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ON THE HARD SPHERE MODEL AND SPHERE PACKINGS IN HIGH DIMENSIONS

Part of: Discrete geometry Equilibrium statistical mechanics

Published online by Cambridge University Press:  14 January 2019

MATTHEW JENSSEN ,
FELIX JOOS  and
WILL PERKINS
MATTHEW JENSSEN
Affiliation:
Mathematics Institute, University of Oxford, UK; matthew.jenssen@maths.ox.ac.uk
FELIX JOOS
Affiliation:
School of Mathematics, University of Birmingham, UK; f.joos@bham.ac.uk
WILL PERKINS
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, USA; math@willperkins.org

Abstract

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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.

MSC classification

Primary: 52C17: Packing and covering in $n$ dimensions
Secondary: 82B21: Continuum models (systems of particles, etc.)
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e1
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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