Exactly Solvable Disordered Sphere-Packing Model in Arbitrary-Dimension Euclidean Spaces

Abstract

We introduce a generalization of the well-known random sequential addition (RSA) process for hard spheres in $d$-dimensional Euclidean space $\mathbb{R}^d$. We show that all of the $n$-particle correlation functions of this nonequilibrium model, in a certain limit called the "ghost" RSA packing, can be obtained analytically for all allowable densities and in any dimension. This represents the first exactly solvable disordered sphere-packing model in arbitrary dimension. The fact that the maximal density $ϕ(\infty)=1/2^d$ of the ghost RSA packing implies that there may be disordered sphere packings in sufficiently high $d$ whose density exceeds Minkowski's lower bound for Bravais lattices, the dominant asymptotic term of which is $1/2^d$. Indeed, we report on a conjectural lower bound on the density whose asymptotic behavior is controlled by $2^{-(0.77865...) d}$, thus providing the putative exponential improvement on Minkowski's 100-year-old bound. Our results suggest that the densest packings in sufficiently high dimensions may be disordered rather than periodic, implying the existence of disordered classical ground states for some continuous potentials.