Abstract
We prove a lower bound on the entropy of sphere packings of Rd of density _.d _ 2􀀀d /. The entropy measures how plentiful such packings are, and our result is significantly stronger than the trivial lower bound that can be obtained from the mere existence of a dense packing. Our method also provides a new, statistical-physics-based proof of the .d _ 2􀀀d / lower bound on the maximum sphere packing density by showing that the expected packing density of a random configuration from the hard sphere model is at least .1 C od .1// log.2=p 3/d _ 2􀀀d when the ratio of the fugacity parameter to the volume covered by a single sphere is at least 3􀀀d=2. Such a bound on the sphere packing density was first achieved by Rogers, with subsequent improvements to the leading constant by Davenport and Rogers, Ball, Vance, and Venkatesh.
Original language | English |
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Article number | e1 |
Number of pages | 19 |
Journal | Forum of Mathematics, Sigma |
Volume | 7 |
Early online date | 14 Jan 2019 |
DOIs | |
Publication status | E-pub ahead of print - 14 Jan 2019 |