维数 $d \ge 3$ 下光滑高斯场存在无界节点超曲面
概率论
2023-07-20 v3
摘要
对于 上 的 Bargmann--Fock 场,我们证明了由超越集 构成的渗流模型的临界水平 严格为正。这意味着对于每一个充分接近 的 (特别是对应于 情形的节点超曲面), 包含一个无界连通分支,其遍历了环境空间的“大部分”。我们的发现实际上对一类具有快速关联衰减的正相关光滑高斯场也成立。本文结果表明,这些高斯场在 ()上的节点超曲面行为与其二维类似物的节点线行为非常不同。
引用
@article{arxiv.2108.08008,
title = {Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension $d \ge 3$},
author = {Hugo Duminil-Copin and Alejandro Rivera and Pierre-François Rodriguez and Hugo Vanneuville},
journal= {arXiv preprint arXiv:2108.08008},
year = {2023}
}
备注
61 pages, 4 figures. The previous version contains one error: Prop 1.12 therein is stated with a sprinkling, which we cannot afford in Sect 5, where it is applied at mesoscopic scales. We circumvent this issue by proving a stronger version of Prop 1.12, which contains no sprinkling and is interesting in its own right. We have also added details with references about proofs relying on Morse theory