## The Annals of Probability

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

### Improper Poisson line process as SIRSN in any dimension

#### 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