Dense packing on uniform lattices

TL;DR

This paper investigates the densest packings in the Hard Core Model on graphs derived from Archimedean tilings, establishing density bounds, optimal configurations, and introducing a probabilistic cellular automaton to generate legal configurations.

Contribution

It provides new density bounds, characterizes optimal packings, and introduces a probabilistic cellular automaton with a packing pressure parameter for generating configurations.

Findings

Density bounds for packings are established.

Optimal configurations are identified for all cases.

A probabilistic cellular automaton with a critical parameter is introduced.

Abstract

We study the Hard Core Model on the graphs ${\rm {\bf \scriptstyle G}}$ obtained from Archimedean tilings i.e. configurations in $\scriptstyle \{0,1\}^{{\rm {\bf G}}}$ 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.