[Paper Review] Bernoulli Percolation on random Tessellations

Sebastian Ziesche|arXiv (Cornell University)|Sep 15, 2016

Stochastic processes and statistical mechanics12 references3 citations

TL;DR

This paper generalizes Bernoulli percolation from regular lattices to random tessellations in ℝᵈ, establishing the uniqueness of the infinite cluster under weak stationarity and moment conditions via an adapted Burton-Keane argument. It proves non-trivial phase transitions for broad classes of random tessellations, including those from determinantal and Gibbsian point processes, by introducing two flexible frameworks based on mixing and cell-size statistics.

ABSTRACT

We generalize the standard site percolation model on the $d$-dimensional lattice to a model on random tessellations of $\mathbb R^d$. We prove the uniqueness of the infinite cluster by adapting the Burton-Keane argument \cite{burton1989density}, develop two frameworks that imply the non-triviality of the phase transition and show that large classes of random tessellations fit into one of these frameworks. Our focus is on a very general approach that goes well beyond the typical Poisson driven models. The most interesting examples might be Voronoi tessellations induced by determinantal processes or certain classes of Gibbs processes introduced in \cite{schreiber2013}. In a second paper we will investigate first passage percolation on random tessellations.

Motivation & Objective

• To extend Bernoulli site percolation from regular lattices to random tessellations of ℝᵈ with minimal symmetry assumptions.
• To establish the uniqueness of the infinite cluster in this generalized model under weak conditions like stationarity and finite cell moment bounds.
• To develop two general frameworks that guarantee a non-trivial phase transition in random tessellations.
• To show that diverse point processes—such as determinantal and Gibbsian processes—generate tessellations satisfying these frameworks.
• To provide a foundational, broadly applicable theory beyond Poisson-driven models, enabling future study of first passage percolation on such tessellations.

Proposed method

• Adapts the Burton-Keane argument to prove uniqueness of the infinite cluster in random tessellations using only stationarity and a finite second moment condition on cell sizes.
• Introduces a framework based on a mixing condition to ensure non-trivial phase transition, generalizing results from i.i.d. or translation-invariant models.
• Proposes a second framework using auxiliary random fields that track the frequency of very small or large cells, ensuring non-triviality under suitable tail behavior.
• Applies these frameworks to prove non-trivial phase transitions for Voronoi tessellations generated by determinantal, Cox, and Poisson cluster processes.
• Uses void probability bounds on the underlying point process to verify the cell-size framework condition (T2), particularly for ν-weakly sub-Poisson processes.
• Employs Borel-Cantelli-type estimates on the number of cells with large or small neighborhoods to control the probability of rare configurations.

Experimental results

Research questions

• RQ1 Under what general conditions on a random tessellation does the infinite cluster in Bernoulli face percolation remain unique?
• RQ2 Can the non-triviality of the phase transition (i.e., 0 < p_c < 1) be guaranteed for a broad class of random tessellations beyond Poisson-driven models?
• RQ3 Do Voronoi tessellations induced by determinantal or Gibbsian point processes exhibit a non-trivial phase transition in Bernoulli percolation?
• RQ4 How can the Burton-Keane argument be adapted to non-transitive, random graph structures arising from random tessellations?
• RQ5 What conditions on the generating point process ensure that the induced tessellation satisfies the criteria for non-trivial phase transition in the proposed frameworks?

Key findings

• The infinite cluster in Bernoulli percolation on random tessellations is unique under stationarity and a finite second moment condition on cell sizes, generalizing the Burton-Keane result to non-lattice settings.
• The critical probability satisfies p_c ≥ 1/2 in the planar case, derived as a corollary of the uniqueness result using Zhang’s argument.
• A mixing condition on the tessellation implies non-trivial phase transition, though a counterexample shows mixing alone is insufficient in general.
• The framework based on auxiliary random fields tracking extreme cell sizes ensures non-trivial phase transition when the underlying point process has sufficiently small void probabilities.
• Voronoi tessellations generated by ν-weakly sub-Poisson point processes, including determinantal processes, satisfy the cell-size framework condition (T2), ensuring non-trivial phase transition.
• Poisson cluster and Cox processes also satisfy the required void probability bounds, confirming their compatibility with the non-trivial phase transition frameworks.