High-intensity Voronoi percolation on manifolds

Abstract

We study Voronoi percolation on a large class of $d$-dimensional Riemannian manifolds, which includes the hyperbolic spaces ${\mathbb{H}}^d$, $d\ge 2$. We prove that as the intensity $\hat{I}»$ of the underlying Poisson point process tends to infinity, both critical parameters $p_c(M,\hat{I}»)$ and $p_u(M,\hat{I}»)$ converge to the Euclidean critical parameter $p_c({\mathbb{R}}^d)$. This extends a recent result of Hansen & Müller in the special case $M={\mathbb{H}}^2$ to a general class of manifolds of arbitrary dimension. A crucial step in our proof, which may be of independent interest, is to show that if $M$ is simply connected and one-ended, then embedded graphs induced by a general class of tessellations on $M$ have connected minimal cutsets. In particular, this result applies to $\epsilon$-nets, allowing us to implement a "fine-graining" argument.

42 pages, 3 figures. Characterization of uniqueness removed due to a mistake