On the hard sphere model and sphere packings in high dimensions

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.