζ) Π z Λ (dζ) .
To ensure that these quantities are well-defined, we impose the following condition on the boundary condition ω. Let ϕ -= (-ϕ) ∨ 0 be the negative part of ϕ.

Definition. A configuration ω ∈ Ω is called admissible for a region Λ ⋐ R d and an activity z > 0 if

H - Λ,ω (ζ) := η∈E Λ (ζ∪ω Λ c ) ϕ -(η, ζ ∪ ω Λ c ) < ∞ for Π z Λ -almost all ζ ∈ Ω Λ
(so that H Λ,ω is almost surely well-defined), and 0 < Z z Λ,ω < ∞. We write Ω Λ,z * for the set of all these ω.

We note that, in contrast to the standard setting of statistical mechanics, the partition function is not automatically positive because, in the present setting, H Λ,ω (∅) is not necessarily finite. For ω ∈ Ω Λ,z * , we can define the Gibbs distribution for (E, ϕ, z) in a region Λ ⋐ R d with boundary condition ω as usual by

(2.6) G z Λ,ω (F ) = Ω Λ 1 F (ζ ∪ ω Λ c ) e -H Λ,ω (ζ) Π z Λ (dζ) Z z Λ,ω ,
where F ∈ F is arbitrary.

Definition. Let E be a hypergraph structure, ϕ a hyperedge potential, and z > 0 an activity. A probability measure P on (Ω, F) is called a Gibbs measure for E, ϕ and z if P (Ω Λ,z * ) = 1 and

(2.7)

f dP = Ω Λ,z * 1 Z z Λ,ω Ω Λ f (ζ ∪ ω Λ c ) e -H Λ,ω (ζ) Π z Λ (dζ) P (dω)
for every Λ ⋐ R d and every measurable f

: Ω → [0, ∞[.
The equations (2.7) are known as the DLR equations (after Dobrushin, Lanford and Ruelle). They express that G z Λ,ω (F ) is a version of the conditional probability P ( F |F Λ c )(ω). We will be particularly interested in Gibbs measures that are stationary, that is, invariant under the shift group Θ = (ϑ x ) x∈R d . We write P Θ for the set of all Θ-invariant probability measures P on (Ω, F) with finite intensity i(P ) = N [0,1] d dP , and G Θ (ϕ, z) for the set of all Gibbs measures for ϕ and z that belong to P Θ . We conclude this section with a discussion of measurability questions.

Remark 2.1. Measurability. The quantities introduced above are not measurable with respect to the underlying σ-algebras defined so far, but only with respect to their universal completion. Specifically, for each σ-algebra A let A * be the associated σ-algebra of all universally measurable sets, i.e., of the sets which belong to the P -completion of A for all probability measures P on A; see [5, pp. 36 & 280]. It is then the case that, for each

Λ ⋐ R d , the Hamiltonian (ζ, ω) → H Λ,ω (ζ) is measurable with respect to (F ′ Λ ⊗ F Λ c ) * . Likewise, the partition function ω → Z z
Λ,ω is measurable with respect to F * Λ c , and Ω Λ,z

Hypotheses and results

Having defined the concept of Gibbs measure for a hyperedge potential we now turn to our main theme, the existence of such Gibbs measures. Let us state the conditions we need.

In the subsequent section we will provide a series of examples for which these conditions are met. We begin with an assumption which says that hyperedges with a large horizon require the existence of a large ball with only a few points. This will imply that the Hamiltonian H Λ,ω depends only on the points of ω in a bounded region ∂Λ(ω), and can be viewed as a sharpening of the finite horizon property (2.1).

(R) The range condition. There exist constants ℓ R , n R ∈ N and δ R < ∞ such that for all (η, ω) ∈ E one can find a horizon ∆ as in (2.1) satisfying the following: For every x, y ∈ ∆, there exist ℓ open balls B 1 , . . . , B ℓ (with ℓ ≤ ℓ R ) such that -the set ∪ ℓ i=1 Bi is connected and contains x and y, and -for each i, either diam

B i ≤ δ R or N B i (ω) ≤ n R .
Note that (R) is trivially satisfied when all horizon sets can be chosen to have uniformly bounded diameters. For instance, this holds in Example 2.1. The use of the range condition (R) will be revealed by Proposition 3.1 below, which states that the following finite range property holds almost surely for nondegenerate P ∈ P Θ .

Definition. Let Λ ⋐ R d be given. We say a configuration ω ∈ Ω confines the range of ϕ from Λ if there exists a set ∂Λ(ω

) ⋐ R d such that ϕ(η, ζ ∪ ωΛ c ) = ϕ(η, ζ ∪ ω Λ c ) whenever ω = ω on ∂Λ(ω), ζ ∈ Ω Λ and η ∈ E Λ (ζ ∪ ω Λ c ).
In this case we write ω ∈ Ω Λ cr . ∂Λ(ω) is called the ω-boundary of Λ, and we use the abbreviation

∂ Λ ω = ω ∂Λ(ω) .
Given any ω ∈ Ω Λ cr , we assume in the following that ∂Λ(ω) = Λ r \ Λ, where Λ r is the closed r-neighborhood of Λ and r = r Λ.ω is chosen as small as possible. Moreover, for ω ∈ Ω Λ cr we have

(3.1) H Λ,ω (ζ) = η∈E Λ (ζ∪∂ Λ ω) ϕ(η, ζ ∪ ∂ Λ ω),
and this sum extends over a finite set. This means that the first assumption in the definition of admissibility for Λ is satisfied. Here is the proposition announced above. It will follow from Proposition 5.4 below.

Proposition 3.1. Under (R), for each Λ ⋐ R d there exists a set ΩΛ cr ∈ F Λ c such that ΩΛ cr ⊂ Ω Λ cr and P ( ΩΛ cr ) = 1 for all P ∈ P Θ with P ({∅}) = 0.

Our next assumption is stability, the standard assumption that ensures the finiteness of all partition functions. In our setting, a somewhat modified definition turns out to be suitable.

(S) Stability. The hyperedge potential ϕ is called stable if there exists a constant c S ≥ 0 such that

(3.2) H Λ,ω (ζ) ≥ -c S #(ζ ∪ ∂ Λ ω) for all Λ ⋐ R d , ζ ∈ Ω Λ and ω ∈ Ω Λ cr .
In Remark 3.4 below we will show that this definition is a natural extension of the familiar concept of stability in statistical mechanics. Complementary to the lower bound provided by stability, we will also need a further condition that provides at least a partial upper bound for the Hamiltonians. This is because in the extreme case when ϕ is constantly equal to ∞ we have Z z Λ, • ≡ 0, so that the definition of Gibbs measures is meaningless. Let M ∈ R d×d be an invertible d × d matrix and consider for each k ∈ Z d the cell

(3.3) C(k) := Mx ∈ R d : x -k ∈ [-1/2, 1/2[ d .
These cells together constitute a periodic partition of R d into parallelotopes. For example, the columns M 1 , . . . , M d of M might form an orthogonal basis of R d or, for d = 2, define the sides of an equilateral triangle. For brevity we write C = C(0). Let Γ be a measurable subset of Ω C \ {∅} and

(3.4) Γ = ω ∈ Ω : ϑ Mk (ω C(k) ) ∈ Γ for all k ∈ Z d
the set of all configurations whose restriction to an arbitrary cell C(k), when shifted back to C, belongs to Γ. We call each ω ∈ Γ pseudo-periodic. The required control of the Hamiltonian from above will then be achieved by the following assumption on the joint behavior of E, ϕ and z. Recall that r Λ,ω was defined before (3.1).

