Equality of critical parameters for percolation of Gaussian free field level sets
Equality of critical parameters for percolation of Gaussian free field level sets
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DOI：
10.1215/00127094-2022-0017
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发表时间：
2020-02
影响因子：
2.5
通讯作者：
H. Duminil-Copin;Subhajit Goswami;Pierre-François Rodriguez;Franco Severo
中科院分区：
文献类型：
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作者：
H. Duminil-Copin;Subhajit Goswami;Pierre-François Rodriguez;Franco Severo
我们考虑高斯自由场在$\mathbb Z^d$上的水平集，对于$d\geq 3$，高于给定的实值高度参数$h$。当h变化时，这定义了一个具有强代数衰减相关性的正则渗流模型。我们证明了与该模型相关的三个自然临界参数，即$h_{**}（d）$，$h_{*}（d）$和$\bar h（d）$，分别描述了一个有序的亚临界相，一个无限簇的出现和一个局部唯一性制度在超临界相中的开始，实际上是一致的，也就是说，对于任意的$d \geq 3$，$h_{*}（d）=h_{*}（d）= \bar h（d）$。在我们的证明的核心在于一个新的插值方案，旨在整合出的高斯自由场的长程依赖。这一策略的成功实施广泛依赖于某些新的重整化技术，特别是控制所谓的大场效应。这种方法开辟了一条道路，完全理解的强相关渗流模型的非临界阶段。
We consider 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, namely $h_{**}(d)$, $h_{*}(d)$ and $\bar h(d)$, 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, i.e. $h_{**}(d)=h_{*}(d)= \bar h(d)$ for any $d \geq 3$. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the Gaussian free field. The successful implementation of this strategy relies extensively on certain novel renormalization techniques, in particular to control so-called large-field effects. This approach opens the way to a complete understanding of the off-critical phases of strongly correlated percolation models.