Estimates of the optimal density of sphere packings in high dimensions

The problem of finding the asymptotic behavior of the maximal density ϕ max 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 ϕ max 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, Exp. Math. 15, 307 (2006)], the putative exponential improvement on ϕ max 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 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 1 ∕ 2 ( 0.77865 ⋯ ) d 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.