Like many discrete statistical mechanics models, stochastic PDEs can exhibit a “critical dimension” beyond which their large-scale behaviour is expected to be trivial (i.e. governed by Gaussian fluctuations). While such sweeping heuristics allow us to formulate rather precise conjectures, there are relatively few cases where these have actually been proven. In this talk, we will mainly focus on the KPZ equation, a standard model of interface fluctuations. There has recently been substantial progress in our mathematical understanding of its large-scale behaviour in the supercritical regime. 
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006