Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

TL;DR

This paper demonstrates that the nodal length distribution of arithmetic random waves on the 2D torus is non-universal and depends on lattice point distribution, with the main contribution being the analysis of the fourth order chaos component.

Contribution

It provides the first explicit derivation of the Wiener-Itô chaos expansion for the nodal length and shows the dominance of the fourth order chaos component, revealing non-Gaussian limiting behavior.

Findings

Nodal length converges to a non-Gaussian distribution.

Fourth order chaos component dominates the nodal length.

Second order chaos component vanishes.

Abstract

"Arithmetic random waves" are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013)). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener-It\^o chaos expansion for the nodal length shows that it is dominated by its $4$th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes). The rest of the argument relies on the precise analysis of the fourth order chaotic component.