Continuum percolation for Cox point processes
We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of non-trivial sub- and super-critical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in large-radius, high-density and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives.
Key words and phrases:Cox processes, percolation, stabilization, large deviations
2010 Mathematics Subject Classification:Primary 60F10; secondary 60K35
Bernoulli bond percolation is one of the most prototypical models for the occurrence of phase transitions. Additionally, as of today, the continuum version of percolation where connections are formed according to distances in a spatial point process, has been investigated intensely in the Poisson case. More recently, the community has started to look at point processes that go far beyond the simplistic Poisson model. In particular, this includes sub-Poisson [4, 5], Ginibre  and Gibbsian point processes [15, 34].
Another stream of research that brought forward a variety of surprising results is the investigation of percolation processes living in a random environment. The seminal work on the critical probability for Voronoi percolation showed that dealing with random environments often requires the development of fundamental new methodological tools [1, 6, 35]. Additionally, recent work on percolation in unimodular random graphs also revealed that fundamental properties of percolation on transitive graphs fail to carry over to the setting of random environments .
In light of these developments it comes as a surprise that continuum percolation for Cox point processes, i.e., Poisson point processes in a random environment, have so far not been studied systematically. In this paper, we rectify this omission by providing conditions for the existence of a phase transition and by investigating the asymptotic behavior of the percolation probability in a number of different limiting regimes.
In addition to this mathematical motivation, our results have applications in the domain of telecommunication. Here, Cox processes are commonly employed for modelling various kinds of networks [33, Chapter 5]. More precisely, for modelling the deployment of a telecommunication network, various random tessellation models for different types of street systems have been developed and tested against real data . The main idea of these models is to generate a random tessellation, with the same average characteristics as the street system, based on a planar Poisson point process. This could be a Voronoi, or Delaunay, or line tessellation, or it could be a more involved model like a nested tessellation .
Once the street system is modelled, it is possible to add wireless users along the streets. The simplest way to do that is to use a linear Poisson point process along the streets. This will give rise to a Cox process. Building the Gilbert graph, i.e., drawing an edge between any two users with distance less than a given connection radius, one can obtain a very simplified model of users communicating via a Device-to-Device mechanism. Then, studying the percolation of this random graph, one could obtain results on the connectivity of the wireless network.
The main results in this paper fall into two large categories: existence of phase transition and asymptotic analysis of percolation probabilities. First, we show that a variant of the celebrated concept of stabilization [31, 28, 17, 26, 27] suffices to guarantee the existence of a sub-critical phase. In contrast, for the existence of a super-critical phase, stabilization alone is not enough since percolation is impossible unless the support of the random measure has sufficiently good connectivity properties itself. Hence, our proof for the existence of a super-critical phase relies on a variant of the notion of asymptotic essential connectedness from .
Second, when considering the Poisson point process, the high-density or large-radius limit of the percolation probability tends to 1 exponentially fast and is governed by the isolation probability. In the random environment, the picture is more subtle since the regime of a large radius is no longer equivalent to that of a high density. Since we rely on a refined large-deviation analysis, we assume that the random environment is not only stabilizing, but in fact -dependent.
Since the high-density and the large-radius limit are no longer equivalent, this opens up the door to an analysis of coupled limits. As we shall see, the regime of a large radius and low density is of highly averaging nature and therefore results in a universal limiting behavior. On the other hand, in the converse limit the geometric structure of the random environment remains visible in the limit. In particular, a different scaling balance between the radius and density is needed when dealing with absolutely continuous and singular random measures, respectively. Finally, we illustrate our results with specific examples and simulations.
2. Model definition and main results
Loosely speaking, Cox point processes are Poisson point processes in a random environment. More precisely, the random environment is given by a random element in the space of Borel measures on equipped with the usual evaluation -algebra. Throughout the manuscript we assume that is stationary, but at this point we do not impose any additional conditions. In particular, could be an absolutely continuous or singular random intensity measure. Nevertheless, in some of the presented results, completely different behavior will appear.
Example 2.1 (Absolutely continuous environment).
Let with a stationary non-negative random field. For example, this includes random measures modulated by a random closed set , [7, Section 5.2.2]. Here, with . Another example are random measures induced by shot-noise fields, [7, Section 5.6]. Here, for some non-negative integrable kernel with compact support and a Poisson point process.
Example 2.2 (Singular environment).
