The problem of finding the asymptotic behavior of the maximal density of sphere packings in high Euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry. One century ago, Minkowski obtained a rigorous lower bound on that is controlled asymptotically by , where is the Euclidean space dimension. An indication of the difficulty of the problem can be garnered from the fact that exponential improvement of Minkowski’s bound has proved to be elusive, even though existing upper bounds suggest that such improvement should be possible. Using a statistical-mechanical procedure to optimize the density associated with a “test” pair correlation function and a conjecture concerning the existence of disordered sphere packings [S. Torquato and F. H. Stillinger, Exp. Math. 15, 307 (2006)], the putative exponential improvement on was found with an asymptotic behavior controlled by . Using the same methods, we investigate whether this exponential improvement can be further improved by exploring other test pair correlation functions corresponding to disordered packings. We demonstrate that there are simpler test functions that lead to the same asymptotic result. More importantly, we show that there is a wide class of test functions that lead to precisely the same putative exponential improvement and therefore the asymptotic form is much more general than previously surmised. This class of test functions leads to an optimized average kissing number that is controlled by the same asymptotic behavior as the one found in the aforementioned paper.
Skip Nav Destination
Article navigation
April 2008
Research Article|
April 03 2008
Estimates of the optimal density of sphere packings in high dimensions
A. Scardicchio;
A. Scardicchio
a)
1Department of Physics, Joseph Henry Laboratories,
Princeton University
, Princeton, New Jersey 08544, USA
2Princeton Center for Theoretical Physics,
Princeton University
, Princeton, New Jersey 08544, USA
Search for other works by this author on:
F. H. Stillinger;
F. H. Stillinger
b)
3Department of Chemistry,
Princeton University
, Princeton, New Jersey 08544, USA
Search for other works by this author on:
S. Torquato
S. Torquato
c)
2Princeton Center for Theoretical Physics,
Princeton University
, Princeton, New Jersey 08544, USA
3Department of Chemistry,
Princeton University
, Princeton, New Jersey 08544, USA
4Program in Applied and Computational Mathematics,
Princeton University
, Princeton, New Jersey 08544, USA
5Princeton Institute for the Science and Technology of Materials,
Princeton University
, Princeton, New Jersey 08544, USA
6
School of Natural Sciences, Institute for Advanced Study
, Princeton, New Jersey, 08540, USA
Search for other works by this author on:
J. Math. Phys. 49, 043301 (2008)
Article history
Received:
September 22 2007
Accepted:
February 15 2008
Citation
A. Scardicchio, F. H. Stillinger, S. Torquato; Estimates of the optimal density of sphere packings in high dimensions. J. Math. Phys. 1 April 2008; 49 (4): 043301. https://doi.org/10.1063/1.2897027
Download citation file:
Sign in
Don't already have an account? Register
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
Could not validate captcha. Please try again.
Sign in via your Institution
Sign in via your InstitutionPay-Per-View Access
$40.00