Arak-Clifford-Surgailis Tessellations. Basic Properties and Variance of the Total Edge Length

Abstract

Non-homogeneous random tessellations in the plane with T-shaped vertices are considered, which are defined as Gibbsian modifications of a Poisson line process with arbitrary locally finite intensity measure. In the homogeneous set-up they can be regarded as a specific case of the general Arak-Surgailis polygonal fields in the plane and share some properties with the iteration stable (STIT) tessellations. In the non-homogeneous and anisotropic environment an explicit expression for the partition function is provided and first- and second-order properties of the random length measure of the tessellations restricted to a convex sampling window are derived. In the more special isotropic regime a closed formula for the pair-correlation function and the variance of the total edge length is obtained.