In this paper we study the nodal lines of random eigenfunctions of the Laplacian on the torus, the so-called ‘arithmetic waves’. To be more precise, we study the number of intersections of the nodal line with a straight interval in a given direction. We are interested in how this number depends on the length and direction of the interval and the distribution of spectral measure of the random wave. We analyse the second factorial moment in the short interval regime and the persistence probability in the long interval regime. We also study relations between the Cilleruelo and Cilleruelo-type fields. We give an explicit coupling between these fields which on mesoscopic scales preserves the structure of the nodal sets with probability close to one.

中文翻译：

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随机场的中小尺度极限定理

在本文中，我们研究了环面上拉普拉斯算子的随机特征函数的节点线，即所谓的“算术波”。更准确地说，我们研究了节线在给定方向上与直线间隔的交点数。我们感兴趣的是这个数字如何取决于区间的长度和方向以及随机波的光谱测量分布。我们分析了短区间机制中的第二阶乘矩和长区间机制中的持续概率。我们还研究了 Cilleruelo 和 Cilleruelo 型场之间的关系。我们给出了这些场之间的显式耦合，它在介观尺度上以接近一的概率保留了节点集的结构。