Related papers: The hard-core model on $\mathbb{Z}^3$ and Kepler's…
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…
We study the Gibbs statistics of high-density hard-core configurations on a unit square lattice $\mathbb{Z}^2$, for a general Euclidean exclusion distance $D$. As a by-product, we solve the disk-packing problem on $\mathbb{Z}^2$ for disks…
We perform a rigorous study of the Gibbs statistics of high-density hard-core random configurations on a unit triangular lattice $\mathbb{A}_2$ and a unit honeycomb graph $\mathbb{H}_2$, for any value of the (Euclidean) repulsion diameter…
We study dense packings of disks and related Gibbs distributions representing high-density phases in the hard-core model on unit triangular, honeycomb and square lattices. The model is characterized by a Euclidean exclusion distance $D>0$…
The hard-sphere model is one of the most extensively studied models in statistical physics. It describes the continuous distribution of spherical particles, governed by hard-core interactions. An important quantity of this model is the…
This is the eighth and final paper in a series giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is…
A system of hard spheres exhibits physics that is controlled only by their density. This comes about because the interaction energy is either infinite or zero, so all allowed configurations have exactly the same energy. The low density…
This is the second in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
This is the first in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the Hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the…
This is the sixth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
This is the fifth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
The problem of finding the most efficient way to pack spheres has an illustrious history, dating back to the crystalline arrays conjectured by Kepler and the random geometries explored by Bernal in the 60's. This problem finds applications…
Hard spheres are ubiquitous in condensed matter: they have been used as models for liquids, crystals, colloidal systems, granular systems, and powders. Packings of hard spheres are of even wider interest, as they are related to important…
We prove a lower bound on the entropy of sphere packings of $\mathbb R^d$ of density $\Theta(d \cdot 2^{-d})$. The entropy measures how plentiful such packings are, and our result is significantly stronger than the trivial lower bound that…
In "Dense Sphere Packings: A Blueprint for Formal Proofs" Hales proves that for every packing of unit spheres, the density in a ball of radius $r$ is at most $\pi/\sqrt{18}+c/r$ for some constant $c$. When $r$ tends to infinity, this gives…
Sphere packing, Hilbert's eighteenth problem, asks for the densest arrangement of congruent spheres in n-dimensional Euclidean space. Although relevant to areas such as cryptography, crystallography, and medical imaging, the problem remains…
The Kepler conjecture asserts that no packing of congruent balls in three-dimensional Euclidean space has density greater than that of the face-centered cubic packing. In 1998, Sam Ferguson and I announced a computer-assisted proof of this…
We consider the hard-core lattice gas model on Z^d and investigate its phase structure in high dimensions. We prove that when the intensity parameter exceeds Cd^{-1/3}(log d)^2, the model exhibits multiple hard-core measures, thus improving…
Packing problems have been a source of fascination for millenia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure…