Poisson–Voronoi Percolation Analysis

• Poisson–Voronoi percolation is a continuum percolation model defined via Poisson point processes and Voronoi tessellations, extending classical lattice systems to random geometric settings.
• The model features critical (p_c) and uniqueness (p_u) thresholds that capture the emergence and consolidation of infinite clusters, influenced by the underlying geometry.
• Recent breakthroughs leverage ideal tessellation properties and local-to-global methods to demonstrate vanishing uniqueness thresholds in nonamenable product spaces, opening new research directions.

Poisson–Voronoi percolation is a paradigmatic model of continuum percolation, generalizing classical lattice percolation to random geometric environments generated by Poisson point processes and their induced tessellations. The theory spans Euclidean, hyperbolic, and higher rank symmetric spaces and has seen recent breakthroughs—especially regarding the behavior of critical and uniqueness thresholds in nonamenable product geometries. This entry provides a comprehensive account of the model, mathematical structure, critical phenomena (including phase transitions and uniqueness thresholds), the role of geometric limits such as the ideal Poisson–Voronoi tessellation, and the implications of new results, notably vanishing-uniqueness thresholds in product and higher-rank settings.

1. Model Definition and Mathematical Setting

Poisson–Voronoi percolation is conducted on a proper geodesic metric space $(M, d, o, \mu)$ equipped with a distinguished origin $o$ and an infinite Radon measure $\mu$.

• Poisson point process: For intensity $\lambda > 0$, consider a Poisson point process $\mathbf{X}^{(\lambda)} = \{X_i^{(\lambda)}\}$ of intensity measure $\lambda\mu$, ordered by increasing $d(o, X_i^{(\lambda)})$.
• Voronoi tessellation: The Voronoi cells are defined as

$$C_i^{(\lambda)} = \{x \in M : d(x, X_i^{(\lambda)}) < d(x, X_j^{(\lambda)})\ \forall j \neq i \}$$

Tie-breaking has no effect on percolation outcomes.

• Random coloring: For $p \in [0,1]$, color each cell black ("open") with probability $p$ and white ("closed") with probability $1-p$, independently for each cell.
• Open region: The continuum percolation configuration is

$$\omega_p^{(\lambda)} = \bigcup_{\text{black } C_i^{(\lambda)}} C_i^{(\lambda)}$$

• Special cases: When $M = \mathbb{R}^d$ with Lebesgue measure, this construction coincides with classical Poisson–Voronoi percolation in Euclidean space.

A connected component of $\omega_p^{(\lambda)}$ is termed an open cluster.

2. Critical and Uniqueness Thresholds

The model is characterized by two primary thresholds:

• Critical threshold $p_c(\lambda)$: The infimum of $p$ for which infinite open clusters appear with positive probability:

$$p_c(\lambda) = \inf\{p : \omega_p^{(\lambda)} \text{ has an infinite cluster with positive probability} \}$$

• Uniqueness threshold $p_u(\lambda)$: The infimum of $p$ for which there is a unique infinite cluster with positive probability:

$$p_u(\lambda) = \inf\{p : \omega_p^{(\lambda)} \text{ has a unique infinite cluster with positive probability}\}$$

• Long-range order characterization: Adapted from Lyons–Schramm, uniqueness can be equivalently characterized via

$$p_u(\lambda) = \inf\left\{p : \inf_{x, y \in M} \mathbb{P}[x \stackrel{\omega_p^{(\lambda)}}{\longleftrightarrow} y] > 0 \right\}$$

where $x \leftrightarrow y$ denotes belonging to the same open cluster.

Critical and uniqueness thresholds depend sharply on geometry and the underlying spatial structure of $(M, d)$. In $\mathbb{R}^d$, $p_c(\lambda) = p_u(\lambda) = \frac{1}{2}$ for all $\lambda$, reflecting self-duality and scale-invariance in the Euclidean setting.

3. Threshold Phenomena in Nonamenable Product Geometries

A recent major advance establishes a robust mechanism in which nonamenable product geometries force the uniqueness threshold $p_u(\lambda)$ to vanish as $\lambda \to 0$ (D'Achille et al., 28 Nov 2025).

Main theorem: For a broad class of non-amenable product spaces (such as $k$-fold products of regular trees and products of hyperbolic spaces),

$\lim_{\lambda \to 0} p_u(\lambda) = 0$

In particular:

• For $G = (\mathbb{T}_d)^k$, the product of $k \geq 2$ copies of the $d$-regular tree ($d \geq 3$):

$p_u(\lambda) \to 0 \quad \text{as } \lambda \to 0$

• For $$M = \mathbb{H}_{d_1} \times \cdots \times \mathbb{H}_{d_k}$$ ($k \geq 2, d_i \geq 2$), Riemannian product of $k$ hyperbolic spaces:

$p_u(\lambda) \to 0 \quad \text{as } \lambda \to 0$

• For the symmetric space of a connected simply-connected higher-rank semisimple real Lie group:

$\lim_{\lambda \to 0} p_u(\lambda) = 0$

In contrast, in classical settings ($\mathbb{R}^d$ or the hyperbolic plane $\mathbb{H}^2$), $p_u(\lambda)$ is independent or remains bounded away from zero at low intensity.

