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 Montecarlo 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 φ ∼ 2 −(0.84±0.06)d , and a competitor, in which their packing fraction decreases super-exponentially, namely φ ∼ d −ad 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.

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“…One branch is that of the geometry of positive definite quadratic forms [40,41]. Utilizing results from this theory and the theory of random walks, Andreanov and Scardicchio [2] in the statistical mechanics community were able to generate very dense lattices. In fact, they were able to recover many of the densest known lattices in certain dimensions with high probability.…”

“…Other techniques have been developed in the statistical mechanics literature, e.g. [24], for generating lattices, but currently [2,31] seem to represent the cutting edge in terms of performance and accuracy. These results are also very recent, having been published only in the past few months.…”

“…The two algorithms that will be focused on in this section are [2,31], both published very recently. The algorithm of [2] is based on the Ryshkov polyhedron described in the previous section. Rather than attempt to enumerate all of the perfect forms as in the Voronoi algorithm, the vertices are transversed via a random walk.…”

“…Very recently, the statistical mechanics community has been interested in finding the most dense lattices of a given dimension. Several algorithms have been proposed that, while not finding the densest lattice, find very dense lattices, in some cases the densest known lattice for a given dimension ( [2,23,31,35,46,47]). Due to time, these algorithms unfortunately have not been implemented to try to find good lattices for our problem, but it is believed that these algorithms could be successful, and thus a brief description of them is included.…”

“…This algorithm can still be useful in that it can generate dense lattices but there is no guarantee about optimality without enumeration of all the perfect forms. Using the algorithm in an exploratory mode such as this reaches a limit around dimension 20 as indicated in [2,31].…”