The Widom-Rowlinson Model on the Delaunay Graph
We establish phase transitions for continuum Delaunay multi-type particle systems (continuum Potts or Widom-Rowlinson models) with infinite range repulsive interaction between particles of different type. Our interaction potential depends solely on the length of the Delaunay edges and is scale invariant up to a parameter replacing the role of inverse temperature. In fact we show that a phase transition occurs for all activities for sufficiently large potential parameter confirming an old conjecture that if phase transition occurs on the Delaunay graph it will be independent of the activity. This is a proof of an old conjecture of Lebowitz and Lieb extended to the Delaunay structure. Our approach involves a Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. The phase transition manifests itself in the mixed site-bond percolation of the corresponding random-cluster model. Our proofs rely mainly on geometric properties of the Delaunay tessellations in and on recent studies [DDG12] of Gibbs measures for geometry-dependent interactions. The main tool is a uniform bound on the number of connected components in the Delaunay graph which provides a novel approach to Delaunay Widom Rowlinson models based on purely geometric arguments.
Key words and phrases:Delaunay tessellation, Widom-Rowlinson, Gibbs measures, Random cluster measures, mixed site-bond percolation, phase transition, coarse graining, multi-body interaction
1. Introduction and results
Although the study of phase transitions is one of the main subjects of mathematical statistical mechanics, examples of models exhibiting phase transition are mainly restricted to lattice systems. In the continuous setting results are much harder to obtain, e.g., the proof of a liquid-vapor phase transition in [LMP99], or the spontaneous breaking of rotational symmetry in two dimensions for a Delaunay hard-equilaterality like interaction [MR09]. These phase transitions manifest themselves in breaking of a continuous symmetry. There is another specific model for which a phase transition is known to occur: the model of Widom and Rowlinson [WR70]. This is a multi-type particle system in , with hard-core exclusion between particles of different type, and no interaction between particles of the same type. The phase transition in this model was stablished by Ruelle [Rue71]. Lebowitz and Lieb [LL72] extended his result by replacing the hard-core exclusion by a soft-core repulsion between unlike particles. Finally, phase transition results for a general class of continuum Potts models in , have been derived in [GH96]. The repulsive interaction between particles of different type in [GH96] is of finite range and a type-independent background potential has been added. The phase transitions for large activities in all these systems reveal themselves in breaking of the symmetry in the type-distribution. The results in [GH96] are based an a random-cluster representation analogous to the Fortuin-Kasteleyn representation of lattice Potts models, see [GHM]. In [BBD04] the soft repulsion in [GH96] between unlike particles has been replaced by another kind of soft repulsion based on the structure of some graph. More precisely, the finite range repulsion between particles of different type acts only on a nearest neighbour subgraph of the Delaunay graph in .
In this paper we establish the existence of a phase transition for a class of continuum Delaunay Widom-Rowlinson (Potts) models in . The repulsive interaction between unlike particles is of infinite range, and it depends on the geometry of the Delaunay tessellation, i.e. length of the edges. The potential is formally given as
where is the length of an Delaunay edge and is the potential parameter serving as an inverse temperature. The main novelty of our paper is a uniform bound on the number of connected components in the Delaunay random cluster model which is purely based on geometrical properties of Delaunay tessellations in two dimensions. We believe that these new techniques can be applied to higher dimensions in the future as well. The potential ensures that for large values of the parameter , Delaunay edges with shorter lengths are more likely to be connected than those with longer edges, enabling us the bound the number of connected components for clouds of points with vanishing point-wise distances. This paper is an extensive development of the recent work [AE16] where all models had an additional background hard-core potential introducing a length scale, whereas the present work has no a priori length scale fixed. Our results extend [BBD04] and [GH96] in two ways by having an infinite range repulsion and a geometry-dependency of the type interaction. Gibbs models on Delaunay structures have been studied in [BBD99, BBD02, BBD04, Der08, DDG12, DG09, DL11], and our results rely on the existence of Gibbs measures for geometry-dependent interactions established in [DDG12]. Our approach is based on a Delaunay random-cluster representation similar to [BBD04] and [GH96], the difference being that we replace edge percolation by an adaptation of lattice hyperedge percolation [Gri94] to our continuum setting. A phase