On the hard sphere model and sphere packings in high dimensions

Abstract

We prove a lower bound on the entropy of sphere packings of ${\mathbb{R}}^d$ of density $\Theta (d\cdot 2^{-d})$. 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 $\Omega (d\cdot 2^{-d})$ 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 $(1+o_d(1))\log (2/\sqrt{3})d\cdot 2^{-d}$ 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.

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Cited by in corpus (4)

• The number of independent sets in an irregular graph
• Perfect simulation of the Hard Disks Model by Partial Rejection Sampling
• Three simple scenarios for high-dimensional sphere packings
• Dynamic Space Packing