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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.10708 |
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| _version_ | 1866913917995843584 |
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| author | Sönmez, Ercan Spanos, Panagiotis Thäle, Christoph |
| author_facet | Sönmez, Ercan Spanos, Panagiotis Thäle, Christoph |
| contents | Consider the $d$-dimensional hyperbolic space $\mathbb{M}_K^d$ of constant curvature $K<0$ and fix a point $o$ playing the role of an origin. Let $\mathbf{L}$ be a uniform random $q$-dimensional totally geodesic submanifold (called $q$-flat) in $\mathbb{M}_K^d$ passing through $o$ and, independently of $\mathbf{L}$, let $\mathbf{E}$ be a random $(d-q+γ)$-flat in $\mathbb{M}_K^d$ which is uniformly distributed in the set of all $(d-q+γ)$-flats intersecting a hyperbolic ball of radius $u>0$ around $o$. We are interested in the distribution of the random $γ$-flat arising as the intersection of $\mathbf{E}$ with $\mathbf{L}$. In contrast to the Euclidean case, the intersection $\mathbf{E}\cap \mathbf{L}$ can be empty with strictly positive probability. We determine this probability and the full distribution of $\mathbf{E}\cap \mathbf{L}$. Thereby, we elucidate crucial differences to the Euclidean case. Moreover, we study the limiting behaviour as $d\uparrow\infty$ and also $K\uparrow 0$. Thereby we obtain a phase transition with three different phases which we completely characterize, including a critical phase with distinctive behavior and a phase recovering the Euclidean results. In the background are methods from hyperbolic integral geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10708 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Intersection probabilities for flats in hyperbolic space Sönmez, Ercan Spanos, Panagiotis Thäle, Christoph Probability Consider the $d$-dimensional hyperbolic space $\mathbb{M}_K^d$ of constant curvature $K<0$ and fix a point $o$ playing the role of an origin. Let $\mathbf{L}$ be a uniform random $q$-dimensional totally geodesic submanifold (called $q$-flat) in $\mathbb{M}_K^d$ passing through $o$ and, independently of $\mathbf{L}$, let $\mathbf{E}$ be a random $(d-q+γ)$-flat in $\mathbb{M}_K^d$ which is uniformly distributed in the set of all $(d-q+γ)$-flats intersecting a hyperbolic ball of radius $u>0$ around $o$. We are interested in the distribution of the random $γ$-flat arising as the intersection of $\mathbf{E}$ with $\mathbf{L}$. In contrast to the Euclidean case, the intersection $\mathbf{E}\cap \mathbf{L}$ can be empty with strictly positive probability. We determine this probability and the full distribution of $\mathbf{E}\cap \mathbf{L}$. Thereby, we elucidate crucial differences to the Euclidean case. Moreover, we study the limiting behaviour as $d\uparrow\infty$ and also $K\uparrow 0$. Thereby we obtain a phase transition with three different phases which we completely characterize, including a critical phase with distinctive behavior and a phase recovering the Euclidean results. In the background are methods from hyperbolic integral geometry. |
| title | Intersection probabilities for flats in hyperbolic space |
| topic | Probability |
| url | https://arxiv.org/abs/2407.10708 |