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摘要翻译：
在高欧氏维数下求球形填料最大密度的渐近性态是离散几何中最吸引人和最具挑战性的问题之一。一个世纪前，Minkowski得到了一个严格的下界，该下界由$1/2^d$渐近控制，其中$d$是欧几里得空间维数。这个问题的困难可以从这样一个事实中得到一个迹象，即Minkowski界的指数改进已经证明是难以捉摸的，尽管现有的上界表明这种改进应该是可能的。利用统计-机械过程优化与“测试”对相关函数有关的密度和关于无序球形填料存在的猜想[S.Torquato和F.H.Stillinger,实验数学。{\bf 15},307(2006)],发现了假定的指数改进，其渐近行为受$1/2^{(0.77865…)d}$控制。利用同样的方法，我们研究了这种指数改进是否可以通过探索与无序包装相关的其他测试对相关函数来进一步改进。我们证明了有更简单的测试函数导致相同的渐近结果。更重要的是，我们证明了有一个广泛的测试函数类导致精确相同的指数改进，因此渐近形式$1/2^{(0.77865...)d}$比以前猜测的更一般。
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英文标题：
《Estimates of the optimal density and kissing number of sphere packings
in high dimensions》
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作者：
A. Scardicchio, F.H. Stillinger, S. Torquato
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最新提交年份：
2007
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分类信息：
一级分类：Physics 物理学
二级分类：Statistical Mechanics 统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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英文摘要：
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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PDF链接：
https://arxiv.org/pdf/705.1482
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