THE POISSON RAIN TESSELLATION: A MODEL OF A RANDOM TESSELLATION IN THE PLANE INDUCED BY A POISSON POINT PROCESS

Tessellations arising from divisions of their 'cells' are quite commonly studied not only out of mathematical interest but also because they serve as models for biological or geological structures. Several mathematical models were introduced and studied by e. g. Miles, Cowan or Mecke/Nagel/Weiss. The model examined in this lecture is derived from a planar Poisson point process and can be described in short as throwing a point onto a convex window in the plane at a random time and drawing a line through the point under a randomly determined direction with that line ending either at the boundary of the window or at a previously drawn line. In the lecture, a formal construction will be given as well as some results for the model. For example, the capacity functional of a compact set depends on its position within the window and is thus not homogeneous in the general case. There is however a case in which the capacity functional is indeed homogeneous (or translation invariant): This is when both the window's sides and the lines drawn within the window are parallel to the axes of a coordinate system. One can also show that under a suitable choice of the intensity measures of a series of windows, in the limit there is homogeneity.