Estimates of the optimal density and kissing number of sphere packings in high dimensions

Details Estimates of the optimal density and kissing number of sphere packings in high dimensions Estimates of the optimal density and kissing number of sphere packings in high dimensions

2007-05-10

arxiv.org/abs/0705.1482v2

Physics

Condensed Matter

Statistical Mechanics

23 pages, 4 figures, submitted to Phys. Rev. E

Scientific paper

10.1063/1.2897027

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 that is controlled asymptotically by $1/2^d$, where $d$ 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, Experimental Math. {\bf 15}, 307 (2006)], the putative exponential improvement was found with an asymptotic behavior controlled by $1/2^{(0.77865...)d}$. Using the same methods, we investigate whether this exponential improvement can be further improved by exploring other test pair correlation functions correponding 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 exponential improvement and therefore the asymptotic form $1/2^{(0.77865...)d}$ is much more general than previously surmised.

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