Continuum percolation for Gibbs point processes

We consider percolation properties of the Boolean model generated by a Gibbs point process and balls with deterministic radius. We show that for a large class of Gibbs point processes there exists a critical activity, such that percolation occurs a.s. above criticality. For locally stable Gibbs point processes we show a converse result, i.e. they do not percolate a.s. at low activity.


Introduction
Let Ξ be a point process in R d , d ≥ 2, and fix a R > 0. Consider the random set Z R (Ξ) = x∈Ξ B R (x), where B R (x) denotes the open ball with radius R around x.Each connected component of Z R (Ξ) is called a cluster.We say that Ξ percolates (or R-percolates) if Z R (Ξ) contains with positive probability an infinite cluster.In the terminology of [9] this is a Boolean percolation model driven by Ξ and deterministic radius distribution R.
It is well-known that for Poisson processes there exists a critical intensity β c such that a Poisson processes with intensity β > β c percolates a.s. and if β < β c there is a.s.no percolation, see e.g.[17], or [14] for a more general Poisson percolation model.
For Gibbs point processes the situation is less clear.In [13] it is shown that for some two-dimensional pairwise interacting models with an attractive tail, percolation occurs if the activity parameter is large enough, see also [1] for a similar result concerning the Strauss hard core process in two dimensions.There is a related work [7], which states conditions on the intensity instead of the activity.Furthermore in [11,16] it is shown that for finite-range pairwise interacting model there is no percolation at low activity.
The aim of this work is to extend the results of [13,1] to any dimension d ≥ 2 and to very general Gibbs point processes.We give a percolation condition on the conditional intensity of a Gibbs process, which is easily understandable and which is satisfied for most Gibbs processes, provided the percolation radius R is large enough.Our main result on percolation, Theorem 3.1 then states, that there exists a critical activity, such that the Gibbs point processes percolate a.s.above criticality.The main idea of the proof is the contour method from statistical physics.
Furthermore we state a result on the absence of percolation for locally stable Gibbs processes.Here the idea of the proof is that a locally stable Gibbs process can be dominated by a Poisson process, and then we use the percolation results available for Poisson processes.
The plan of the paper is as follows.In Section 2 we introduce the Gibbsian formalism and give the necessary notations.Our main results are stated in Section 3, and in Section 4 they are applied to pairwise interaction processes.Finally, Section 5 contains the proofs of our main results.
Let (N, N ) denote the space of locally finite point measures on R d equipped with the σ-algebra generated by the evaluation maps [N ∋ ξ → ξ(Λ) ∈ Z + ] for bounded Borel sets Λ ⊂ R d .A point process is just a Nvalued random element.We assume the point processes to be simple, i.e. do not allow multi-points.Thus we can use set notation, e.g.x ∈ ξ means that the point x lies in the support of the measure ξ and |ξ| = ξ(R d ) denotes the total number of points in ξ.For a measurable Λ ⊂ R d let (N Λ , N Λ ) be the space of locally finite point measures restricted to Λ and its canonical σalgebra, respectively.Denote ξ| Λ for the restriction of a point configuration ξ ∈ N onto Λ.
To define Gibbs processes on R d we use the Dobrushin-Lanford-Ruelleapproach of local specifications, see e.g.[5,12,15].Fix a bounded measurable Λ ⊂ R d and a configuration ω ∈ N Λ c , called the boundary condition, where Λ c = R d \Λ.Furthermore fix a measurable function Φ : N → R∪{∞}, Define a probability measure on N by where is called the partition function.Note that the sum in (2.1) and the partition function may not exist.However we tacitly assume that the potential Φ is chosen such that µ Λ,ω is well-defined for all bounded measurable Λ ⊂ R d and for all ω ∈ N Λ c ; we refer the reader to [15] or [4] for such and related questions.A probability measure µ on N is called a Gibbs measure, if it satisfies the Dobrushin-Lanford-Ruelle equation For x ∈ R d and ξ ∈ N define the conditional intensity λ, see [12], as Fix a bounded domain Λ ⊂ R d and a boundary condition ω ∈ N Λ c .Then we get from (2.1) for all x ∈ Λ and for all ξ ∈ N Λ such that x / ∈ ξ and u Λ,ω (ξ) > 0. Let µ ∈ G(Φ) and define and by (2.4) we get µ(N Λ ) = 0 as well.Thus (2.6) holds for µ-a.e.ξ ∈ N with x / ∈ ξ.Note that if we are interested in Gibbs point processes on a bounded domain with empty boundary condition (ω = ∅), Equation (2.6) coincides, up to the null-set N Λ , with the definition of the conditional intensity commonly used in spatial statistics, see e.g.[10,Def. 6.1].Roughly speaking, the conditional intensity is the infinitesimal probability that Ξ has a point at x, given that Ξ coincides with the configuration ξ everywhere else.
