  Hard core lattice gas models are minimal models to study entropy driven phase
transitions. In the $k$-NN lattice gas, a particle excludes all sites upto the
$k$-th next-nearest neighbors from being occupied by another particle. As $k$
increases from one, it extrapolates from nearest neighbor exclusion to the hard
sphere gas. In this paper, we study the model on the triangular lattice for
$k\leq 7$ using a flat histogram algorithm that includes cluster moves. Earlier
studies had focused on $k\leq 3$. We show that for $4\leq k\leq 7$, the system
undergoes a single phase transition from a low-density fluid phase to a
high-density sublattice-ordered phase. Using partition function zeros and
non-convexity properties of the entropy, we show that the transitions are
discontinuous. The critical chemical potential, coexistence densities, and
critical pressure are determined accurately.

Statistical Mechanics