Kepler's conjecture and phase transitions in the high-density hard-core model on ${\mathbb{Z}}^3$

Abstract

We perform a rigorous study of the identical sphere packing problem in ${\mathbb{Z}}^3$ and of phase transitions in the corresponding hard-core model. The sphere diameter $D>0$ and the fugacity $u\gg 1$ are the varying parameters of the model. We solve the sphere packing problem for values $D^2=2,3,4,5,6,8,9,10,11,12,2{\ell }^2$, $\ell \in \mathbb{N}$. For values $D^2=2,3,5,8,9,10,12,2{\ell }^2$, $\ell \in \mathbb{N}$ and $u>u^0(D)$ we establish the diagram of periodic pure phases, completely or partially. For the case $D^2=2{\ell }^2$, $\ell \in \mathbb{N}$ we use results from Hales' proof of Kepler's conjecture.