transition for our Delaunay Potts models follows if we can show that the corresponding percolation process contains an infinite cluster. A similar program was carried out by Chayes et al. in [CCK95] for the hard-core Widom-Rowlinson model. In that case, the existence of infinite clusters follows from a stochastic comparison with the Poisson Boolean model of continuum percolation, while our framework uses a coarse graining method to derive a stochastic comparison with mixed site-bond percolation on . Our results are extension of [LL72] and [CCK95] to the Delaunay structure replacing hard-core constraint by our soft-core repulsion. In particular we obtain phase transition for all activities once the interaction parameter (inverse temperature) is sufficiently large. This confirms the conjecture that phase transitions on the Delaunay structure should once they occur be independent of the activity. This idea actually goes back to [Hag00] in which percolation for the Poisson Voronoi model has been established. We note that our random-cluster representation requires the symmetry of the type interaction. In the non-symmetric Widom-Rowlinson models, the existence of a phase transition has been established by Bricmont et al. [BKL], and recently by Suhov et al. [MSS].
1.2. Remarks on Delaunay tessellations
We conclude with some remarks on the particular features models defined on the Delaunay hypergraph structure show. The most simple Delaunay Potts model would be one with no more interaction than constant Delaunay edge interaction as then the percolation would be independent of the activity parameter. Our triangle model is a step towards understanding this system. We draw attention to some differences between geometric models on the Delaunay hypergraph structure and that of classical models such as the Widom-Rowlinson model and its soft-core variant of Lebowitz and Lieb [LL72]. The first is that edges and triangles in the Delaunay hypergraph are each proportional in number to the number of particles in the configuration. However, in the case of the complete hypergraph the number of edges is proportional to the number of particles squared and the number of triangles is proportional to the number of particles cubed. Secondly, in the complete graph of the classical models, the neighbourhood of a given point depends only on the distance between points and so the number of neighbours increases with the activity parameter of the underlying point process. This means that the system will become strongly connected for high values of . This is not the case for the Delaunay hypergraphs which exhibit a self-similar property. Essentially, as the activity parameter increases, the expected number of neighbours to a given point in the Delaunay hypergraph remains the same, see [Mø94]. Therefore, in order to keep a strong connectivity in our geometric models on Delaunay hypergraphs, we use a type interaction between particles of a hyperedge with a non-constant mark. Finally, and perhaps most importantly, is the question of additivity. Namely, suppose we have an existing particle configuration and we want to add a new particle to it. In the case of classical many-body interactions, this addition will introduce new interactions that occur between and the existing configuration . However, the interactions between particles of remain unaffected, and so classical many-body interactions are additive. On the other hand, in the Delaunay framework, the introduction of a new particle to an existing configuration not only creates new edges and triangles, but destroys some too. The Delaunay interactions are therefore not additive, and for this reason, attractive and repulsive interactions are indistinct. In the case of a hard exclusion interaction, we arrive at the possibility that a configuration is excluded, but for some , is not. This is called the non-hereditary property [DG09], which seems to rule out using techniques such as stochastic comparisons of point processes [GK97].
We consider configurations of points in with internal degrees of freedom, or marks. Let , be the finite set of different marks. That is, each marked point is represented by a position and a mark , and each marked configuration is a countable subset of having a locally finite projection onto . We denote by the set of all marked configurations with locally finite projection onto . We will sometimes identify with a vector of pairwise disjoint locally finite sets in (we write for the set of all locally finite configurations in ). Any is uniquely determined by the pair , where is the set of all occupied positions, and where the mark function is defined by if . For each measurable set in the counting variable on gives the number of marked particles such that the pair (position, mark) belongs to . We equip the space with the -algebra generated by the counting variables and the space of locally finite configurations with the -algebra generated by the counting variables for where we write for any bounded . As usual, we take as reference measure on the marked Poisson point process with intensity measure where is an arbitrary activity, is the Lebesgue measure in , and is the uniform probability measure on .