For the rest of this paper we assume the potential Φ to be constant for one-point configurations, i.e.Φ {x} The constant β is called activity parameter or simply activity.To be able to keep track of β in our main results, we will mainly use the notation β λ for the conditional intensity.
In order to describe Gibbs processes by the DLR-approach, one can equivalently characterize them through the conditional intensity, see [12].Therefore, denote G(β, λ) as the set of Gibbs measures corresponding to the conditional intensity β λ.
A Gibbs point process Ξ ∼ µ ∈ G(Φ) is called a pairwise interaction process if for every configuration ξ ∈ N with |ξ| ≥ 3 we have Φ(ξ) = 0.By denoting ϕ(x, y) = exp − Φ {x, y} the conditional intensity simplifies to for all x ∈ R d and for all ξ ∈ N, with x / ∈ ξ.With a slight abuse of notation, we shall use G(β, ϕ) for the set of the corresponding Gibbs measures.
One of the most important point processes is surely the Poisson process.Here, we concentrate only on homogeneous Poisson processes, which can be defined as pairwise interaction processes with ϕ ≡ 1.For Poisson processes the parameter β is called intensity.The more common definition, however, is the following.A point process Π is called a Poisson process with intensity β if for bounded and pairwise disjoint sets Λ 1 , . . ., Λ n the random variables Π(Λ 1 ), . . ., Π(Λ n ) are independent and Poisson distributed with mean β|Λ i |, for i = 1, . . ., n.It is an easy exercise to show the equivalence of the two definitions using (2.2).
Proposition 2.1.Let Π be a Poisson process with intensity β.For all R > 0 there exists a critical intensity 0 < β c < ∞ such that Z R (Π) contains a.s.only finite clusters if β < β c and Z R (Π) has an infinite cluster a.s.if β > β c .
If for all β > 0, all Gibbs processes with distribution in G(β, λ) satisfy condition (P), then we say that λ satisfies condition (P) itself.
A physical interpretation of condition (P) could be the following.Let ξ ∈ N and choose a x ∈ R d such that its nearest point in ξ is at a distance at least r.Assume that the point process Ξ coincides with ξ everywhere except at x. Then the condition (P) states that the energy cost of adding a point at x is bounded from above by an universal constant which does not depend on the location x nor on the configuration ξ.
The next theorem is our main result on percolation.
Theorem 3.1.Let λ satisfy condition (P) with constants r and δ.Then for all R > r there exist a β + < ∞ such that for all β > β + and for all Gibbs processes Ξ with distribution in G(β, λ), the set Z R (Ξ) contains a.s.an infinite cluster.
Remark 3.2.In fact, the proof of Theorem 3.1 yields a slightly stronger percolation result.Let Ξ be as in Theorem 3.1, and let H ⊂ R d be any affine subspace of dimension at least two.Then Z R (Ξ) ∩ H contains a.s.an infinite cluster.
For our result on non-percolation we need the following stability assumption.Let Ξ be a Gibbs process with distribution µ and conditional intensity λ.Then Ξ is called locally stable if there exists a constant c * such that λ(x | ξ) ≤ c * for all x ∈ R d and for µ-a.e.ξ ∈ N. If all Gibbs point processes corresponding to the conditional intensity λ are locally stable, we call λ itself locally stable.Most Gibbs point processes considered in spatial statistics are locally stable, see [10, p. 84] and [8, p. 850].However the most important example in statistical physics, the Lennard-Jones process, is not locally stable.Theorem 3.3.Let β λ be locally stable.Then for all R > 0 there exists a β − > 0 such that for all β < β − and for all Gibbs processes Ξ with distribution in G(β, λ), the set Z R (Ξ) contains a.s.only finite clusters.
Remark 3.4.The proof of Theorem 3.3 relies on the fact that a locally stable Gibbs process can be dominated by a Poisson process, and then Proposition 2.1 is applied.However for Poisson processes there are more general percolation results available, e.g. the balls could be replaced by some random geometric objects.Thus Theorem 3.3 may be generalized along the lines of [14].
