High-dimensional limits arising from hyperbolic Poisson k-plane processes

Abstract

We consider a stationary Poisson process of $k$-planes in the $d$-dimensional hyperbolic space ${\mathbb{H}}^d$ of constant curvature $-1$, with $d\ge 4$ and $1\le k\le d-1$. It is known that, after centring and normalization, the total $k$-volume of all intersections of $k$-planes with a geodesic ball of radius $R$ converges in distribution, as $R\to \infty$, to a non-Gaussian infinitely divisible random variable $Z_{d,k}$ whenever $2k>d+1$. We investigate the distributional behaviour of $Z_{d,k}$ in the high-dimensional regime $d\to \infty$ and depending on how fast $k$ grows in relation to $d$. We derive precise conditions for the variance normalized sequence to converge in law to a standard Gaussian random variable or to a degenerate law, respectively, and show that an alternative rescaling of the Lévy measures yields an explicit non-Gaussian infinitely divisible limit for fixed codimension $d-k$ and a standard Gaussian limit for $d-k\to \infty$.