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

TL;DR

This paper investigates the asymptotic behavior of intersection volumes of Poisson k-flats in hyperbolic space, determining their limit distributions, convergence rates, and covariance properties for all parameter ranges.

Contribution

It provides the first complete characterization of the limit distributions for intersection volumes of Poisson k-flats in hyperbolic space across all parameters.

Findings

Limit distributions are fully characterized for all d,k,m.

Explicit convergence rates in the Kolmogorov distance are established.

Asymptotic covariance matrices have full rank or rank one depending on parameters.

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 $\eta$ consists of all intersections of distinct flats $E_1,\ldots,E_m \in \eta$ 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 $2k \geq d+1$.