论文标题
高斯自由场级别渗透的关键参数的平等性
Equality of critical parameters for percolation of Gaussian free field level-sets
论文作者
论文摘要
我们将高斯免费字段的上层集合在$ \ mathbb z^d $上，对于$ d \ geq 3 $，高于给定的实值高度参数$ h $。随着$ H $的变化，这定义了具有强，代数衰减相关的规范渗透模型。我们证明，与该模型相关的三个自然临界参数分别描述了有序的亚临界相，无限群集的出现以及在超临界阶段中局部唯一性制度的发作，实际上是重合的。我们证明的核心是一种新的插值方案，旨在整合高斯自由领域的远程依赖性。由于相关性的强度，其成功的实施要求我们在有效的关键政权中工作。我们的分析广泛依赖于某些新型的重新归一化技术，这些技术同时发挥了所有相关量表。本文中的方法铺平了对密切相关的无序系统的非临界阶段的完整理解的道路。
We consider upper level-sets of the Gaussian free field on $\mathbb Z^d$, for $d\geq 3$, above a given real-valued height parameter $h$. As $h$ varies, this defines a canonical percolation model with strong, algebraically decaying correlations. We prove that three natural critical parameters associated to this model, respectively describing a well-ordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the Gaussian free field. Due to the strength of correlations, its successful implementation requires that we work in an effectively critical regime. Our analysis relies extensively on certain novel renormalization techniques that bring into play all relevant scales simultaneously. The approach in this article paves the way to a complete understanding of the off-critical phases for strongly correlated disordered systems.
