Percolation results for the Continuum Random Cluster Model
The continuum random cluster model is a Gibbs modification of the standard Boolean model with intensity and law of radii .
unnormalized density is given by where is a fixed parameter and is the number of connected components in the random structure.
We prove for a large class of parameters that percolation occurs for large enough and does not occur for small enough.
An application to the phase transition of the Widom-Rowlinson model with random radii is given.
Our main tools are stochastic domination properties, a fine study of the interaction of the model and a Fortuin-Kasteleyn representation.
Keywords: Gibbs point process ; Widom-Rowlinson model ; stochastic comparison ; Fortuin-Kasteleyn representation.
AMS MSC 2010: 60D05 ; 60G10 ; 60G55 ; 60G57 ; 60G60 ; 60K35 ; 82B21 ; 82B26 ; 82B43.
This paper is interested in the Continuum Random Cluster Model (called CRCM in the following), which is defined on a bounded window as a penalized Poisson Boolean model with intensity and law of radii . The unnormalized density is where is a positive real number and denotes the number of connected components of the random closed set considered. In the infinite volume regime a global density is meaningless and a definition of the CRCM using Gibbs modifications is required. In this paper we prove the existence of a subcritical and supercritical phase for the percolation of the CRCM.
The original random cluster model was introduced as a lattice model in the late 1960’s by Fortuin and Kasteleyn to unify models of percolation as Ising and Potts models. Properties and results about this model, such as existence of random cluster model on infinite graphs, percolation and phase transition properties can be found in [georgii_haggstrom_maes, grimmett_livre_rcm]. In the continuum setting the CRCM is directly linked to the well-known Widom-Rowlinson multi-type model by the so-called Fortuin-Kasteleyn representation (also known as grey representation or random cluster representation). This representation was used in the 1990’s to give a new proof of the phase transition for the Widom-Rowlinson model [chayes_kotecky], or for the more general class of continuum Potts models [georgii_haggstrom]. The CRCM is also studied in stochastic geometry and spatial statistics as an interacting random germ-grain model [moller_helisova08, moller_helisova10]. For a suitable parameter the CRCM fits as best as possible the clustering of the real dataset.
The existence of the infinite volume CRCM, defined throught standard DLR formalism, was recently investigated in [dereudre_houdebert] which gives an existence result in cases like or unbounded radii.
Percolation refers to the existence of at least one unbounded (or infinite) connected component in the random structure. For the Poisson Boolean model percolation is well-understood, see [meester_roy], and we have the existence of a positive threshold for the intensity, such that percolation occurs for and not for . Because of its strong independence properties, the Poisson Boolean model is sometimes irrelevant for applications in physics or biology.
In recent works [jansen_2016, stucki_2013] percolation was proved to occur in the case of Boolean model with deterministic radii and germs driven by a Gibbs point process. Both proofs strongly rely on the independent marking of the Boolean model and don’t apply for the CRCM.
Moreover, in statistical mechanics the percolation of the CRCM leads to the phase transition of the Widom-Rowlinson model. This was already done in the case of deterministic radii in [chayes_kotecky] and [georgii_haggstrom]. Percolation is also related to uniqueness of the Gibbs measure, with the method of disagreement percolation, which is based on constructing a coupling with good properties, see [georgii_haggstrom_maes, vandenberg_maes].
The aim of the present paper is to provide percolation results for the CRCM with parameters as general as possible. The two cases considered are (C1) , and (C2) , for some . In both cases we prove the absence of percolation for small and the existence of percolation for large . Part of the result is trivially obtained using a stochastic domination result of Georgii and Küneth [georgii_kuneth]. The challenging part, proving the existence of percolation for large in (C1) and the absence of percolation for small in (C2), relies on a discrete stochastic domination result proved by Liggett, Schonmann and Stacey in [liggett_schonmann_stacey]. This result is stated in Proposition 4.2. This method was already successfully used in [coupier_dereudre] to prove a percolation result for the continuum Quermass-interaction model.
The paper is organized as follows. Section 2 is devoted to the introduction of the model and the notations. Section 3.1 contains the main results of the paper, concerning percolation for the CRCM. In Section 3.2 we present a phase transition result for the Widom-Rowlinson model with random radii. This result is a non-trivial extension of the result in [chayes_kotecky] and [georgii_haggstrom], relying on an infinite-volume FK-representation. In Section 4 we introduce basic notions of stochastic domination and we prove, as a direct consequence of Theorem 1.1 of [georgii_kuneth], the absence of percolation for small activities in the case (C1) and the percolation for large activities in the case (C2). Section 5 is devoted to the proof of the percolation for large activities in the case (C1) and Section 6 deals with the proof of the absence of percolation for small activities in the case (C2). Finally Section 7 contains a sketch of the proof of standard results involving the uniqueness of the unbounded cluster and the infinite-volume FK-representation.
