Combinatorial principles in stochastic geometry in the plane: A review
The author gives a review on some combinatorial principles in Stochastic Geometry in the plane; he mainly underlines some results, obtained in Yerevan, concerning tomography of random processes of lines, random tessellations and Boolean models in the plane. The required invariance is either with respect to the group \(T\) of translations or to the group \(M\) of the motions of \(\mathbb{R}^2\). After recalling basic concepts on translation invariant line processes, the modified Palm formulas and the method of invariant embedding are described. The author also states the condition implying Poissonity on the point process of intersections induced on a test line. Moreover, the method of integration of a combinatorial decomposition is described and the method based on summation of Pleijel-type identities is applied to random Boolean models invariant with respect to the group \(M\) of the Euclidean motions in \(\mathbb{R}^2\). An example of application of the method of invariant embedding, consisting of integration a combinatorial decomposition and derivation of a differential identity for the probability distribution of the number of hits of a test segment by the random lines, is also given.
scientific article; zbMATH DE number 5345590
scientific article