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

Abstract

Let $\eta$ be an isometry invariant Poisson process of $k$-flats, $0\le k\le d-1$, in $d$-dimensional hyperbolic space. For $d-m(d-k)\ge 0$, the $m$-th order intersection process of $\eta$ consists of all nonempty intersections of distinct flats $E_1,\dots ,E_m\in \eta$. Of particular interest is the total volume $F_r^{\left(m\right)}$ of this intersection process in a ball of radius $r$. For $2k>d+1$, we determine the asymptotic distribution of $F_r^{\left(m\right)}$, as $r\to \infty$, previously known only for $m=1$, and derive rates of convergence in the Kolmogorov distance. Properties of the non-Gaussian limit distribution are discussed. We further study the asymptotic covariance matrix of the vector ${(F_r^{\left(1\right)},\dots ,F_r^{\left(m\right)})}^{\top }$.