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From random lines to metric spaces

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Kendall, W. S. (2017) From random lines to metric spaces. Annals of Probability, 45 (1). pp. 469-517. doi:10.1214/14-AOP935

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Official URL: http://dx.doi.org/10.1214/14-AOP935

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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; indeed that this produces a parameter-dependent random geodesic metric for d-dimensional space (d≥2), 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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH): Metric spaces, Poisson processes, Geodesics (Mathematics)
Journal or Publication Title: Annals of Probability
Publisher: Institute of Mathematical Statistics
ISSN: 0091-1798
Official Date: 26 January 2017
Dates:
DateEvent
26 January 2017Available
29 April 2014Accepted
March 2014Submitted
Volume: 45
Number: 1
Page Range: pp. 469-517
DOI: 10.1214/14-AOP935
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Funder: Engineering and Physical Sciences Research Council (EPSRC)
Grant number: EP/K013939
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