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Voronoi Percolation: Topological Stability and Giant Cycles

Benjamin Schweinhart, Morgan Shuman

TL;DR

This work advances the understanding of Voronoi percolation by establishing a finite-volume, higher-dimensional sharp threshold for the emergence of giant homological cycles on the $2i$-torus, showing that a modest increase in $p$ enables a discretization that preserves key topological events with high probability. The authors develop a robust coupling between continuous Voronoi percolation and a discrete model, leveraging Delaunay stability results and a Friedgut–Kalai–style sharp-threshold framework to transfer the phase transition to the continuous setting. Central contributions include a generalized stability lemma for topological properties under discretization, a detailed analysis of δ-bad and δ-unstable configurations, and a complete argument that ties primal/dual homology events to a universal $p_c=1/2$ threshold in the appropriate dimension. The results deepen the connection between geometric random complexes and topological phase transitions, with implications for understanding high-dimensional percolation phenomena on tori and related random complexes.

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 Bollobás 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.

Voronoi Percolation: Topological Stability and Giant Cycles

TL;DR

This work advances the understanding of Voronoi percolation by establishing a finite-volume, higher-dimensional sharp threshold for the emergence of giant homological cycles on the -torus, showing that a modest increase in enables a discretization that preserves key topological events with high probability. The authors develop a robust coupling between continuous Voronoi percolation and a discrete model, leveraging Delaunay stability results and a Friedgut–Kalai–style sharp-threshold framework to transfer the phase transition to the continuous setting. Central contributions include a generalized stability lemma for topological properties under discretization, a detailed analysis of δ-bad and δ-unstable configurations, and a complete argument that ties primal/dual homology events to a universal threshold in the appropriate dimension. The results deepen the connection between geometric random complexes and topological phase transitions, with implications for understanding high-dimensional percolation phenomena on tori and related random complexes.

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 Bollobás 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.
Paper Structure (20 sections, 43 theorems, 64 equations, 6 figures)

This paper contains 20 sections, 43 theorems, 64 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Delaunay and Voronoi
  4. A crash course in homology
  5. Voronoi Percolation
  6. Homological Percolation and Discretization
  7. Probabilistic tools
  8. Paper outline
  9. Asymptotics of bad points
  10. Delaunay Triangulations
  11. Poisson-Delaunay preliminaries
  12. Unstable points and bad clusters
  13. Topology of Voronoi Cells
  14. Background
  15. Stability of Voronoi cells
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\epsilon_0 > 0$ and set $\delta_0 = N^{-\epsilon_0}$. Then there exists a coupling of $\left( Z_1, R_1(p) \right)$ with $\left( Z_2, R_2(p + \epsilon) \right)$ so that there is a $\delta_0$-good subset $W\subset Z_2$ so that $W\subset R_2$ and with high probability.

Figures (6)

  • Figure 1: Voronoi percolation at $p = 0.5$.
  • Figure 2: A giant cycle in $2$-dimensional plaquette percolation. Taken from duncan2025homological.
  • Figure 3: A percolation where $A$ occurs, but not $S$. Both the red and white Voronoi cells contains a giant $1$-cycle. The giant $1$-cycle in the red cells is highlighted.
  • Figure 4: A Poisson-Voronoi mosaic
  • Figure 5: Voronoi diagram with Delaunay complex superimposed.
  • ...and 1 more figures

Theorems & Definitions (101)

  • Theorem 1
  • Theorem 2
  • Definition 3: Polytope and Polyhedral complex
  • Definition 4: Simplex and simplicial complex
  • Definition 5: Delaunay ball
  • Definition 6: Delaunay triangulation
  • Lemma 7
  • proof
  • Definition 8: Star
  • Definition 9: $n$-chain
  • ...and 91 more