Geodesics in planar Poisson roads random metric
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913425483890688 |
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| author | Blanc, Guillaume Curien, Nicolas Kahn, Jonas |
| author_facet | Blanc, Guillaume Curien, Nicolas Kahn, Jonas |
| contents | We study the structure of geodesics in the fractal random metric constructed by Kendall from a self-similar Poisson process of roads (i.e, lines with speed limits) in $\mathbb{R}^2$. In particular, we prove a conjecture of Kendall stating that geodesics do not pause en route, i.e, use roads of arbitrary small speed except at their endpoints. It follows that the geodesic frame of $\left(\mathbb{R}^2,T\right)$ is the set of points on roads. We also consider geodesic stars and hubs, and give a complete description of the local structure of geodesics around points on roads. Notably, we prove that leaving a road by driving off-road is never geodesic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07887 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geodesics in planar Poisson roads random metric Blanc, Guillaume Curien, Nicolas Kahn, Jonas Probability Metric Geometry We study the structure of geodesics in the fractal random metric constructed by Kendall from a self-similar Poisson process of roads (i.e, lines with speed limits) in $\mathbb{R}^2$. In particular, we prove a conjecture of Kendall stating that geodesics do not pause en route, i.e, use roads of arbitrary small speed except at their endpoints. It follows that the geodesic frame of $\left(\mathbb{R}^2,T\right)$ is the set of points on roads. We also consider geodesic stars and hubs, and give a complete description of the local structure of geodesics around points on roads. Notably, we prove that leaving a road by driving off-road is never geodesic. |
| title | Geodesics in planar Poisson roads random metric |
| topic | Probability Metric Geometry |
| url | https://arxiv.org/abs/2407.07887 |