The volume of hyperbolic Poisson zero cells: critical divergence and exact second moment

We investigate the second volume moment of the zero cell Zₒ of a Poisson hyperplane tessellation with intensity $γ$ in the d-dimensional hyperbolic space. We focus on the phase transition at the critical intensity γc⁽ᵈ⁾, the minimum value for which Zₒ is almost surely bounded. In the critical regime γ=γc⁽ᵈ⁾, we show that the second volume moment of the restricted zero cell Zₒ ∩ BR, where BR is a hyperbolic ball of radius R centred at o, diverges in any dimension at the universal rate R³ as R → ∞. In the supercritical case γ> γc⁽ᵈ⁾, we prove that the full second volume moment is finite. Using tools from harmonic analysis in hyperbolic space, we derive an exact expression for this moment in terms of the Meijer G-function. Furthermore, we determine the asymptotic behaviour of the second moment as γ→ ∞ and as γ↓ γc⁽ᵈ⁾, facilitating a direct comparison with the corresponding Euclidean values as well as the mean-field universality class of percolation theory.