Asymptotic distribution of Nodal Intersections for ARW against a Surface
Abstract
We investigate Gaussian Laplacian eigenfunctions (Arithmetic Random Waves) on the threedimensional 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 (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 twodimensional unit sphere in the highenergy limit. He also noticed that there are ``special'' surfaces, socalled 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 nonCentral Limit Theorem and a precise asymptotic law for the variance of the nodal intersection length, conditioned on the existence of socalled wellseparated sequences of Laplacian eigenvalues. It turns out that, in this regime, the nodal area investigated by Cammarota (2019) is asymptotically fully correlated with the length of the nodal intersections against any sphere. The main ingredients for our proofs are the KacRice formula for moments, the chaotic decomposition for square integrable functionals of Gaussian fields, and some arithmetic estimates that may be of independent interest.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.08218
 Bibcode:
 2021arXiv211008218M
 Keywords:

 Mathematics  Probability;
 Mathematics  Number Theory;
 60G60;
 60B10;
 60D05;
 35P20;
 58J50;
 11P21