Quasicrystalline ground states without matching rules

A simple, local cluster interaction is presented, which has as (only) ground states perfectly quasicrystalline tilings from a single local isomorphism class. Since these tilings do not allow for any perfect matching rules, it is thereby shown that the class of structures which are the ground state of some finite range interaction is considerably larger than previously anticipated. Cluster interactions having a quasicrystalline ground state turn out to be simple and robust, and therefore provide an attractive explanation for the existence of quasicrystals. A simplified version of our cluster interaction is found to have super-tile random tiling ground states. Due to the large size of the super-tiles, these random tilings still look perfect on a local scale.