(U) Upper regularity. M and Γ can be chosen so that the following holds.

(U1) Uniform confinement: Γ ⊂ Ω Λ cr for all Λ ⋐ R d , and

r Γ := sup Λ⋐R d sup ω∈Γ r Λ,ω < ∞. (U2) Uniform summability: c + Γ := sup ω∈Γ η∈E(ω): η∩C =∅ ϕ + (η, ω) #(η) < ∞,
where η := {k ∈ Z d : η ∩ C(k) = ∅} and ϕ + is the positive part of ϕ.

(U3) Strong non-rigidity:

e z|C| Π z C (Γ) > e c Γ
, where c Γ is defined as in (U2) with ϕ in place of ϕ + . Hypothesis (U1) states that the configurations in Γ confine the range of ϕ in a uniform way. So, for ω ∈ Γ, the ω-boundary ∂Λ(ω

) of a set Λ ⋐ R d is contained in the r Γ -boundary ∂ Γ Λ := Λ r Γ \ Λ, and the cardinality of ∂ Λ ω ist not larger than that of ∂ Γ Λ ω := ω ∂ Γ Λ .
On the other hand, condition (U2) provides a uniform upper bound for the local Hamiltonians H Λ,• on Γ. This bound is of order c Γ |Λ|, cf. (5.8) below. Finally, hypothesis (U3) holds for all z in an interval of the form ]z 0 , ∞[, provided that Π z C (Γ) > 0 for some (and thus all) z > 0. Indeed, since ∅ / ∈ Γ it follows that

(3.5) e z|C| Π z C (Γ) = ∞ k=1 z k k! C • • • C 1 Γ ({x 1 , . . . , x k }) dx 1 . . . dx k
is then a strictly increasing function of z. We emphasize that condition (U) imposes conflicting demands on Γ. While (U1) and (U2) suggest to choose the set Γ as small as possible, (U3) requires that Γ is not too small. The point will be to choose a set Γ that satisfies all requirements simultaneously. Here is our main existence theorem.

Theorem 3.2. For every hypergraph structure E, hyperedge potential ϕ and activity z > 0 satisfying (S), (R) and (U) there exists at least one Gibbs measure P ∈ G Θ (ϕ, z).

In some cases it is difficult to satisfy hypothesis (U3) when z is small. This occurs, for example, when a hard-exclusion hyperedge interaction enforces a minimal number of points per cell; see Proposition 4.2. But a slight variation of proof allows to establish the existence of a Gibbs measure for every z > 0 also in this case. Assumptions (U1) and (U3) are replaced by conditions ( Û1) and ( Û3) as follows.

( Û) Alternative upper regularity. M and Γ can be chosen so that the following holds.

( Û1) Lower density bound:

There exist constants a, b > 0 such that #(ζ) ≥ a|Λ|-b whenever ζ ∈ Ω f is such that H Λ,ω (ζ) < ∞ for some ζ ⊂ Λ ⋐ R d and some ω ∈ Γ. ( Û2) = (U2) Uniform summability. ( Û3) Weak non-rigidity: Π z C (Γ) > 0.
Here is the modified existence theorem.

Theorem 3.3. A Gibbs measure P ∈ G Θ (ϕ, z) exists also under the hypotheses (S), (R) and ( Û).

It is often natural to choose Γ as the set of all configurations that consist of a single point in some Borel set A ⊂ C. So we define

(3.6) Γ A = ζ ∈ Ω C : ζ = {x} for some x ∈ A .
For Γ = Γ A , the assumptions (U) and ( Û) are respectively called (U A ) and ( ÛA ) . In particular, (U2 A ) and (U3 A ) take the simpler form

(U2 A ) c + A := sup ω∈Γ A η∈E(ω): η∩C =∅ ϕ + (η, ω) #(η) < ∞ and (U3 A ) z|A| > e c A .
We then have the following corollary of Theorems 3.2 and 3.3.

Corollary 3.4. A Gibbs measure P ∈ G Θ (ϕ, z) exists under the hypotheses (S), (R) and either (U A ) or ( ÛA ).

We conclude this section with a series of comments on our assumptions and on the extension to marked particles. Remark 3.1. Bounded horizons. The conditions (R) and (U A ) hold as soon as ϕ({0}, {0}) is finite and (E, ϕ) has bounded horizons, in that there exists some r ϕ < ∞ such that r η,ω ≤ r ϕ for all (η, ω) ∈ E. (The notation r η,ω was introduced after (2.1).) Indeed, condition (R) holds trivially with δ R = 2r ϕ . As for (U A ), let M = aE, where a > 2r ϕ and E is the identity matrix. Let A = B(0, b) be a centered ball of radius b < a/2r ϕ . For Γ = Γ A , condition (U1) holds with r Γ = r ϕ . Moreover, by the choice of a and b, each η ∈ E(ω) with ω ∈ Γ

A must be a singleton {x}, so that ϕ(η, ω) = ϕ({x}, {x}). In view of the shift-invariance of ϕ, this means that (U2 A ) holds with c + A = ϕ + ({0}, {0}) < ∞. Finally, (U3 A ) holds if a and b are in fact chosen so large that also πzb 2 > e ϕ({0},{0}) . Remark 3.2. Scale-invariant potentials. Suppose E and ϕ are scale-invariant in the sense that (rη, rω) ∈ E and ϕ(rη, rω) = ϕ(η, ω) for all (η, ω) ∈ E and r > 0. Here, rη = {rx : x ∈ η} and rω = {rx : x ∈ ω}. Then Theorem 3.2 is still valid when assumption (U3) is replaced by ( Û3). Indeed, the scale invariance of ϕ implies that the image of a Gibbs measure for ϕ and z under the rescaling ω → rω is a Gibbs measure for ϕ and zr -d . So, it is sufficient to have the existence of a Gibbs measure for large z, and this follows from the remark around (3.5).

Remark 3.3. Stability via sublinearity of the hypergraph. We say that a hypergraph E is sublinear if there exists a constant C < ∞ such that #E(ω) ≤ C #(ω) for every finite configuration ω. In this case, the stability is ensured by requiring that the hyperedge potential ϕ is bounded below, in that

(3.7) ϕ(η, ω) ≥ -c ϕ
for some c ϕ < ∞. If the sublinearity of the hypergraph structure fails, the stability can simply be achieved by requiring that the potential ϕ is nonnegative (i.e., c ϕ = 0). For example, for d = 2 it follows from Euler's formula that the cardinalities of the Delaunay edges and triangles are sublinear [START_REF] Dereudre | Variational characterisation of Gibbs measures with Delaunay triangle interaction[END_REF], so that the stability follows directly from (3.7). Remark 3.6. Upper regularity in Delaunay models. For potentials acting on the Delaunay graph, the matrix M and the set Γ in hypotheses (U A ) and ( ÛA ) will be chosen as follows.

