On the number of excursion sets of planar Gaussian fields
arXiv:1807.10209 · doi:10.1007/s00440-020-00984-9
Abstract
The Nazarov-Sodin constant describes the average number of nodal set components of 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.
arXiv comment38 pages, 14 figures. Updated to match published version and correct several typos. Probab. Theory Relat. Fields (2020)
References in corpus (1)
Cited by in corpus (5)
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- Smoothness and monotonicity of the excursion set density of planar Gaussian fields
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- A central limit theorem for the number of excursion set components of Gaussian fields
- A covariance formula for the number of excursion set components of Gaussian fields and applications