174 research outputs found
The hard-core model on and Kepler's conjecture
We study the hard-core model of statistical mechanics on a unit cubic lattice
, which is intrinsically related to the sphere-packing problem
for spheres with centers in . The model is defined by the sphere
diameter which is interpreted as a Euclidean exclusion distance between
point particles located at spheres centers. The second parameter of the
underlying model is the particle fugacity . For the ground states of
the model are given by the dense-packings of the spheres. The identification of
such dense-packings is a considerable challenge, and we solve it for as well as for , where
. For the former family of values of our proofs are
self-contained. For our results are based on the proof of
Kepler's conjecture. Depending on the value of , we encounter three
physically distinct situations: (i) finitely many periodic ground states, (ii)
countably many layered periodic ground states and (iii) countably many not
necessarily layered periodic ground states. For the first two cases we use the
Pirogov-Sinai theory and identify the corresponding periodic Gibbs
distributions for and , ,
in a high-density regime , where the system is ordered and tends to
fluctuate around some ground states. In particular, for only a finite
number out of countably many layered periodic ground states generate pure
phases
Minimal Area of a Voronoi Cell in a Packing of Unit Circles
We present a new self-contained proof of the well-known fact that the minimal
area of a Voronoi cell in a unit circle packing is equal to , and
the minimum is achieved only on a perfect hexagon. The proof is short and, in
our opinion, instructive
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