# Asymptotic distribution of nodal intersections for arithmetic random waves

## TL;DR

This paper investigates the distribution of nodal intersections of arithmetic random waves with curves, proving a CLT for generic curves and a non-Gaussian limit for special static curves, revealing detailed fluctuation behaviors.

## Contribution

It establishes the first CLT for nodal intersections on generic curves and characterizes non-Gaussian limits for static curves, advancing understanding of eigenfunction nodal patterns.

## Key findings

- CLT proven for generic curves' nodal intersections.
- Non-Gaussian limit distribution for static curves.
- Asymptotic behavior of fluctuations determined for most eigenvalues.

## Abstract

We study the nodal intersections number of random Gaussian toral Laplace eigenfunctions ("arithmetic random waves") against a fixed smooth reference curve. The expected intersection number is proportional to the the square root of the eigenvalue times the length of curve, independent of its geometry. The asymptotic behaviour of the variance was addressed by Rudnick-Wigman; they found a precise asymptotic law for "generic" curves with nowhere vanishing curvature, depending on both its geometry and the angular distribution of lattice points lying on circles corresponding to the Laplace eigenvalue. They also discovered that there exist peculiar "static" curves, with variance of smaller order of magnitude, though did not prescribe what the true asymptotic behaviour is in this case. In this paper we study the finer aspects of the limit distribution of the nodal intersections number. For "generic" curves we prove the Central Limit Theorem (at least, for "most" of the energies). For the aforementioned static curves we establish a non-Gaussian limit theorem for the distribution of nodal intersections, and on the way find the true asymptotic behaviour of their fluctuations, under the well-separatedness assumption on the corresponding lattice points, satisfied by most of the eigenvalues.