Let M be such that |M i | = a > 0 for i = 1, . . . , d and ∢(M i , M j ) = π/3 for i = j, and let Γ = Γ A with A = B(0, b). If b ≤ ̺ 0 a for some sufficiently small constant ̺ 0 > 0, the Delaunay neighborhood of a point x in a configuration ω ∈ Γ contains a minimal number of points denoted by γ d . For d = 2 one can take ̺ 0 = √ 3/6 and has γ 2 = 6, and the Delaunay neighborhood of the unique point x k in ω ∩ C(k) consists of the unique points x l in ω ∩ C(l) with lk ∈ {(-1, 0), (-1, 1), (0, 1), (1, 0), (1, -1), (0, -1)}. For d > 2, γ d is less easy to determine but it is clearly not larger than 3 d -1, the value corresponding to the case M = aE. Remark 3.7. Extension to the marked case. The preceding results can be easily extended to the case of particles with internal degrees of freedom, or marks. Let Σ be an arbitrary Polish space with Borel σ-algebra S and reference probability measure µ. Σ serves as the space of marks. That is, each marked point is represented by a position x ∈ R d and a mark σ ∈ Σ, and each configuration ω is a countable subset of R d × Σ having a locally finite projection onto R d . The role of the reference measure on the configuration space Ω is taken over by the Poisson point process Π z with intensity measure zλ ⊗ µ, where λ is Lebesgue measure on R d . The translations ϑ x act only on the positions of the particles and leave their marks untouched. We do not discuss the further formal details here, which are standard and can be found in [START_REF] Matthes | Infinitely Divisible Point Processes[END_REF] or [START_REF] Georgii | Large deviations and the maximum entropy principle for marked point random fields[END_REF], for example. What we want to emphasize here is that all definitions and results above carry over to this setting without any change, provided it is understood that all regions Λ or ∆ in R d always refer to the positional part of a configuration. For example: the notation ω ∆ now stands for ω ∩ (∆ × Σ); a set ∆ ⋐ R d is the horizon of (η, ω) ∈ E if (η, ω) ∈ E and ϕ(η, ω) = ϕ(η, ω) whenever ω = ω on ∆ × Σ; and the condition in Remark 3.1 should now read sup σ∈S ϕ({(0, σ), {(0, σ)}) < ∞ for some Borel set S ⊂ Σ with µ(S) > 0.

Examples

In this section we present a series of examples to illustrate our general existence results. These examples satisfy the assumptions of Theorem 3.2 or 3.3 and many of them have been introduced in practical or theoretical papers without justification or with justification in some partial cases.

To sort the examples of this section we will distinguish whether or not the potential ϕ(η, ω) depends explicitly on ω. If not, we speak of a pure hyperedge potential. Otherwise, the finite horizon property implies that ϕ(η, ω) actually depends only on some points of ω \ η close to η, which is expressed by speaking of a neighborhood-dependent hyperedge potential. But note that this distinction is merely a matter of how the interaction is represented. In the pure case, the extended potential (η, ω) → 1 E (η, ω) ϕ(η) on Ω f × Ω clearly does depend on ω, while in the other case one can often include the neighboring points of a hyperedge into an enlarged hyperedge to obtain a pure hyperedge potential.

Most of the following examples are based on the Delaunay graph. For simplicity, we will then often confine ourselves to the case d = 2, in which the stability is ensured by Remark 3.3 as soon as ϕ is bounded from below. But the reader should note that analogous results hold also in higher dimensions when ϕ is nonnegative, so that (S) is trivial.

Pure hyperedge interactions

In this subsection we consider examples of hyperedge potentials ϕ which only depend on the first parameter, so that ϕ(η, ω) = ϕ(η, η) =: ϕ(η).

Many-body interactions of finite range

Let r > 0 and E = LC r be the locally complete graph of Example 2.1, and suppose that ϕ(η, ω) = ϕ(η). Remark 3.1 then shows that a Gibbs measure exists as soon as the potential ϕ is stable and ϕ({0}) < ∞. By Remark 3.4, the first condition is equivalent to the classical stability assumption, and the second is necessary for defining Gibbs measures. So, as was observed first in [14, Remark 4.2], the techniques used here allow to weaken the superstability assumption of Ruelle's classical existence result [START_REF] Ruelle | Superstable interactions in classical statistical mechanics[END_REF], provided the interaction has finite range. But our techniques neither allow to treat the case of infinite range nor to rederive Ruelle's probability estimates.

An example of a stable but not superstable many-body interaction is the so-called quermass interaction. In space dimension d = 2, the associated Hamiltonian for a configuration ω is a linear combination of area, perimeter and Euler-Poincaré characteristic of the union of all discs of fixed radius that are centered at the points of ω. By the additivity of the Minkowski functionals, this Hamiltonian can be expressed in terms of a many-body interaction of finite range. Its stability is proved in [START_REF] Kendall | Quermass-interaction processes: conditions for stability[END_REF], but the superstability fails. The existence of Gibbs processes for this interaction has been proved in [START_REF] Dereudre | The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains[END_REF] by the same methods used here.

Delaunay edge interactions

Here we consider two classes of potentials ϕ on the hypergraph structure Del 2 in R 2 which are bounded below and depend only on the length of the Delaunay edges, in that ϕ({x, y}) := φ(|x-y|) when {x, y} ∈ Del 2 . Such potentials have been studied in [START_REF] Bertin | Existence of "nearest-neighbour" spatial Gibbs models[END_REF][START_REF] Bertin | Existence of Delaunay pairwise Gibbs point process with superstable component[END_REF][START_REF] Bertin | k-nearest-neighbour Gibbs point processes[END_REF][START_REF] Bertin | R-local Delaunay inhibition model[END_REF]. We improve the existence results of these papers.

Polynomially increasing edge interactions. Suppose that (4.1)

φ(ℓ) ≤ κ 0 + κ 1 ℓ α for some constants κ 0 ≥ 0, κ 1 ≥ 0 and α > 0 .

Recall the definition of ̺ 0 in Remark 3.6.

Proposition 4.1. Let d = 2 and ϕ be a pure edge potential on Del 2 which is bounded below and satisfies (4.1). Then there exists at least one Gibbs measure for ϕ and every activity z > (1+2̺ 0 )e 3κ 0 (3αe 2 κ 1 /2) 1/α /(π̺ 2 0 ).

Proof. We apply Corollary 3.4 with the assumption (U A ). The stability condition (S) holds because of Remark 3.3. We know further from Example 2.2 that every edge η ∈ Del 2 (ω) has the finite horizon B(η, ω). This shows that assumption (R) is satisfied.

Concerning assumption (U A ), let M and Γ A be defined as in Remark 3.6 with b = ̺ 0 a and a to be chosen later. The assumptions (U1 A ) and (U2 A ) are then trivial. We also find that c A ≤ 3 κ 0 + κ 1 a α (1+2̺ 0 ) α . So, (U3 A ) holds as soon as z > K 0 e K 1 a α /a 2 , where K 0 = e 3κ 0 /(π̺ 2 0 ) and K 1 = 3κ 1 (1+2̺ 0 ) α . Optimizing over a we find the value a = (2/αK 1 ) 1/α , which gives the sufficient condition z > K 0 (αK 1 e 2 /2) 1/α . The bound for z in Proposition 4.1 is certainly not the best possible. However, in the bounded case κ 1 = 0 it reduces to the trivial condition z > 0. As for condition (4.1), at first sight it might seem surprising that φ(ℓ) does not necessarily converge to zero when ℓ tends to infinity, but is even allowed to converge to ∞. However, φ(ℓ) should not be compared with a pairwise interaction that must decay over large distances, but with a nearest-neighbor interaction between oscillators that form an elastic lattice. A potential like the harmonic interaction φ(ℓ) = ℓ 2 is therefore quite natural.

Long-edge exclusion.

Suppose there are constants 0 ≤ ℓ 0 < ℓ 1 ≤ ℓ 2 such that (4.2) sup

ℓ 0 ≤ℓ≤ℓ 1 φ(ℓ) < ∞ and φ(ℓ) = ∞ if ℓ > ℓ 2 .
Proposition 4.2. Let d = 2 and ϕ be a pure edge potential on Del 2 which is bounded below and satisfies (4.2). Then there exists at least one Gibbs measure for ϕ and every z > 0.

