On the number of excursion sets of planar 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 London; King's College London; University of London; Queen Mary University London
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00984-9
• 发表日期: 2020
• 页码: 655-698
• 关键词: constant; points

摘要

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.