论文标题
关于高斯随机零集的拓扑
On the topology of Gaussian random zero sets
论文作者
论文摘要
我们研究了数量，贝蒂数量和同位素类别的渐近定律，这些组件是零的真实高斯随机场的零组成分，其中随机零组几乎肯定由大于或等于一个的代码符号组成。我们的结果包括“随机结”作为特殊情况。我们的工作与Berry在[4,5]中提出的一系列问题密切相关。特别是，我们的结果适用于出现在复杂算术随机波（示例1.5），Bargmann-Fock模型（示例1.1），黑体辐射（示例1.2）和Berry的单色随机波中的随机结的合并。我们的证明结合了针对随机标量值函数的水平集引入的技术，以及来自差异几何和差异拓扑的方法。
We study the asymptotic laws for the number, Betti numbers, and isotopy classes of connected components of zero sets of real Gaussian random fields, where the random zero sets almost surely consist of submanifolds of codimension greater than or equal to one. Our results include `random knots' as a special case. Our work is closely related to a series of questions posed by Berry in [4,5]; in particular, our results apply to the ensembles of random knots that appear in the complex arithmetic random waves (Example 1.5), the Bargmann-Fock model (Example 1.1), Black-Body radiation (Example 1.2), and Berry's monochromatic random waves. Our proofs combine techniques introduced for level sets of random scalar-valued functions with methods from differential geometry and differential topology.
