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Vanishing uniqueness thresholds in Voronoi percolation on products

Matteo D'Achille, Jan Grebík, Ali Khezeli, Konstantin Recke, Amanda Wilkens

TL;DR

The paper develops a robust local-to-global framework for Poisson--Voronoi and Bernoulli--Voronoi percolation on non-amenable product spaces, proving that the uniqueness threshold $p_u(λ)$ vanishes as $λ\to0$ in a broad class of discrete and continuum settings. It introduces the unbounded borders phenomenon for ideal Voronoi tessellations and leverages it, together with product structure, to replace property (T) and obtain long-range order from local information. The main contributions include proving vanishing $p_u(λ)$ for k-fold tree products and for products of hyperbolic spaces and higher rank symmetric spaces (including cases without property (T)), plus new FIID sparse unique infinite cluster examples in both discrete and continuum contexts. These results yield new non-amenable Cayley graphs with FIID sparse infinite clusters and provide a flexible, broadly applicable method for establishing global connectivity from local data in complex geometric settings.

Abstract

We study Poisson--Voronoi percolation and its discrete analogue Bernoulli--Voronoi percolation in spaces with a non-amenable product structure. We develop a new method of proving smallness of the uniqueness threshold $p_u(λ)$ at small intensities $λ>0$ based on the unbounded borders phenomenon of their underlining ideal Poisson--Voronoi tessellation. We apply our method to several concrete examples in both the discrete and the continuum setting, including $k$-fold graph products of $d$-regular trees for $k\ge2,d\ge3$ and products of hyperbolic spaces $\mathbb H_{d_1}\times \ldots \times \mathbb H_{d_k}$ for $k\ge2, d_i\ge2$, complementing a recent result of the second and fourth author for symmetric spaces of connected higher rank semisimple real Lie groups with property (T). We also provide new examples of non-amenable Cayley graphs with the FIID sparse unique infinite cluster property, answering positively a recent question of Pete and Rokob.

Vanishing uniqueness thresholds in Voronoi percolation on products

TL;DR

The paper develops a robust local-to-global framework for Poisson--Voronoi and Bernoulli--Voronoi percolation on non-amenable product spaces, proving that the uniqueness threshold vanishes as in a broad class of discrete and continuum settings. It introduces the unbounded borders phenomenon for ideal Voronoi tessellations and leverages it, together with product structure, to replace property (T) and obtain long-range order from local information. The main contributions include proving vanishing for k-fold tree products and for products of hyperbolic spaces and higher rank symmetric spaces (including cases without property (T)), plus new FIID sparse unique infinite cluster examples in both discrete and continuum contexts. These results yield new non-amenable Cayley graphs with FIID sparse infinite clusters and provide a flexible, broadly applicable method for establishing global connectivity from local data in complex geometric settings.

Abstract

We study Poisson--Voronoi percolation and its discrete analogue Bernoulli--Voronoi percolation in spaces with a non-amenable product structure. We develop a new method of proving smallness of the uniqueness threshold at small intensities based on the unbounded borders phenomenon of their underlining ideal Poisson--Voronoi tessellation. We apply our method to several concrete examples in both the discrete and the continuum setting, including -fold graph products of -regular trees for and products of hyperbolic spaces for , complementing a recent result of the second and fourth author for symmetric spaces of connected higher rank semisimple real Lie groups with property (T). We also provide new examples of non-amenable Cayley graphs with the FIID sparse unique infinite cluster property, answering positively a recent question of Pete and Rokob.

Paper Structure

This paper contains 23 sections, 26 theorems, 139 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Vanishing uniqueness thresholds in products
  3. Remarks on the discrete case
  4. Applications to FIID sparse unique infinite clusters
  5. Bernoulli--Voronoi percolation on graphs
  6. Description of the model
  7. The uniqueness threshold
  8. Description of the continuum model
  9. From local to global uniqueness in products
  10. Non-amenable graph products
  11. Beyond Cayley graphs
  12. Products of symmetric spaces
  13. Poisson--Voronoi percolation on general products
  14. Local uniqueness for products of trees
  15. The space of distance-like functions
  16. ...and 8 more sections

Key Result

Theorem 1

Let $k\ge 2$, $d\ge 3$ and let $G\coloneqq\mathbb T_d\times\ldots\times \mathbb T_d$ be the direct product of $k$ copies of the $d$-regular tree $\mathbb{T}_d$. Then $p_u(\lambda)\to 0$ as $\lambda\to 0$.

Figures (1)

  • Figure 1: Portrait of a "good" event $A(\lambda,p,R,v)$.

Theorems & Definitions (56)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4: Products of symmetric spaces
  • Remark 5
  • Proposition 7
  • Theorem 8: Uniqueness and long-range order
  • Proposition 9: Phase transition of Voronoi percolation
  • Theorem 10
  • Theorem 11: Threshold for infinite clusters
  • ...and 46 more