  Disagreement percolation connects a Gibbs lattice gas and i.i.d. site
percolation on the same lattice such that non-percolation implies uniqueness of
the Gibbs measure. This work generalises disagreement percolation to the
hard-sphere model and the Boolean model. Non-percolation of the Boolean model
implies the uniqueness of the Gibbs measure and exponential decay of pair
correlations and finite volume errors. Hence, lower bounds on the critical
intensity for percolation of the Boolean model imply lower bounds on the
critical activity for a (potential) phase transition. These lower bounds
improve upon known bounds obtained by cluster expansion techniques. The proof
uses a novel dependent thinning from a Poisson point process to the hard-sphere
model, with the thinning probability related to a derivative of the free
energy.