Proof. We apply Corollary 3. Then ( Û2 A ) follows, and ( Û3 A ) is obvious.

Since φ(ℓ) may be equal to ∞ for ℓ less than some r 0 < ℓ 0 , the present example includes the classical case of a hard-core interaction which forces the points to keep distance at least r 0 > 0 from each other. Remark 4.1. Let us consider whether or not Propositions 4.1 and 4.2 remain valid when the Delaunay graph is replaced by a subgraph, as was discussed in Remark 3.5. One possible subgraph is the Gabriel graph. By definition, the Gabriel edge set Gab 2 (ω) consists of all edges {x, y} ∈ Del 2 (ω) for which the open disc with center (x + y)/2 and radius |x -y|/2 contains no point of ω. Since Gab 2 is shift-invariant and local, the restriction of ϕ to Gab 2 remains a shift-invariant hyperedge potential satisfying the finite horizon property (2.1). By Remark 3.3, (S) is inherited when we pass from Del 2 to Gab 2 . In view of our choice of M and Γ, the constant c Γ remains unchanged, so that (U3) is also inherited without modification of the valid range for z. This means that Proposition 4.1 holds also for the Gabriel graph. Unfortunately, condition ( Û1 A ) is lost, so that Proposition 4.2 does not carry over to the Gabriel graph. Another example of a Delaunay subgraph is the minimum spanning tree graph. This is tailor-made to be non-local, so that our results cannot be applied.

Delaunay tile interactions

In this subsection we deal with potentials on the hypergraph structure Del 3 of all Delaunay triangles in R 2 , and we still assume that ϕ is bounded below and depends only on the hyperedge and not on the remaining configuration. Such models have been considered recently in [START_REF] Dereudre | Gibbs Delaunay tessellations with geometric hard core conditions[END_REF] and [START_REF] Dereudre | Variational characterisation of Gibbs measures with Delaunay triangle interaction[END_REF]. We improve the existence results given there.

Polynomially increasing triangle interactions.

The first example is specified by assuming that the interaction of a Delaunay triangle η ∈ Del 3 is controlled by its size as in Proposition 4.1:

(4.3) ϕ(η) ≤ κ 0 + κ 1 δ(η) α for some constants κ 0 ≥ 0, κ 1 ≥ 0 and α > 0,
where δ(η) is the diameter of the circumcircle of η. We then have a similar result.

Proposition 4.3. Let d = 2 and ϕ be a pure triangle potential on Del 3 which is bounded below and satisfies (4.3). Then there exists at least one Gibbs measure for ϕ and every z > (2/ √ 3)+2̺ 0 e 2κ 0 (αe 2 κ 1 ) 1/α /(π̺ 2 0 ).

If ϕ is bounded above, so that κ 1 = 0, we recover the existence result of [START_REF] Dereudre | Variational characterisation of Gibbs measures with Delaunay triangle interaction[END_REF].

Proof. The proof is identical to that of Proposition 4.1. One simply has to note that δ(η) ≤ ((2/ √ 3)+2̺ 0 a when η ∈ Del 3 (ω) for some ω ∈ Γ.

Shape-dependent triangle interactions.

The shape of a Delaunay triangle can be captured by its angles. So let β(η) and γ(η) respectively denote the smallest and the largest interior angle of a triangle η. We assume that ϕ has the form Then there exists at least one Gibbs measure for ϕ and every z > 0.

Proof. We can again apply Corollary 3.4 with the assumption (U A ). The hypotheses (S) and (R) hold for the same reasons as in Proposition 4.1. The matrix M and the set Γ A can be defined as before, except that b is now chosen so small that β(η) > (π/3)δ for all η ∈ Del 3 (ω) with ω ∈ Γ A . Conditions (U1 A ) and (U2 A ) are then obvious. By Remark 3.2, it only remains to check ( Û3 A ) because ϕ is scale invariant. But this is trivial.

The conditions of Proposition 4.4 include the hard-equilaterality model, in which

(4.5) φ(β, γ) = ∞ if β ≤ (π/3) -δ, 0 otherwise
for some 0 < δ < π/3. The associated Gibbs measure produces a random Delaunay triangulation for which each triangle is almost equilateral in the sense that every angle exceeds π/3δ. This model is considered in [START_REF] Dereudre | Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction[END_REF] (Model 1, Section 2.3) as the crystallized triangulation model.

Neighborhood-dependent hyperedge interactions

We now turn to examples of potentials ϕ for which ϕ(η, ω) does not only depend on η but also on the points of ω in some neighborhood of η. In two of these examples, the underlying hypergraph structure is based on the singleton graph

(4.6) SG = ({x}, ω) : x ∈ ω ∈ Ω
for which all hyperedges are singletons. Since this might seem trivial, we add immediately that more complex hyperedges will be hidden in the horizons of the potentials.

Forced-clustering k-nearest neighbor interactions

In the k-nearest neighbors model on the space R d of any dimension d, each point x ∈ ω interacts with the k points of ω that are closest to x. This model was introduced in [START_REF] Bertin | Existence of "nearest-neighbour" spatial Gibbs models[END_REF][START_REF] Bertin | k-nearest-neighbour Gibbs point processes[END_REF].

We focus here on a non-hereditary variant, in which the k-nearest neighbors of x are forced to keep within distance δ > 0 from each other. This model was mentioned in [START_REF] Dereudre | Campbell equilibrium equation and pseudo-likelihood estimation for non-hereditary Gibbs point processes[END_REF] without precise justification of the proof, which is given here.

Let k ≥ 1 be some fixed positive integer. For each integer 1 ≤ i ≤ k, each configuration ω ∈ Ω with #(ω) ≥ k+1, and each x ∈ ω we let x i:ω be the i-th nearest neighbor of x in ω. More precisely, x i:ω is defined as the i-th element of ω in the total order < x on R d , in which y 1 < x y 2 if and only if

|y 1 -x| < |y 2 -x| or |y 1 -x| = |y 2 -x| and y 1 < d y 2 .
Here, < d stands for the lexicographic order on R d . We also set x 0:ω = x. Now let SG k = ({x}, ω) ∈ SG : #(ω) ≥ k+1 . We consider hyperedge potentials ϕ on SG k of the form

(4.7) ϕ({x}, ω) = φ(x 0:ω , . . . , x k:ω ) if diam{x 0:ω , . . . , x k:ω } < δ, ∞ otherwise,
where φ : (R d ) k+1 → R is any bounded measurable function and δ > 0 a fixed constant. It is clear that this interaction is non-hereditary in the sense that the removal of a particle from an allowed configuration can lead to a forbidden configuration. Since no kind of symmetry is required of φ, the present formalism includes both the directed and undirected k-nearest neighbor graphs.

Proposition 4.5. In the forced-clustering k-nearest neighbor model described above, there exists at least one Gibbs measure for ϕ and every z > 0.

