Infinite Volume Continuum Random Cluster Model
The continuum random cluster model is defined as a Gibbs modification of the stationary Boolean model in with intensity and the law of radii . The formal unormalized density is given by where is a fixed parameter and the number of connected components in the random germ-grain structure. In this paper we prove the existence of the model in the infinite volume regime for a large class of parameters including the case or distributions without compact support. In the extreme setting of non integrable radii (i.e. ) and is an integer larger than 1, we prove that for small enough the continuum random cluster model is not unique; two different probability measures solve the DLR equations. We conjecture that the uniqueness is recovered for large enough which would provide a phase transition result. Heuristic arguments are given. Our main tools are the compactness of level sets of the specific entropy, a fine study of the quasi locality of the Gibbs kernels and a Fortuin-Kasteleyn representation via Widom-Rowlinson models with random radii.
Keywords. Gibbs point process ; phase transition ; specific entropy ; Boolean model ; Widom-Rowlinson model ; Fortuin-Kasteleyn representation
In this paper we are interested in a continuum version of the random cluster model usually defined on a deterministic graph. The reference model is the stationary Poisson Boolean model with intensity and the law of radii a probability measure on . It is built by union of balls in centred to the points of a stationary Poisson point process with intensity and with random independent radii following the distribution . The finite volume continuum random cluster model is then defined as a penalized Boolean model in some bounded window . The unormalized density is given by where is a positive real number and denotes the number of connected components of the random closed set considered. For this model, the mean number of connected components is increasing with respect to which provides a clear interpretation of this parameter. For we recover the standard Poisson Boolean model. In the infinite volume regime a global density is senseless and a definition of the continuum random cluster model (called CRCM in the following) via Gibbs modifications is required. Precisely a CRCM is a solution of the standard DLR equations (5). Existence, uniqueness and non-uniqueness questions arise.
Originally the random cluster model is a lattice model introduced in the late 1960’ by Fortuin and Kasteleyn to unify the models of percolation as Ising and Potts models. Most properties and results about this model, such as existence of random cluster model on infinite graphs, percolation property and phase transition property can be found in [ghm, grimbook]. In the continuum setting the CRCM has been also introduced for its relations with the continuum Potts model and the Widom-Rowlinson model. It led to new proofs of phase transition for those models, see [cck] and [gh]. The CRCM is also studied in stochastic geometry and spatial statistics as an interacting random germ-grain model [A-Moller08]. For a suitable parameter the CRCM fits as best as possible the clustering of the real dataset. The estimation of the parameter and the law of radii is studied in [A-MollerHel10].
All these works, including those in statistical mechanics, involve only the finite volume CRCM. The infinite volume version has not really been studied and highlighted as in its analogous on deterministic graphs. However its interests are numerous in statistical physics and spatial statistics. Involving physical considerations, phase transition phenomenons are observable from infinite volume CRCM; it is believed that the uniqueness of the CRCM would be violated for special critical values of , . Many conjectures and open questions for the lattice models concern the continuum case as well. In stochastic geometry, the infinite volume CRCM provides a more relevant model than the Boolean model for the applications in material science, microemulsion modelling, etc. Its macroscopic properties (mean value, conductivity, permeability) can be studied via stationary tools as Palm theory and ergodic theory. Finally in spatial statistics, the existence of models in infinite volume regime enables the study of the asymptotic properties of estimators, functionals, etc. For example the maximum likelihood estimator of the parameter along a sequence of increasing observable windows requires the existence of the model in the whole space.
