论文标题
高斯字段的游览集组件数量的中心限制定理
A central limit theorem for the number of excursion set components of Gaussian fields
论文作者
论文摘要
对于$ \ Mathbb {r}^d $和级别$ \ ell \ in \ Mathbb {r} $的平稳固定高斯字段，我们考虑了偏移的连接组件的数量$ \ \ \ {f \ ge \ ell \ el \ el \ el \ el \ el \ el \ el \ el \ el \ el \ el \ ell clupp $ \ \ \ \ \ \ \ \ el \ ell \ ell \ el \ \ ell \} $ concorment counce councation counce concoment。已知该数量的平均值可以像域中的一般假设下的域的体积一样扩展。我们证明，假设相关性的足够衰减（例如，bargmann fock磁场），则中心极限定理具有体积级缩放。以前，仅针对使用Hermite膨胀的“添加”几何函数（例如，体积或Euler特性）建立了这样的结果。基于Martingale分析，我们的方法更强大，可以推广到更广泛的拓扑功能。证明中的主要成分是临界点上的第三刻，这是独立的。
For a smooth stationary Gaussian field on $\mathbb{R}^d$ and level $\ell \in \mathbb{R}$, we consider the number of connected components of the excursion set $\{f \ge \ell\}$ (or level set $\{f = \ell\}$) contained in large domains. The mean of this quantity is known to scale like the volume of the domain under general assumptions on the field. We prove that, assuming sufficient decay of correlations (e.g. the Bargmann-Fock field), a central limit theorem holds with volume-order scaling. Previously such a result had only been established for `additive' geometric functionals of the excursion/level sets (e.g. the volume or Euler characteristic) using Hermite expansions. Our approach, based on a martingale analysis, is more robust and can be generalised to a wider class of topological functionals. A major ingredient in the proof is a third moment bound on critical points, which is of independent interest.
