Seeing Through Hyperbolic Space: Visibility for $λ$-Geodesic Hyperplanes
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918368549797888 |
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| author | Kabluchko, Zakhar Mattutat, Vanessa Thaele, Christoph |
| author_facet | Kabluchko, Zakhar Mattutat, Vanessa Thaele, Christoph |
| contents | We study visibility from a fixed point in the presence of a Poisson process of $λ$--geodesic hyperplanes in a $d$-dimensional hyperbolic space. The family of $λ$--geodesic hyperplanes interpolates between totally geodesic hyperplanes and horospheres. Our main result establishes a universality principle for this model: we prove that the fundamental visibility properties are invariant with respect to the parameter $λ\in[0,1]$. Namely, there is a critical intensity $γ_{\mathrm{crit}}>0$ such that the visible region is unbounded with positive probability for $γ< γ_{\mathrm{crit}}$ and almost surely bounded for $γ> γ_{\mathrm{crit}}$. For $d=2$ we establish almost sure boundedness also at criticality. The value for $γ_{\mathrm{crit}}$ is explicit and does not depend on $λ$. In the bounded phase, we show that the mean visible volume is identical with the known formula for $λ=0$. The key integral-geometric step is an explicit computation showing that the measure of $λ$-geodesic hyperplanes hitting a geodesic segment is a linear function of the length of the segment, independent of~$λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20935 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Seeing Through Hyperbolic Space: Visibility for $λ$-Geodesic Hyperplanes Kabluchko, Zakhar Mattutat, Vanessa Thaele, Christoph Probability Metric Geometry 51M10, 52A22, 53C65, 60D05 We study visibility from a fixed point in the presence of a Poisson process of $λ$--geodesic hyperplanes in a $d$-dimensional hyperbolic space. The family of $λ$--geodesic hyperplanes interpolates between totally geodesic hyperplanes and horospheres. Our main result establishes a universality principle for this model: we prove that the fundamental visibility properties are invariant with respect to the parameter $λ\in[0,1]$. Namely, there is a critical intensity $γ_{\mathrm{crit}}>0$ such that the visible region is unbounded with positive probability for $γ< γ_{\mathrm{crit}}$ and almost surely bounded for $γ> γ_{\mathrm{crit}}$. For $d=2$ we establish almost sure boundedness also at criticality. The value for $γ_{\mathrm{crit}}$ is explicit and does not depend on $λ$. In the bounded phase, we show that the mean visible volume is identical with the known formula for $λ=0$. The key integral-geometric step is an explicit computation showing that the measure of $λ$-geodesic hyperplanes hitting a geodesic segment is a linear function of the length of the segment, independent of~$λ$. |
| title | Seeing Through Hyperbolic Space: Visibility for $λ$-Geodesic Hyperplanes |
| topic | Probability Metric Geometry 51M10, 52A22, 53C65, 60D05 |
| url | https://arxiv.org/abs/2602.20935 |