Let where denotes the one-dimensional Hausdorff measure and is a stationary segment process in . That is, is a stationary point process in the space of line segments [7, Chapter 8]. For example consider to be a Poisson-Voronoi, Poisson-Delaunay or a Poisson line tessellation.
Then, let be a Cox process in with stationary intensity measure where and . That is, conditioned on , the point process is a Poisson point process with intensity measure , see Figure 1.
To study continuum percolation on , we work with the Gilbert graph on the vertex set where two points are connected by an edge if their distance is less than a connection threshold . The graph percolates if it contains an infinite connected component.
2.1. Phase transitions
First, we establish sufficient criteria for a non-trivial phase transition of continuum percolation in Cox processes. More precisely, we let
denote the critical intensity for continuum percolation. In contrast to the Poisson case, in the Cox setting the non-triviality of the phase transition, i.e., , may fail without any further assumptions on [4, Example 4.1]. For our results we therefore assume that exhibits weak spatial correlations in the spirit of stabilization . To make this precise, we write to indicate the restriction of the random measure to the set . Further write
for the cube with side length centered at and put . We write to denote the distance between sets .
The random measure is stabilizing, if there exists a random field of stabilization radii defined on the same probability space as such that
are jointly stationary,
for all , the random variables
are independent for all bounded measurable functions and finite with for all .
A strong form of stabilization is given if is -dependent in the sense that and are independent whenever . The above Example 2.1 is -dependent for example in cases where is the classical Poisson-Boolean model and for the shot-noise field. Further, the Poisson-Voronoi tessellation, as in Example 2.2, is stabilizing, as we see in Section 3.
In order to avoid confusion, note that the literature contains several different forms of stabilization. Our definition is in the spirit of internal stabilization . Loosely speaking, the configuration of the measure in a neighborhood of does not depend on the configuration of the measure in the neighborhood of points with . The notion of external stabilization would additionally include that conversely the configuration of the measure around does not affect the measure around .
Stabilization implies the existence of sub-critical phase.
If and is stabilizing, then .
For the existence of a super-critical regime, the condition that is stabilizing is not sufficient. For example, the measure is stabilizing, but for every . Consequently, we rely on the idea of asymptotic essential connectedness (see ) to introduce a sufficient condition for the existence of a super-critical phase. To state this succinctly, we write
for the support of a measure .
A stabilizing random measure with stabilization radii is asymptotically essentially connected if for all , whenever we have that
is contained in a connected component of .
Example 2.1 for the Poisson-Boolean model and with as well as Example 2.2 for the Poisson-Voronoi and Poisson-Delaunay tessellation are asymptotically essentially connected. For the Poisson-Boolean model this is clear and for the Poisson-Voronoi tessellation case we provide a detailed proof in Section 3. The shot-noise field is not asymptotically essentially connected in general. Under the assumption of asymptotic essential connectedness, there is a non-trivial super-critical phase.
If and is asymptotically essentially connected, then .
2.2. Asymptotic results on the percolation probability
In classical continuum percolation based on a homogeneous Poisson point process, the critical intensity is characterized via the percolation probability in the sense that
where denotes the event that is contained in an infinite connected component of the Gilbert graph . The reason for this identity is the equivalence of the Poisson point process and its reduced Palm version . This is no longer true for general Cox processes and therefore, the proper definition of the percolation probability relies on Palm calculus. The Palm version of is a point process whose distribution is defined via
where is any bounded measurable function acting on the set of -finite counting measures . In particular, . Now,
denotes the percolation probability (of the origin). With this definition we recover the identity . Indeed, if , then and hence . Conversely, if , then and hence by stationarity.
Note that the Palm version is the union of the origin and another Cox process defined via the Palm version of the original random measure, see . Finally, the translation operator is defined by for all measurable . The distribution of is given by
where is any bounded measurable function.
2.2.1. Large-radius and high-density limits
In the Poisson-Boolean model, the percolation probabilities approach 1 exponentially fast as the radius grows large . More precisely,
where denotes the volume of the -dimensional ball with radius centered at . For -dependent Cox processes, the exponentially fast convergence remains valid with a -dependent rate.
If , then
If, additionally, is -dependent and has all exponential moments, then the limit
The above result is applicable to Example 2.1 in case of the Poisson-Boolean model and the shot-noise field. To the best of our knowledge, for the classical tessellation processes, as in Example 2.2, the large-deviation behavior for the total edge length has not been derived in literature yet. Therefore, in this situation, the above result only gives a lower bound. Although computing the limiting Laplace transform is difficult in general, we show in Section 3 that for the shot-noise field, the original expression simplifies substantially.
