On the hard sphere model and sphere packings in high dimensions - Jenssen, Matthew; Joos, Felix; Perkins, Will (Forum of mathematics. Sigma, 2019)
We prove a lower bound on the entropy of sphere packings of RdRd\mathbb{R}^{d} of density Θ(d⋅2−d)𝛩(d⋅2−d)\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 Ω(d⋅2−d)𝛺(d⋅2−d)\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+od(1))log(2/3-√)d⋅2−d(1+od(1))log⁡(2/3)d⋅2−d(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/23−d/23^{-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.
Gesehen am 27.07.2022