The existence of the infinite volume CRCM has not been proved in a general setting of random radii, continuum parameter and . However it is known that it could be constructed via a colour-blind Widom Rowlinson in the setting where is an integer values and the radii are not random [cck]. The aim of this paper is to provide the existence of the model for the larger class of parameters as possible. In some case, the non uniqueness is also proved. Our first theorem gives the existence of the CRCM for any distribution with compact support, any and . In the case of unbounded radii, the existence is proved if has a -moment (i.e. ), and . In the case where does not have a -moment, the existence is trivial since the Poisson Boolean model is a CRCM itself. However in a second theorem we prove the existence of another CRCM leading to a non uniqueness Gibbs measures phenomenon. This result is obtained in the case where is an integer and is small enough. Using a Pirogov-Sinai approach, this non uniqueness result could provide an interesting tool for proving a phase transition phenomenon in the approximation setting . However we think that it has its own interest as well since non uniqueness results are quite rare for continuum models. We conjecture that for large enough the uniqueness of Gibbs measures is recovered in the non-integrable case. It would provide a phase transition where the uniqueness is lost only for small enough. This behaviour is unusual for Gibbs point processes where the uniqueness is in general lost for large enough. Heuristic arguments of the conjecture are given.
The proof of the first theorem is based on the compactness of the level set of the specific entropy and a fine study of the quasi locality of the Gibbs kernels. This strategy have already successfully applied for proving the existence of several Gibbs models [david, dereudredrouilhetgeorgii, gh]. In the present paper the very long range dependence is our major problem. Indeed the radii are not bounded and the influence of a ball can be felt far away if it splits a large connected component when it is removed. Such long range dependence were not dealt in the papers mentioned above. In the extreme setting of the second Theorem, we did not succeed to manage the long range dependence of the interaction as in the the first theorem. The non-integrability of the radii may produce balls with too large radii. So we turned to a Fortuin-Kasteleyn representation of the CRCM via a colour-blind Widom-Rowlinson model as in [cck, ghm]. In this setting the DLR equations are simpler to obtain since the non overlapping assumption for balls with different colours confines naturally the range of the interaction. In this setting represents the number of colors and it is the reason why is an integer.
Finally note that the standard FKG inequalities, which are abundantly used for the random cluster models on graphs, are not satisfied in the present setting. In particular the thermodynamic limit of finite volume Gibbs measures to the infinite volume Gibbs measure can not be proved. Only the convergence of the empirical field of the finite volume Gibbs measures is obtained.
In Section 2 we introduce the notations and give the formal definition of the CRCM using the DLR formalism. Then we give both main theorems mentioned above in the Section 3 devoted to the results. The proof of the first existence theorem is given in Section 4 and the second theorem in Section 5. The Heuristic arguments of the conjecture are presented in Section 6.
2 Notations and Results
2.1 State space and reference measure
For at least 2, denotes the space endowed with the Borel -algebra. stands for the set of non negative integer-valued measures on with finite mass on set for any bounded set . An element of is called "configuration" and can be represented as for a finite or infinite sequence of points in without accumulation points for the sequence . is equipped with the classical -algebra generated by the counting variables where is a bounded Borel subset of . We denote by the subset of finite configurations. For a subset of , the configuration restricted to is defined by and is the sub -algebra of generated by the counting variables where is a bounded subset of . We write if . For a configuration and a subset of , denotes the number of points such that . At each configuration we associate its germ-grain structure
where is the Euclidean closed ball of center and radius .
For a positive and a probability measure on , let be the distribution on of the Poisson point process of intensity measure . It is the distribution of the homogeneous Poisson point process on with independent marks distributed by . For , denotes the projection of on . The random closed set under the law is the so-called Poisson Boolean model with intensity and law of radii .
In the following a probability on is called stationary if it is invariant under the translations by vectors in . A definition with translations by vectors in could have been considered as well.
For any configuration , the connected components in are defined via the graph of connections where the vertices are and the edges . A connected component in is defined as the union of balls for in a connected component of . Note that it could be different from a topological connected component in . For instance the configuration has two connected components in and only one topological connected component in . For finite configurations, both definitions are equivalent.
For fixed, the interaction between the particles is given by the unnormalized density
where denotes the number of connected components of (or equivalently in ). This density is well defined only for finite configurations. As usual, for infinite configurations we define a local conditional density.