In classical continuum percolation, the scaling invariance of the Poisson-Boolean model makes it possible to translate limiting statements for into statements for . This is no longer the case for Cox processes. In the high-density regime, the connectivity structure of the support of becomes apparent. Loosely speaking, here the rate function is given by the most efficient way to avoid percolation. For a given realization of , percolation can be avoided by finite clusters at the origin such that there are no points at distance from the cluster. In a second step, we optimize over , see the right side of Figure 2.
More precisely, let the family consist of all compact sets that contain the origin and are -connected. That is, of all compact such that and such that the points at distance at most from form a connected subset of . Moreover, let
denote the -boundary of . The next result characterizes the asymptotic behavior of the percolation probability for stabilizing random measures that are supported on .
Let . Then,
If, additionally, is -dependent and for every , then
In general it is not true that the lower bound given by the isolation probability describes the true rate of decay of . Indeed, if the support of does not percolate, then . Nevertheless, for Example 2.1 in case of the Poisson-Boolean model with the above right-hand sides are optimal for . For the singular examples and for the shot-noise example the condition for every is not satisfied. The right-hand side of the lower bound can be computed for the Poisson-Voronoi tessellation case and equals .
2.2.2. Coupled limits
Theorems 2.7 and 2.8 describe the limiting behavior w.r.t. and separately. Now, we present three results about coupled limits. First if and are such that is constant, then by [8, Theorem 11.3.III], the rescaled Cox process converges weakly to a homogeneous Poisson point process with intensity , see Figure 3. This gives rise to the following statement about the percolation probabilities, where denotes the continuum percolation probability associated with a homogeneous Poisson point process with intensity and connection radius 1.
Let . Then,
If is stabilizing, then
For the converse limit , again with , one cannot hope for such a universal result since the structure of becomes prominent also in the limit. In particular, completely different scaling limits emerge for absolutely continuous and singular random intensity measures.
Let us start with the absolutely continuous case where with a stationary, non-negative random field as in Example 2.1. In this case, , where is the -size-biased version of . Since , that is,
where is any bounded measurable function. Let
denote the superlevel set of at level where is the critical intensity of the classical Poisson-Boolean model associated to . The strict superlevel set is defined accordingly. Then, similarly to the setting analyzed in , the percolation probability is asymptotically governed by a local and a global constraint, see the left side of Figure 1. Locally, the connected component must leave a small neighborhood around the origin, an event with probability . Globally, it must be possible to reach infinity along a path of super-critical intensity in , which in the following we denote as .
The next result describes the asymptotic behavior of the percolation probability under the assumption that is highly connected.
Let and be upper semicontinuous. If, with probability 1,
is continuous at , and
the intersection of any connected component of with remains connected,
If, additionally, is stabilizing and with a probability tending to 1 in , the set contains a compact interface separating from . Then,
In Example 2.1, for the Poisson-Boolean model, the upper semicontinuity is satisfied for . Further, assumption (1) – (3) are satisfied as long as . The additional assumption on the existence of the interface can only be guaranteed for sufficiently high intensity of the underlying Poisson process, . For the shot-noise field similar sufficient conditions can be formulated.
Next, we consider a singular setting as in Example 2.2. The scaling relation in Theorem 2.10 was chosen in such a way, that the expected number of neighboring Cox points remains constant. If we were to apply this scaling also in the singular case, then the scaling limit would be trivial. Indeed, with high probability on the majority of all edges some subsequent Cox points are separated by gaps of size at least , so that no percolation can occur. This is not a problem in case of absolutely continuous measures, since the continuous support of the underlying random intensity measure allows for percolation in all directions. Hence, we consider the more appropriate limit where the expected number of gaps per edge remains constant, see the right side of Figure 1.
In this regime, the limiting behavior is governed by an inhomogeneous Bernoulli bond percolation model on the Palm version for the segment system, where the probability for an edge of length to be open is given by for a suitable . For the Poisson-Delaunay tessellation, a homogeneous version was considered for example in . We write for the resulting percolation probability. The next result makes this precise under the assumption that the expected number of gaps per edge is small.
Let . Then,
If is essentially asymptotically connected and is sufficiently small, then
The rest of this paper is structured as follows. First in the remainder of the present section, we outline the proofs of the main results. Next, in Section 3, we provide examples. In Section 4, we present numerical simulations for the percolation probability. Section 5 contains the proofs for non-trivial phase transitions. Finally, Section 6 is devoted to the large-radius and high-density limit and in Section 7 we deal with coupled limits.
