Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Intersections of Poisson k-flats in hyperbolic space: completing the picture

Tillmann Bühler, Daniel Hug

Abstract

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 $2k \geq d+1$.

Intersections of Poisson k-flats in hyperbolic space: completing the picture

Abstract

In recent years there has been a lot of interest in the study of isometry invariant Poisson processes of -flats in -dimensional hyperbolic space , for . A phenomenon that has no counterpart in euclidean geometry arises in the investigation of the total -dimensional volume of the process inside a spherical observation window of radius when one lets tend to infinity. While is asymptotically normally distributed for , it has been shown to obey a nonstandard central limit theorem for . The intersection process of order , for , of the original process consists of all intersections of distinct flats with . For this intersection process, the total -dimensional volume of the process in , again as , is of particular interest. For it has been shown that is again asymptotically normally distributed. For , the limit is so far unknown, although it has been shown for certain and that it cannot be a normal distribution. We determine the limit distribution for all values of . 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 has full rank when and rank one when .

Paper Structure

This paper contains 12 sections, 14 theorems, 120 equations, 1 figure.

Table of Contents

  1. MSC-classes 2020.
  2. Keywords.
  3. Introduction
  4. Preliminaries
  5. Comparison with the euclidean case
  6. Determination of the limit distribution
  7. Convergence rates
  8. Bounds on g and g_r
  9. Proof of Theorem \ref{['thm:convergence_rates_1']}
  10. Convergence of intersection processes
  11. Numerical calculations
  12. Asymptotic covariance matrix

Key Result

Lemma 1

If $r\geq 1$, $k\in\{0,\ldots,d-1\}$ and $i\in\{1,\ldots,m\}$, then

Figures (1)

  • Figure 1: Left: Real part (red) and imaginary part (blue) of $\widetilde{\psi}_{d,k}$. The dotted line is the (entirely real) characteristic function of a standard normal distribution. Right: Density of $Z_{d,k}^*$ (red) and density of a standard normal distribution (dotted).

Theorems & Definitions (27)

  • Lemma 1
  • Proposition 2
  • proof
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 17 more