Critical Poisson hyperplane percolation in hyperbolic space has no unbounded cells

TL;DR

This paper proves that in hyperbolic space, Poisson hyperplane tessellations do not have unbounded cells at the critical intensity, extending previous results and fully characterizing the phase transition of the model.

Contribution

It extends the understanding of hyperbolic Poisson hyperplane percolation by establishing the absence of unbounded cells at criticality in dimensions three and higher.

Findings

No unbounded cells at critical intensity in hyperbolic space.

Existence of infinitely many unbounded cells below critical intensity.

Complete phase transition description for the model.

Abstract

We show that tessellations of hyperbolic space by isometry-invariant Poisson processes of $(d-1)$-dimensional hyperplanes do not have an unbounded cell at the critical intensity. This extends a result by Porret-Blanc for the hyperbolic plane (C. R. Acad. Sci. Paris, Ser. I, Vol. 344 (2007)) to dimensions $d\ge3$. We also show that for intensities strictly below the critical intensity, infinitely many unbounded cells exist, while for intensities larger than or equal to the critical intensity, no unbounded cell exists. This completely describes the basic phase transition of this continuum percolation model. Our proof uses a method from discrete percolation theory which we adapt to the continuum and combine with specific computations for Poisson hyperplane processes.