The hard-core model on Z3\mathbb{Z}^3 and Kepler's conjecture

We study the hard-core model of statistical mechanics on a unit cubic lattice $\mathbb{Z}^3$, which is intrinsically related to the sphere-packing problem for spheres with centers in $\mathbb{Z}^3$. The model is defined by the sphere diameter $D>0$ 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 $u$. For $u>1$ 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 $D^2=2, 3, 4, 5, 6, 8, 9, 10, 11, 12$ as well as for $D^2=2\ell^2$, where $\ell\in\mathbb{N}$. For the former family of values of $D^2$ our proofs are self-contained. For $D^2=2\ell^2$ our results are based on the proof of Kepler's conjecture. Depending on the value of $D^2$, 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 $D^2=2,3,5,8,9,10,12$ and $D^2=2\ell^2$, $\ell\in\mathbb{N}$, in a high-density regime $u>u_*(D^2)$, where the system is ordered and tends to fluctuate around some ground states. In particular, for $D^2=5$ 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 $2\sqrt{3}$, and the minimum is achieved only on a perfect hexagon. The proof is short and, in our opinion, instructive

High-density hard-core model on Z2\mathbb{Z}^2 and norm equations in ring Z[−16]\mathbb{Z} [{\sqrt[6]{-1}}]

We study the Gibbs statistics of high-density hard-core configurations on a square lattice $\mathbb{Z}^2$, for a general Euclidean exclusion distance $D$. The key point is an analysis of solutions to norm equations in the ring $\mathbb{Z}[{\sqrt[6]{-1}}]$. We describe the ground states in terms of M-triangles, i.e., non-obtuse $\mathbb{Z}^2$-triangles of a minimal area with the side-lengths $\geq D$. Further, we identify $\mathbb{Z}^2$-triangles as elements of $\mathbb{Z}[{\sqrt[6]{-1}}]$. First, there is a finite class (Class S) formed by values $D^2$ generating sliding, a phenomenon leading to countable families of periodic ground states. We identify all $D^2$ with sliding. Each of the remaining classes is proven to be infinite; they are characterized by uniqueness or non-uniqueness of a minimal triangle for a given $D^2$, up to $\mathbb{Z}^2$-congruencies. For values of $D^2$ with uniqueness (Class A) we describe the periodic ground states as admissible triangular sub-lattices $\mathbb{E}\subset\mathbb{Z}^2$ of maximum density. By using the Pirogov-Sinai theory, it allows us to identify the extreme Gibbs measures (pure phases) for large values of fugacity and describe symmetries between them. Next, we analyze the values $D^2$ with non-uniqueness. For some $D^2$ all M-triangles are ${\mathbb{R}}^2$-congruent but not $\mathbb{Z}^2$-congruent (Class B0). For other values of $D^2$ there exist non-${\mathbb{R}}^2$-congruent M-triangles, with different collections of side-lengths (Class B1). Moreover, there are values $D^2$ for which both cases occur (Class B2). The phase diagram for Classes B0, B1, B2 is determined by dominant ground states. Classes A, B0-B2 are described in terms of cosets in $\mathbb{Z}[{\sqrt[6]{-1}}]$ by the group of units.Comment: Supplemen

Observation of Anomalous Internal Pair Creation in 8^8Be: A Possible Signature of a Light, Neutral Boson

Electron-positron angular correlations were measured for the isovector magnetic dipole 17.6 MeV state ($J^\pi=1^+$, $T=1$) $\rightarrow$ ground state ($J^\pi=0^+$, $T=0$) and the isoscalar magnetic dipole 18.15 MeV ($J^\pi=1^+$, $T=0$) state $\rightarrow$ ground state transitions in $^{8}$Be. Significant deviation from the internal pair creation was observed at large angles in the angular correlation for the isoscalar transition with a confidence level of $> 5\sigma$. This observation might indicate that, in an intermediate step, a neutral isoscalar particle with a mass of 16.70$\pm0.35$ (stat)$\pm 0.5$ (sys) MeV$/c^2$ and $J^\pi = 1^+$ was created.Comment: 5 pages, 5 figure