2.3. Outline of proofs
2.3.1. Phase transition
The proof is based on a renormalization argument. More precisely, the stabilization condition makes it possible to create a suitable -dependent auxiliary percolation process, which in turn can be analyzed using the techniques from . The existence of a sub-critical phase is easier to establish, since it suffices to create large regions without any points. For the super-critical regime, more care must be taken in order to produce appropriate connected components. It is at this point that we use the assumption on the asymptotic essential connectedness.
2.3.2. Large-radius and high-density limits
For the lower bounds in Theorem 2.7 and 2.8 we consider isolation probabilities of -connected sets containing the origin. If is such that , then the points in are contained in a different connected component of than the points in . In particular,
For any , this expression is at least
which gives the lower bound in Theorem 2.8. On the other hand, choosing in (2), taking the logarithm, dividing by and sending gives the lower bound in Theorem 2.7. The upper bounds in both Theorem 2.7 and 2.8 follow from a Peierls argument in Section 6.
2.3.3. Coupled limits
The proofs of Theorems 2.9, 2.10 and 2.11 are all based on two meta-results. The upper bounds rely on convergence in finite domains. To make this precise, we say that in if the connected component of in is not contained in . Moreover, we put
Now, convergence in finite domains provides the desired upper bounds by virtue of the following elementary upper-semicontinuity result.
Proposition 2.12 (Upper bound via convergence in bounded domains).
Let be monotone and an increasing function. Let and be a decreasing function such that for every ,
For the lower bound, we use a tightness condition based on a renormalized percolation process.
Proposition 2.13 (Lower bound via tightness).
Let and be as in Proposition 2.12 and assume that is stabilizing. Let denote the event that
contains a unique connected component intersecting both and , and
this component also intersects .
There exists a constant , only depending on the dimension, such that if for all sufficiently large ,
3.1. Stabilization and asymptotic essential connectedness
Let us discuss the case of Poisson-Voronoi tessellations in order to show that the assumptions of stabilization and asymptotic essential connectedness are indeed satisfied by a large class of Cox processes. The case of Poisson-Delaunay tessellations can be dealt with in a similar fashion.
Let where is the Poisson-Voronoi tessellation based on the homogeneous Poisson point process with intensity . More precisely, to each we associate the cell
consisting of all points in having as the closest neighbor. Then, is the union of the one-dimensional facets of the collection of cells .
We claim that the random measure in Example 3.1 is asymptotically essentially connected. Let us start by verifying the stabilization. We define the radius of stabilization
to be the nearest neighbor distance in the underlying Poisson point process. Let us check the conditions. First, by stationarity of , also is translation invariant. Second, let be a sub-division of into congruent sub-cubes of side length . Then,
where the right hand side tends to 0 exponentially fast as . This property is referred to in the literature as exponential stabilization, see . Finally, for almost all realizations of and for all , by the definition of the Poisson-Voronoi tessellation, there exist unique points such that . The number of such points depends on the dimension of the facet of containing . Under the event we thus have . In particular, a change in the configuration leaves unaffected and thus is independent of . Since also the event is independent of , the last condition is verified.
The Poisson-Voronoi tessellation is also asymptotically essentially connected, which we show now. In order to show the non-emptyness of the support, note that emptyness of the support, i.e. , implies that is contained in the Voronoi cell of some . Choose any two points with . Then, under the event , the points must have some distinct Poisson points with and and hence,
a contradiction. As for the connectedness, denote by the connected components in . By the definition of the Voronoi tessellation, every void space , which separates two of the connected components in , must be the intersection of one Voronoi cell with . Let denote the boundary of . We claim that, under the event , we have that is connected in . Indeed, let be such that then, since contains points in , we have . If then since for all we have which implies that . Hence, in this case the associated disconnected components in must be connected in . On the other hand, if then there exists a chord in starting at and crossing completely. But any such chord has maximal length , and thus again the associated disconnected components in must be connected in . Since the argument holds for any void space , we have connectedness of all in the larger volume .
3.2. Computation of the rate function in Theorem 2.7 for the shot-noise field
Let be the shot-noise field with for some non-negative integrable with compact support and be a Poisson point process with intensity . Then, for we can calculate,
By separating the domain of integration into and , we arrive at
We claim that the second and third summand in the last line are of order . Indeed, let be large enough such that the support of is contained in . Then, for the second summand, we have for that
For the third summand, using similar arguments,
In order to provide numerical illustrations for the main mathematical theorems, we estimated the actual percolation probability for a variety of parameters via Monte Carlo simulations. More precisely, is assumed to be the random measure given by the edge length of a planar tessellation as in Example 2.2. Here, we consider either Poisson-Voronoi tessellation or Poisson-Delaunay tessellation, and fix the length intensity .
