The Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave.

中文翻译：

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关于平面高斯场的偏移集数

Nazarov-Sodin 常数描述了大尺度平滑高斯场的节点集分量的平均数量。我们将其概括为描述任意级别的相应数量的级别集组件的函数。使用莫尔斯理论的结果，我们将这个泛函表示为不同类型临界点的水平密度的积分，从而推导出随着水平变化的泛函的绝对连续性。我们进一步给出了上限和下限，表明对于某些各向同性场，泛函至少是双峰的，包括随机平面波的重要特例。