Proof. Let us apply Theorem 3.2. The hypergraph structure SG k is clearly sublinear and ϕ is bounded below. The stability assumption (S) thus follows from Remark 3.3. Next, for given ({x}, ω) ∈ SG k , the closed ball B(x, |x k:ω -x|) with center x and radius |x k:ω -x| can serve as a horizon of ({x}, ω). Since the corresponding open ball contains at most k points of ω, assumption (R) follows immediately. Concerning (U), let b > 0 be a number to be specified later and M = aE as in Remark 3.1 for some a > 2(b+δ). We set

Γ = ω = {x 0 , x 1 , . . . , x k } : x 0 ∈ B(0, b), x i ∈ B(x 0 , δ/2) ∀ i = 1, . . . , k .
For each ω ∈ Γ and x ∈ ω it then follows that |xx k:ω | < δ. So (U1) holds with r Γ = δ, and (U2) follows with c + Γ ≤ (k+1) φ ∞ . Finally, we have

e z|C| Π z C (Γ) = z k+1 (k+1)! C . . . C 1 Γ ({x 0 , . . . , x k }) dx 0 . . . dx k ≥ z k+1 (k+1)! ν k+1 d b d (δ/2) kd ,
where ν d is the volume of the unit ball in R d . This shows that (U3) holds as soon as b is chosen large enough.

Voronoi cell interactions

Here we consider potentials that depend on the structure of the Voronoi cells. This type of model was introduced first by Ord; see the discussion in [START_REF] Ripley | Modelling spatial patterns (with discussion)[END_REF]. The spatial dimension d is arbitrary here. Specifically, let The following proposition includes a result of [START_REF] Bertin | Existence of "nearest-neighbour" spatial Gibbs models[END_REF].

SG b = ({x}, ω) ∈ SG : Vor ω (x)
Proposition 4.6. In the Voronoi cell interaction model with bounded φ, there exists at least one Gibbs measure for ϕ and every z > 0.

As in the case of Proposition 4.6, this example can easily be extended to the case when φ is polynomially increasing in the diameter of the associated doubleflower, or when φ exhibits a hard exclusion that permits the configurations in Γ

A for a suitable choice of A.

A particular adjacent Voronoi cell interaction with hard exclusion was proposed in [START_REF] Dereudre | Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction[END_REF], Model 3, Section 2.3. In this model, a hard exclusion forces the cells to be neither too small nor too large, and a smooth contribution induces a competition between the areas of adjacent cells.

Conclusion

The preceding series of examples presents only a selection of possible models and could easily be extended. For instance, having dealt with interactions acting on single Voronoi cells or pairs of adjacent Voronoi cells, we could proceed to triples of Voronoi cells with a common point, which are indexed by Delaunay triangles, or even larger clusters of Voronoi cells. On the other hand, the preceding interactions can be combined (i.e., added up) to obtain models with a richer interaction structure. A further universe of models opens up if one passes to marked configurations as in Remark 3.7. Future will show which kind of interaction will prove suitable for modeling geometric structures in the plane or in space that occur in the sciences. In any case, it should be evident that the conditions of our theorems are flexible enough to guarantee the existence of Gibbsian point processes for a large variety of geometric interactions.

Proofs of the theorems

Let E, ϕ and z be fixed throughout this section. Before we enter into the proofs of the theorems, it is necessary to verify a basic ingredient of the theory of Gibbs measures, namely the consistency of the finite-volume Gibbs distributions.

Lemma 5.1. Let Λ ⊂ ∆ ⋐ R d and ω ∈ Ω ∆,z * . Then G z ∆,ω Ω Λ,z * = 1 and f dG z ∆,ω = f dG z Λ,ω G z ∆,ω (dω)
for all measurable functions f : Ω → [0, ∞[.

Proof. Let Λ, ∆ and ω be fixed. Since G z ∆,ω does not depend on ω ∆ , we can assume for notational convenience that ω ⊂ ∆ c . Consider any two configurations ζ ∈ Ω Λ and ξ ∈ Ω ∆\Λ . It follows straight from the definition that

E Λ (ζ ∪ ξ ∪ ω) ⊂ E ∆ (ζ ∪ ξ ∪ ω). Hence, H ∆,ω (ζ ∪ ξ) is the sum of H Λ,ξ∪ω (ζ) and a term in R ∪ {∞} which does not depend on ζ. So, H - Λ,ξ∪ω (ζ) < ∞ when H - ∆,ω (ζ ∪ ξ) < ∞, and H Λ,ξ∪ω (ζ) < ∞ when H ∆,ω (ζ ∪ ξ) < ∞.
On the other hand, it follows that

H ∆,ω (ζ ∪ ξ) + H Λ,ξ∪ω (ζ ′ ) = H ∆,ω (ζ ′ ∪ ξ) + H Λ,ξ∪ω (ζ)
for all ζ ′ ∈ Ω Λ . Taking the negative exponential and integrating over ζ ′ we thus find that (5.1)

e -H ∆,ω (ζ∪ξ) Z z Λ,ξ∪ω = e -H Λ,ξ∪ω (ζ) Z z Λ,∆,ω (ξ) ,
where Z z Λ,∆,ω (ξ) = e -H ∆,ω (ζ ′ ∪ξ) Π z Λ (dζ ′ ) is the partial partition function for which the configuration ξ is held fixed. Since G z ∆,ω is concentrated on the set

{H - ∆,ω < ∞, H ∆,ω < ∞} which is contained in {H - Λ, • ∪ω < ∞, H Λ, • ∪ω < ∞}, we can conclude that G z ∆,ω Z z Λ, • = 0 = G z ∆,ω • pr -1 ∆\Λ Z z Λ,∆,ω (•) = 0 = (Z z ∆,ω ) -1 Z z Λ,∆,ω (•) 1 {Z z Λ,∆,ω (•)=0} dΠ z ∆\Λ = 0 . Likewise, G z ∆,ω Z z Λ, • = ∞ = 0 because Z z Λ,∆,ω (•) dΠ z ∆\Λ = Z z Λ,ω < ∞.
Combining these results we obtain the first assertion of the lemma. As a consequence, we can divide Eq. (5.1) by Z z ∆,ω Z z Λ,ξ∪ω to obtain the consistency equation

G z ∆,ω = G z Λ,ω G z ∆,ω (dω)
.

We now turn to the proof of Theorem 3.2. So we assume that the hypotheses (S), (R) and (U) are satisfied. As usual, we construct a Gibbs measure as a limit of Gibbs distributions in suitable boxes. We choose M and Γ as in hypothesis (U) and consider for each n ≥ 1 the parallelotope

Λ n = k∈Ln C(k) ,
where L n = {-n, . . . , n} d . Let ω ∈ Γ be a fixed pseudo-periodic configuration with sup k∈Z d N C(k) (ω) < ∞; the last condition can be satisfied by letting ω be periodic. Assumption (U1) implies that ω ∈ Ω Λn cr . Combined with (3.1), (S) and (U2&3), this shows that ω is admissible for Λ n and z; cf. (5.8) below. So we can define the Gibbs distribution

G n = G z Λn,ω • pr -1
Λn in Λ n with boundary condition ω and activity z, projected to Λ n . Since we aim at constructing a shift-invariant Gibbs measure, we will introduce a spatial averaging of G n , and it is convenient to work directly on the set of all shift-invariant probability measures on Ω. So, let P n be the probability measure on (Ω, F) relative to which the configurations in the disjoint blocks Λ n + (2n+1)Mk, k ∈ Z d , are independent with identical distribution G n . We consider the averaged measure We will show that the sequence ( Pn ) has an accumulation point in a suitable topology. As in [START_REF] Georgii | Phase transition in continuum Potts models[END_REF], we will take the required compactness from the compactness of the level sets of the specific entropy. Let us recall the necessary concepts.

