A CENTRAL LIMIT THEOREM FOR THE NUMBER OF EXCURSION SET COMPONENTS OF GAUSSIAN FIELDS
成果类型：
Article
署名作者：
Beliaev, Dmitry; Mcauley, Michael; Muirhead, Stephen
署名单位：
University of Oxford; University of Helsinki; University of Melbourne
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/23-AOP1672
发表日期：
2024
页码：
882-922
关键词：
lipschitz-killing curvatures
percolation
crossings
moments
points
zeros
摘要：
3 School of Mathematics and Statistics, University of Melbourne, c smui@unimelb.edu.au For a smooth stationary Gaussian field f on R d and level is an element of R, we consider the number of connected components of the excursion set { f >= } (or level set { f = } ) 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.