On the asymptotic representation of average distances in a random point set on the plane

Abstract

In this paper, we give a solution to a general problem in geometric probability theory, which is the subject of a number of issues in modern technology (aviation agricultural technology, visibility in turbid environments, etc.). Let us consider a set A of points A₀, A..., An, randomly distributed in a circle K of radius R. We assume that the occurrence of each individual point of this set in parts of the circle K equal in area is equally probable (the law of equal probability). Let, further, the number of points in the set A be related to the value of the radius R such that the limit of the ratio exists and is finite. In other words, the average concentration of points in the circle K, with n and R increasing without limit, tends to a finite limiting concentration. We will study the random variable r, which is the smallest of the distances of an arbitrary point A₀ of set A, with its random position in the circle K, to the other points of set A. The random variable under consideration will obviously be equal to the radius of a circle with its center at point A₀, which does not contain other points of set A and has at least one point of this set on its boundary.