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We study the $k$-NN hard core lattice gas model in which the first $k$ nextnearest neighbor sites of a particle are excluded from occupation by otherparticles on a two dimensional square lattice. This model is the latticeversion of the hard disc system with increasing $k$ corresponding to decreasinglattice spacing. While the hard disc system is known to undergo a two stepfreezing process with increasing density, the lattice model has been known toshow only one transition. Here, based on Monte Carlo simulations and highdensity expansions of the free energy and density, we argue that for$k=4,10,11,14...$, the lattice model undergoes multiple transitions withincreasing density. Using Monte Carlo simulations, we confirm the same for$k=4,\ldots, 11$. This, in turn, resolves an existing puzzle as to why the 4-NNmodel has a continuous transition against the expectation of a first ordertransition.

Multiple Phase Transitions in Extended Hard Core Lattice Gas Models in Two Dimensions