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

6 Stochastic Geometry and Quantum Gravity: Some Rigorous Results 327We call ˆQ ρ,D the Poisson–Delaunay surface with intensity ρ. We know alreadythat ˆQ ρ,D ∈ P 0 Ɣ. Moreover, ˆQ ρ,D is of first order with respect to the barycentresince P ρ has this property.Finally one hasLemma 4 ˆQ ρ,D is mixing, i.e.lim|a|→+∞ˆQ ρ,D (A ∩ T a B) = ˆQ ρ,D (A) · ˆQ ρ,D (B), A, B ∈ F . Ɣ .This implies immediately the ergodicity of ˆQ ρ,D , i.e. ˆQ ρ,D (A) ∈{0, 1} for anytranslation-invariant event A ∈ F . Ɣ .Proofs of these assertions are contained in [16] (Theorem 6.4.2 of the Germanedition) resp. [5].6.3.3 Scholion: The Voronoi Cluster PropertyHere we present shortly the alternative construction of Delaunay tesselations bymeans of Voronoi tesselations as one can find them in the book of Schneider/Weil[16]. Given a ∈ R d and η ∈ C . the associated Voronoi polytope is defined byV (a,η)={b ∈ R d ∣ ∣ d(b, a) ≤ d(b,η)}.Let E V (a,η) denote the set of its extreme points. The Voronoi cell belonging to(a,η) is given by y + δ a ,thesety augmented by a, where y = E V (a,η).TheVoronoi cluster property is then defined by the following measurable subset D v ofR d × M . f× M . :((a, x,η)∈ D v iff η ∈ C . , a ∈ x, x − δ a = E V (a, η)).We call then x −δ a ,thesetx without its element a,theVoronoi cluster for η centredin a. The Voronoi configuration belonging to η is η ∗ = supp ∑ a ∈ η b ∈ E ∑V (a,η)δ b .D v is stationary and locally finite. Moreover, one can show that the countingvariablecd Dv : η → ∑ ∑1 Dv (a, x,η)a∈η a∈x⊆η ∗ +δ asatisfies P z·λ {cd Dv ≥ 1} > 0. Thus given ρ = z·λ⊗τ, the associated Voronoi clusterprocess Q = Q ρ,Dv is well defined as the image of P ρ,Dv under the measurabletransformationγ Dv : ν ↦→ κ =∑ ∑1 Dv ((a, t), z,ν)· δ (q(z−δ(a,t) ),t).(a,t)∈ν (a,t)∈z⊆ν ∗ +δ (a,t)