GEODESICS AND FLOWS IN A POISSONIAN CITY

• 成果类型: Article
• 署名作者: Kendall, Wilfrid S.
• 署名单位: University of Warwick
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/10-AAP724
• 发表日期: 2011
• 页码: 801-842
• 关键词: limit-theorems; perpetuities; functionals; polygons; PARADOX

摘要

The stationary isotropic Poisson line process was used to derive upper bounds on mean excess network geodesic length in Aldous and Kendall [Adv. in Appl. Probab. 40 (2008) 1-21]. The current paper presents a study of the geometry and fluctuations of near-geodesics in the network generated by the line process. 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 behavior at the center 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.