Igor Wigman: Nodal intersections of random toral eigenfunctions with a test curve

This talk is based on joint works with Zeev Rudnick, and Maurizia
Rossi. 

We investigate the number of nodal intersections of random Gaussian
Laplace eigenfunctions on the standard 2-dimensional flat torus
("arithmetic random waves") with a fixed reference curve. The expected
intersection number is universally proportional to the length of the
reference curve, times the wavenumber, independent of the geometry. 

Our first result prescribes the asymptotic behaviour of the nodal
intersections variance for generic 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. For these
curves we can prove the Central Limit Theorem. In a work in progress
we construct some exceptional examples of curves where the variance is
of smaller order of magnitude, and the limit distribution is
non-Gaussian.