For any and bounded, the following limit
exists and is called local number of connected components in . The limit is taken along any increasing sequence of sets .
For a given we are interested in the quantity , where is a given increasing sequence converging to . Since the quantity has integer values, the sequence converges if and only if it is constant for large enough. For a subset , a connected component of is called a -component of if it is connected to . For any , any let us introduce the quantity
which gives the variation of the number of connected components when the ball is added. It is not difficult to see that may be not zero only if one of the two following situations occurs
is connected to at least two -components of ,
intersects one ball of without intersecting any -component of connected to (this case happens in particular when does not intersects any -component of ).
Show that there exists , which may depend on , such that none of the two situations occurs for any . It ensures that the sequence is constant for . Since the number of -components is finite we can choose large enough such that the number of -components in is equal to the number of -components in . In other words the -components in are identifiable in . Now it remains the problem that a ball outside may be connected to without intersecting any -component of . But the number of such balls is finite. So for large enough this situation does not occur.
Let us point out that, even if depends only on , the determination of involves a global knowledge of the configuration . This long range dependence is the major problem in the present paper.
The local number of connected components satisfies the following additivity properties which is a direct consequence of (1). For any couple of bounded sets in , there exists a function such that, for all in
The function depends only on the configurations outside . It is a crucial point for the compatibility of the Gibbs Kernels. Let us finish this section in giving useful bounds for .
For any configuration and any bounded set
Moreover, for any there exists such that for any configuration satisfying for all points , then
For any subset the difference is obviously smaller than and so its limit when tends to as well. The first inequality (3) follows.
To get the lower bound for , we first note that the worst case occurs when has one connected component which intersects a lot of connected components of . So let us control this number of connected components. We consider such that intersects a connected component of . Since all balls of have a radius smaller than we have
where is the volume of the unit ball in dimension . So the number of connected components of which are connected to is bounded from above by . Taking into consideration the balls in we have
and (4) follows. ∎
2.3 Continuum Random Cluster Model
The continuum random cluster model is defined via standard DLR formalism which requires that the probability measure satisfies equilibrium equations based on Gibbs kernels (see equations (5)). Before giving these equations we need to assume that these kernels are well-defined which is the case if for any bounded set and any configuration the partition function
is non degenerate which means that . As usual, for any configuration , . For the other bound, the following assumption is required
Under the assumption (A), for any configuration and any bounded set the partition function is finite.
We are now in position to give the definition of a continuum random cluster model.
Under the assumption (A), a probability measure on is called a continuum random cluster model for parameters , and (CRCM()) if for all bounded and all bounded measurable functions we have
Equivalently, for -almost every the conditional law of given is absolutely continuous with respect to with density
These equations, for all , are called DLR (Dobrushin, Lanford, Ruelle) equations. The existence of such Gibbs measures is the main question of the present paper. The non uniqueness is also considered.
Our first result theorem ensures the existence of at least one for the larger class of parameters as possible.
If has a bounded support, i.e there exits such that , then for all and there exists at least one stationary CRCM().
If is finite, then for all and there exists at least one stationary CRCM().
The proof of this theorem is based on the compactness of the level set of the specific entropy (see Proposition 4.1.1). This tightness tool allows to build a limit point of a sequence of stationary empirical field coming from the finite volume Gibbs measures. Then the main difficulty is to prove that this limit point satisfies the DLR equations. This strategy has already been successfully applied for proving the existence of several Gibbs models [david, dereudredrouilhetgeorgii, gh]. In the present context of continuum random cluster model, the strong non-locality of the interaction is our major problem. Indeed the radii are not bounded which produce a long range dependency. Moreover the contribution of each ball in the interaction can be long range if the ball is "pivotal" in the sense that it plays a crucial role in the determination of . The size of the connected components has also an influence on the range of the interaction. In particular, for proving the DLR equations, we need to prove that the limit point has, a priori, at most one infinite connected component. The proof of this theorem is given in Section 4.
