齐性空间上的不变高斯场：显式构造和平均节点体积
Invariant Gaussian fields on homogeneous spaces: explicit constructions and mean nodal volume
摘要：
本文在假定概率分布在齐次空间的等距群下不变的条件下，研究了齐次空间上光滑高斯随机场的一些性质。我们首先给出了一个阐述，建立在早期的结果Yaglom的方式，其中表示论和相关的特殊函数使人们有可能给完全明确的描述这些领域在许多情况下的利益。然后我们转向零集合的期望大小：扩展了光学和神经科学的二维结果，我们证明了每个不变场都有一个自然的体积单位（根据场中的几何冗余来定义），相对于该零集，零集的平均大小由仅取决于源空间和目标空间的维度的普适常数给出，而不是场所表现出的精确对称性。
英文摘要：
We review and study some of the properties of smooth Gaussian random fields defined on a homogeneous space, under the assumption that the probability distribution is invariant under the isometry group of the space. We first give an exposition, building on early results of Yaglom, of the way in which representation theory and the associated special functions make it possible to give completely explicit descriptions of these fields in many cases of interest. We then turn to the expected size of the zero-set: extending two-dimensional results from Optics and Neuroscience, we show that every invariant field comes with a natural unit of volume (defined in terms of the geometrical redundancies in the field) with respect to which the average size of the zero-set is given by a universal constant depending only on the dimension of the source and target spaces, and not on the precise symmetry exhibited by the field.
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