On the hard sphere model and sphere packings in high dimensions
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 m...
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| Main Authors: | , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
14 January 2019
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| In: |
Forum of mathematics. Sigma
Year: 2019, Volume: 7, Pages: 1-19 |
| ISSN: | 2050-5094 |
| DOI: | 10.1017/fms.2018.25 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1017/fms.2018.25 Verlag, lizenzpflichtig, Volltext: https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/on-the-hard-sphere-model-and-sphere-packings-in-high-dimensions/230F9C40FC941DA38505046530C2F2A1 
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| Author Notes: | Matthew Jenssen, Felix Joos and Will Perkins |
MARC
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| 520 | |a We prove a lower bound on the entropy of sphere packings of RdRd\mathbb{R}^{d} of density Θ(d⋅2−d)(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)(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. | ||
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