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.