ON THE HARD SPHERE MODEL AND SPHERE PACKINGS IN HIGH DIMENSIONS

We prove a lower bound on the entropy of sphere packings of R-d of density Theta (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 Omega(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 + o(d)(1)) log(2/root 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.