[Paper Review] A new lower bound for sphere packing

Marcelo Campos, Matthew Jenssen+2 more|arXiv (Cornell University)|Dec 15, 2023

Point processes and geometric inequalities4 citations

TL;DR

This paper establishes a new asymptotic lower bound for sphere packing density in high-dimensional Euclidean space, achieving a factor of order $d\log d$ improvement over Rogers' 1947 bound. The authors construct highly disordered sphere packings via a recursive random process on a modified Poisson point process, demonstrating that the density exceeds $(1-o(1))\frac{d\log d}{2^{d+1}}$ as $d\to\infty$, marking the first such improvement beyond the linear factor in decades.

ABSTRACT

We show there exists a packing of identical spheres in $\mathbb{R}^d$ with density at least \[ (1-o(1))\frac{d \log d}{2^{d+1}}\, , \] as $d o\infty$. This improves upon previous bounds for general $d$ by a factor of order $\log d$ and is the first asymptotically growing improvement to Rogers' bound from 1947.

Motivation & Objective

• To establish a new asymptotic lower bound for the sphere packing density $\theta(d)$ in high-dimensional Euclidean space $\mathbb{R}^d$.
• To overcome the long-standing barrier of linear improvements over Minkowski's bound by introducing a method that yields a super-linear $d\log d$ factor improvement.
• To demonstrate that disordered, non-lattice sphere packings can outperform structured lattice-based constructions in terms of asymptotic density.
• To extend the method to spherical codes, providing a new lower bound for the size of spherical codes with given angular separation.
• To address the open question of whether high-dimensional sphere packings exhibit phase transitions from disordered to ordered structures, as suggested in statistical physics.

Proposed method

• Construct a random sphere packing in $\mathbb{R}^d$ by recursively adding spheres centered on a modified Poisson point process in $\mathbb{R}^d$.
• Use a randomized, iterative construction process that avoids overlaps by rejecting spheres that would intersect existing ones.
• Apply probabilistic methods to control the expected number of overlapping spheres and ensure a high-density packing with high probability.
• Leverage concentration inequalities and tail bounds to show that most points in the Poisson process have few neighbors within the sphere radius, ensuring sparsity of conflicts.
• Adapt the method to spherical codes by applying it on the unit sphere $S^{d-1}$, using angular separation constraints to define spherical caps.
• Use a Poisson process of intensity $\lambda = \left(\frac{\sqrt{d}}{2\log d}\right)^d$ on $S^{d-1}$ and delete points with high local density or excessive overlap to obtain a valid code.

Experimental results

Research questions

• RQ1 Can a disordered, non-lattice construction achieve a higher asymptotic sphere packing density than known lattice-based constructions in high dimensions?
• RQ2 What is the maximal possible growth rate of the sphere packing density $\theta(d)$ as $d \to \infty$, beyond the linear factor in Rogers' bound?
• RQ3 Can the method used for sphere packing be adapted to yield improved lower bounds for spherical codes with angular separation $\theta$?
• RQ4 Does the exponential gap between upper and lower bounds for $\theta(d)$ persist, and can it be reduced by non-lattice constructions?
• RQ5 Can random, amorphous sphere packings model phase transitions in physical systems, such as from gas-like to lattice-like behavior in high dimensions?

Key findings

• The paper establishes a new lower bound for sphere packing density: $\theta(d) \geq (1-o(1))\frac{d\log d}{2^{d+1}}$ as $d \to \infty$, improving Rogers' bound by a factor of order $d\log d$.
• This improvement is the first asymptotically growing enhancement over Rogers' 1947 bound, which had stood for over 70 years.
• The construction is highly disordered and non-lattice-based, contrasting with all prior improvements that relied on structured lattices.
• For spherical codes, the method yields $A(d,\theta) \geq (1-o(1))\frac{d\log d}{2s_d(\theta)}$ as $d \to \infty$, where $s_d(\theta)$ is the normalized spherical cap area.
• In the case $\theta = \pi/3$, this implies a new lower bound for the kissing number: $K(d) \geq (1-o(1))\sqrt{\frac{\pi}{8}}\left(\frac{2}{\sqrt{3}}\right)^{d-1}d^{3/2}\log d$.
• The proof relies on a Poisson process on $S^{d-1}$ with intensity $\lambda = \left(\frac{\sqrt{d}}{2\log d}\right)^d$, followed by deletion of points with high local overlap to ensure independence.