Combinatorial principles in stochastic geometry in the plane: A review

Abstract

The paper contains a review of the main results of Yerevan research group in planar stochastic geometry, in particular the second order random geometrical processes using the methods of integration of combinatorial decompositions and invariant imbedding. For the present review we have chosen the results concerning tomography (or stereology) of random processes of lines, random tessellations (mosaics) and Boolean models (not necessarily Poisson) in the plane. The results are obtained either under the assumption of invariance with respect to the group T of translations, or the group M of the Euclidean motions of ℝ2. In each case the marked point processes {P i ,Ψ i } of intersections induced by the random pattern in question on the test lines are studied, where {P i } is the point process of intersections induced on a test line of direction α, the mark Ψ i is the angle at which the intersection with the test line of direction α occurs at point P i . In the case of random line processes, these approaches lead to differential relations between the joint probabilities for the numbers of intersections that occur in disjoint intervals on a test line having direction α, and the first and the second order Palm probabilities of the similar events. By an analysis of these relations, conditions of Poissonity of n-dimensional distributions for any n ≥ 1 are derived. In the case of random tessellations the distribution of the length of so-called typical edge of direction α is obtained in terms of the probability distribution of the corresponding {P i ,Ψ i }. As regards random M-invariant Boolean models, the present review includes a description of the method based on summation of Pleijel-type identities for the part of the realization within a finite disc and subsequent calculation of the limit when the radius of the disc tends to infinity. An alternating process of marked intervals on a line is called black-recurrent if the black marked intervals are independent and the lengths of the white intervals constitute an independent sequence of independent random variables. One of the earliest results in the field states that black-recurrence implies an exponential distribution for the length of the typical white interval on a test line.