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July 2016 Improper Poisson line process as SIRSN in any dimension
Jonas Kahn
Ann. Probab. 44(4): 2694-2725 (July 2016). DOI: 10.1214/15-AOP1032


Aldous has introduced a notion of scale-invariant random spatial network (SIRSN) as a mathematical formalization of road networks. Intuitively, those are random processes that assign a route between each pair of points in Euclidean space, while being invariant under rotation, translation, and change of scale, and such that the routes are not too long and mainly lie on “main roads”.

The only known example was somewhat artificial since invariance had to be added using randomization at the end of the construction. We prove that the network of geodesics in the random metric space generated by a Poisson line process marked by speeds according to a power law is a SIRSN, in any dimension.

Along the way, we establish bounds comparing Euclidean balls and balls for the random metric space. We also prove that in dimension more than two, the geodesics have “many directions” near each point where they are not straight.


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Jonas Kahn. "Improper Poisson line process as SIRSN in any dimension." Ann. Probab. 44 (4) 2694 - 2725, July 2016.


Received: 1 March 2015; Revised: 1 April 2015; Published: July 2016
First available in Project Euclid: 2 August 2016

zbMATH: 1366.60027
MathSciNet: MR3531678
Digital Object Identifier: 10.1214/15-AOP1032

Primary: 60D05
Secondary: 51F99 , 60G55 , 90B15

Keywords: $\Pi$-geodesic , many directions , Poisson line process , Random metric space , scale-invariant random spatial network , SIRSN , spatial network , Stochastic geometry

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 4 • July 2016
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