Abstract : The emergence of randomness from chaotic dynamical systems is a well established idea. Most results in this direction describe the statistics of scalar observables along trajectories of some dynamical system on a manifold or a discrete space. I will describe instead the statistics of a functional observable along the flow of a partial differential equation.

More precisely, I will prove that, for a wide class of initial conditions, after evolving for a long time under the Schr\”odinger equation on a compact manifold of negative curvature, a high-energy state converges (in the Benjamini-Schramm sense) to a smooth Gaussian field. This is a first, modest, step towards a broad statement conjectured by M. V. Berry in 1977.

This is joint work with Maxime Ingremeau.