A measurable function f : Ω → R is called local and tame if In view of (U2) and translation invariance, the first term on the right is not larger than c Γ #L n . Likewise, the second term is dominated by c + Γ #(L n+m \ L n ). This proves (5.8) and leads us to the estimate (5.9)

f (ω) = f (ω Λ ) and |f (
Z z Λn,ω ≥ 1 Γ ( ζ) e -H Λn,ω (ζ) Π z Λn (dζ) ≥ e -c Γ #Ln-o(vn) Π z C (Γ) #Ln .
Combining this with (5.5) to (5.7) we end up with (5.4).

The two propositions above imply that the sequence ( Pn ) admits a subsequence that converges to some P ∈ P Θ in the L-topology. The limit P is non-degenerate, in that P = δ ∅ . Indeed, in view of the lower semicontinuity of I z (implied by Proposition 5.2) and the continuity of the intensity i we obtain from (5.4) that (5.10)

I z ( P ) -c S i( P ) ≤ |C| -1 c Γ -ln Π z C (Γ) .
But hypothesis (U3) ensures that the quantity on the right-hand side is strictly less than

z = I z (δ ∅ ) -c S i(δ ∅ ).
It is natural to expect that P is the Gibbs measure we are looking for. Unfortunately, however, we are unable to show that P is concentrated on the admissible configurations. However, since P is non-degenerate, we can consider the conditioned measure P = P ( • |{∅} c ) ∈ P Θ with P ({∅}) = 0 and apply Proposition 3.1. Let us give a more precise statement of this proposition.

Let ℓ R , n R , δ R be the constants introduced in condition (R). Also, let δ -and δ + be the diameters of the largest open ball in C and of the smallest closed ball containing C, respectively. Fix an integer m ≥ 6ℓ R δ + /δ -. For each n ≥ 1, we decompose the parallelotope Λn := Λ n+(2n+1)m into the (2m+1) d translates Λ k n := Λ n + (2n+1)Mk of Λ n , where k ∈ L m . For any Λ ⋐ R d let n Λ ≥ 1 be the smallest number with Λ n Λ ⊃ Λ and n Λ ≥ δ R /6δ + . For all n ≥ n Λ we consider the events (5.11) ΩΛ,n cr = min

0 =k∈Lm N Λ k n > n R ∈ F Λn\Λ as well as ΩΛ cr = n≥n Λ ΩΛ,n cr ∈ F Λ c .
We then have the following result. Proof. Let Λ and n ≥ n Λ be fixed and consider any ω ∈ Ω for which the range of ϕ from Λ is not confined within Λn . Then there exists a configuration ω ∈ Ω with ω = ω on Λn \ Λ, a configuration ζ ∈ Ω Λ and a hyperedge η

∈ E Λ (ζ ∪ ω Λ c ) such that ϕ(η, ζ ∪ ω Λ c ) = ϕ(η, ζ ∪ ωΛ c ). So, every horizon ∆ of (η, ζ ∪ ω Λ c
) as in (2.1) hits Λc n . By (2.4), ∆ hits Λ too. Now let ∆ be chosen as in (R). We pick some x ∈ Λ ∩ ∆ and y ∈ ∆ \ Λn and, as in (R), a chain of ℓ ≤ ℓ R balls that hit each other successively and run from x to y.

There is a first ball B ℓ ′ hitting Λc n . Shrinking B ℓ ′ if necessary, we find a connected chain B 1 , . . . , B ℓ ′ of at most ℓ R balls that are all contained in Λn , connect x to ∂ Λn , and have either diameter at most δ R or contain at most n R particles of ζ ∪ ω Λ c . Since the distance between x and ∂ Λn is at least m(2n+1)δ -, at least one ball B i has a diameter exceeding 3(2n+1)δ + . As n ≥ δ R /6δ + , this bound is larger than both δ R and 3 diam Λ n . So there exist at least two indices k, k ′ ∈ L m such that Λ k n and Λ k ′ n are included in B i and thus hold at most n R points of ζ ∪ ω Λ c . At least one of these parallelotopes is different from Λ n , say Λ k n . This proves that ω / ∈ ΩΛ,n cr and completes the proof of the first statement. To prove the second claim let P ∈ P Θ be such that P ({∅}) = 0. We have

1 -P ( ΩΛ cr ) = P n≥n Λ ( ΩΛ,n cr ) c ≤ inf n≥n Λ 0 =k∈Lm P N Λ k n ≤ n R .
By translation invariance, the last expression is equal to

(#L m -1) inf n≥n Λ P N Λn ≤ n R .
But this term vanishes because

P N Λn ≤ n R → P N R d ≤ n R = P ({∅}) = 0
as n → ∞. The next to last identity comes from the well-known fact [19, 6.1.3] that P (0 < N R d < ∞) = 0 when P is translation invariant. The proof is therefore complete.

The final step in the proof of Theorem 3.2 is as follows.

Proposition 5.5. The conditional probability P = P ( • |{∅} c ) ∈ P Θ is a Gibbs measure for E, ϕ and z.

Proof. Since P ∈ P Θ with P ({∅}) < 1, P is well-defined and belongs to P Θ . To show that P is a Gibbs measure we fix some Λ ⋐ R d and consider the sets ΩΛ,p cr defined in (5.11) for p ≥ n Λ . We also set ΩΛ,≤p

f : Ω → [0, 1] is a local function and p ≥ n Λ is so large that f is F Λp -measurable.
Here, f Λ is defined by

f Λ (ω) := f dG Λ,ω .
Indeed, letting p → ∞ and setting f = 1 we then find that P ( ΩΛ cr ∩Ω Λ,z * ) = P ( ΩΛ cr ) and thus P (Ω Λ,z * ) = 1 by Proposition 5.4. For arbitrary f we obtain further that P = G z Λ,ω P (dω). Since Λ is arbitrary, this means that P is a Gibbs measure.

To prove (5.12) let f and p ≥ n Λ be fixed. It will be convenient to replace the sequence ( Pn ) introduced in (5.2) by an alternative sequence of measures with the same limit P . Suppose n is so large that Λp ⊂ Λ n and let

Λ • n = {x ∈ R d : Λp + x ⊂ Λ n }
be the " Λp -interior" of Λ n , which coincides with the closure of Λ n-p-(2p+1)m . We define the (subprobability) measure

Ḡn := 1 v n Λ • n G z Λn,ω • ϑ -1 x dx = 1 v n Λ • n G z Λn-x,ϑxω dx ;
the equality comes from the shift-invariance of ϕ and the symmetry of Λ • n . The argument in [START_REF] Georgii | Large deviations and the maximum entropy principle for marked point random fields[END_REF]Lemma 5.7] then shows that f d Pnf d Ḡn → 0 for all f ∈ L. This means that P can also be viewed as an accumulation point of the sequence ( Ḡn ). Now let x ∈ Λ • n , so that Λp ⊂ Λ nx. Using the consistency lemma 5.1 and the fact that ΩΛ,≤p

cr ∈ F Λp\Λ ⊂ F (Λn-x)\Λ we find ΩΛ,≤p cr f dG z Λn-x,ϑxω = Ω Λ,z * ∩ ΩΛ,≤p cr f dG z Λ,ω G z Λn-x,ϑxω (dω) ,
and averaging over x yields (5.13)

ΩΛ,≤p cr f d Ḡn = Ω Λ,z * ∩ ΩΛ,≤p cr f Λ d Ḡn .
The integrand of the integral on the left is measurable with respect to F Λp\Λ and thus belongs to L. By (A.4) in the appendix, the integrand on the right of (5.13) is measurable with respect to the universal completion F * Λp\Λ and thus can be squeezed between two functions in L which coincide P -almost surely; cf. [START_REF] Cohn | Measure Theory[END_REF], Proposition 2.2.3. So, (5.13) gives (5.12) in the limit when n runs through a subsequence for which Ḡn tends to P in the L-topology.

