Voronoi Percolation: Topological Stability and Giant Cycles

TL;DR

This paper investigates the topological stability of Voronoi percolation in higher dimensions, demonstrating how small parameter changes preserve topological features and establishing a phase transition for giant cycles.

Contribution

It extends existing theorems to higher dimensions and proves a sharp phase transition for the emergence of giant cycles in Voronoi percolation.

Findings

Discretization preserves topological properties with high probability

Generalization of Bollobás and Riordan's theorem to higher dimensions

Sharp phase transition for giant cycles in 2i-dimensional torus

Abstract

We study the topological stability of Voronoi percolation in higher dimensions. We show that slightly increasing p allows a discretization that preserves increasing topological properties with high probability. This strengthens a theorem of Bollob\'as and Riordan and generalizes it to higher dimensions. As a consequence, we prove a sharp phase transition for the emergence of i-dimensional giant cycles in Voronoi percolation on the 2i-dimensional torus.