Boundedness of level lines for two-dimensional random fields

• 成果类型: Article
• 署名作者: Alexander, KS
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1996
• 页码: 1653-1674
• 关键词: infinite clusters; percolation; uniqueness; graphs

摘要

Every two-dimensional incompressible flow follows the level lines of some scalar function psi on R(2); transport properties of the flow depend in part on whether all level lines are bounded. We study the structure of the level lines when psi is a stationary random held. We show that under mild hypotheses there is only one possible alternative to bounded level lines: the ''treelike'' random fields, which, for some interval of values of a, have a unique unbounded level line at each level a, with this line ''winding through every region of the plane.'' If the random field has the FKG property, then only bounded level lines are possible. For stationary C-2 Gaussian random fields with covariance function decaying to 0 at infinity, the treelike property is the only alternative to bounded level lines provided the density of the absolutely continuous part of the spectral measure decays at infinity ''slower than exponentially,'' and only bounded level lines are possible if the covariance function is nonnegative.