We let (resp. ) denote the set of all finite configurations. For each we write for the set of configurations in , for the projection from to (similarly for unmarked configurations), for the trace -algebra of on , and for the -algebra of all events that happen in only. The reference measure on is . In a similar way we define the corresponding objects for unmarked configurations, , and . Finally, let be the shift group, where is the translation of the spatial component by the vector . Note that by definition, for all .
The interaction between the points depend son the geometry of their location. We describe this in terms of hypergraph structures. A hypergraph structure is a measurable set of (resp. ). We outline the definitions for the unmarked configurations first with obvious adaptations to the case of marked point configurations. The set of Delaunay hyperedges consist of all pairs with for which there exits an open ball with that contains no points of . For , we write for the set of Delaunay simplices with vertices. Given a configuration the set of all Delaunay hyperedges with is denoted by . It is possible that consists of four or more points on a sphere with no points inside. In fact, for this not to happen, we must consider configurations in general position as in [Mø94]. More precisely, this means that no four points lie on the boundary of a circle and every half-plane contains at least one point. Fortunately, this occurs with probability one for our Poisson reference measures, and in fact, for any stationary point process. Note that the open ball is only uniquely determined when and is affinely independent. Henceforth, for each configuration we have an associated Delaunay triangulation
of the plane, where is the unique open ball with . The set in (1.1) is uniquely determined and defines a triangulation of the convex hull of whenever is in general position ([Mø94]). In a similar way one can define the marked Delaunay hyperedges, i.e., and as measurable sets in where the Delaunay property refers to the spatial component only.
Given a configuration (or ) we write (resp. ) for the set of configurations which equal off . For any edge we denote its length by , i.e., if . We write in the following when is non-empty and finite, i.e., . The interaction is given by the following Hamiltonian in with boundary condition ,
where . Here is a measurable function of the length of an edge defined for any ,
Note the following scaling relation for the potential
Following [DDG12] we say a configuration (or ) is admissible for and activity if is -almost surely well-defined and , where the partition function is defined as
We denote the set of admissible configurations by . The Gibbs distribution for , and in with admissible boundary condition is defined as
It is evident from (1.5) that, for fixed , the conditional distribution of the marks of relative to is that of a discrete Potts model on embedded in the Delaunay triangulation with position-dependent interaction between the marks. This justifies calling our model Delaunay Potts model or Delaunay Widom-Rowlinson model.
A probability measure on is called a Gibbs measure for the Delaunay Potts model with activity and interaction potentials and if and
for every and every measurable function .
1.4. Results and remarks
We outline the results for our triangle and edge models all of which are generic examples of a more general class of coarse grain ready potentials, see Section 2.2 for further details.
Proposition 1.2 (Existence of Gibbs measures).
For any there exist at least one Gibbs measure for the Delaunay Widom-Rowlinson (Potts) model with parameter .
Remark 1.3 (Gibbs measures).
The proof is using the so-called pseudo-periodic configurations (see Appendix A or [DDG12]). Large activities then ensure that these pseudo-periodic configurations have sufficient mass under the reference process. Existence of Gibbs measures for related Delaunay models have been obtained in [BBD99, Der08, DG09]. Note that for our models have no marks and Gibbs measures do exist as well ([DDG12]).
Gibbs measures for the Delaunay Potts model do exist for activity . A phase transition is said to occur if there exists more than one Gibbs measure for the Delaunay Potts model. The following theorem shows that this happens for all activities and sufficiently large parameter depending on . Note that is a parameter for the type interaction and not the usual inverse temperature.
Theorem 1.4 (Phase transition).
