From random lines to metric spaces

Abstract

Consider an improper Poisson line process, marked by positive speeds so as to satisfy a scale-invariance property (actually, scale-equivariance). The line process can be characterized by its intensity measure, which belongs to a one-parameter family if scale and Euclidean invariance are required. This paper investigates a proposal by Aldous, namely that the line process could be used to produce a scale-invariant random spatial network (SIRSN) by means of connecting up points using paths which follow segments from the line process at the stipulated speeds. It is shown that this does indeed produce a scale-invariant network, under suitable conditions on the parameter; in fact, it then produces a parameter-dependent random geodesic metric for $d$-dimensional space ($d\geq2$), where geodesics are given by minimum-time paths. Moreover, in the planar case, it is shown that the resulting geodesic metric space has an almost everywhere unique-geodesic property that geodesics are locally of finite mean length, and that if an independent Poisson point process is connected up by such geodesics then the resulting network places finite length in each compact region. It is an open question whether the result is a SIRSN (in Aldous’ sense; so placing finite mean length in each compact region), but it may be called a pre-SIRSN.