Dense Packing on Uniform Lattices

We study the Hard Core Model on the graphs G obtained from Archimedean tilings i.e. configurations in {0,1} ^G with the nearest neighbor 1’s forbidden. Our particular aim in choosing these graphs is to obtain insight to the geometry of the densest packings in a uniform discrete set-up. We establish density bounds, optimal configurations reaching them in all cases, and introduce a probabilistic cellular automaton that generates the legal configurations. Its rule involves a parameter which can be naturally characterized as packing pressure. It can have a critical value but from packing point of view just as interesting are the noncritical cases. These phenomena are related to the exponential size of the set of densest packings and more specifically whether these packings are maximally symmetric, simple laminated or essentially random packings. Research partially supported by The Finnish Academy of Science and Letters.