Subcritical Boolean percolation on graphs of bounded degree

TL;DR

This paper investigates Boolean percolation on bounded degree graphs, providing conditions for the existence of a subcritical phase where all components are finite and decay exponentially in size.

Contribution

It establishes sufficient conditions on the radius distribution for subcritical behavior and exponential decay in Boolean percolation on graphs of bounded degree.

Findings

Identifies conditions for subcritical phase existence

Provides criteria for exponential decay of component sizes

Shows when the model does not exhibit a phase transition

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.