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Subcritical Boolean percolation on graphs of bounded degree

Corentin Faipeur

TL;DR

The paper investigates subcritical Boolean percolation on graphs of bounded degree by examining a long-range site percolation model where each vertex hosts a random-radius ball and is active with probability $p$. It derives sufficient conditions, expressed through moments of the radius distribution, ensuring a subcritical phase where all wet components are finite and establishing exponential decay of component sizes under stronger moment assumptions. The analysis leverages a Peierls-type argument and a coupling to subcritical Galton–Watson processes via two Poisson point-process realizations: one on vertex-radius pairs and another on marked boundary events, enabling a layer-wise exploration that bounds growth of the wet set. The results generalize known continuum and discrete Boolean percolation criteria to graphs of bounded degree (including directed graphs) and provide explicit moment-based conditions that guarantee non-percolation for small $p$ and controlled tail behavior of component sizes.

Abstract

In this paper, we study a model of long-range site percolation on graphs of bounded degree, namely the Boolean percolation model. In this model, each vertex of an infinite connected graph is the center of a ball of random radius, and vertices are said to be active independently with probability $p \in [0, 1]$. We consider $W$ to be the reunion of random balls with an active center. In certain circumstances, the model does not exhibit a phase transition, in the sense that $W$ almost surely contains an infinite component for all $p > 0$, or even $W$ covers the entire graph. In this paper, we give a sufficient condition on the radius distribution for the existence of a subcritical phase, namely a regime such that almost surely all the connected components of $W$ are finite. Additionally, we provide a sufficient condition for the exponential decay of the size of a typical component.

Subcritical Boolean percolation on graphs of bounded degree

TL;DR

The paper investigates subcritical Boolean percolation on graphs of bounded degree by examining a long-range site percolation model where each vertex hosts a random-radius ball and is active with probability . It derives sufficient conditions, expressed through moments of the radius distribution, ensuring a subcritical phase where all wet components are finite and establishing exponential decay of component sizes under stronger moment assumptions. The analysis leverages a Peierls-type argument and a coupling to subcritical Galton–Watson processes via two Poisson point-process realizations: one on vertex-radius pairs and another on marked boundary events, enabling a layer-wise exploration that bounds growth of the wet set. The results generalize known continuum and discrete Boolean percolation criteria to graphs of bounded degree (including directed graphs) and provide explicit moment-based conditions that guarantee non-percolation for small and controlled tail behavior of component sizes.

Abstract

In this paper, we study a model of long-range site percolation on graphs of bounded degree, namely the Boolean percolation model. In this model, each vertex of an infinite connected graph is the center of a ball of random radius, and vertices are said to be active independently with probability . We consider to be the reunion of random balls with an active center. In certain circumstances, the model does not exhibit a phase transition, in the sense that almost surely contains an infinite component for all , or even covers the entire graph. In this paper, we give a sufficient condition on the radius distribution for the existence of a subcritical phase, namely a regime such that almost surely all the connected components of are finite. Additionally, we provide a sufficient condition for the exponential decay of the size of a typical component.

Paper Structure

This paper contains 10 sections, 6 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Notations and statement of the main results
  4. Graph terminology
  5. Phase transition of the Boolean percolation model
  6. Reformulation of the main results
  7. Proof of Theorem \ref{['boolean_perco']}
  8. Realisation of the model as a PPP on $V \times \mathbf{N}^*$
  9. Realisation of the model as PPP of marked balls
  10. Comparison with a Galton-Watson process

Key Result

Theorem 1

Let $G$ be a graph of bounded degree, and $R$ distributed according to $\nu$. For all $n \geq 1$, let $\varphi(n):= \Delta \sum_{r=1}^n c_r c_r^\top$. If $\mathbf{E}[\varphi(R)] < \infty$, then

Theorems & Definitions (12)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • proof : Proof of Theorem \ref{['thm_pc>0']}
  • proof : Proof of Theorem \ref{['thm_expo_decay']}
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 2 more