Intersections of Poisson k-flats in hyperbolic space: completing the picture
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2024
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| _version_ | 1866912072426586112 |
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| author | Bühler, Tillmann Hug, Daniel |
| author_facet | Bühler, Tillmann Hug, Daniel |
| contents | In recent years there has been a lot of interest in the study of isometry invariant Poisson processes of $k$-flats in $d$-dimensional hyperbolic space $\mathbb{H}^d$, for $0\le k\le d-1$. A phenomenon that has no counterpart in euclidean geometry arises in the investigation of the total $k$-dimensional volume $F_r$ of the process inside a spherical observation window $B_r$ of radius $r$ when one lets $r$ tend to infinity. While $F_r$ is asymptotically normally distributed for $2k\leq d+1$, it has been shown to obey a nonstandard central limit theorem for $2k>d+1$.
The intersection process of order $m$, for $d-m(d-k) \geq 0$, of the original process $η$ consists of all intersections of distinct flats $E_1,\ldots,E_m \in η$ with $\dim(E_1\cap\ldots\cap E_m) = d-m(d-k)$. For this intersection process, the total $d-m(d-k)$-dimensional volume $F^{(m)}_r$ of the process in $B_r$, again as $r \to \infty$, is of particular interest. For $2k \leq d+1$ it has been shown that $F^{(m)}_r$ is again asymptotically normally distributed. For $m \geq 2$, the limit is so far unknown, although it has been shown for certain $d$ and $k$ that it cannot be a normal distribution.
We determine the limit distribution for all values of $d,k,m$. In addition, we establish explicit rates of convergence in the Kolmogorov distance and discuss properties of the limit distribution. Furthermore we show that the asymptotic covariance matrix of the vector $(F^{(1)}_r,\ldots,F^{(m)}_r)^\top$ has full rank when $2k < d+1$ and rank one when $2k \geq d+1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_10413 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Intersections of Poisson k-flats in hyperbolic space: completing the picture Bühler, Tillmann Hug, Daniel Probability Metric Geometry Primary: 60D05, 53C65, 52A22, Secondary: 52A55, 60F05 In recent years there has been a lot of interest in the study of isometry invariant Poisson processes of $k$-flats in $d$-dimensional hyperbolic space $\mathbb{H}^d$, for $0\le k\le d-1$. A phenomenon that has no counterpart in euclidean geometry arises in the investigation of the total $k$-dimensional volume $F_r$ of the process inside a spherical observation window $B_r$ of radius $r$ when one lets $r$ tend to infinity. While $F_r$ is asymptotically normally distributed for $2k\leq d+1$, it has been shown to obey a nonstandard central limit theorem for $2k>d+1$. The intersection process of order $m$, for $d-m(d-k) \geq 0$, of the original process $η$ consists of all intersections of distinct flats $E_1,\ldots,E_m \in η$ with $\dim(E_1\cap\ldots\cap E_m) = d-m(d-k)$. For this intersection process, the total $d-m(d-k)$-dimensional volume $F^{(m)}_r$ of the process in $B_r$, again as $r \to \infty$, is of particular interest. For $2k \leq d+1$ it has been shown that $F^{(m)}_r$ is again asymptotically normally distributed. For $m \geq 2$, the limit is so far unknown, although it has been shown for certain $d$ and $k$ that it cannot be a normal distribution. We determine the limit distribution for all values of $d,k,m$. In addition, we establish explicit rates of convergence in the Kolmogorov distance and discuss properties of the limit distribution. Furthermore we show that the asymptotic covariance matrix of the vector $(F^{(1)}_r,\ldots,F^{(m)}_r)^\top$ has full rank when $2k < d+1$ and rank one when $2k \geq d+1$. |
| title | Intersections of Poisson k-flats in hyperbolic space: completing the picture |
| topic | Probability Metric Geometry Primary: 60D05, 53C65, 52A22, Secondary: 52A55, 60F05 |
| url | https://arxiv.org/abs/2410.10413 |