Owing to the co-existence of multiple physical scales, wave propagation through highly heterogeneous (random) media is an inherently complex physical phenomenon. A well-known conjecture in physical literature states that high frequency waves propagating over long distances through random environments eventually follow Gaussian statistics. While consistent with experimental observations, this conjecture is not entirely supported by any detailed mathematical analysis.

In this talk, I will discuss some recent results settling the Gaussian conjecture in a weak-coupling regime of the paraxial approximation. The paraxial approximation takes the form of a Schrödinger equation with a random potential and is often used to model laser propagation through random media. In particular, I will describe a diffusive scaling where the multiscale wavefield converges to a mean zero complex Gaussian process over the space of Hölder continuous functions. The limiting wavefield is then completely described by its correlation function, which follows an anomalous diffusion. Numerical simulations illustrate theoretical results.