Estimates of the optimal density and kissing number of sphere packings in high dimensions

TL;DR

This paper investigates the asymptotic behavior of the maximal density of high-dimensional sphere packings, showing that a broad class of disordered packings can achieve an exponential density bound of approximately 1/2^{0.77865d}, improving upon Minkowski's classical bound.

Contribution

The authors demonstrate that the exponential density bound for high-dimensional sphere packings is more general than previously thought, applying to a wide class of disordered packings and test functions.

Findings

Exponential improvement over Minkowski's bound is achievable.

A broad class of test functions yields the same asymptotic density bound.

The asymptotic form 1/2^{0.77865d} is robust across different disordered packings.

Abstract

The problem of finding the asymptotic behavior of the maximal density of sphere packings in high Euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry. One century ago, Minkowski obtained a rigorous lower bound that is controlled asymptotically by $1/2^d$, where $d$ is the Euclidean space dimension. An indication of the difficulty of the problem can be garnered from the fact that exponential improvement of Minkowski's bound has proved to be elusive, even though existing upper bounds suggest that such improvement should be possible. Using a statistical-mechanical procedure to optimize the density associated with a "test" pair correlation function and a conjecture concerning the existence of disordered sphere packings [S. Torquato and F. H. Stillinger, Experimental Math. {\bf 15}, 307 (2006)], the putative exponential improvement was found with an asymptotic behavior controlled by $1/2^{(0.77865...)d}$. Using the same methods, we investigate whether this exponential improvement can be further improved by exploring other test pair correlation functions correponding to disordered packings. We demonstrate that there are simpler test functions that lead to the same asymptotic result. More importantly, we show that there is a wide class of test functions that lead to precisely the same exponential improvement and therefore the asymptotic form $1/2^{(0.77865...)d}$ is much more general than previously surmised.