Nodal intersections of random eigenfunctions against a segment on the 2-dimensional torus

Abstract: 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.