We consider the ensemble of random Gaussian Laplace eigenfunctions on T 3 = R 3 /Z 3 ('3d arithmetic random waves'), and study the distribution of their nodal surface area. The expected area is proportional to the square root of the eigenvalue, or 'energy', of the eigenfunction. We show that the nodal area variance obeys an asymptotic law. The resulting asymptotic formula is closely related to the angular distribution and correlations of lattice points lying on spheres.

We consider random Gaussian eigenfunctions of the Laplacian on the three-dimensional flat torus, and investigate the number of nodal intersections against a straight line segment. The expected intersection number, against any smooth curve, is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. We found an upper bound for the nodal intersections variance, depending on the arithmetic properties of the straight line. The considerations made establish a close relation between this problem and the theory of lattice points on spheres.

We consider random Gaussian eigenfunctions of the Laplacian on the standard torus, and investigate the number of nodal intersections against a line segment. The expected intersection number, against any smooth curve, is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. We found an upper bound for the nodal intersections variance, depending on whether the slope of the straight line is rational or irrational. Our findings exhibit a close relation between this problem and the theory of lattice points on circles.

Given the ensemble of random Gaussian Laplace eigenfunctions on the three-dimensional torus ('3d arithmetic random waves'), we investigate the 1-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 nonvanishing 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.

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