In the extreme setting of non-integrable radii (i.e. ). First we note that the existence of a is obvious since the Poisson point process solves the DLR equations (5).
If then the Poisson process is a .
It is well known that, for any bounded set and -almost all , the set covers the full space [chiu2013]. Therefore the function is identically null for -almost every outside configuration . The DLR equations follows easily. ∎
Our second theorem ensures the existence of another different from when is an integer and is small enough. It is a non uniqueness result which proves that the simplex of is not reduced to a singleton.
If and if is an integer larger than 2, there exists such that, for all , there exists a stationary CRCM() different from .
The reason why must be an integer comes from the FK representation we used in the proof. Indeed we are not able to extend the proof of Theorem 1 to the case . The influence of large balls centred far away is too difficult to control and we do not succeed to prove that the limit point satisfied the DLR equations. Using the representation of the CRCM as a Widom-Rowlinson model (a model of non overlapping balls with different colors) as in [cck, ghm], the existence problem becomes simpler. Actually the DLR equations of the Widom-Rowlinson are more "local" since balls with different colors are not allowed to overlap. It produces a natural locality of the interaction. However we think that the assumption is only technical and could be relaxed by .
Involving the parameter we believe that the assumption small enough is crucial. In our proof, it ensures that the CRCM() we build is different from . It is based on specific entropy inequalities which ensure the discrimination for small enough. From a general point of view, we conjecture that for large enough there exists an unique CRCM() which is . The uniqueness would be recovered for large enough leading to a phase transition phenomenon.
If , there exists such that, for all , there exists an unique stationary CRCM() which is .
Note also that it is unusual in statistical mechanics that the non uniqueness result is obtained for small (and not large). The proof of the conjecture would reinforce this curious behaviour. Let us finish this section by giving an interpretation of the phase transition conjecture as a competition between the Poisson process and the energy density. Recall that the CRCM on a finite window is a Poisson process with the unormalized density . On one hand, since the Poisson process covers completely the space , it influences the CRCM to have an unique connected component which annihilates the energy contribution. The CRCM tends to be a Poisson process and more is large more this influence is strong. On the other hand the energy density influences the CRCM to have several connected components which tends to seperate the CRCM from the Poisson process. This competition between the Poisson point process and the energy is called Entropy-Energy competition in statistical Physics. We prove in Theorem 2 that the competition is well balanced for small enough. Both forces can influence the infinite volume phase. We believe that the Poisson process dominates the competition when is large enough and it is the sense of the conjecture. Heuristic arguments are given in Section 6.
4 Proof of Theorem 1
In Section 4.1 we construct a sequence of finite-volume CRCM from which we extract an accumulation point . The compactness (for the local convergence) of level sets of the specific entropy is the main tool here. Then it remains to prove that satisfies the DLR equations. To this end we need to show first that has at most one unique infinite connected component. This question is addressed in Section 4.2. Finally in Section 4.3 the DLR equations are proved. The idea is simple, since satisfies the DLR equations and that tends to for the local convergence, we get the DLR equations for in passing through the limit. However the Gibbs kernels are not local and so a sequence of localizing events has to be introduced.
4.1 Existence of a limit point
For a positive integer, we set and we define the finite-volume Gibbs measure with free boundary condition as follow
where is the normalizing constant. We need to define a stationary version of . Let be the translation of vector . Then we define as the probability measure and finally
where . Then is invariant under the translations (i.e. is stationary). Our aim is to find an accumulation point of the sequence for the suitable local convergence topology.
A function is local if there exists a bounded set such that for all configurations in . A sequence of measures converges to for the local convergence topology if, for all bounded local functions we have
The specific entropy is a powerful tool for proving the tightness for such topology. Let and be two probability measures on . The relative entropy of with respect to on the set is defined by
where means that is absolutely continuous with respect to .
Let be a stationary probability measure on . Then
is the specific entropy of with respect to .