This vanishing uniqueness threshold is proven via a two-step strategy:

• Local-to-global uniqueness: At small intensity and fixed $p$, if every ball of radius $R$ contains at most one cluster up to a larger radius $3R$ (local uniqueness), then with high probability, one obtains long-range order: $\inf_{x,y} \mathbb{P}[x \leftrightarrow y] > 0$.
• Verification via ideal tessellation borders: For sufficiently small intensity, the Poisson–Voronoi tessellation converges to the ideal Poisson–Voronoi tessellation (IPVT), in which the unbounded borders phenomenon (see below) enables the local-uniqueness condition.

4. The Ideal Poisson–Voronoi Tessellation and Unbounded Borders

In the low-intensity regime of such product spaces, the random Poisson–Voronoi tessellation converges (Fell topology) to an ideal Poisson–Voronoi tessellation (IPVT). Key features:

• Unbounded cells: Each cell in the IPVT is unbounded.
• Unbounded borders: Every pair of distinct cells shares an unbounded boundary. In the Delaunay graph of the IPVT, adjacency forms a complete countable graph.
• Unbounded stabilizers (Proposition 4.2): Distinct limiting horofunctions from different boundary data in at least two factors are stabilized by an unbounded subgroup of automorphisms.
• Exponential correlation decay (Lemma 4.3): The probability that two cells both approach fixed heights along two distant vertices decays exponentially in distance.
• Discrete analog (tree products): For tree products, for each $R > 1$, any two cells $C_i, C_j$ have an infinite set of vertices on their border well-separated from all other cells by at least $R$.

These properties are pivotal for the control of unique infinite clusters in low-intensity percolation, underpinning the new thresholds.

5. Concrete Examples and Applications

The methods and phenomena described have sharp implications for a range of structures:

Graph products of trees: In $G = (\mathbb{T}_d)^k$, the metric is the $L^1$ product of tree distances, and IPVT arises via horofunctions on each factor tree.

Products of hyperbolic spaces: For $$M = \mathbb{H}_{d_1} \times \cdots \times \mathbb{H}_{d_k}$$, the Riemannian product structure and prior convergence results for symmetric spaces ensure the vanishing uniqueness threshold regime.

Higher-rank symmetric spaces: Combining the new local-to-global method with earlier work, all symmetric spaces of higher rank (with or without Kazhdan’s property (T)) exhibit vanishing uniqueness thresholds.

FIID sparse unique infinite cluster: Each such Cayley-graph example (from products above) admits an invariant percolation built as a factor of IID with a unique infinite cluster of arbitrarily small density, affirmatively settling a question of Pete–Rokob.

6. Methodological Innovations and Proof Techniques

The analysis draws on several advanced methodologies:

• Use of Lyons–Schramm-type long-range order for the uniqueness threshold.
• Sprinkling arguments and coupling strategies to bridge local events to global percolation.
• Employment of the Mecke–Palm formula and Kochen–Stone lemma to establish that borders between IPVT cells are almost surely infinite.
• Construction of local uniqueness events and control via explicit lemmas (e.g., Lemmas 4.5, 4.6) leveraging typical cell counts in finite balls.
• Use of “ideal tessellation” geometry for verifying local events and propagating to global conclusions.

These arguments together establish a robust geometric mechanism—unbounded cell borders in IPVTs—that forces uniqueness thresholds to vanish in nonamenable products at low intensity.

7. Open Problems and Future Directions

Several open questions and further directions arose from these results (D'Achille et al., 28 Nov 2025):

• Behavior for strong ($L^\infty$) products: For $\mathbb{T}_d \boxtimes \mathbb{Z}$, the IPVT degenerates into tubes $\{c\} \times \mathbb{Z}$, so the unbounded border phenomenon only holds for cells truly touching. Whether $\liminf_{\lambda \to 0} p_u(\lambda) = 0$ persists remains open.
• Formulation of general criteria: Seeking general sufficient conditions, beyond Lie-group or tree-product cases, for the almost-sure unbounded-borders property in the IPVT.
• Possible discontinuity of $p_u(\lambda)$: Does $p_u(\lambda)$ jump discontinuously at $\lambda = 0$—i.e., is $\lim_{\lambda \to 0} p_u(\lambda) = 0$ but $p_u(0) > 0$ for Bernoulli–Voronoi on the IPVT graph?
• Sensitivity to generating set: Determining how the vanishing of $p_u(\lambda)$ depends on the quasi-isometry class (i.e., the choice of generating set) for a Cayley-graph factor.

These directions target both rigorous identification of the geometric features enabling the IPVT border phenomenon, and the potential extension to broader classes of random geometric graphs and percolation models.