For all there is such that for all there exit at least different Gibbs measures for the Delaunay Widom Rowlinson (Potts) model.
Remark 1.5 (Free energy).
One may wonder if the phase transitions manifest itself thermodynamically by a non-differentiability (”discontinuity”) of the free energy (pressure). Using the techniques from [Geo94] and [DG09], it should be possible to obtain a variational representation of the free energy, see also [ACK11] for free energy representations for marked configurations. Then a discontinuity of the free energy can be established using our results above. For continuum Potts models this has been established in [GH96, Remark 4.3]. For a class of bounded triangle potential [DG09] shows that the Gibbs measures are minimisers of the free energy. The free energy is the level-3 large deviation rate functional for empirical processes, see [Geo94, ACK11].
Remark 1.6 (Uniqueness of Gibbs measures).
To establish uniqueness of the Gibbs measure in our Delaunay Potts models one can use the Delaunay random-cluster measure , to be defined in (2.2) below. In [GHM, Theorem 6.10] uniqueness is established once the probability of an open connection of the origin to infinity is vanishing for the limiting lattice version of the random-cluster measure, that is, for some set containing the origin,
for a sequence of boxes with as . One way to achieve this, is to obtain an stochastic domination of the Delaunay random-cluster measure by the so-called random Delaunay edge model of hard-core particles. Using [BBD02] we know that the critical probabilities for both, the site and bond percolation on the Delaunay graph, are bounded from below. Extension to our tile (hyperedge) percolation using [Gri94] can provide a corresponding lower bound as well. Thus, if our parameter is chosen sufficiently small, then there is no percolation in our Delaunay random-cluster measures and therefore uniqueness of the Gibbs measure.
It goes without saying that the results addressed here are merely a first step towards a closer study of phase transitions with geometry-dependent interactions. The study for Widom-Rowlinson or Potts models with geometry-dependent interaction is by far not complete, one may wish to extend the single tile (edge or triangle) interaction to mutual adjacent Voronoi cell interaction. The common feature of all these ’ferromagnetic’ systems is that phase transitions are due to breaking the symmetry of the type distribution. Breaking of continuous symmetries is a much harder business, see e.g. [MR09], which shows a breaking of rotational symmetry in two dimensions, and which can be seen as a model of oriented particles in with a Delaunay hard-equilaterality interaction. Models studied in [DDG12] may be natural candidates for the existence of a crystallisation transition. A recent ground state study (zero temperature) [BPT] for multi-body interactions of Wasserstein types shows optimality of the triangular lattice. It seems to be promising to analyse this model for non-zero temperature in terms of crystallisation transition. We will investigate this further.
Another direction for the class of ’ferromagnetic’ models with geometry-dependent interaction is to analyse closer the percolation phenomena, and in particular to study the conformal invariance of crossing probabilities as done for the Voronoi percolation ([BS98, BR06], and [T14]).
Last but not least it is interesting to study the higher dimensional cases in , as well (see [GH96]). We believe that our results still hold for these cases but there are some technical issues related to the general quadratic position such that we defer that analysis for future study.
The rest of the paper is organised a follows. In Section 2.1 we define the Delaunay random-cluster measure for edge configruations, and in Section 2.2 . The main novelty is the extensive proof of the uniform bound on the number of connected components using purely geometric properties in Section 3. Finally, in Section 4 we gives details of our remaining proofs.
2. The random cluster method
In this Section the main method for proving existence of multiple Gibbs measures is outlined. In Section 2.1 we introduce the Delaunay random cluster model and show percolation for this model in Section 2.2. We conclude in Section 2.3 with our proof of Theorem 1.4. The key step of the proof of percolation for the Delaunay random cluster model is the novel uniform estimate of the number of connect components in Section 3. The uniform bound on the number of connected components uses solely geometric arguments and constitutes a major part of this work.