Note that the limit above always exists. We refer to [g] for a general presentation. The following proposition is our tightness tool.
Proposition 4.1.1 (Proposition 2.6 [Gz]).
For every the set
where is the mean number of points in the box for the probability measure , is compact and sequentially compact for the local convergence topology.
So by Proposition 4.1.1, to ensure the existence of an accumulation point for the sequence , we just have to prove an uniform bound for the specific entropy .
For all we have,
First, it is straightforward that by Proposition 15.52 in [g]
The existence of a an accumulation point follows and for simplicity we write that the sequence converges to in place of a subsequence.
For technical reasons involving the DLR() equation, the sequence has to be modified by the the sequence ;
This is no longer a probability measure sequence but the Proposition 4.1.3 below shows that the local convergence to holds as well. Moreover each satisfies the DLR() equation.
For all local bounded functions we have
and for all
The proof of the first part is given in [david] Lemma 3.5. The proof of the DLR() equation for is a standard consequence of the compatibility equations (2).
4.2 Uniqueness of the infinite connected component
For in , we denote by (respectively ) the event of configurations having (respectively no more than ) infinite connected component(s). This section is devoted to the proof of the following proposition.
Under the assumption we have
The proof is based on a local modification property which claims that the configurations in a finite box can be modified with positive probability.
Proposition 4.2.2 (Local modification).
Under the assumption , for all bounded, all satisfying and all satisfying we have
First for any real number , let be the event By the monotone convergence Theorem, there is a finite such that . Since it is sufficient to prove the proposition in the special case and that is what we do. By a martingale theorem, we have . Moreover the function , that we denote by , is local and the local convergence can be applied.
The second and third equalities are obtained by Proposition 4.1.3.
From now on we have to separate the cases and .
which gives .
Using the Proposition 4.2.2, we are now in position to prove Proposition 4.2.1 in following a standard strategy in percolation theory. We just give a sketch of the proof and we refer to [meeroy] for details. First we represent as a mixture of extremal ergodic stationary probability measures where each satisfies the local modification (10) property. We show now that for -a.s. all , . By ergodicity of , the number of infinite connected components is -almost surely constant. The case of a finite number, larger than one, infinite connected components is excluded thanks to the local modification property. The case of an infinite number of infinite connected components is also excluded by a Burton and Keane argument [burkea].
4.3 DLR equations
In this section, we fix the bounded set and we show the DLR() equation. To this end sequences and of events are defined on which the variable is local and such that the probabilities and tend to one when and tend to infinity in a good way. Without loss of generality we assume that the function in the DLR() equation is local and satisfies, for a finite , as soon as there is in with . The general case is obtained by standard approximations.
Let . For we define
the event of in having at most one connected component of which intersects and , where is the set .
Before investigating the probability of those events, the next lemma shows the "localization" of the functional .
For all large enough (depending on and ) and for all in
Since is in , each balls of does not hit and does not hit two or more -components of . Therefore
The probability of the events and now have to be controlled. Involving the events we have the following proposition.
We have that
which is a direct contradiction of Proposition 4.2.1.
The control of the probability of is a bit harder to obtain. Since is not a local event, the probabilities need to be controlled uniformly on .
Under the assumptions of Theorem 1, meaning bounded radii or and , then for all
The case of bounded radii is quite simple since for , and . In the case and , we use stochastic comparison results in [gk] to compare with respect to . Recall standard definitions on stochastic domination for point processes. An event is called increasing if for any and any configuration , then as soon as . If and are two probability measures on , we say that dominates if for all increasing set .
For any and any finite configuration the difference is at most one. Therefore, thanks to Theorem 1.1 in [gk], is stochastically dominated by , and for any increasing event we have . Since the event is increasing, we have the inequality
In considering the events
It is well-known that the number of balls in a Poisson boolean model (with intensity measure ) which intersects a bounded set is a Poisson random variable with parameter (See [chiu2013] for instance). Since