Lattice-Packing by Spheres and Eutactic Forms

Abstract

We consider a semi-random walk on the space X of lattices in Euclidean n-space which attempts to maximize the sphere-packing density function Φ. A lattice (or its corresponding quadratic form) is called “sticky” if the set of directions in X emanating from it along which Φ is infinitesimally increasing has measure 0 in the set of all directions. Thus the random walk will tend to get “stuck” in the vicinity of a sticky lattice. We prove that a lattice is sticky if and only if the corresponding quadratic form is semi-eutactic. We prove our results in the more general setting of self-adjoint homogeneous cones. We also present results from our experiments with semi-random walks on X. These indicate some idea about the landscape of eutactic lattices in the space of all lattices.

Keywords:

• sphere-packing
• eutactic lattices

2010 Mathematics Subject Classification:

• Primary 11H31
• secondary 37B30
• 82C41

Acknowledgements

We wish to thank Roland Bacher, Jacques Martinet, Gabriele Nebe, and Christophe Soulé for help in tracking down recent results about the number of eutactic lattices in various dimensions. We thank the referee of an earlier version of this paper for many helpful comments.