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 → ∞.