Coverage of the unit cube by dynamic Boolean models
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915362184888320 |
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| author | Döring, Hanna de Jonge, Lianne Yang, Xiaochuan |
| author_facet | Döring, Hanna de Jonge, Lianne Yang, Xiaochuan |
| contents | Motivated by peer-to-peer telecommunication, we study a dynamic Boolean model. We define a Poisson number of random lines through the $(d-1)$-dimensional base of a $d$-dimensional unit cube and dilate them to define cylinders. Letting $ρ$ be the expected number of cylinders, the random variable of interest is the coverage radius $R_ρ$, which is the cylinder radius required to cover the $d$-dimensional unit cube. We show that $R_ρ^{d-1}$ is of the order $\log ρ/ ρ$ with high probability as $ρ$ tends to infinity.
We also consider alternative dynamics resulting in generalized cylinders that are generated by dilating the trajectories of stochastic processes, in particular Brownian motions. This leads to a coverage radius of the same order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22286 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coverage of the unit cube by dynamic Boolean models Döring, Hanna de Jonge, Lianne Yang, Xiaochuan Probability 60F05, 60D05, 60K35 Motivated by peer-to-peer telecommunication, we study a dynamic Boolean model. We define a Poisson number of random lines through the $(d-1)$-dimensional base of a $d$-dimensional unit cube and dilate them to define cylinders. Letting $ρ$ be the expected number of cylinders, the random variable of interest is the coverage radius $R_ρ$, which is the cylinder radius required to cover the $d$-dimensional unit cube. We show that $R_ρ^{d-1}$ is of the order $\log ρ/ ρ$ with high probability as $ρ$ tends to infinity. We also consider alternative dynamics resulting in generalized cylinders that are generated by dilating the trajectories of stochastic processes, in particular Brownian motions. This leads to a coverage radius of the same order. |
| title | Coverage of the unit cube by dynamic Boolean models |
| topic | Probability 60F05, 60D05, 60K35 |
| url | https://arxiv.org/abs/2506.22286 |