Poisson-Voronoi percolation in the hyperbolic plane with small intensities

TL;DR

This paper investigates percolation on Poisson-Voronoi tessellations in the hyperbolic plane, revealing that the critical probability approaches a specific asymptotic value as the intensity diminishes, thus answering a longstanding question.

Contribution

It establishes the asymptotic behavior of the critical percolation probability for small intensities in hyperbolic Poisson-Voronoi tessellations, addressing a question posed by Benjamini and Schramm.

Findings

Critical probability asymptotically equals πλ/3 as λ→0

Provides a rigorous answer to an open question in hyperbolic percolation

Enhances understanding of geometric percolation in non-Euclidean spaces

Abstract

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster is asymptotically equal to $\pi \lambda/3$ as $\lambda\to0.$ This answers a question of Benjamini and Schramm.