Infinite ground-state degeneracy of a two-dimensional athermal lattice-gas

I studied the ground state properties and phase behavior of a two-dimensional lattice gas in which hard-core particles can have at most one nearest neighboring occupied site on the square lattice. Monte Carlo simulations in the grand-canonical ensemble showed no apparent signature of singular thermodynamic behavior when the chemical potential was increased. The absence of an ordering phase transition is traced to the large number of ground state configurations the model is endowed, which is due to the impossibility of satisfying simultaneously closest packing around a vacancy and around a particle. Numerical simulations confirm that the ground state entropy is proportional to the square root of system size.