Random nodal lengths and Wiener chaos
vendredi 10 novembre 2017, 9h30 - 10h30
In this talk we present some of the recent results on the « nodal geometry » of random eigenfunctions on Riemannian surfaces. We focus on the asymptotic behavior, for high energy levels, of the nodal length of Gaussian Laplace eigenfunctions on the torus (arithmetic random waves) and on the sphere (random spherical harmonics). We give some insight on both Berry?s cancellation phenomenon and the nature of nodal length second order fluctuations (non-Gaussianity on the torus and Gaussianity on the sphere) in terms of chaotic components. Finally we consider the general case of monochromatic random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian surface with frequencies from a short interval, whose scaling limit is Berry?s Random Wave Model. For the latter we present some recent results on the asymptotic distribution of its nodal length in the high energy limit (equivalently, for growing domains).