论文标题
强烈相关的高斯田地的渗透II。相变的清晰度
Percolation of strongly correlated Gaussian fields II. Sharpness of the phase transition
论文作者
论文摘要
我们在$ \ mathbb {z}^d $或$ \ mathbb {r}^d $，$ d \ ge 2 $上建立了广泛的高斯渗滤模型的相过渡的清晰度，相关性衰减至少与代数$α> 0 $，包括$α> $ divet geege gee gee geege geege n = d \ d \ d \ d \ d \ d $ d c.膜模型（$ d \ ge 5，α= d -4 $），以及许多其他示例，既离散又连续。特别是我们不假定正相关。 This result is new for all strongly correlated models (i.e. $α\in (0,d]$) in dimension $d \ge 3$ except the Gaussian free field, for which sharpness was proven in a recent breakthrough by Duminil-Copin, Goswami, Rodriguez and Severo; even then, our proof is simpler and yields new near-critical information on the percolation density. 对于连续且相关的平面场，我们通过利用新的“弱混合”特性来建立渗透密度的更尖锐的界限，以实现强烈相关的高斯领域。作为副产品，我们建立了具有独立关注的节点集的盒装属性。 这是一系列两篇论文中的第二篇研究，研究了密切相关的高斯田地的水平集渗透，可以独立读取。
We establish the sharpness of the phase transition for a wide class of Gaussian percolation models, on $\mathbb{Z}^d$ or $\mathbb{R}^d$, $d \ge 2$, with correlations decaying at least algebraically with exponent $α> 0$, including the discrete Gaussian free field ($d \ge 3, α= d-2$), the discrete Gaussian membrane model ($d \ge 5, α= d - 4$), and many other examples both discrete and continuous. In particular we do not assume positive correlations. This result is new for all strongly correlated models (i.e. $α\in (0,d]$) in dimension $d \ge 3$ except the Gaussian free field, for which sharpness was proven in a recent breakthrough by Duminil-Copin, Goswami, Rodriguez and Severo; even then, our proof is simpler and yields new near-critical information on the percolation density. For planar fields which are continuous and positively-correlated, we establish sharper bounds on the percolation density by exploiting a new `weak mixing' property for strongly correlated Gaussian fields. As a byproduct we establish the box-crossing property for the nodal set, of independent interest. This is the second in a series of two papers studying level-set percolation of strongly correlated Gaussian fields, which can be read independently.
