Asymptotic distribution of Nodal Intersections for ARW against a Surface

TL;DR

This paper studies the asymptotic distribution of nodal intersection lengths of Gaussian eigenfunctions on a 3D torus against a surface, establishing CLT for generic surfaces and detailed laws for special static surfaces.

Contribution

It provides the first CLT for generic surfaces and a precise asymptotic law for static surfaces, advancing understanding of nodal intersection fluctuations in high-energy limits.

Findings

CLT for generic surfaces in high-energy limit

Non-Central Limit Theorem for static surfaces

Asymptotic correlation between nodal area and intersection length

Abstract

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 (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 (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.