2.1. Delaunay Random Cluster measure
For and parameters and we define a joint distribution of the Delaunay Potts model and an edge (tile) process which we call Delaunay random-cluster model. The basic idea is to introduce random edges between points in the plane. Let
be the set of all possible edges of points in , likewise, let be the set of all edges in and for the set of edges in . We identify with and . This allows only monochromatic boundary conditions whereas the general version involves the so-called Edwards-Sokal coupling (see [GHM] for lattice Potts models). We restrict ourself to the former case for ease of notation. We write
for the set of all locally finite edge configurations.
The joint distribution is built from the following three components.
The point distribution is given by the Poisson process for any admissible boundary condition and activity .
The type picking mechanism for a given configuration is the distribution of the mark vector . Here are independent and uniformly distributed random variables on with for all . The latter condition ensures that all points outside of carry the given fixed mark.
The tile drawing mechanism. Given a point configuration , we let be the distribution of the random edge configuration with the edge configuration having probability
The measure is a point process on . Note that and are probability kernels (see [Eye14]). Let the measure
be supported on the set of all with and . We shall condition on the event that the marks of the points are constant on each connected component in the graph . Two distinct vertices and are adjacent to one another if there exists such that . A path in the graph is an alternating sequence of distinct vertices and edges such that for all . We write
for the set of marked point configurations such that all vertices of the edges carry the same mark. The set is measurable which one can see from writing the condition in the following way
and using the fact that , are measurable (see [GH96, Chapter 2]). Furthermore, , which follows easily observing where is the configuration which equals outside of and which is empty inside . Henceforth, the random-cluster measure
is well-defined. As in [GH96] we obtain the following two measures from the random-cluster measure , namely if we disregard the edges we obtain the Delaunay Gibbs distribution in (1.5) (see [Eye14]). For the second measure consider the mapping from onto where . For each with we let denote the number of connected components in the graph . The Delaunay random-cluster distribution on is defined by
where is the Poisson process with activity replacing and where
is the normalisation. It is straightforward to show that (see [Eye14]).
For our main proofs we need to investigate the geometry of the Delaunay triangulation, and in particular what happens to it when we augment with a new point . Some hyperedges may be destroyed, some are created, and some remain. This process is well described in [Lis94]. We give a brief account here for the convenience of the reader. We insert the point into one of the triangles in . We then create three new edges that join to each of the three vertices of . This creates three new triangles, and destroys one. We now need to verify that the new triangles each satisfy the Delaunay condition (1.1), that is, that their circumscribing balls contain no points of . If this condition is satisfied the new triangle remains, if it is not satisfied, then there is a point inside the circumscribing ball. We remove the edge not connected to , and replace it by an edge connecting and . This results in the creation of two new triangles. Each of these triangles must be checked as above and the process continues. Once all triangles satisfy the Delaunay condition, we arrive at the Delaunay triangulation and their edge part . Let
be the set of exterior, created, and destroyed Delaunay edges respectively, see figure 1.
Note that any new triangle must contain , i.e.,
We let , and be the edge drawing mechanisms on , and , respectively, which are derived from the edge drawing measure above. To add to the above sets of Delaunay edges, we also define the neighbourhood of a point . For any the neighbourhood graph of a point is the following random graph where is the set of points that share an edge with in and is the set of edges in that have both endpoints in , more precisely.
The graph splits the plane into two regions, The region containing we call the neighbourhood of and is called the boundary of the neighbourhood of . Having the edge drawing mechanism define
The main task is the find an uniform upper bound (independent of ) for the expected number of connected components of that intersect the boundary of the neighbourhood , where is sampled from . This bound will enable us to bound certain conditional Papangelou intensities from below and above enabling us to work with no background potential as in recent work ([BBD04, AE16]). Given and and the Delaunay graph we denote the number of connected components that intersect .
Theorem 2.1 (Number of connected components).
For all there exists such that
for all and and for all with .
