Nodal area distribution for arithmetic random waves
We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on $ mathbb{T}^3= mathbb{R}^3/ mathbb{Z}^3$ (three-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 the two-dimensional case addressed by Marinucci et al., [Geom. Funct. Anal. 26 (2016), pp.