The next example combines Theorem 3.1 and Theorem 3.3 to get a similar result as in Proposition 2.1.
Example 3.5.A Gibbs point process is called an area interaction process, see [2], if its conditional intensity is given by for some γ, r 0 > 0. One easily gets the estimates Thus λ satisfies condition (P) for all r > 0 with δ = min{1, γ −α d r 0 d } and it is locally stable with constant c * = β max{1, γ −α d r 0 d }.Then the Theorems 3.1 and 3.3 yield that for all R > 0 there exists two constants 0 < β − ≤ β + < ∞ such that all area interaction processes with β < β − do not R-percolate a.s. and all area interaction processes with β > β + do R-percolate a.s.It remains open whether β − = β + , as for the Poisson process, see Proposition 2.1.

Pairwise interaction processes
In this section we consider pairwise interaction processes and compare the results of [13] with ours.The next proposition shows which pairwise interaction processes satisfy condition (P).
Proposition 4.1.Let r > 0. Assume that the interaction function ϕ satisfies one of the following conditions.
(i) For all x, y ∈ R d with x − y ≥ r we have ϕ(x, y) ≥ 1.
(ii) There exist constants δ > 0 and r max < ∞ such that Then there exists a δ > 0 such that all Gibbs processes with distribution in G(β, ϕ) satisfy condition (P) with constants r and δ.
For a bounded Λ ⊃ B rmax (x) and any boundary condition ω ∈ N Λ c we get µ Λ,ω (N x ) = 0 and by (2.4) also µ(N x ) = 0. Furthermore there exists a constant m < ∞ such that for all ξ ∈ N \ N x we have ξ B rmax (x) ≤ m, and m does not depend on x (e.g.m can be chosen as the maximal number of balls with radius r/2 which can be placed in a ball with radius r max + r/2).Thus condition (P) is satisfied with δ = δm for all x ∈ R d and for µ-a.e.ξ ∈ N.
In [13] there are various assumptions on ϕ including our condition (i) of Proposition 4.1 and an attraction condition.Namely, ϕ is said to have an attractive tail if there exists two constants r a < r ′ a such that ϕ(x, y) > 1, whenever r a ≤ x − y ≤ r ′ a .Furthermore the authors percolation radius R has to be greater than √ 2r.We could reduce this bound by a factor √ 2, but in some cases, e.g.
for a Strauss hard core process (ϕ(x, y) = ½{ x − y ≥ r}), one would expect a lower bound on the percolation radius of r/2.The main difference is however, that the percolation result of [13] is valid only in two dimension, whereas our result holds in any dimension d ≥ 2.
A pairwise interaction process is locally stable in the following two cases.Firstly if the interaction function ϕ(x, y) is bounded by one; such a process is called inhibitory.Secondly if it has a hard core radius r, i.e. ϕ(x, y) = 0 whenever x − y ≤ r, and ϕ(x, y) → 1 fast enough as x − y → ∞.The non-percolation result in [13] treats only the hard core case.However, unless for the percolation result, the authors proof is quite different from ours.

Proofs
The main idea for the proof of percolation is based on techniques close to the contour method in lattice models.In particular, it is a modification of the arguments in [13] and [1].Let condition (P) be satisfied with constants r, δ > 0, and let R > r.Choose a m ∈ N such that m > √ d/(R − r).Divide R d into cubes of length 1/m.For this sake define where we use the maximum norm The cardinality of S is denoted by |S|.Lemma 5.1.There exists a constant c > 0 such that for all S ⊂ 1 m Z d with |S| < ∞, there exists a S ′ ⊂ S with |S ′ | ≥ c|S| and for each pair z, z Proof.We can assume |S| ≥ 1 and then obviously there exists a S ′ with cardinality one.To identify the constant c, consider the following inductive procedure.Choose an arbitrary point z 1 ∈ S and draw a ball around z 1 of radius r + √ d/m.Exclude all points of S which are contained in this ball and denote by S 1 the remaining points of S. Note that dist(Q z 1 , Q z ) ≥ r for all z ∈ S 1 .Continue this procedure and define S ′ = {z 1 , . . ., z n }, the set of the chosen points.At each step we exclude at most α d (r + 3 where ⌊x⌋ denotes the greatest integer less than or equal to x. Choosing c = 1/(2α d (r + 3 The following Lemma is the key ingredient for the proof of Theorem 3.1. Lemma 5.2.Assume Ξ ∼ µ ∈ G(β, λ) satisfies condition (P) with constants r and δ.
where c is the same constant as in Lemma 5.1.
Proof.By Lemma 5.1 choose a S ′ ⊂ S with |S ′ | ≥ c|S| and such that for each pair z, z Choose a bounded Λ ⊂ R d , such that the distance between S and the boundary of Λ is at least r + √ d/m and fix a boundary condition ω ∈ N Λ c .
Then for µ-a.e.ξ ∪ {x 1 , . . ., x n } ∈ B S,r we have by (2.6) and condition (P) that (5. 2) The partition function can then be bounded from below as where k!/(k − n)! is the number of possibilities to choose a set of k − n elements and n sets of one elements out of a set of k elements, and the last inequality follows by (5.2), by integrating over the cubes {Q z i , i = 1, . . ., n} and by the change of variable j = k − n.Thus, by (2.2) .
Remainder of the proof of Theorem 3.1.We call a set L ⊂ 1 m Z d a loop if |L| < ∞, L is connected and each cube {Q z , z ∈ L} has exactly two neighbours.We say L ⊂ 1 m Z d is a loop around the origin if the origin is contained in the convex hull of L. Denote by L 0 the set of all loops around the origin.
Consider the loop L = {z 1 , . . ., z k } ∈ L 0 such that Q z 1 and Q z k are neighbours and for i = 1, . . ., k − 1 the cubes Q z i and Q z i+1 are neighbours.Since the origin is contained in the convex hull of L, the point z 1 lies necessarily in the big cube [−k/m, k/m] d , which gives (2k + 1) d possibilities.For i = 1, . . ., k − 1, the cube Q z i+1 is a neighbour of Q z i , thus there are at most 2 d possibilities to place the cube Q z i+1 .We conclude, that there are at most (2k + 1) d 2 d(k−1) loops in L 0 with length k.Consider the event Then for β ≥ m d /δ Lemma 5.2 yields The first Borel-Cantelli Lemma together with (5.4) then yield that there are a.s.only finitely many loops L ∈ L 0 with dist Ξ, z∈L Q z ≥ r.
The following trick allows us to reduce the problem to the two-dimensional case.Consider the set Z where we identify R 2 as the plane spanned by the first two canonical basis vectors in R d .A loop is said to be a R 2 -loop if the centres of its cubes lie in R 2 .
The parameter m is chosen such that if two points x and x ′ lie in the same connected component of R 2 \ Z R (Ξ) which connects x with x ′ and choose all cubes which intersect with this path.Obviously the cubes are connected and the length of the diagonal of a cube is less than R − r.Thus, by the reverse triangle inequality, the selected cubes and Ξ are separated by a distance at least r. Figure 1 shows a graphical illustration of this procedure.
Assume that Z R (Ξ) does not percolate, i.e. there exists a.s.only finite clusters.Obviously the clusters in Z (2) R (Ξ) are also finite, and each cluster can be surrounded by a R 2 -loop L ∈ L 0 with dist Ξ, z∈L Q z ≥ r.For large β this eventually leads to a contradiction.
Note that the choice of a basis in R d is by no means important.Thus, the statement of Remark 3.  It is a known fact that every locally stable Gibbs process on a bounded domain can be obtained as a dependent random thinning of a Poisson process; see [8,Remark 3.4].In particular, there exists a Poisson process Π n with intensity c * and a Gibbs process Ξn ∼ G β λ(• | • ∪ ω) on Λ n , such that Ξn ⊂ Π n a.s.Let Ξ n be the point process obtained replacing Ξ| Λn with Ξn , i.e.Ξ n = Ξ| Λ c n ∪ Ξn .Since Ξ| Λn and Ξn have the same conditional distribution µ Λn,ω , Equation (2.4) yields that Ξ and Ξ n have the same distribution for all n ∈ N. A standard result ([3, Theorem 11.1.VII]) then yields that as n → ∞ the sequence Ξ n has a weak limit Ξ ′ with the same distribution as Ξ.By the same result Π n → Π as n → ∞, where Π is a Poisson process on R d with intensity c * .Furthermore, since Ξn ⊂ Π n a.s.for all n ∈ N, we conclude Ξ ′ ⊂ Π a.s.

Figure 1 :
Figure 1: Here we choose d = 2, r = 0.2, R = 0.3 and m = 15.The grey area corresponds to Z R (Ξ) and the chain of shaded boxes shows one possibility to separate the two components of Z R (Ξ).Note that the boxes do not intersect with the circles of radius r.