In Figure 4, we present the estimated percolation probability as a function of the parameter for three choices of the radii: , and . In Theorem 2.9 we have seen that in the asymptotic setting of a large radius and small intensity, the percolation probability does not depend on the choice of the random intensity measure. It converges to the percolation probability of the Poisson-Boolean model. This behavior is reflected in the right-most panel of Figure 4 where , as there is very little difference between the percolation probability in the Voronoi or Delaunay setting. For , we see that the geometry of the random intensity measure influences substantially the percolation probability. This is even more prominent for smaller radii, such as . Indeed, Theorem 2.11 describes the behavior for small radii, also in the asymptotic regime, but here the dependence of the percolation probability on the underlying random intensity measure is not lost in the limit.
5. Proof of phase transitions
The main idea is to introduce a renormalization scheme reducing the continuum percolation problem to a dependent lattice percolation problem. To make this work, we rely crucially on the stabilization assumption. It allows us to make use of the standard -dependent percolation arguments presented in [18, Theorem 0.0].
5.1. Existence of sub-critical phase
In the renormalization we single out large regions that do not contain any Cox points and where one has good control over the spatial dependencies induced by . In the following, we put , where is the stabilization radius associated to .
Proof of Theorem 2.4.
A site is -good if
A site is -bad if it is not -good. By property (2), percolation of the Gilbert graph implies percolation of -bad sites. Hence, it suffices to verify that -bad sites do not percolate if is sufficiently small.
The process of -bad sites is -dependent as can be seen from the definition of stabilization. Moreover,
By [18, Theorem 0.0], we conclude that the process of -bad sites is stochastically dominated by a sub-critical Bernoulli percolation process. In particular, with probability one, there is no infinite path of -bad sites. ∎
5.2. Existence of super-critical phase
This time, our goal is to identify large regions where the support of is well-connected and the Cox points are densely distributed on the support of in these regions.
Proof of Theorem 2.6.
A site is -good if
every are connected by a path in .
We claim that if there exists an infinite connected component of -good sites, then percolates. Indeed, let and be such that , . If is any finite path connecting and , by property (2) we can choose points for every . Using property (3), we see that these points as well as and are contained in a connected component in . This gives the existence of an infinite cluster.
It remains to show that the process of -good sites is supercritical for sufficiently large . The process of -good sites is -dependent as can be seen from the definition of stabilization. Moreover, writing for the events (1), (2) and (3) we have
where by stabilization. Further, by condition (1) in Definition 2.5 and dominated convergence,
Finally, by asymptotic essential connectedness, if , then there exists a connected component with . Note that the support of will be filled with Cox points in the limit and thus any two points in must be connected eventually. More precisely, let be a finite set of points in such that where . Further, let
denote the event that every contains at least one Cox point. Then, yields that . In addition, by asymptotic essential connectedness,
Now using dominated convergence,
Again by [18, Theorem 0.0], the process of -good sites is stochastically dominated from below by a super-critical Bernoulli percolation process, as required. ∎
6.1. Proof of Theorem 2.7
As explained in Section 2.3.2, we only need to prove the upper bound together with the existence of the limiting cumulant generating function . We prove the assertion for finite domains and then rely on a Peierls argument to establish that the (as defined in (3)), form exponentially good approximations of in the sense of [9, Definition 4.2.10]. We put . To prove Theorem 2.7, we proceed in two steps:
for every and ,
for every and all sufficiently large ,
The idea for the upper bound on finite domains is to consider the convex hull of the cluster at the origin. In particular, there are no Cox points in a forbidden volume formed by all points outside the convex hull but within distance at most of one of its vertices. By Steiner’s formula from convex geometry [29, Theorem 1.1], the volume of this set is at least , so that we arrive at the desired upper bound for finite .
To prove the exponentially good approximation, note that if the cluster at the origin is finite but percolates beyond , then we can define a substantially larger forbidden volume, giving rise to a faster exponential decay.
The main ingredient in the proof is a large deviation formula for the Laplace transform of the random measure in a large set.
Let and assume that is -dependent and has all exponential moments.
Then the following limit exists
Let be compact with and . Then,