Nodal intersections for random eigenfunctions on the torus

We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard two-dimensional flat torus (``arithmetic random waves'') with a fixed smooth reference curve with nonvanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Our main result prescribes the asymptotic behavior of the nodal intersections variance for smooth curves in the high energy limit; remarkably, it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber, and the geometry of the given curve. In particular, this implies that the nodal intersection number admits a universal asymptotic law with arbitrarily high probability.