--- title: Bernoulli–Voronoi Percolation url: https://www.emergentmind.com/topics/bernoulli-voronoi-percolation type: topic --- # Bernoulli–Voronoi Percolation 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 study 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 $\lambda$ [1708.03054]. ### 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 $\lambda\searrow 0$, $$ p_c(\lambda)\sim \frac{\pi}{3}\lambda $$ [2111.04299]. - As $\lambda\to\infty$, $p_c(\mathbb{H}^2,\lambda)\to1/2$, converging to the Euclidean threshold [2004.01464, 2503.21737]. - 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 $\lambda\to\infty$ [2503.21737]. ### 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 $\lambda\to0$: $$ \lim_{\lambda\to0} p_u(\lambda) = 0\quad\text{for }\,M=\mathbb H_{d_1}\times\cdots\times\mathbb H_{d_k},\,k\ge2 $$ [2511.23317]. 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** ([2503.21737]): 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** ([2503.21737, 2511.23317]): 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 $p_u(\lambda)\to0$ [2511.23317]. ## 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 [2503.21737]. - **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 [2503.21737]. - **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 [2503.21737]. - **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