Percolation on stationary tessellations: models, mean values, and second-order structure

Abstract

We consider a stationary face-to-face tessellationXofRdand 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 probabilitypand white otherwise. We are interested in geometric properties of the unionZof 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 focus on cell percolation on normal tessellations and study asymptotic covariances of intrinsic volumes ofZ∩W, where the observation windowWis assumed to be a convex body. Special emphasis is given to the planar case where the formulae become more explicit, though we need to assume the existence of suitable asymptotic covariances of the face processes ofX. We check these assumptions in the important special case of a Poisson-Voronoi tessellation.