New Conjectural Lower Bounds on the Optimal Density of Sphere Packings

@article{Torquato2006NewCL,
  title={New Conjectural Lower Bounds on the Optimal Density of Sphere Packings},
  author={Salvatore Torquato and Frank H. Stillinger},
  journal={Experimental Mathematics},
  year={2006},
  volume={15},
  pages={307 - 331}
}
Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a hundred-year-old lower bound on the maximal packing density due to Minkowski in d-dimensional Euclidean space ℝ d . The asymptotic behavior of this bound is controlled by 2-d in high dimensions. Using an optimization procedure that we introduced earlier [Torquato and Stillinger 02] and a… Expand
Estimates of the optimal density of sphere packings in high dimensions
The problem of finding the asymptotic behavior of the maximal density ϕmax of sphere packings in high Euclidean dimensions is one of the most fascinating and challenging problems in discreteExpand
Sphere packing and quantum gravity
We establish a precise relation between the modular bootstrap, used to constrain the spectrum of 2D CFTs, and the sphere packing problem in Euclidean geometry. The modular bootstrap bound for chiralExpand
Densest local sphere-packing diversity: general concepts and application to two dimensions.
TLDR
This paper finds the putative densest packings and corresponding Rmin(N) for selected values of N up to N=348 and uses this knowledge to construct such a realizability condition and an upper bound on the maximal density of infinite sphere packings in Rd. Expand
Efficient linear programming algorithm to generate the densest lattice sphere packings.
  • É. Marcotte, S. Torquato
  • Mathematics, Medicine
  • Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2013
TLDR
The Torquato-Jiao packing algorithm is applied, which is a method based on solving a sequence of linear programs, to robustly reproduce the densest known lattice sphere packings for dimensions 2 through 19 and it is shown that the TJ algorithm is appreciably more efficient at solving these problems than previously published methods. Expand
Packing hyperspheres in high-dimensional Euclidean spaces.
TLDR
Although the stable phase at high density is a crystalline solid, nucleation appears to be strongly suppressed with increasing dimension, consistent with a recently proposed "decorrelation principle". Expand
Random perfect lattices and the sphere packing problem.
TLDR
It is found that, even at infinite temperature, the typical perfect lattices are considerably denser than known families (like A(d) and D(d)), and a competitor in which their packing fraction decreases superexponentially is proposed, namely, φ~d(-ad) but with a very small coefficient a=0.04. Expand
Hyperuniformity order metric of Barlow packings.
TLDR
The geometry of three classes of Barlow packings, which are the infinitely degenerate densest packings of identical rigid spheres that are distinguished by their stacking geometries, is described and it is found that the value of Λ[over ¯] of all BarlowPackings is primarily controlled by the local cluster geometry. Expand
Spherical codes, maximal local packing density, and the golden ratio
The densest local packing (DLP) problem in d-dimensional Euclidean space Rd involves the placement of N nonoverlapping spheres of unit diameter near an additional fixed unit-diameter sphere such thatExpand
ON THE HARD SPHERE MODEL AND SPHERE PACKINGS IN HIGH DIMENSIONS
We prove a lower bound on the entropy of sphere packings of $\mathbb{R}^{d}$ of density $\unicode[STIX]{x1D6E9}(d\cdot 2^{-d})$ . The entropy measures how plentiful such packings are, and our resultExpand
Cavity approach to sphere packing in Hamming space.
TLDR
It is shown that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically the same as the lower bound of Gilbert and Varshamov. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 83 REFERENCES
Exactly solvable disordered sphere-packing model in arbitrary-dimensional Euclidean spaces.
TLDR
The results suggest that the densest packings in sufficiently high dimensions may be disordered rather than periodic, implying the existence of disordered classical ground states for some continuous potentials. Expand
Finite and Uniform Stability of Sphere Packings
TLDR
It is shown that many of the usual best-known candidates, for the most dense packings with congruent spherical balls, have the property of being uniformly stable, i.e., for a sufficiently small ε > 0 every finite rearrangement of the balls of this packing, where no ball is moved more than ε, is the identity rearrangements. Expand
Sphere packings, I
  • T. Hales
  • Mathematics, Computer Science
  • Discret. Comput. Geom.
  • 1997
TLDR
A program to prove the Kepler conjecture on sphere packings is described and it is shown that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture. Expand
The Kepler conjecture
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 thanExpand
A linear programming algorithm to test for jamming in hard-sphere packings
Jamming in hard-particle packings has been the subject of considerable interest in recent years. In a paper by Torquato and Stillinger [J. Phys. Chem. B 105 (2001)], a classification scheme of jammedExpand
Jamming in hard sphere and disk packings
Hard-particle packings have provided a rich source of outstanding theoretical problems and served as useful starting points to model the structure of granular media, liquids, living cells, glasses,Expand
Sphere Packings, Lattices and Groups
  • J. Conway, N. Sloane
  • Mathematics, Computer Science
  • Grundlehren der mathematischen Wissenschaften
  • 1988
The second edition of this book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also continue toExpand
Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings.
TLDR
The computational data unambiguously separate the narrowing delta -function contribution to g(2) due to emerging interparticle contacts from the background contribution due to near contacts and find that ordering has a significant impact on the shape of P(f) for small forces. Expand
Aspects of correlation function realizability
The pair-correlation function g2(r) describes short-range order in many-particle systems. It must obey two necessary conditions: (i) non-negativity for all distances r, and (ii) non-negativity of itsExpand
Controlling the Short-Range Order and Packing Densities of Many-Particle Systems†
Questions surrounding the spatial disposition of particles in various condensed-matter systems continue to pose many theoretical challenges. This paper explores the geometric availability ofExpand
...
1
2
3
4
5
...