New Paths Towards Quantum Gravity (Lecture Notes in Physics, 807)

6 Stochastic Geometry and Quantum Gravity: Some Rigorous Results 325()P qρ lim sup A n(t) = 1.n→∞Since limn→∞ sup A n(t) ⊆ C . , the assertion follows.It remains to show that P qρ {cd D ′ ≥ 1} > 0, where D ′ is defined by(x,η)∈ D ′ iff x ∈ A ḋ ,η( . K (x)) = 0.This follows by a similar argument: Choose t as above and ε>0 small enough suchthatE n (t) ={x ∈ M . 3∣ x(B ε (a)) = 1 for any a ∈ t} ⊆A . 2 .Then choose a ball B containing all B ε (a), a ∈ t, and let B ′ := B ∪ a∈t B ε (a).Then the event {ζ B ′ = 0,ξ Bε (a) = 1 for any a ∈ t} is contained in {cd D ′ ≥ 1} andhas positive probability with respect to P qρ .qedAs a consequence of Theorem 3 the associated cluster process Q ρ,Del of P ρof type D el is well defined. We call this process the Poisson–Delaunay process,PD-process for short.The following result is well known. A proof can be found in Chap. 6 of theGerman edition of Schneider/Weil [16].Lemma 3 For P ρ – almost any ν ∈ M the random cluster measure κ = γ D(ν),İ,Dhaving Q ρ, Del as its distribution, has the following support properties:(1) If z, z ′ ∈ κ are distinct with 〈x〉 ∩〈x ′ 〉 ̸= ∅(where x = qz, x ′ = qz ′ ), thenx ∩ x ′ ̸= ∅,x x ′ and x ′ x are nonempty and situated on opposite sides ofsome hyperplane containing x ∩ x ′ .(2) Every (d − 1) face of an element z ∈ κ is shared by another element z ′ ∈ κ.(3) κ is a tesselation of R d ; in particular one has⋃〈x〉 =R d .x∈qν:(x,qν)∈DThus a PD-process typically realizes tesselations built on Delaunay triangleswhose vertices are marked by strictly positive radii.We remark as an aside: If we add more restrictions to the definition of D el ,then κ = γ Del (ν) may show a qualitatively different geometrical and topologicalbehaviour. To be more precise: Given two parameter 0 < r < R < +∞, considerthe following cluster property in M . :(x,η)∈ D el (r, R) iff (x,η)∈ D el ,η ∈ M . (r), diam x < R.Here η ∈ M . (r) iff η ∈ M. s.th. (a, b ∈ η, a ̸= b ⇒‖a − b‖ ≥r) and diam xdenotes the diameter of x being defined as the diameter of K (x). Note that r = 0and R =+∞corresponds to the former case.