Phase transitions and coarse-graining for a system of particles in the continuum

We revisit the proof of the liquid-vapor phase transition for systems with finite-range interaction by Lebowitz et al. (J. Stat. Phys. 94(5-6), 955-1025, 1999 [1]) and extend it to the case where we additionally include a hard-core interaction to the Hamiltonian. We establish the phase transition for the mean field limit and then we also prove it when the interaction range is long but finite, by perturbing around the mean-field theory. A key step in this procedure is the construction of a density (coarse-grained) model via cluster expansion. In this note we present the overall result but we mainly focus on this last issue.