We need to study the change of when adding the point . Adding a point to without considering the change to will always increase the number of connected components by one. On the other hand, the augmentation of a single triangle to can result in the connection of a maximum of three different connected components, leaving one. Therefore,
We establish the existence of edge percolation for the Delaunay random-cluster measure when and the parameter are sufficiently large. Note that for any we write
The key step in our results is the following percolation result.
Suppose all the assumptions hold and that and are sufficiently large. Suppose that is a finite union of cells defined in (A.1) in Appendix A. Then there exists such that
for any cell , any finite union of cells and any admissible pseudo-periodic boundary condition .
Proof of Proposition 2.2: We split the proof in several steps and Lemmata below. Our strategy to establish percolation in the Delaunay random-cluster model is to compare it to mixed site-bond percolation on (se appendix C on site-bond percolation). We entend here the strategy in [GH96] to our case in the following steps similar to [AE16]. First we employ a coarse-graining strategy to relate each site to a cell which is a union of parallelotopes. The second step is to consider the links (bonds) of two good cells. In order to establish mixed site-bond percolation we need to define when cells are good (open) and when two neighbouring cells are linked once they are open which happens when the corresponding link (bond) is open as well. This link establishes then an open connection in our Delaunay graph structures . In order to apply mixed site-bond percolation we need to substantially adapt the coarse graining method used in [AE16].
Step 1: Coarse graining.
Let be the parallelotope given as the finite union of cells (A.1) with side length , i.e.,
where are parallelotopes with side length and each parallelotope has side length where the coordinate systems is the canonical one, that is is the paralleloptope in the bottom right corner. The union of the smaller parallelotopes towards the centre of is denoted
see Figure 2 These sites will act as the sites in the mixed site-bond percolation model on . Finally, we define the link boxes between and as
which act as the bonds in the mixed site-bond percolation model on , see Figure. This completes the coarse grain procedure. When we establish percolation in the mixed site-bond percolation model on , i.e., the existence of an infinite chain of open sites and open bonds, we would like to relate it to the existence of an infinite connected component of open edges in . This infinite connected components will connect withe the complement of any finite boxes, and thus this connected components corresponds to an infinite connected component of edges where all the sites carry the mark agreed for the boundary condition, that is, an infinite connected component in an appropriate Delaunay site percolation model defined in (2.17) below. To do this, we define to be the straight line segment between the centres of the parallelotopes and and let
We need to consider the distribution of the points given by the marginal distribution
of the Delaunay random-cluster measure on . Note that (2.2) can be written as
We define to be the Radon-Nikodym density of with respect to , i.e., for ,
In the following lemma we derive a bound for the Papangelou conditional intensity of .
For any and any admissible boundary and -almost all and a point with ,
where is given in Theorem 2.1.
due to the fact that adding any edge from will only fuse connected components of the remaining graph. The second inequality is just Jensen’s inequality applied to the convex function . Note that new edges from , made by the insertion of to the configuration , are edges connecting to points in and are open with respect to , and therefore
for any and any , and thus by Theorem 2.1 we conclude with the statement. ∎
An important component of our coarse graining method is to estimate the conditional probability that at least one point lies inside some . In the following we denote by the conditional distribution (relative to ) in given that the configuration in is equal to with . The details of the construction of the regular conditional probability distribution can be found in [Eye14] or Appendix C. Having a uniform lower bound for the quotient allows to exhibit some control over the distribution for any . In the following we write for any cell . We fix
where is the critical probability for site percolation on .
For all there exist such that for all admissible pseudo-periodic boundary conditions ,
for all cells , and for any admissible boundary condition with .
Proof. The statement follows immediately from
where the inequality follows from Lemma 2.3. It follows that
for . ∎
Note that one can fix the length scale for the coarse graining partitions in Lemma 2.4. Then the same results holds for all .
Let be an element of of partitioning of and define to be the event that each of the smaller boxes that are not in the centre region , contain at least one point. We call the elements in this event ’well-behaved’ configurations,
For any and any admissible boundary and -partitioning of and configuration and a point with for any ,
where is given in Theorem 2.1.