各向同性中心极限随机场场的期望Minkowski泛函的渐近展开
Asymptotic expansion of the expected Minkowski functional for isotropic central limit random fields
摘要：
闵可夫斯基泛函，包括欧拉示性数统计，是宇宙学形态学分析的标准工具。受宇宙研究的启发，我们研究了各向同性中心极限随机场的漂移集的Minkowski泛函，其$k$点相关函数（$k$阶累积量）具有与宇宙研究中假设的相同的结构.利用3点和4点的相关函数，我们得到了欧拉示性数密度的渐近展开，它是Minkowski泛函的基石。由此产生的公式揭示了闵可夫斯基泛函无法捕获的非高斯性类型。作为一个例子，我们考虑了各向同性卡方随机场，并确认渐近展开准确地逼近真实的欧拉示性数密度。
英文摘要：
The Minkowski functionals, including the Euler characteristic statistics, are standard tools for morphological analysis in cosmology. Motivated by cosmic research, we examine the Minkowski functional of the excursion set for an isotropic central limit random field, the $k$-point correlation functions ($k$th order cumulants) of which have the same structure as that assumed in cosmic research. Using 3- and 4-point correlation functions, we derive the asymptotic expansions of the Euler characteristic density, which is the building block of the Minkowski functional. The resulting formula reveals the types of non-Gaussianity that cannot be captured by the Minkowski functionals. As an example, we consider an isotropic chi-square random field and confirm that the asymptotic expansion accurately approximates the true Euler characteristic density.
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