2 Notations and definitions
2.1 State space and reference measure
For at least 2, we denote by the space endowed with the Borel -algebra. A configuration is a non-negative integer-valued measure on , which can be represented as for a finite or infinite sequence , with finite mass on for every bounded subset of . The configuration set is equipped with the classical -algebra generated by the counting variables where is a bounded Borel subset of . The configuration restricted to a subset of is defined by and is the sub -algebra of generated by the counting variables where is a bounded subset of . We write if . To each configuration is associated the germ-grain structure
where is the Euclidean closed ball centred in with radius .
For a positive and a probability measure on , denotes the distribution on of the homogeneous Poisson point process with spatial intensity and 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 radii distribution .
For every configuration , the connected components in are defined as followed. Two points are in the same connected component of if there is a finite number of points in such that
This connectivity is the usual one defined with the Gilbert graph. However it is different from the topological one for some infinite configurations.
The random cluster interaction between the particles in a finite configuration is given by the unnormalized density , where denotes the number of connected components of . This density is well-defined only for finite configurations. As usual, for infinite configurations we define a local conditional density.
For a bounded subset of , the -local number of connected components of a configuration (finite or infinite) is given by
where the limit, taken along any increasing sequence, is well defined, see [dereudre_houdebert, Prop. 2.1].
2.3 Continuum Random Cluster Model
The continuum random cluster model is defined using standard DLR formalism which requires that the probability measure satisfies equilibrium equations based on Gibbs kernels, see equation (2.1).
A probability measure on is a continuum random cluster model for parameters , and (CRCM()) if it is stationary and if for every bounded subset of and every bounded measurable function we have
where is the partition function which is assumed to be non-degenerate.
Equivalently, for -almost every configuration the conditional law of given is absolutely continuous with respect to with density
These equations, for every bounded , are called DLR (Dobrushin, Lanford, Ruelle) equations. Existence and (non)-uniqueness of the CRCM are standard but complex questions of statistical mechanics. In a recent paper [dereudre_houdebert], the existence was proved for a large class of parameters such as the case or unbounded radii. In this paper we are interested in the two following cases.
and satisfying .
and satisfying for a given .
For both cases the existence of a CRCM was proved in [dereudre_houdebert].
3.1 Percolation results
A configuration is said to percolate if contains at least one unbounded connected component. We say a probability measure percolates (respectively does not percolate) if the probability of the percolation event is (respectively ). The first natural question is about the number of infinite connected components, which is answered by the following Theorem 1.
For every CRCM, almost surely the number of unbounded connected components is at most 1.
The proof of this result uses classical percolation techniques and is developed in Section 7.
The main question of the present paper is the classical question : “Does percolation occur?”. One trivial case is when and there is trivially no percolation since the CRCM is just a Poisson point process. In the following this case is omitted. The following theorems are the main results of the present paper. These theorems prove the existence, for both cases (C1) and (C2), of a subcritical phase for small activities where no percolation occurs and a supercritical phase for large activities where percolation occurs.
In the case (C1),
there exists positive such that for every , no percolates;
with the additional assumption , there exists finite such that for every , every percolates.
In the case (C2),
there exists finite such that for every , every percolates;
there exists positive such that for every , no percolates.
The proof of both Theorem 2 and Theorem 3 are based on stochastic domination techniques which enable us to compare the CRCM to more simple models, namely the Poisson Boolean model and the Bernoulli percolation model. Proofs of those theorems are done in sections 4, 5 and 6.
For both Theorem 2 and Theorem 3, we get the existence of . The existence of a threshold, meaning that is a natural and interesting question still open for the CRCM and many other models. Indeed the only existing technique for this question is to prove stochastic monotonicity with respect to the parameter . This is proved for continuum models using the result of Georgii and Küneth [georgii_kuneth], which does not apply for the CRCM.
