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Hard core lattice gas models are minimal models to study entropy driven phasetransitions. 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 hardsphere 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. Earlierstudies had focused on $k\leq 3$. We show that for $4\leq k\leq 7$, the systemundergoes a single phase transition from a low-density fluid phase to ahigh-density sublattice-ordered phase. Using partition function zeros andnon-convexity properties of the entropy, we show that the transitions arediscontinuous. The critical chemical potential, coexistence densities, andcritical pressure are determined accurately.

The freezing phase transition in hard core lattice gases on triangular lattice with exclusion up to seventh next-nearest neighbor