The proof of Theorem 3.3 requires only two minor observations. First, we note that ( Û1) and ( Û2) together with (R) imply (U1). Indeed, let ω ∈ Γ. Condition ( Û2) then shows that H Λm,ω (ω Λm ) < ∞ for all m ≥ 1. If m is so large that av mb > n R for the constant n R in (R), the lower density bound ( Û1) thus gives that each translate Λ k m , k ∈ Z d , contains more than n R points of ω. Hence, every ball with at most n R points has a diameter no larger than 2 diam(Λ m ). Invoking the range condition (R), we can therefore conclude that (U1) holds with r Γ ≤ ℓ R max(δ R , 2 diam(Λ m )).

Next we note that the non-rigidity condition in its strong form (U3) was only used below (5.10) when we showed that the accumulation point P is non-degenerate; for all other purposes, the weak form ( Û3) was sufficient. However, the non-degeneracy of P is trivial under ( Û1) because i( Pn ) ≥ ab/v n and thus i( P ) ≥ a > 0 by the continuity of i. This completes the proof of Theorem 3.3.

Appendix: Measurability

Here we collect and prove the measurability properties we have used and add a further comment on measurability. Let Λ ⋐ R d be fixed. Indeed, consider the measurable functions f Λ (η, ζ, ω) = (η, ζ ∪ ω Λ c ) and g(η, ζ, ω) = (η, ω) from Ω f × Ω Λ × Ω to Ω f × Ω. Since E and ϕ are measurable by assumption, the event ĒΛ := {ϕ • g = ϕ • f Λ } then belongs to F f ⊗ F Λ ⊗ F, and E Λ is equal to the projection image g( ĒΛ ). Since F Λ is known [START_REF] Kallenberg | Random measures[END_REF][START_REF] Matthes | Infinitely Divisible Point Processes[END_REF] to be the Borel σ-algebra for a Polish topology on Ω Λ , one can apply Prop. 8.4.4 of [START_REF] Cohn | Measure Theory[END_REF] to conclude that E Λ is universally measurable, as claimed.

Claim A.2. The functions (ζ, ω) → H Λ,ω (ζ) and (ζ, ω) → H - Λ,ω (ζ) are measurable with respect to (F ′ Λ ⊗ F Λ c ) * . To see this, we observe first that the mapping f Λ in Claim A.1 is measurable from F f ⊗F ′ Λ ⊗ F Λ c to F f ⊗F, and therefore also from (F f ⊗F ′ Λ ⊗F Λ c ) * to (F f ⊗F) * ; see [START_REF] Cohn | Measure Theory[END_REF], Lemma 8.4.6. In view of Claim A.1, this means that the indicator function 1 E Λ (η, ζ ∪ ω Λ c ) is measurable with respect to (F f ⊗ F ′ Λ ⊗ F Λ c ) * . Given any probability measure on Ω f × Ω Λ × Ω Λ c , we can therefore squeeze this indicator function between two F f ⊗ F ′ Λ ⊗ F Λ c -measurable functions which coincide almost surely; cf. Proposition 2.2.3 of [START_REF] Cohn | Measure Theory[END_REF]. Writing

H Λ,ω (ζ) = η⊂ω 1 E Λ (η, ζ ∪ ω Λ c ) ϕ(η, ζ ∪ ω Λ c ) ,
applying Theorem 5.1.2 of [START_REF] Matthes | Infinitely Divisible Point Processes[END_REF] repeatedly when the indicator function is replaced by one of the squeezing functions and using Proposition 2.2.3 of [START_REF] Cohn | Measure Theory[END_REF] in the converse direction we get the result. ) to (Ω, F).

Let P be an arbitrary probability measure on F Λ c . As in Claim A.2, we can squeeze the function e -H Λ,ω (ζ) between two F ′ Λ ⊗ F Λ c -measurable functions which coincide Π z Λ ⊗ P -almost surely. Integrating these functions over ζ with respect to Π z Λ we obtain two functions of ω, which squeeze Z z Λ,ω , are F Λ c -measurable by the measurability part of Fubini's theorem, and coincide P -almost surely. As P was arbitrary, the first result follows.

In the same way one finds that the function ω → Π z Λ (H - Λ,ω < ∞) is F * Λ c -measurable. Hence

Ω Λ,z * = ω ∈ Ω : Π z Λ (H - Λ,ω < ∞) = 1, 0 < Z z Λ,ω < ∞ ∈ F * Λ c .
One also finds that the integral in (2.6) depends F * Λ c -measurably on ω, which proves the last statement. The counterpart of Claim A.2 is therefore obvious, and the analog of Claim A.3 follows as before.

Claim A.5. The use of universal measurability could be avoided by modifying the definition of E Λ .

Namely, E Λ (ω) could be defined as the set of all η ∈ E(ω) for which either η ∩ Λ = ∅ or

Π z Λ ⊗ Π z Λ (ζ 1 , ζ 2 ) ∈ Ω 2 Λ : ϕ(η, ζ 1 ∪ ω Λ c ) = ϕ(η, ζ 2 ∪ ω Λ c ) > 0 .
Then E Λ ∈ F f ⊗ F by the measurability part of Fubini's theorem, and Claims A.2, A.3 and A.4 would follow without the stars referring to universal extensions. This modified definition, however, is less intuitive and destroys the simple monotonicity of E Λ (ω) in Λ which was used in the proof of Lemma 5.1. One can still show that the required monotonicity holds for Π z ∆\Λ -almost all ξ, but this is more involved. It is also necessary to redefine Ω Λ cr to obtain Proposition 5.4. We therefore decided to make use of universal measurability.

  4 with the assumption ( ÛA ). Hypotheses (S) and (R) follow as in Proposition 4.1. Condition ( Û1 A ) is satisfied since the long-edge exclusion condition ensures a minimal density of points. The values a and b in the definition of M and Γ are now chosen such that each {x, y} ∈ Del 2 (ω) (with ω ∈ Γ A ) satisfies ℓ 0 < |x -y| < ℓ 1 .

(4. 4 )Proposition 4 . 4 .

 444 ϕ(η) := φ(β(η), γ(η)) . Let d = 2 and ϕ be a pure triangle potential on Del 3 bounded below and satisfying(4.4). Suppose there exists some δ > 0 such that sup γ≥β>(π/3)-δ φ(β, γ) < ∞ .

4

 4 is bounded be the hypergraph structure of singletons with bounded Voronoi cells and ϕ be of the formϕ({x}, ω) = φ Vor ω (x) for ({x}, ω) ∈ SG b .For instance, φ might depend on the number of faces, the volume, or the surface area of the Voronoi cells. The necessity of passing from SG to SG b was discussed in Example 2.2. Each ({x}, ω) ∈ SG b has as finite horizon the bounded Voronoi flower (

  Λn P n • ϑ -1 x dx, where v n = |Λ n | is the volume of Λ n . By the periodicity of P n , Pn is shift-invariant. Moreover, the intensity i( Pn ) = N Λn dG n /v n of Pn is finite because N Λn e -H Λn,ω dΠ z Λn ≤ e c S #(∂ Λn ω) N Λn e c S N Λn dΠ z Λn < ∞ by (3.1) and (S). So, Pn ∈ P Θ .

