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

We consider a stationary Poisson process of k-planes in the d-dimensional hyperbolic space Hd of constant curvature −1, with d≥4 and 1≤k≤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 a non-Gaussian infinitely divisible random variable Zd,k whenever 2k>d+1. We investigate the distributional behaviour of Zd,k in the high-dimensional regime d→∞ 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→∞.