On the visibility window for Brownian interlacements, Poisson cylinders and Boolean models
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909660783575040 |
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| author | Mu, Yingxin Sapozhnikov, Artem |
| author_facet | Mu, Yingxin Sapozhnikov, Artem |
| contents | We study visibility inside the vacant set of three models in $\mathbb R^d$ with slow decay of spatial correlations: Brownian interlacements, Poisson cylinders and Poisson-Boolean models. Let $Q_x$ be the radius of the largest ball centered at $x$ every point of which is visible from $0$ through the vacant set of one of these models. We prove that conditioned on $x$ being visible from $0$, $Q_x/δ_{\|x\|}$ converges weakly, as $x\to\infty$, to the exponential distribution with an explicit intensity, which depends on the parameters of the respective model. The scaling function $δ_r$ is the visibility window introduced in arXiv:2304.10298, a length scale of correlations in the visible set at distance $r$ from $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21516 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the visibility window for Brownian interlacements, Poisson cylinders and Boolean models Mu, Yingxin Sapozhnikov, Artem Probability We study visibility inside the vacant set of three models in $\mathbb R^d$ with slow decay of spatial correlations: Brownian interlacements, Poisson cylinders and Poisson-Boolean models. Let $Q_x$ be the radius of the largest ball centered at $x$ every point of which is visible from $0$ through the vacant set of one of these models. We prove that conditioned on $x$ being visible from $0$, $Q_x/δ_{\|x\|}$ converges weakly, as $x\to\infty$, to the exponential distribution with an explicit intensity, which depends on the parameters of the respective model. The scaling function $δ_r$ is the visibility window introduced in arXiv:2304.10298, a length scale of correlations in the visible set at distance $r$ from $0$. |
| title | On the visibility window for Brownian interlacements, Poisson cylinders and Boolean models |
| topic | Probability |
| url | https://arxiv.org/abs/2506.21516 |