In this talk we will first survey recent developments on the study of random data problems for nonlinear (dispersive) PDEs. Then we will explain how the correct scaling heuristics for the critical regularity of wellposedness works in the probabilistical context and how it opened the door to unveil the random structures of nonlinear waves as they propagate. This is joint work with Yu Deng (UChicago) and Andrea Nahmod (UMass Amherst).
