A central limit theorem for the number of excursion set components of Gaussian fields

Abstract

For a smooth stationary Gaussian field on ${\mathbb{R}}^d$ and level $\ell \in \mathbb{R}$, we consider the number of connected components of the excursion set $\{f\ge \ell \}$ (or level set $\{f=\ell \}$) contained in large domains. The mean of this quantity is known to scale like the volume of the domain under general assumptions on the field. We prove that, assuming sufficient decay of correlations (e.g. the Bargmann-Fock field), a central limit theorem holds with volume-order scaling. Previously such a result had only been established for `additive' geometric functionals of the excursion/level sets (e.g. the volume or Euler characteristic) using Hermite expansions. Our approach, based on a martingale analysis, is more robust and can be generalised to a wider class of topological functionals. A major ingredient in the proof is a third moment bound on critical points, which is of independent interest.

arXiv comment36 pages, 1 figure. Published version

References in corpus (10)

• Level crossings and other level functionals of stationary Gaussian processes
• Nodal portraits of quantum billiards: Domains, lines, and statistics
• The sharp phase transition for level set percolation of smooth planar Gaussian fields
• Unexpected Topology of the Temperature Fluctuations in the Cosmic Microwave Background
• On the second moment of the number of crossings by a stationary Gaussian process
• Fluctuations of the number of excursion sets of planar Gaussian fields
• On the number of excursion sets of planar Gaussian fields
• Fluctuations in the number of nodal domains
• Smoothness and monotonicity of the excursion set density of planar Gaussian fields
• On the expected Betti numbers of the nodal set of random fields

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