Random perfect lattices and the sphere packing problem

Area of Science:

• Mathematics
• Geometry
• Computer Science

Background:

• The problem of finding the best lattice sphere packings in high-dimensional Euclidean spaces is a longstanding challenge.
• Perfect lattices are crucial for optimal sphere packing and are relatively easy to generate.
• The number of perfect lattices grows superexponentially with dimension, necessitating new study approaches.

Purpose of the Study:

• To investigate the properties of perfect lattices in moderately large dimensions (up to d=19).
• To explore a randomized algorithm for generating perfect lattices and define a random ensemble with an effective temperature.
• To analyze the distribution of packing fractions and kissing numbers within these ensembles.

Main Methods:

• A randomized version of the perfect lattice generating algorithm was employed.
• A random ensemble with an effective temperature was defined, simulating Monte Carlo methods.
• Distributions of packing fractions and kissing numbers were studied as temperature varied.

Main Results:

• Decreasing temperature in the ensemble recovered known best-known lattice sphere packers.
• Typical perfect lattices, even at infinite temperature, are denser than established families like A(d) and D(d).
• Two hypotheses regarding packing fraction improvement were proposed: one improving the Minkowski bound, another with superexponential decrease.

Conclusions:

• Randomized perfect lattices offer a promising avenue for discovering denser sphere packings.
• The study suggests potential improvements to the Minkowski bound for sphere packing.
• Properties observed in the random walk dynamics hint at glassy system behavior in moderately small dimensions.