Bernoulli–Voronoi Percolation

• Bernoulli–Voronoi percolation is a continuum model that combines Poisson–Voronoi tessellations with independent Bernoulli site percolation to analyze infinite clusters.
• It reveals critical behavior where thresholds for percolation (pₙ and pᵤ) vary with the underlying geometry, exhibiting fixed values in Euclidean spaces and variable limits in hyperbolic or non-amenable settings.
• Methodological innovations such as local Euclidean coupling, ε-net fine graining, and sharp phase transition analysis provide insights into connectivity and cluster uniqueness.

Bernoulli–Voronoi percolation is a continuum percolation model that combines the random spatial structure of the Poisson–Voronoi tessellation with independent Bernoulli site percolation. In this setting, one studies the emergence and properties of infinite clusters in randomly colored Voronoi tessellations constructed over a broad class of ambient metric spaces, including Euclidean spaces, Riemannian manifolds, and non-amenable structures such as products of trees and hyperbolic spaces. The primary control parameters are the Poisson intensity $\lambda$ and the Bernoulli coloring probability $p$, leading to the paper of critical phenomena and uniqueness issues for unbounded connected components as these parameters vary.

1. Construction of the Bernoulli–Voronoi Percolation Model

Let $(M,d,\mu)$ denote a proper geodesic metric space or a Riemannian manifold (with the associated metric $d_M$ and measure $\mu_M$). The construction proceeds as follows:

• Poisson Sampling: A Poisson point process $\eta_\lambda$ is realized on $M$ with intensity measure $\lambda\mu_M$, where each Borel set $A\subset M$ receives a Poisson-distributed number of points with mean $\lambda\mu_M(A)$, independently for disjoint sets.
• Voronoi Tessellation: Each point $x\in\eta_\lambda$ generates a cell

$$C(x;\eta_\lambda)=\{y\in M : d_M(y,x)\leq d_M(y,\eta_\lambda\setminus\{x\})\}.$$

For manifolds with bounded sectional curvatures and positive injectivity radius, these cells are compact and partition $M$ up to null sets.

• Bernoulli Coloring: Each cell $C(x)$ is independently colored “white” with probability $p$ and “black” with probability $1-p$.
• Clusters: The union of cells of a given color forms a random closed set, with connectivity defined via adjacency of Voronoi cells (i.e., cells sharing a boundary).

The critical percolation threshold $p_c(M,\lambda)$ is the infimum of $p$ for which there exists almost surely at least one unbounded white cluster. The uniqueness threshold $p_u(M,\lambda)$ is the infimum $p$ for which almost surely there is exactly one unbounded white cluster.

2. Critical and Uniqueness Thresholds across Geometries

Euclidean Case

In $\mathbb{R}^d$, by scaling and self-duality, the critical parameter for planar Voronoi percolation is exactly $p_c(\mathbb{R}^2)=1/2$ and does not depend on $\lambda$ (Ahlberg et al., 2017).

Hyperbolic and Non-Euclidean Geometries

In spaces with nonzero (specifically negative) curvature, such as hyperbolic spaces $\mathbb{H}^d$, the critical threshold $p_c(\lambda)$ exhibits strong dependence on both geometry and Poisson intensity:

• In the hyperbolic plane, as $\lambda\searrow 0$,

$p_c(\lambda)\sim \frac{\pi}{3}\lambda$

(Hansen et al., 2021).

• As $\lambda\to\infty$, $p_c(\mathbb{H}^2,\lambda)\to1/2$, converging to the Euclidean threshold (Hansen et al., 2020, Bühler et al., 27 Mar 2025).
• For general Riemannian manifolds with suitable geometric properties (simply connected, one-ended, curvature bounds, global bi-infinite log-expanding path), both $$p_c(M,\lambda), p_u(M,\lambda)\to p_c(\mathbb{R}^d)$$ as $\lambda\to\infty$ (Bühler et al., 27 Mar 2025).

Non-Amenable Product Geometries

In product spaces with non-amenable geometry, notably products of regular trees or products of hyperbolic spaces, an anomalous phenomenon occurs:

• The uniqueness threshold $p_u(\lambda)$ vanishes as $\lambda\to0$:

$$\lim_{\lambda\to0} p_u(\lambda) = 0\quad\text{for }\,M=\mathbb H_{d_1}\times\cdots\times\mathbb H_{d_k},\,k\ge2$$

(D'Achille et al., 28 Nov 2025). This is intimately connected to the unbounded borders phenomenon of the ideal Poisson–Voronoi tessellation (IPVT) at low intensity, in which every pair of distinct cells shares unbounded common boundaries, resulting in strong local merging of clusters and vanishing uniqueness thresholds.

3. Main Locality and Asymptotic Theorems

The convergence of percolation thresholds in high-intensity regimes is captured by the following result:

High-Intensity Locality Theorem (Bühler et al., 27 Mar 2025): If $M$ is a simply connected, one-ended Riemannian manifold with bounded sectional curvature, positive global injectivity radius, and a log-expanding bi-infinite path, then

$$\lim_{\lambda\to\infty} p_c(M,\lambda) = p_c(\mathbb{R}^d),\quad \lim_{\lambda\to\infty} p_u(M,\lambda) = p_c(\mathbb{R}^d).$$

Two-sided error estimates of order $\varepsilon(\lambda)\to0$ as $\lambda\to\infty$ apply.

