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

TL;DR

This paper studies the high-dimensional behavior of the total volume of intersections of hyperbolic Poisson k-planes, revealing conditions under which the distribution converges to Gaussian or non-Gaussian limits as the dimension grows.

Contribution

It provides the first detailed analysis of the asymptotic distributional limits of hyperbolic Poisson k-plane processes in high dimensions, including explicit conditions for Gaussian and non-Gaussian limits.

Findings

Convergence to Gaussian law when the codimension grows fast.

Non-Gaussian infinitely divisible limits for fixed codimension.

Precise conditions for variance-normalized sequence convergence.

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 L\'evy measures yields an explicit non-Gaussian infinitely divisible limit for fixed codimension $d-k$ and a standard Gaussian limit for $d-k \to \infty$.