Nodal lengths of arithmetic random waves
mardi 27 mars 2018, 13h30 - 14h30
« Arithmetic random waves » are the Gaussian Laplace eigenfunctions on the two-dimensional torus. In this talk we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles.
Our argument has two main ingredients. An explicit derivation of the Wiener-Ito chaos expansion for the nodal length shows that it is dominated by its 4th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes). The rest of the argument relies on the precise analysis of the fourth order chaotic component. Based on a joint work with D. Marinucci, G. Peccati and I. Wigman.