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A point process is R-dependent, if it behaves independently beyond theminimum distance R. This work investigates uniform positive lower bounds on theavoidance functions of R-dependent simple point processes with a commonintensity. Intensities with such bounds are described by the existence ofShearer's point process, the unique R-dependent and R-hard-core point processwith a given intensity. This work also presents several extensions of theLov\'asz Local Lemma, a sufficient condition on the intensity and R toguarantee the existence of Shearer's point process and exponential lowerbounds. Shearer's point process shares combinatorial structure with thehard-sphere model with radius R, the unique R-hard-core Markov point process.Bounds from the Lov\'asz Local Lemma convert into lower bounds on the radius ofconvergence of a high-temperature cluster expansion of the hard-sphere model.This recovers a classic result of Ruelle on the uniqueness of the Gibbs measureof the hard-sphere model via an inductive approach \`a la Dobrushin.

Shearer's point process, the hard-sphere model and a continuum Lov\'asz Local Lemma