  ω)| ≤ a N Λ (ω) + b where the error term is uniform in ζ. Indeed, since η ∈ E Λn ( ζ) when η ∈ E( ζ) and k ∈ η ∩ L n , we can write H Λn,ω (ζ) = k∈Ln η∈E( ζ): η∋k ϕ(η) #η + k∈L n+m \Ln η∈E Λn ( ζ): η∋k ϕ(η) #η .

Proposition 5 . 4 .

 54 Given any Λ ⋐ R d , we have ΩΛ cr ⊂ Ω Λ cr and ∂Λ(ω) ⊂ Λn when ω ∈ ΩΛ,n cr for some n ≥ n Λ . Moreover, P ( ΩΛ cr ) = 1 for all P ∈ P Θ with P ({∅}) = 0.

Claim A. 1 .

 1 E Λ := {(η, ω) : η ∈ E Λ (ω)} ∈ (F f ⊗ F) * .

Claim A. 3 .

 3 The partition functionω → Z z Λ,ω is measurable with respect to F * Λ c , Ω Λ,z * ∈ F * Λ c , and G z Λ,ω (F ) is a probability kernel from (Ω Λ,z * , F * Λ c | Ω Λ,z *

Claim A. 4 .

 4 Let p ≥ n Λ be fixed and suppose condition (R) holds. Claims A.2 and A.3 remain valid with F Λp\Λ in place of F Λ c as soon as all quantities are restricted to the set ΩΛ,p cr defined in(5.11). In particular, Ω Λ,z * ∩ ΩΛ,p cr ∈ F *Λp\Λ, andG z Λ,ω (F ) is a probability kernel from (Ω Λ,z * ∩ ΩΛ,p cr , F * Λp\Λ | Ω Λ,z * ∩ ΩΛ,p cr ) to (Ω, F).Indeed, by Proposition 5.4 and (3.1) we haveH Λ,ω (ζ) = η∈E Λ (ζ∪ω Λp\Λ )ϕ(η, ζ ∪ ω Λp\Λ ) when ω ∈ ΩΛ,p cr .

  Remark 3.4. Stability: comparison with the classical case. Consider the hypergraph structure E = LC r of Example 2.1 describing many-body interactions of range r. In contrast to Example 2.2 where the Delaunay tiles depend on the presence of further particles, it is then meaningful to define the energy of a finite configuration ζ ∈ Ω f by H(ζ) = η∈E(ζ) ϕ(η).The classical stability condition asserts that H(ζ) ≥ -c S #(ζ) for all ζ ∈ Ω f ; see[START_REF] Ruelle | Statistical Mechanics. Rigorous results[END_REF], for example. This follows from (S) by choosing Λ ⊃ ζ and ω = ∅. Conversely, the Hamiltonian (2.5) is equivalent to the Hamiltonian HΛ,ω (ζ) := H(ζ ∪ ∂ Λ ω) in the sense that the associated Gibbs distributions coincide (at least when ϕ < ∞), and the classical stability assumption for H gives (S) for HΛ,ω . This shows that hypothesis (S) is essentially equivalent to the classical concept of stability. Remark 3.5. Sub-hypergraph potentials. Consider a shift-invariant sub-hypergraph structure E ′ ⊂ E of E and a hyperedge potential ϕ on E, and let ϕ ′ be its restriction to E ′ . In general, ϕ ′ does not satisfy the finite horizon property, but let us assume it does. Which of the assumptions (R), (S), (U) and ( Û) on ϕ are inherited by ϕ ′ ? It is clear that assumptions (R), (U1), (U2) and ( Û3) are hereditary. Assumption (U3) remains also valid for ϕ ′ , but for a different range of values of z because the constant c Γ is different in general. Assumption (S) is lost, but a positive exception is the case of Remark 3.3 when stability follows from the sublinearity of E and the lower boundedness of ϕ; these properties are obviously inherited by ϕ ′ . Assumption ( Û1) is lost in general.

Proof. Let us apply Corollary 3.4 again. The stability comes from the sublinearity of SG b (ω) and the lower boundedness of ϕ, cf. Remark 3.3. The proof of (U A ) is essentially the same as in Proposition 4.1 with κ 1 = 0. Concerning the range condition (R), let {x} ∈ SG b (ω) and y 1 , y 2 any two points of the set ∆ defined in (4.8). Then there exist ξ 1 , ξ 2 ∈ Del(ω) such that y 1 ∈ B(ξ 1 , ω), y 2 ∈ B(ξ 2 , ω) and x ∈ B(ξ 1 , ω) ∩ B(ξ 2 , ω). The latter means that the union of these two balls is connected. By definition, B(ξ 1 , ω) and B(ξ 2 , ω) contain no point of ω. So, condition (R) holds with ℓ R = 2, n R = 0 and arbitrary δ R .

In the above, the boundedness of φ was only assumed for simplicity. In analogy to Propositions 4.1 and 4.3, one can consider potentials ϕ that are polynomially increasing in the diameter of the cell's flower (4.8). The existence result then holds only for sufficiently large activity z. It is also straightforward to consider hard-exclusion models as in Propositions 4.2 and 4.4. For example, let d = 2 and (4.9)

φ Vor ω (x) = 0 if Vor ω (x) has six edges (and vertices), ∞ otherwise.

It is then clear that Theorem 3.2 applies, and by scale invariance a Gibbsian point process exists for all z > 0. The typical configurations of such a Gibbs process preserve the topology of the triangular lattice but are less regular than those of the hard-equilaterality model (4.5) for small δ. The model (4.9) may therefore be called the randomly distorted triangular lattice.

Adjacent Voronoi cell interactions

Here we reconsider the potential presented in Example 2.2, which describes an interaction between two adjacent Voronoi cells. That is, let d = 2 and the set of hyperedges of a configuration ω ∈ Ω be given by For instance, ϕ(η, ω) can either depend on the length of the common edge or on the area ratio of the cells Vor ω (x) and Vor ω (y). As noticed before, the Voronoi "doubleflower"

then serves as finite horizon of (η, ω) ∈ Del 2,b . In this setting we have the following result which can be proved in the same way as Proposition 4.6.

Proposition 4.7. In the adjacent Voronoi cell interaction model in two dimensions with bounded φ, there exists at least one Gibbs measure for ϕ and every z > 0.

for all ω ∈ Ω, some Λ ⋐ R d and suitable constants a, b ≥ 0. Let L be the set of all local and tame functions. The topology of local convergence, or L-topology, on P Θ is then defined as the weak* topology induced by L, i.e., as the smallest topology for which the mappings P → f dP (with f ∈ L) are continuous. Note that the intensity P → i(P ) is continuous in the L-topology.

Next, for any P ∈ P Θ let P Λn = P • pr -1 Λn be the projection of P to Ω Λn and

the relative entropy of P Λn with respect to Π z Λn ; here, "≪" stands for absolute continuity. The specific entropy of P (relative to Π z ) is then defined by (5.3)

see [START_REF] Georgii | Gibbs Measures and Phase Transitions[END_REF] and [START_REF] Georgii | Large deviations and the maximum entropy principle for marked point random fields[END_REF] for the existence of the limit and further properties of I z . Our key tool is the following result of [START_REF] Georgii | Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction[END_REF]Lemma 3.4], which is based on [15, Prop. 2.6].

Proposition 5.2. For all c 1 , c 2 ≥ 0 and z > 0, the set

is relatively sequentially compact in the L-topology.

In view of this fact, the following entropy bound implies that the sequence ( Pn ) has a convergent subsequence.