Fluctuations of λ-geodesic Poisson hyperplanes in hyperbolic space

Abstract Poisson processes of so-called λ -geodesic hyperplanes in d -dimensional hyperbolic space are studied for 0 ≤ λ ≤ 1. The case λ = 0 corresponds to genuine geodesic hyperplanes, the case λ = 1 to horospheres and λ ∈ (0, 1) to λ -equidistants. In the focus are the fluctuations of the centred and normalized total surface area of the union of all λ -geodesic hyperplanes in the Poisson process within a hyperbolic ball of radius R centred at some fixed point, as R → ∞. It is shown that for λ < 1 these random variables satisfy a quantitative central limit theorem precisely for d = 2 and d = 3. The exact form of the non-Gaussian, infinitely divisible limiting distribution is determined for all higher space dimensions d ≥ 4. The special case λ = 1 is in sharp contrast to this behaviour. In fact, for the total surface area of Poisson processes of horospheres, a non-standard central limit theorem with limiting variance 1/2 is established for all space dimensions d ≥ 2. We discuss the analogy between the problem studied here and the Random Energy Model whose partition function exhibits a similar structure of possible limit laws.