Geodesic
ON THE HARD SPHERE MODEL AND SPHERE PACKINGS IN HIGH DIMENSIONS
Forum of Mathematics, Sigma, Tome 7 (2019)

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We prove a lower bound on the entropy of sphere packings of $\mathbb{R}^{d}$ of density $\unicode[STIX]{x1D6E9}(d\cdot 2^{-d})$. 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 $\unicode[STIX]{x1D6FA}(d\cdot 2^{-d})$ 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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     author = {MATTHEW JENSSEN and FELIX JOOS and WILL PERKINS},
     title = {ON {THE} {HARD} {SPHERE} {MODEL} {AND} {SPHERE} {PACKINGS} {IN} {HIGH} {DIMENSIONS}},
     journal = {Forum of Mathematics, Sigma},
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     doi = {10.1017/fms.2018.25},
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MATTHEW JENSSEN; FELIX JOOS; WILL PERKINS. ON THE HARD SPHERE MODEL AND SPHERE PACKINGS IN HIGH DIMENSIONS. Forum of Mathematics, Sigma, Tome 7 (2019). doi: 10.1017/fms.2018.25

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