# Non-triviality of the vacancy phase transition for the Boolean model

## TL;DR

This paper proves that in a spherical Poisson Boolean model with finite $d$-th moment radius distribution, there exists a positive intensity at which the vacant region percolates, highlighting a non-trivial phase transition.

## Contribution

It establishes the existence of a non-trivial vacancy phase transition in the Boolean model under finite $d$-th moment conditions.

## Key findings

- Vacant region percolates at positive intensity
- Phase transition is non-trivial under finite $d$-th moment
- Results apply for Euclidean $d$-space with $d \\geq 2"

## Abstract

In the spherical Poisson Boolean model, one takes the union of random balls centred on the points of a Poisson process in Euclidean $d$-space with $d \geq 2$. We prove that whenever the radius distribution has a finite $d$-th moment, there exists a strictly positive value for the intensity such that the vacant region percolates.