Uniqueness Characterization (Bühler et al., 27 Mar 2025, D'Achille et al., 28 Nov 2025): For many settings, the uniqueness threshold admits a connectivity characterization:

$$p_u(\lambda) = \inf\left\{p : \inf_{x,y\in M} \mathbb{P}_p^{(\lambda)}(x \leftrightarrow y) > 0\right\}.$$

This “long-range order” (LRO) criterion is fundamental in distinguishing uniqueness from mere existence of infinite clusters.

In non-amenable products such as graph products of trees or products of hyperbolic spaces, local uniqueness at small $\lambda$ propagates to global uniqueness as a function of local cluster merging—enabled by the unbounded borders of the IPVT—yielding $p_u(\lambda)\to0$ (D'Achille et al., 28 Nov 2025).

4. Methodological Innovations and Proof Strategies

A suite of techniques underpins the analysis of Bernoulli–Voronoi percolation:

• Local Euclidean Coupling: On small scales, Riemannian metrics are $C^2$-close to Euclidean, allowing coupling between Poisson–Voronoi tessellations on $M$ and $\mathbb{R}^d$ to transfer local percolation behavior (Bühler et al., 27 Mar 2025).
• Fine-Graining via $\varepsilon$-nets: The use of $\varepsilon$-nets and corresponding connectivity graphs enables propagation of local crossing/non-crossing events to global percolation, even without global periodic tilings (Bühler et al., 27 Mar 2025).
• Connected Minimal Cutsets: Key combinatorial input is the connectivity of minimal vertex cutsets in adjacency graphs of tessellations on simply connected, one-ended manifolds. This property is essential for applying Peierls-type renormalization and bounding uniqueness thresholds (Bühler et al., 27 Mar 2025).
• Annealed Exploration and Exploration Schemes: “Annealed” algorithms that reveal Poisson points and their colors sequentially can be leveraged to establish monotonicity of the uniqueness phase. Explorations couple percolation at parameters $p<p'$ on a common Poisson configuration (Bühler et al., 27 Mar 2025).
• Unbounded Borders in Ideal Tessellations: In non-amenable product geometries, the emergence of unbounded shared boundaries between Voronoi cells in the IPVT ensures mixing of local clusters and the vanishing uniqueness window for low $\lambda$ (D'Achille et al., 28 Nov 2025).
• Sharp Phase Transition Theory: Russo's formula, OSSS inequality, and decision-tree techniques adapted to continuum and hyperbolic settings establish exponential decay of connectivity below $p_c$ and linear mean-field lower bounds above $p_c$ (Li et al., 2021).

5. Extensions, Generalizations, and Applications

• Other Tessellations: Fine-graining and stabilization methods extend to partition models generated by Poisson–Delaunay and weighted Voronoi (Laguerre, Johnson–Mehl) tessellations, provided key properties such as stabilization and asymptotic essential connectedness are preserved (Gall et al., 2019).
• Broader Metric Spaces: The general theory includes symmetric spaces of higher rank, non-amenable Cayley graphs, and complements recent rigidity results for Lie groups with property (T) (D'Achille et al., 28 Nov 2025).
• Quantitative and Numerical Results: In the planar case, the site percolation threshold for vertices of Poisson–Voronoi tessellation is $p^*\approx0.713$ (Gall et al., 2019). Numerical studies also yield bond thresholds and boundary curves for more complex models.
• Continuum Percolation in Communication Models: Cox point process models supported on Voronoi–Delaunay infrastructure under line-of-sight constraints provide a rigorous framework for line-of-sight percolation in random environments mimicking telecommunication networks (Gall et al., 2019).

6. Open Problems and Future Directions

Several fundamental questions remain:

• Precise Critical Exponents and Universality: Determining the universality class of Bernoulli–Voronoi percolation, scaling limits, and critical exponents, especially in non-Euclidean settings, remains open (Hansen et al., 2021, Hansen et al., 2020).
• Sharp Phase Transitions in Higher Dimensions and Complex Geometries: Extending sharpness results and quantitative thresholds to a wider class of symmetric spaces and their products.
• Noise Sensitivity and Dynamical Features: Investigation of noise sensitivity, spectral analysis, and dynamical percolation in both Euclidean and non-Euclidean Bernoulli–Voronoi percolation, building on stratified Wiener–Itô expansions and pivotal processes (Bhattacharjee et al., 18 Jul 2024).
• Further Applications of the IPVT Unbounded Borders: Exploiting the combinatorial and ergodic properties revealed by the IPVT in probabilistic and geometric group theory, including constructions of FIID sparse unique infinite cluster processes in group-invariant settings (D'Achille et al., 28 Nov 2025).

7. Summary Table: Critical and Uniqueness Threshold Behavior

Geometry/Class	$\lim_{\lambda\to 0} p_c(\lambda)$	$\lim_{\lambda\to \infty} p_c(\lambda)$	$\lim_{\lambda\to 0} p_u(\lambda)$
$\mathbb{R}^2$ (Euclidean)	$1/2$	$1/2$	$1/2$
$\mathbb{H}^2$ (Hyperbolic plane)	$0$	$1/2$	$1-p_c(\lambda)$ for each $\lambda$
Products of Trees/Hyperbolic	—	—	$0$

The convergence and vanishing phenomena highlight the intricate relationship between geometry, combinatorics of tessellations, and local-to-global cluster structure in Bernoulli–Voronoi percolation across different ambient spaces (D'Achille et al., 28 Nov 2025, Bühler et al., 27 Mar 2025, Hansen et al., 2021, Hansen et al., 2020).