论文标题
固定关键和超临界膜模型
Pinning for the critical and supercritical membrane model
论文作者
论文摘要
膜模型是一种高斯界面模型，其涉及界面高度的第二个衍生物的哈密顿量。我们考虑尺寸$ \ mathsf {d} \ ge4 $的模型，在$δ$ - 强度$ \ varepsilon $的影响下。众所周知，此固定潜在的设法可以将任何$ \ varepsilon> 0 $的接口定位。我们通过建立差异的$ \ varepsilon $依赖性来完善这一结果，以及小$ \ varepsilon $的协方差的指数衰减率（类似于布尔托森 - 维雷尼克（Bolthausen-Velenik）的离散高斯自由场的相应结果）。我们还显示了该场热力学极限的存在。这些结论改善了Bolthausen-Cipriani-Kurt和Sakagawa的早期作品。 这个问题与随机穿孔域中椭圆运算符的均质化相似，而我们的证明从这种联系中汲取了灵感。主要的新想法是一组固定点的相关性不平等，以及依赖离散的多极性耐铁 - 雷利希不平等和多规模参数来构建合适的测试功能的概率的Widman孔填充参数。
The membrane model is a Gaussian interface model with a Hamiltonian involving second derivatives of the interface height. We consider the model in dimension $\mathsf{d}\ge4$ under the influence of $δ$-pinning of strength $\varepsilon$. It is known that this pinning potential manages to localize the interface for any $\varepsilon>0$. We refine this result by establishing the $\varepsilon$-dependence of the variance and of the exponential decay rate of the covariances for small $\varepsilon$ (similar to the corresponding results for the discrete Gaussian free field by Bolthausen-Velenik). We also show the existence of a thermodynamic limit of the field. These conclusions improve upon earlier works by Bolthausen-Cipriani-Kurt and by Sakagawa. The problem has similarities to the homogenization of elliptic operators in randomly perforated domains, and our proof takes inspiration from this connection. The main new ideas are a correlation inequality for the set of pinned points, and a probabilistic Widman hole filler argument which relies on a discrete multipolar Hardy-Rellich inequality and on a multi-scale argument to construct suitable test functions.
