随机球谐函数的亏损方差
The defect variance of random spherical harmonics
摘要：
函数$f:M\rightarrow \mathbb{R}$的缺陷被定义为正区域和负区域的度量之间的差。本文首先分析了随机高斯球谐函数的缺陷分布。通过一个简单的论点，缺陷是非平凡的，只有偶数次和期望值总是为零。我们的主要结果是获得的缺陷方差的渐近形状，在高频率限制。与随机本征函数的其他几何泛函一样，该亏损函数可以用来研究球形随机场的统计性质，这是现代宇宙学数据分析的一个重要课题.
英文摘要：
The defect of a function $f:M\rightarrow \mathbb{R}$ is defined as the difference between the measure of the positive and negative regions. In this paper, we begin the analysis of the distribution of defect of random Gaussian spherical harmonics. By an easy argument, the defect is non-trivial only for even degree and the expected value always vanishes. Our principal result is obtaining the asymptotic shape of the defect variance, in the high frequency limit. As other geometric functionals of random eigenfunctions, the defect may be used as a tool to probe the statistical properties of spherical random fields, a topic of great interest for modern Cosmological data analysis.
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