Nodal area distribution for arithmetic random waves

TL;DR

This paper studies the distribution of the nodal area of random eigenfunctions on a 3D torus, showing it converges to a universal, non-Gaussian limit as the eigenspace multiplicity increases, using Wiener chaos expansion and lattice point equidistribution.

Contribution

It establishes the limiting distribution of the nodal area for 3D arithmetic random waves and demonstrates universality and non-Gaussianity in the limit.

Findings

Nodal area converges to a universal, non-Gaussian distribution.

Fluctuations are dominated by the fourth-order chaotic component.

Results extend to higher dimensions.

Abstract

We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on $\mathbb{T}^3= \mathbb{R}^3/ \mathbb{Z}^3$ ($3$-dimensional 'arithmetic random waves'). We prove that, as the multiplicity of the eigenspace goes to infinity, the nodal area converges to a universal, non-Gaussian, distribution. Universality follows from the equidistribution of lattice points on the sphere. Our arguments rely on the Wiener chaos expansion of the nodal area: we show that, analogous to (Marinucci, Peccati, Rossi and Wigman, 2016), the fluctuations are dominated by the fourth-order chaotic component. The proof builds upon recent results in (Benatar and Maffiucci, 2017) that establish an upper bound for the number of non-degenerate correlations of lattice points on the sphere. We finally discuss higher-dimensional extensions of our result.