# Existence of an unbounded vacant set for subcritical continuum percolation

## TL;DR

This paper proves that in subcritical Poisson Boolean percolation in the plane, the thresholds for the emergence of unbounded occupied and vacant regions are the same, extending to higher dimensions with finite moments.

## Contribution

It establishes the equality of thresholds for unbounded occupied and vacant sets in subcritical continuum percolation, linking finite moments to phase transition existence.

## Key findings

- Thresholds for unbounded occupied and vacant sets coincide.
- Finite second moment of radii distribution is crucial in 2D.
- Finite d-th moment is necessary and sufficient in higher dimensions.

## Abstract

We consider the Poisson Boolean percolation model in $\mathbb{R}^2$, where the radii of each ball is independently chosen according to some probability measure with finite second moment. For this model, we show that the two thresholds, for the existence of an unbounded occupied and an unbounded vacant component, coincide. This complements a recent study of the sharpness of the phase transition in Poisson Boolean percolation by the same authors. As a corollary it follows that for Poisson Boolean percolation in $\mathbb{R}^d$, for any $d\ge2$, finite moment of order $d$ is both necessary and sufficient for the existence of a nontrivial phase transition for the vacant set.