[Paper Review] Sharpness of the phase transition for continuum percolation in R^2

Daniel Ahlberg, Vincent Tassion+1 more|arXiv (Cornell University)|May 19, 2016

Stochastic processes and statistical mechanics32 references10 citations

TL;DR

This paper establishes the sharpness of the phase transition in two-dimensional continuum percolation with unbounded, possibly heavy-tailed radii distributions in a Poisson Boolean model. Using a finite-size criterion and Russo-Seymour-Welsh theory, it proves that both the occupied and vacant sets undergo a sharp transition at a unique critical intensity λc, with exponential decay of one-arm probabilities in the subcritical and supercritical phases, and polynomial decay at criticality under sharp moment conditions on the radius distribution.

ABSTRACT

We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood when the radii are uniformly bounded from above. In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter $λ_c$. Moreover, - For $λ< λ_c$, the vacant set has a unique unbounded connected component and we give precise bounds on the one-arm probability for the occupied set, depending on the radius distribution. - At criticality, we establish the box-crossing property, implying that no unbounded component can be found, neither in the occupied nor the vacant sets. We provide a polynomial decay for the probability of the one-arm events, under sharp conditions on the distribution of the radius. - For $λ> λ_c$, the occupied set has a unique unbounded component and we prove that the one-arm probability for the vacant decays exponentially fast. The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation.

Motivation & Objective

• To establish the sharpness of the phase transition in Poisson Boolean percolation with unbounded radii in R².
• To analyze the behavior of the occupied and vacant sets across subcritical, critical, and supercritical regimes.
• To extend the sharp threshold framework beyond bounded radii to heavy-tailed distributions.
• To demonstrate that the critical parameter λc is a single point, not an interval, under mild moment conditions on the radius distribution.
• To show that the techniques apply to related models such as Poisson Voronoi and confetti percolation.

Proposed method

• Develops a finite-size criterion to link crossing probabilities in rectangles to exponential decay of connection probabilities.
• Applies Russo-Seymour-Welsh theory to control crossing probabilities across different aspect ratios, enabling uniform bounds.
• Uses a spatial mixing argument based on large deviations to control long-range dependencies in the model.
• Establishes moment conditions on the radius distribution to ensure polynomial decay at criticality.
• Applies a coupling argument to relate the behavior of the occupied and vacant sets, especially under self-duality.
• Adapts techniques from discrete percolation to the continuum setting, particularly for unbounded radii.

Experimental results

Research questions

• RQ1 Does the phase transition in Poisson Boolean percolation with unbounded radii remain sharp, as in the bounded case?
• RQ2 What is the decay rate of the one-arm probability for the occupied set in the subcritical regime under heavy-tailed radius distributions?
• RQ3 Does the box-crossing property hold at criticality, and what decay rate does it imply for one-arm events?
• RQ4 Can the critical parameter λc be shown to be a single point rather than an interval, even with heavy-tailed radii?
• RQ5 To what extent can the methods be extended to other continuum percolation models like Poisson Voronoi or confetti percolation?

Key findings

• For λ < λc, the vacant set has a unique unbounded connected component, and the one-arm probability for the occupied set decays as r−10 under mild moment conditions.
• At criticality λ = λc, the box-crossing property holds, implying no unbounded components in either the occupied or vacant sets, with one-arm probability decaying polynomially as r−α for some α > 0.
• For λ > λc, the occupied set has a unique unbounded component, and the one-arm probability for the vacant set decays exponentially fast.
• The critical parameter λc is strictly between 0 and ∞, and is a single point, not an interval, under the given moment conditions.
• The results extend to Poisson Voronoi and confetti percolation under analogous assumptions, with similar decay and critical behavior.
• Under self-duality (G0 = G1), the critical parameter satisfies λc = 1/2, recovering known results in symmetric settings.