3.2 Phase transition for the Widom-Rowlinson model
In this section the Widom-Rowlinson model ([widom_rowlinson]) is introduced and a phase transition result is exhibited as a direct consequence of Theorem 2. This section gives a non-trivial generalisation of the results in [chayes_kotecky] and [georgii_haggstrom] to the case of unbounded radii. It also gives a more direct proof relying on an infinite-volume Fortuin-Kasteleyn representation, Proposition 3.1. Let be an integer larger than . The Widom-Rowlinson model is defined on the space of coloured configuration in . The marks in are called colours. The definitions of the -field and the coloured Poisson point process with uniform colour marks are the natural extension of the definitions in Section 2 and are omitted. Let be the event of coloured configuration such that any two balls of two different colours do not overlap.
A probability measure on () is a Widom-Rowlinson model of parameters (WR()) if it is stationary and if, for every bounded set and every bounded measurable function ,
For , and a positive integer, the existence is well-known and can be seen as a consequence of the existence of the CRCM and the Fortuin-Kasteleyn representation Proposition 3.1.
For every integer larger than and every probability measure satisfying and , we have the existence of such that, for every , there exists at least ergodic WR().
The proof of this theorem is based on an infinite-volume Fortuin-Kasteleyn representation given in Proposition 3.1.
Let be a CRCM(). We build a probability measure on by colouring each finite connected component independently and uniformly over the colours . If it exists, the (unique) infinite connected component is assigned the colour .
Then the measure is a WR(). Moreover if is ergodic, the same is true for .
Using Theorem 2 we have for that every CRCM() percolates. Choose an ergodic , which can be done since every CRCM() is a mixing of ergodic CRCM(), see [georgii_livre]. From Proposition 3.1 one can build ergodic WR which are distinct since the unbounded connected component does not have the same colour. This proves Theorem 4.
4 Stochastic domination
Stochastic domination is at the core of the proof of Theorem 2 and Theorem 3. Let us recall some basic definitions on stochastic domination. We say that is smaller than , and we write , if for every Borel set of . An event in is said to be increasing if for every and every , we have . One example of increasing event is the percolation event . Finally if and are two probability measures on , we say that stochastically dominates , writing , if for every increasing event . One way of proving stochastic domination between Gibbs point processes is the Theorem 1.1 in [georgii_kuneth] which relies on an inequality between the Papangelou intensities of the processes considered. This leads to the following proposition.
Let be a CRCM().
If , then ;
if , then .
The Papangelou intensity of a CRCM() is , where counts the number of connected components of overlapping the ball . Since the function is non-negative, the Theorem 1.1 in [georgii_kuneth] directly gives the result. ∎
Since the percolation event is increasing, using Proposition 4.1 and the existence of subcritical and supercritical phases for the Poisson Boolean model of radii law satisfying , see [gouere_2008], we get the following corollary proving the existence of a subcritical phase (respectively a supercritical phase) for the CRCM in the case (C1) (respectively (C2)).
Let be the percolation threshold of the Poisson point process , then
in the case (C1), for every , no CRCM() percolates;
in the case (C2), for every , each CRCM() percolates.
The Corollary 4.1 gives the existence of the constant of Theorem 2 and the constant of Theorem 3. In some cases like deterministic radii we have and Theorem 1.1 in [georgii_kuneth] can be used to get the existence of the others constants. This argument was already used in [chayes_kotecky] and [georgii_haggstrom].
To this end we introduce the Proposition 4.2 which provides a good stochastic domination between the law of a dependent family of random variables in and the Bernoulli (with parameter ) product measure on . This type of argument was already used to prove percolation results for continuum models, see [coupier_dereudre].
Proposition 4.2 (Liggett, Schonmann and Stacey [liggett_schonmann_stacey]).
Let be a (dependent) family of -valued random variables of joint law . Let and assume that for every vertex ,
where is the infinite norm on and is a positive fixed constant.
Then is stochastically dominated by the distribution of , where is a deterministic function depending on such that .
We assume here that conditions (C1) are satisfied: and satisfying . We are also making the extra assumption that . Let be a . The idea of the proof is to construct a family such that the conditional probability as in Proposition 4.2 is as large as we need and such that if the family percolates, the same is true for our model. To this end we look into the probability of covering small cubes. Intuitively if the family of covered cubes percolates, then the same is true for the underlying configuration. Let be such that and let be the cube . To look into the probability of covering the cube , for a radius let be the event of configurations such that contains at least one ball of radius larger than .
