Extreme lattices: symmetries and decorrelations

Abstract

We study statistical and structural properties of extreme lattices, which are the local minima in the density landscape of lattice sphere packings, in $d$-dimensional Euclidean space $\mathbb{R}^d$. Specifically, we ascertain the distributions of densities, kissing numbers and numbers of symmetries of the packings across a wide range of dimensions using the stochastic Voronoi algorithm. The degree to which the packings decorrelate as well as the correlations between the density maxima as the space dimension increases is also investigated. We find that the extreme lattices decorrelate with increasing dimension, the least symmetric lattices decorrelate faster. The extreme lattices in a fixed dimension of space $d$ ($d\geq 8$) are dominated by typical lattices that have similar packing properties, like packing densities and kissing numbers, while the best and the worst packers are in the long tails of the distribution of the extreme lattices.