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.

摘要

我们考虑在欧几里得n空间中的格子空间X上进行半随机游走，它试图使球体堆积密度函数 Φ 最大化。如果从它发出的X中的一组方向，Φ 沿着它无限增加的方向在所有方向的集合中测量为 0，则一个格（或其相应的二次形式）被称为“粘性” 。因此，随机游走往往会“卡”在粘性格子附近。我们证明格是粘性的当且仅当对应的二次形式是半正规的。我们在自伴同质锥的更一般设置中证明了我们的结果。我们还展示了我们在X上进行半随机游走的实验结果. 这些表明了一些关于在所有格子空间中正构格子的景观的想法。