论文标题
单色连通性度量的有效界限
Effective bounds for monochromatic connectivity measures in two dimensions
论文作者
论文摘要
我们在Sarnak和Wigman引入的二维中建立了单色连通性度量的数值下限。该度量决定了随机平面波的节点域中，比例有任何给定数量的孔以及它们的嵌套方式。我们的边界为简单连接域的数量和包含一个孔的人的数量提供了第一个有效估计。证明的确定性方面是找到一个具有规定的零集的单个函数，并使用隐式函数定理的定量形式来争辩说，相同的配置发生在任何足够接近此函数的任何足够接近的近似值中。概率方面是量化随机波足够接近此功能的可能性。
We establish numerical lower bounds for the monochromatic connectivity measure in two dimensions introduced by Sarnak and Wigman. This measure dictates among the nodal domains of a random plane wave what proportion have any given number of holes, and how they are nested. Our bounds provide the first effective estimate for the number of simply connected domains and for those that contain a single hole. The deterministic aspect of the proof is to find a single function with a prescribed zero set and, using a quantitative form of the implicit function theorem, to argue that the same configuration occurs in the zero set of any sufficiently close approximation to this function. The probabilistic aspect is to quantify the likelihood of a random wave being close enough to this function.
