Nodal Intersections for Arithmetic Random Waves Against a Surface

Given the ensemble of random Gaussian Laplace eigenfunctions on the three-dimensional torus (‘3d arithmetic random waves’), we investigate the one-dimensional Hausdorff measure of the nodal intersection curve against a compact regular toral surface (the ‘nodal intersection length’). The expected length is proportional to the square root of the eigenvalue, times the surface area, independent of the geometry. Our main finding is the leading asymptotic of the nodal intersection length variance, against a surface of non-vanishing Gauss–Kronecker curvature. The problem is closely related to the theory of lattice points on spheres: by the equidistribution of the lattice points, the variance asymptotic depends only on the geometry of the surface.