标题： 关于高斯随机零点集的拓扑结构 显示英文标题

标题： On the topology of Gaussian random zero sets

摘要： 我们研究实高斯随机场零点集的连通分支的数量、贝蒂数和同伦类的渐近规律，其中随机零点集几乎必然由余维大于或等于一的子流形组成。 我们的结果包括“随机扭结”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，我们的结果适用于复数算术随机波（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扭结集合。 我们的证明结合了用于随机标量值函数水平集的技术以及微分几何和微分拓扑的方法。 显示英文摘要

摘要： 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.