The giant component of excursion sets of spherical Gaussian ensembles: existence, uniqueness, and volume concentration

• 成果类型: Article; Early Access
• 署名作者: Muirhead, Stephen; Wigman, Igor
• 署名单位: University of Melbourne; University of London; King's College London; Monash University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01367-8
• 发表日期: 2025
• 关键词: large deviations; free-field; percolation; probability; discrete

摘要

We establish the existence and uniqueness of a well-concentrated giant component in the supercritical excursion sets of three important ensembles of spherical Gaussian random fields: Kostlan's ensemble, band-limited ensembles, and the random spherical harmonics. Our main results prescribe quantitative bounds for the volume fluctuations of the giant that are essentially optimal for non-monochromatic ensembles, and suboptimal but still strong for monochromatic ensembles. Our results support the emerging picture that giant components in Gaussian random field excursion sets have similar large-scale statistical properties to giant components in supercritical Bernoulli percolation. The proofs employ novel decoupling inequalities for spherical ensembles which are of independent interest.