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

TL;DR

This paper analyzes the second volume moment of hyperbolic Poisson zero cells, revealing a phase transition at a critical intensity and deriving exact formulas and asymptotics using hyperbolic harmonic analysis.

Contribution

It provides the first exact expression for the second volume moment of hyperbolic Poisson zero cells and characterizes its divergence at the critical phase transition.

Findings

Second moment diverges at rate R^3 at criticality in any dimension.

Full second moment is finite in the supercritical regime.

Derived an exact formula involving Meijer G-function for the second moment.

Abstract

We investigate the second volume moment of the zero cell $Z_o$ of a Poisson hyperplane tessellation with intensity $\gamma$ in the $d$-dimensional hyperbolic space. We focus on the phase transition at the critical intensity $\gamma_c^{(d)}$, the minimum value for which $Z_o$ is almost surely bounded. In the critical regime $\gamma=\gamma_c^{(d)}$, we show that the second volume moment of the restricted zero cell $Z_o \cap B_R$, where $B_R$ is a hyperbolic ball of radius $R$ centred at $o$, diverges in any dimension at the universal rate $R^3$ as $R \to \infty$. In the supercritical case $\gamma > \gamma_c^{(d)}$, 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 $\gamma \to \infty$ and as $\gamma \downarrow \gamma_c^{(d)}$, facilitating a direct comparison with the corresponding Euclidean values as well as the mean-field universality class of percolation theory.