Author ORCID Identifier

0000-0003-2486-931X

Document Type

Article

Disciplines

1.1 MATHEMATICS, Probability

Publication Details

https://arxiv.org/abs/2205.09085

Dmitry Beliaev, Michael McAuley, Stephen Muirhead. "A central limit theorem for the number of excursion set components of Gaussian fields." The Annals of Probability, 52(3) 882-922 May 2024.

DOI:10.1214/23-AOP1672

Abstract

For a smooth stationary Gaussian field f on Rd and level ℓ ∈ R, we consider the number of connected components of the excursion set {f ≥ ℓ} (or level set {f = ℓ}) 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.

DOI

https://doi.org/10.1214/23-AOP1672

Funder

The second author was supported by the European Research Council (ERC) Advanced Grant QFPROBA (Grant number 741487) and completed part of this work while affiliated with the School of Mathematics and Statistics at Technological University Dublin. The third author was supported by the Australian Research Council (ARC) Discovery Early Career Researcher Award DE200101467, and also acknowledges the hospitality of the Statistical Laboratory, University of Cambridge, where part of this work was carried out.

Creative Commons License

Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License
This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.

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