Existence of a plane without edge crossings in projections of the random geometric graph

Abstract

Consider a random geometric graph $G$ with a vertex set defined by a Poisson point process with intensity $t>0$ in a convex body. We can generate a drawing of the graph by projecting the construction onto some plane $L$. Choosing different planes leads to different drawings, and in particular, potentially more or fewer edge crossings. In this paper, we prove that if the connection radius is smaller than a given threshold, the probability that there exists a plane with zero crossings tends to one as $t\to \infty$. We also state the asymptotic probability that such a plane is found after considering a given number of randomly chosen planes.

12 pages. Version 2 includes arbitrary dimension $d$; additional author