论文标题
平滑高斯字段级的无界组件的独特性
Uniqueness of unbounded component for level sets of smooth Gaussian fields
论文作者
论文摘要
对于$ \ mathbb {r}^d $上的一个固定连续的高斯田地$ f $，包括bargmann-fock和cauchy田地，我们证明，最多有一个无限的连接组件在级别集合$ \ \ \ \ \ ell \ ell \} $（exculse $ \ eell）$ \ e \ eell for \ eell for \ ge \ e el for \ eell（级$ \ ell \ in \ mathbb {r} $，从而证明了杜米尼尔·孔宾（Duminil-Copin），里维拉（Rivera），罗德里格斯（Rodriguez）和瓦内维尔（Vanneuville）提出的猜想。由于所考虑的田地通常非常僵化（例如〜肯定是肯定的），因此没有可用的有限能量属性，并且证明唯一性的经典方法很难实施。我们使用基于Cameron-Martin定理的软移参数绕过了这一难度。
For a large family of stationary continuous Gaussian fields $f$ on $\mathbb{R}^d$, including the Bargmann-Fock and Cauchy fields, we prove that there exists at most one unbounded connected component in the level set $\{f=\ell\}$ (as well as in the excursion set $\{f\geq\ell\}$) almost surely for every level $\ell\in \mathbb{R}$, thus proving a conjecture proposed by Duminil-Copin, Rivera, Rodriguez & Vanneuville. As the fields considered are typically very rigid (e.g.~analytic almost surely), there is no sort of finite energy property available and the classical approaches to prove uniqueness become difficult to implement. We bypass this difficulty using a soft shift argument based on the Cameron-Martin theorem.
