Random processes of linear segments and graphs

Abstract

By a graph G we understand a finite set of points (vertices)together with the line segments which unites some pairs of distinct points of the set. Sets of congruent graphs are considered. The position of a graph on the plane is defined by the position of one of its vertices P and a rotation ϕ about P. Assuming P Poisson distributed on the plane and ϕ uniformly distributed over 0 <=ϕ <= 2п, we extend to graph processes some known properties of line segment processes (Coleman, Parker and Cowan). We find the probability that the distance from a point chosen at random independently of the process of graph to the nearest vertice of a graph or to the nearest graph exceed u. Some of the results are also extended from the Euclidean plane to surfaces (sets of geodesic segments and sets of geodesic graphs, for instance to the sphere and to the hyperbolic plane