The Annals of Probability

Improper Poisson line process as SIRSN in any dimension

Jonas Kahn

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Abstract

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.

Article information

Source
Ann. Probab., Volume 44, Number 4 (2016), 2694-2725.

Dates
Received: March 2015
Revised: April 2015
First available in Project Euclid: 2 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1470139152

Digital Object Identifier
doi:10.1214/15-AOP1032

Mathematical Reviews number (MathSciNet)
MR3531678

Zentralblatt MATH identifier
1366.60027

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 90B15: Network models, stochastic 51F99: None of the above, but in this section 60G55: Point processes

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

Citation

Kahn, Jonas. Improper Poisson line process as SIRSN in any dimension. Ann. Probab. 44 (2016), no. 4, 2694--2725. doi:10.1214/15-AOP1032. https://projecteuclid.org/euclid.aop/1470139152


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References

  • Aldous, D. (2014). Scale-invariant random spatial networks. Electron. J. Probab. 19 no. 15, 41.
  • Aldous, D. and Ganesan, K. (2013). True scale-invariant random spatial networks. Proc. Natl. Acad. Sci. USA 110 8782–8785.
  • Aldous, D. J. and Kendall, W. S. (2008). Short-length routes in low-cost networks via Poisson line patterns. Adv. in Appl. Probab. 40 1–21.
  • Kalapala, V., Sanwalani, V., Clauset, A. and Moore, C. (2006). Scale invariance in road networks. Phys. Rev. E (3) 73 026130.
  • Kendall, W. S. (2015). From random lines to metric spaces. Ann. Probab. To appear.
  • Le Gall, J. F. (2014). Random geometry on the sphere. Preprint. Available at arXiv:1403.7943.
  • Møller, J. and Waagepetersen, R. P. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall, Boca Raton.
  • Slivnyak, I. M. (1962). Some properties of stationary flows of homogeneous random events. Theory Probab. Appl. 7 336–341.
  • Stoyan, D., Kendall, W. S. and Mecke, J. (1996). Stochastic Geometry and Its Applications. Wiley, New York.