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Paper No. 09-43

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WS Kendall

Geodesics and flows in a Poissonian city

Date: October 2009

Abstract: The stationary isotropic Poisson line network was used to derive upper bounds on mean excess network-geodesic length in Aldous and Kendall (2008). This new paper presents a study of the geometry and fluctuations of near-geodesics in such a network. The notion of a “Poissonian city” is introduced, in which connections between pairs of nodes are made using simple “no-overshoot” paths based on the Poisson line process. Asymptotics for geometric features and random variation in length are computed for such near-geodesic paths; it is shown that they traverse the network with an order of efficiency comparable to that of true network geodesics. Mean characteristics and limiting behaviour at the centre are computed for a natural network flow. Comparisons are drawn with similar network flows in a city based on a comparable rectilinear grid. A concluding section discusses several open problems. MSC 2000 subject classifications: Primary 60D05, 90B15

Keywords: Dufresne integral; frustrated optimization; geometric spanner network; growth process; improper anisotropic Poisson line process; Lamperti transformation; Laplace exponent; Lévy process; logarithmic excess; Manhattan city network; Mills ratio; mark distribution; martingale central limit theorem; network geodesic; Palm distribution; perpetuity; Poisson line process; Poissonian city network; Slivynak theorem; spanner; spatial network; subordinator; traffic flow; uniform integrability.