[Paper Review] Absence of percolation in the Bernoulli Boolean model

Cristian F. Coletti, Sebastian P. Grynberg|arXiv (Cornell University)|Feb 13, 2014

Stochastic processes and statistical mechanics4 references3 citations

TL;DR

This paper establishes conditions for the absence of percolation in the discrete Bernoulli Boolean model on $\mathbb{Z}^d$, proving that connected components are almost surely finite when the intensity $p$ is small enough and the radius distribution has finite $d$-th moment. The result is extended to non-homogeneous $p_x$ and used to construct interacting particle systems with infinite-range interactions via a Harris graphical procedure under Kalikow-type decomposition assumptions.

ABSTRACT

We consider the Bernoulli Boolean discrete percolation model on the d-dimensional integer lattice. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the intensity of the underlying point process is small enough. We also study a Harris graphical procedure to construct, forward in time, particle systems with interactions of infinite range under the assumption that the corresponding generator admits a Kalikow-type decomposition. We do so by using the subcriticality of the boolean model of discrete percolation.

Motivation & Objective

• To establish sufficient conditions for the absence of percolation in the discrete Bernoulli Boolean model on $\mathbb{Z}^d$.
• To determine the role of the $d$-th moment of the radius distribution in subcritical behavior.
• To extend the subcriticality result to non-homogeneous retention probabilities $p_x$.
• To apply the subcriticality result to the graphical construction of interacting particle systems with infinite-range interactions.
• To provide a constructive method for such systems using a Harris graphical procedure under Kalikow-type decomposition.

Proposed method

• Define the discrete Bernoulli Boolean model with i.i.d. radii $R_x$ and retention probability $p_x$ on $\mathbb{Z}^d$.
• Use coupling arguments to extend the subcriticality result from constant $p_x = p$ to non-constant $p_x$.
• Apply geometric estimates on $L_1$-balls and spheres, particularly $S_{nr} \subset \bigcup_{x \in S_n} B(rx, \frac{d}{2}r)$, to control cluster growth.
• Prove that if $\mathbb{E}[R^d] < \infty$, then for small enough $p$, the occupied clusters are almost surely finite.
• Use the subcriticality of the Boolean model to construct a graphical coupling via Poisson processes on finite islands $C_\ell$.
• Construct the infinite-volume process inductively over ordered jump times $\tau_k$ using state updates at each site.

Experimental results

Research questions

• RQ1 Under what conditions on the radius distribution does the discrete Bernoulli Boolean model exhibit no percolation?
• RQ2 What is the critical role of the $d$-th moment of the radius in determining subcritical behavior?
• RQ3 Can the subcriticality result be extended to non-homogeneous retention probabilities $p_x$?
• RQ4 How can the subcriticality of the Boolean model be leveraged to construct interacting particle systems with infinite-range interactions?
• RQ5 What conditions on the generator ensure a Kalikow-type decomposition enabling such graphical constructions?

Key findings

• If $p_x = p \in (0,1)$ for all $x$ and $\mathbb{E}[R^d] < \infty$, then the connected components of the Boolean model are almost surely finite for sufficiently small $p$.
• If $\mathbb{E}[R^d] = \infty$, then percolation occurs regardless of how small $p$ is, indicating that the $d$-th moment is a sharp threshold.
• The subcriticality result extends to non-constant $p_x$ via a coupling argument, provided the radius distribution has finite $d$-th moment.
• The graphical construction of interacting particle systems with infinite-range interactions is possible under the assumption that the generator admits a Kalikow-type decomposition.
• The construction is achieved by partitioning $\mathbb{Z}^d$ into finite islands $C_\ell$ with no interaction between them, enabling inductive simulation on each component.
• The method relies on ordering jump times from Poisson processes and updating the state at each site using the corresponding rate and environment.