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Using Microwave and Macroscopic Samples of Dielectric Solids to Study the Photonic Properties of Disordered Photonic Bandgap Materials
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Random perfect lattices and the sphere packing problem.

A Andreanov1, A Scardicchio

  • 1Abdus Salam ICTP, Strada Costiera 11, 34151, Trieste, Italy.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 11, 2012
PubMed
Summary
This summary is machine-generated.

Researchers explored random perfect lattices to find optimal sphere packings. They discovered denser lattices than previously known, potentially improving packing efficiency in high-dimensional spaces.

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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.