NODAL AREA DISTRIBUTION FOR ARITHMETIC RANDOM WAVES

We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on T-3 = R-3/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. 926-960] the fluctuations are dominated by the fourth-order chaotic component. The proof builds upon recent results from Benatar and Maffiucci [Int. Math. Res. Not. IMRN (to appear)] that establish an upper bound for the number of nondegenerate correlations of lattice points on the sphere. We finally discuss higher-dimensional extensions of our result.