We investigate Gaussian Laplacian eigenfunctions (Arithmetic Random Waves) on the three-dimensional standard flat torus, in particular the asymptotic distribution of the nodal intersection length against a fixed regular reference surface. Expectation and variance have been addressed by Maffucci [Ann. Henri Poincaré 20(11), 3651–3691 (2019)] who found that the expected length is proportional to the square root of the eigenvalue times the area of the surface, while the asymptotic variance only depends on the geometry of the surface, the projected lattice points being equidistributed on the two-dimensional unit sphere in the high-energy limit. He also noticed that there are “special” surfaces, so-called static, for which the variance is of smaller order; however he did not prescribe the precise asymptotic law in this case. In this paper, we study second order fluctuations of the nodal intersection length. Our first main result is a Central Limit Theorem for “generic” surfaces, while for static ones, a sphere or a hemisphere e.g., our main results are a non-Central Limit Theorem and a precise asymptotic law for the variance of the nodal intersection length, conditioned on the existence of so-called well-separated sequences of Laplacian eigenvalues. It turns out that, in this regime, the nodal area investigated by Cammarota [Trans. Am. Math. Soc. 372(5), 3539–3564 (2019)] is asymptotically fully correlated with the length of the nodal intersections against any sphere. The main ingredients for our proofs are the Kac-Rice formula for moments, the chaotic decomposition for square integrable functionals of Gaussian fields, and some arithmetic estimates that may be of independent interest.

中文翻译：

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ARW 相对于表面的节点交点的渐近分布

我们研究三维标准平面环面上的高斯拉普拉斯特征函数（算术随机波），特别是节点相交长度相对于固定规则参考表面的渐近分布。Maffucci [Ann. Henri Poincaré 20(11), 3651–3691 (2019)]谁发现预期长度与特征值的平方根乘以表面面积成正比，而渐近方差仅取决于表面的几何形状，在高能极限下，投影晶格点均匀分布在二维单位球面上。他还注意到存在“特殊”表面，即所谓的静态表面，其方差较小。然而，他没有在这种情况下规定精确的渐近定律。在本文中，我们研究节点交叉长度的二阶波动。我们的第一个主要结果是“通用”表面的中心极限定理，而对于静态表面、球体或半球，例如，我们的主要结果是非中心极限定理和节点交叉长度方差的精确渐近定律，以所谓的拉普拉斯特征值的良好分离序列的存在为条件。事实证明，在这种情况下，Cammarota [Trans.] 调查的节点区域。是。数学。苏克。372(5), 3539–3564 (2019)] 与任何球体的节点相交长度渐近完全相关。我们证明的主要成分是矩的 Kac-Rice 公式、高斯场平方可积泛函的混沌分解，以及一些可能具有独立意义的算术估计。