2013/12/22 by Günter Last, Eva Ochsenreither, Last, Günter +1
Mathematics · #60D05 #Census and Population Estimation #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60D05
paper · pdf · doi:10.48550/arxiv.1312.6366
arxiv created 2013/12/22 · openalex publication_date 2013/12/22 · arxiv updated 2013/12/24 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We consider a stationary face-to-face tessellation X of ℝd and introduce several percolation models by colouring some of the faces black in a consistent way. Our main model is cell percolation, where cells are declared black with probability p and white otherwise. We are interested in geometric properties of the union Z of black faces. Under natural integrability assumptions we first express asymptotic mean-values of intrinsic volumes in terms of Palm expectations associated with the faces. In the second part of the paper we study asymptotic covariances of intrinsic volumes of Z∩ W, where the observation window W is assumed to be a polytope. Here we need to assume the existence of suitable asymptotic covariances of the face processes of X. We check these assumptions in the important special case of a Poisson Voronoi tessellation. In the case of cell percolation on a normal tessellation, especially in the plane, our formulae simplify considerably.