We define the random variable equal to if and otherwise. To use Proposition 4.2 we need to prove that
where is a bounded set containing and which will be defined shortly. By construction we have
The goal now is to find a good bound for , uniform with respect to , which tends to when grows to infinity. To control the quantity , we need to introduce a “protective layer”, hence the , and a good event on which is defined below. We have
5.1 Bound for the quantity
Thanks to the DLR equations (2.1) on , we have
where denotes the number of balls of a finite configuration and where the second line comes from the trivial inequality . From (5.3) one can understand the condition . Now we need to give a precise definition of and . To this end let be a positive real number such that , which is possible by the choice of . We define and we take such that . Finally take such that . The last thing to do is to define as the event of configurations having many small balls centred inside ,
where is the ceiling function. The following Lemma 5.1 gives a good lower bound for .
For we have for a constant .
Proof of the lemma.
We first realize that the worst case scenario occurs when has a single connected component which intersects many connected components of . Since , the radii of the balls of are bounded by . Note that the number of connected components intersecting is smaller or equal to the number of disjoint balls in that can intersect . By the choice of , a ball in which intersects satisfies and hence for a given positive constant . In the same manner, any ball of with satisfies . So the number of such balls is at most . To get the bound on , we just need to take into consideration the balls of with radius smaller than , which number is not greater than since . Hence the result holds by taking . ∎
To get an upper bound for the integral in (5.3) we consider the conditional probability
5.2 Bound for the quantity
Now we need to control the quantity in (5). Using Theorem 1.1 of [georgii_kuneth], the probability measure stochastically dominates , and since the event is increasing, we have
where the last line comes from the Lagrange inequality and the Stirling formula. By the choice of the parameters, we have
5.3 Construction of the dependent family
To each we associate the small cube and the large cube , where stands for standard Minkowski sum.
We define as we did for , meaning that is equal to if , and otherwise. Finally take such that if , then . For example works. For every , since is stationary we have
But using the convergence (5.1), we have for any the existence of such that, for every and every
Using Proposition 4.2, is stochastically dominated by the law of . The parameter can be chosen such that is larger than any given percolation threshold. But it is clear that if the family percolates, with respect to the cubic lattice, the same is true for the configuration . Therefore if the activity is large enough, then is larger than the percolation threshold of and percolates. Theorem 2 is proved.
The assumption was used in the proof to ensure that constants and are well defined. It seems possible to do the same construction for radii law having a small atom in , but since the construction of depends on all the others constants introduced in the proof, it is impossible to derive from the proof presented a general condition in that case.
We assume here that conditions (C2) are satisfied: and there exists such that . Let be a . Without loss of generality we are making the proof in the specific case . The idea is, as in the proof of Theorem 2, to build a good family in order to apply Proposition 4.2.
Let and . We define equal to when and otherwise. As the radii are bounded by , depends only on the restricted configuration .
We want to prove that
For a configuration , let denote the number of connected components of which overlap . The random variables and are strongly related since if and only if .
There exists such that
This lemma is proved at the end of the section. First let us see how Lemma 6.1 leads to the wanted result. We have
Now consider the family of variables defined as , meaning that if , where . Take and let . For every vertex we have
and using the convergence (6.1), we have for any the existence of such that, for every and every ,
Using Proposition 4.2, stochastically dominates the law of . The parameter can be chosen such that is lower than any given percolation threshold.
As in Section 5 the absence of percolation of the family , with respect to the cubic latice, leads to the absence of percolation in the configuration . Therefore for small activities , does not percolate.
Proof of Lemma 6.1.
We use the GNZ equations satisfied by the (conditional) probability measure .
Lemma 6.2 (GNZ equations).
For every bounded measurable function ,
where is a marked point, where with the Lebesgue measure on and where is the function defined in the proof of Proposition 4.1 as the number of connected components of which overlap the ball . In the integrals stands for .
The proof of this lemma is exactly the same as the proof of the well-known Slivnyak-Mecke formula and is omitted. To use the GNZ equations let us define a function such that is equal to if the following conditions are fulfilled and otherwise.
The connected component of in intersects .
is one of the balls of its connected component of which minimize the quantity .
By the GNZ equations applied to the bounded function , we have
In the following we find an upper bound for the quantity when . To this end consider a ball and let us count the maximum number of disjoint balls of radius not smaller than overlapping . By a volume argument, this quantity is no more than . To obtain this value we just look at the intersection of each disjoint ball with the ball . The volume of each intersection is at least where is the volume of the unit ball in dimension . This bound is not sharp but is enough for the following, since it does not depend on .
By contradiction assume that in there is a connected component (with this notation is a restriction of ) which overlaps and such that .
Let such that overlaps . Such a ball exists by hypothesis on and we have