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

Abstract

We consider a stationary face-to-face tessellation $X$ of $\mathbb{R}^d$ 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 $Z\cap W$, 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.