Random perfect lattices and the sphere packing problem

Abstract

Motivated by the search for best lattice sphere packings in Euclidean spaces of large dimensions we study randomly generated perfect lattices in moderately large dimensions (up to d=19 included). Perfect lattices are relevant in the solution of the problem of lattice sphere packing, because the best lattice packing is a perfect lattice and because they can be generated easily by an algorithm. Their number however grows super-exponentially with the dimension so to get an idea of their properties we propose to study a randomized version of the algorithm and to define a random ensemble with an effective temperature in a way reminiscent of a Monte-Carlo simulation. We therefore study the distribution of packing fractions and kissing numbers of these ensembles and show how as the temperature is decreased the best know packers are easily recovered. We find that, even at infinite temperature, the typical perfect lattices are considerably denser than known families (like A_d and D_d) and we propose two hypotheses between which we cannot distinguish in this paper: one in which they improve Minkowsky's bound phi\sim 2^{-(0.84+-0.06) d}, and a competitor, in which their packing fraction decreases super-exponentially, namely phi\sim d^{-a d} but with a very small coefficient a=0.06+-0.04. We also find properties of the random walk which are suggestive of a glassy system already for moderately small dimensions. We also analyze local structure of network of perfect lattices conjecturing that this is a scale-free network in all dimensions with constant scaling exponent 2.6+-0.1.

19 pages, 22 figures

References in corpus (7)

• Robust Algorithm to Generate a Diverse Class of Dense Disordered and Ordered Sphere Packings via Linear Programming
• Amorphous packings of hard spheres in large space dimension
• Complexity and algorithms for computing Voronoi cells of lattices
• Estimates of the optimal density and kissing number of sphere packings in high dimensions
• On the most compact regular lattice in large dimensions: A statistical mechanical approach
• Application of Edwards' statistical mechanics to high dimensional jammed sphere packings
• High-dimensional generalizations of the kagome and diamond crystals and the decorrelation principle for periodic sphere packings

Cited by in corpus (6)

• Basic Understanding of Condensed Phases of Matter via Packing Models
• Statistical mechanics of the lattice sphere packing problem
• A geometric-Structure Theory for maximally Random Jammed Packings
• An Efficient Linear Programming Algorithm to Generate the Densest Lattice Sphere Packings
• Packing hard spheres with short-range attraction in infinite dimension: Phase structure and algorithmic implications
• An upper bound on the number of perfect quadratic forms