EXISTENCE OF AN UNBOUNDED NODAL HYPERSURFACE FOR SMOOTH GAUSSIAN FIELDS IN DIMENSION d ≥ 3

• 成果类型: Article
• 署名作者: Duminil-Copin, Hugo; Rivera, Alejandro; Rodriguez, Pierre-Francois; Vanneuville, Hugo
• 署名单位: University of Geneva; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Imperial College London; Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-APP1594
• 发表日期: 2023
• 页码: 228-276
• 关键词: percolation; phase; set

摘要

For the Bargmann-Fock field on R-d with d >= 3, we prove that the critical level l(c) (d) of the percolation model formed by the excursion sets {f >= l} is strictly positive. This implies that for every l sufficiently close to 0 (in particular for the nodal hypersurfaces corresponding to the case l = 0), {f = l} contains an unbounded connected component that visits most of the ambient space. Our findings actually hold for a more general class of positively correlated smooth Gaussian fields with rapid decay of correlations. The results of this paper show that the behavior of nodal hypersurfaces of these Gaussian fields in R-d for d >= 3 is very different from the behavior of nodal lines of their 2-dimensional analogues.