Low-temperature stability of quasicrystals

We discuss classical lattice-gas models without periodic ground states. To prove the thermodynamic stability of such systems we have to construct nonperiodic equilibrium states which are small perturbations of ground states. We review partial results in this direction and discuss possibilities of constructing a model with nonperiodic equilibrium states. It would be a microscopic (toy) model of a quasicrystal, a system of interacting particles with a stable nonperiodic equilibrium state.